CN100468254C - Industrial product design method, numerical control method and device using clothoid curve - Google Patents

Abstract

A. Adopt the three-dimensional curve (called three-dimensional clothoid curve) that the inclination angle and the deflection angle in the tangential direction are respectively given by the curve length or the quadratic equation of the curve length variable, and design the track of the motion of the mechanical element. B. The inclination angle and deflection angle in the tangential direction are respectively given by the three-dimensional curve (called three-dimensional clothoid curve) given by the quadratic formula of the curve length or the curve length variable to represent the tool trajectory or the contour shape of the workpiece, and control the tool through the three-dimensional curve exercise.

Description

Industrial product design method, numerical control method and device using clothoid curve

technical field

In the following specification, A is appended to the explanation about "the design method of the industrial product using the clothoid curve and the industrial product designed by the design method" (hereinafter, simply referred to as the design method of the industrial product using the clothoid curve) , add note B in the explanation about "Numerical control method and device using clothoid curve" (hereinafter, simply referred to as numerical control method using clothoid curve).

A. Design method of industrial products using clothoid curve

The present invention relates to a method for designing the shape of an industrial product using a clothoid curve, and more particularly to a method for designing a motion path for smooth movement of a mechanical element in a machine including a mechanism for moving the mechanical element with mass.

B. Numerical control method using clothoid curve

In addition, the present invention relates to a numerical control method and apparatus for controlling the movement of tools (including grips such as handles or various tools) in working machines (referred to as robots, etc.) such as robots, machine tools, assembly machines, and inspection machines using a clothoid curve .

Background technique

A. Design method of industrial products using clothoid curve

With the miniaturization and high precision of machinery, the mechanism of high-speed moving mechanical components becomes important. It is strongly required to design a mechanically reasonable and smooth motion trajectory, reduce vibration or motion errors, suppress aging changes or damage, and achieve high-speed, high-precision motion.

As for the design method of the free motion trajectory, a method of connecting analytical curves such as straight lines or arcs, or spline interpolation (a method of interpolating a given point sequence with a clothoid curve) has been conventionally used (see Non-Patent Document 1).

B. Numerical control method using clothoid curve

In a numerically controlled robot that performs welding, painting, adhesive application, etc., the input pattern is generally input as discrete point sequence data. Therefore, to generate a continuous graph, any method is required to interpolate the point columns.

As a method of interpolating between arbitrarily given point sequences, there are known methods of rounding the corners of polylines, B-spline interpolation, cubic spline interpolation, etc., but as methods that can strictly pass each given point As an interpolation method, cubic spline interpolation is known (see Non-Patent Document 1).

However, the cubic spline interpolation has a big defect that the relationship between the independent variable and the geometric quantities of the curve is not stable because it expresses an independent variable that does not have a geometric meaning as a parameter. This cubic spline interpolation has a complex relationship between the moving distance from the starting point and the curvature, and is not suitable for controlling the linear velocity to be constant.

Non-Patent Document 1: "Integrated CAD/CAM System" Ohm Press, 1997

Non-Patent Document 2: Qiu Shiyu, Makino Hiroshi, Suda Daharu, Yokoyama Kyoo, "Free Curve Interpolation Using Convolution" (Journal of Robotics Society of Japan, Volume 8, No. 6, pp40-47)

Non-Patent Document 3: Li Guiqing, Li Xianmin, Li Hua, "3D Discrete ClothoidSplines", (CGI’01, pp321-324)

Contents of the invention

A. Design method of industrial products using clothoid curve

In the method of connecting analytical curves such as straight lines and circular arcs, it is difficult to continuously connect curvatures at the connecting points of straight lines and circular arcs. If the curvature can be continuously connected according to the method of spline interpolation, but because the relationship between the moving distance from the starting point and the curvature is complicated, it is difficult to design a mechanically reasonable curvature distribution along the track, and good motion cannot be obtained. track.

Therefore, it is an object of the present invention to provide a method of designing a motion path for smooth motion of a mechanical element in a machine including a mechanism for moving a mechanical element having mass. This method is a new and novel method proposed by the present inventors.

Here, the so-called smooth means that the change of the tangent, contact plane (normal) or curvature of the track is continuous along the track, so the force acting on the mechanical element moving along the track changes continuously.

However, the shape of the return path of the ball screw, which is often used in robots, machine tools, assembly machines, inspection machines, etc., is connected by a straight line or an arc, the tangent or curvature of the curve is not continuous, and the degree of freedom in track design is insufficient.

Another object of the present invention is to reduce the loss of kinetic energy on the circulation path of the ball screw in the design of the ball circulation path of the ball screw, and to prevent damage to components along the circulation path, and to establish the structure of the circulation path. A design method for circular paths with continuous tangent or curvature and smooth curvature changes. The design method of the circulation path of the ball screw is an application example of the method of designing a motion track so that the motion of the machine element is smooth.

B. Numerical control method using clothoid curve

As an interpolation method passing through given points in two dimensions, a convolutional interpolation method proposed by the inventors is known, which enables smooth interpolation (see Non-Patent Document 2). Therefore, it is believed that if the three-dimensional expansion of the clothoid curve is used for the interpolation of the free point sequence, compared with the characteristics of the clothoid curve expressed as a function of the length of the curve, the control of keeping the linear speed constant or changing the linear speed according to the length of the line can be easily realized. . In addition, since the length of the curve is used as a parameter, unlike other methods, there is also an advantage that there is no need to obtain the length of the line later. It is hoped that the three-dimensional expanded clothoid curve will be useful in the fields of numerical control and the like. Conventionally, "3D Discrete ClothoidSplines" by Li et al. (refer to Non-Patent Document 3) etc. have been known about the three-dimensional dilated clothoid curves, but no three-dimensional dilated clothoids in the form of equations have been found. Expansion in the form of an equation is advantageous in that it is easy to calculate each value.

Therefore, the object of the present invention is to provide a new definition of a three-dimensional clothoid curve for numerically controlling the movement of a tool, and its curvature change graph relative to the independent variable can replace the characteristics of a simple two-dimensional clothoid curve as much as possible. In addition, an object of the present invention is to interpolate a point sequence using the three-dimensional clothoid curve.

A. Design method of industrial products using clothoid curve

Hereinafter, the invention of the method of designing a clothoid industrial product according to claims 1 to 10 will be described.

The first invention, in order to solve the above-mentioned problems, provides a kind of design method of industrial product, it is characterized in that: the inclination angle (pitch angle) and deflection angle (yaw angle) of tangential direction are given by the quadratic formula of curve length or curve length variable ) of their respective three-dimensional curves (called three-dimensional clothoid curves) to design the shape of industrial products.

The second invention is the design method of the industrial product as described in the first invention, wherein the industrial product is a machine including a mechanism for moving a mechanical element with mass; the three-dimensional curve (called a three-dimensional clothoid curve) ) to design the motion track of the mechanical element.

The third invention is the method of designing an industrial product according to the second invention, wherein the machine is a screw device including a mechanism for moving balls as the mechanical element; The screw shaft of the spiral rolling body raceway groove and the load rolling body raceway groove opposite to the rolling body raceway groove on the inner peripheral surface have one end connected to the load rolling body raceway groove and the nut of the return path at the other end, and a plurality of rolling elements arranged between the rolling element raceway groove of the screw shaft and the loaded rolling element raceway groove of the nut and on the return path ; Design the regression path of the screw device by using the three-dimensional curve (called a three-dimensional clothoid curve).

A fourth invention is the method for designing an industrial product according to any one of inventions 1 to 3, wherein the three-dimensional clothoid curve is defined by the following formula.

[Formula 1]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^{\mathrm{the\; s}}\mathrm{uds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S\mathrm{wxya},$ 0≤s≤h, $0\&le;S=\frac{\mathrm{the\; s}}{h}\&le;1$

(1)

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}$

(2)

α＝a ₀ +a ₁ S+a ₂ S ² (3)

β=b ₀ +b ₁ S+b ₂ S ² (4)

here,

[Formula 2]

$\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Respectively represent the position vector of the point on the three-dimensional clothoid curve and its initial value.

Let s be the length of the curve from the start point, and let its total length (the length from the start point to the end point) be h. Let S denote the value of dividing s by h. S is a value of a dimensionless quantity, which is referred to as a curve length variable.

i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively.

u is a unit vector representing the tangential direction of the curve on the point P, and is given by Equation (2). E ^kβ and E ^jα are rotation matrices, and represent the rotation of the angle β of the k-axis system and the rotation of the angle α of the j-axis system, respectively. The former is called yaw rotation, and the latter is called pitch rotation. Equation (2) indicates that the tangent vector u is obtained by first rotating the unit vector in the i-axis by only α on the j-axis, and then only rotating by β on the k-axis.

a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ are constants.

The fifth invention is the method for designing an industrial product according to the fourth invention, wherein a plurality of spatial points are specified in three-dimensional coordinates, and the industrial product is designed by interpolating these spatial points using the three-dimensional clothoid curve. shape.

The 6th invention is the design method of an industrial product according to the 5th invention, characterized in that: at the plurality of space points, a three-dimensional clothoid segment (a unit curve constituting a curve group generated by interpolation) and the following A three-dimensional clothoid segment (constituting the unit curve of the curve group generated by interpolation), and the position, tangent method, normal direction and curvature of the two are connected, and the seven parameters a ₀ and a ₁ of the three-dimensional clothoid segment are calculated. , a ₂ , b ₀ , b ₁ , b ₂ , h.

The 7th invention is the design method of an industrial product according to the 6th invention, characterized in that: designate the tangent direction, normal direction and curvature of the start point and the end point among the plurality of spatial points; Re-insert the interpolation object between the spatial points, so that the conditional expressions of the tangent direction, the normal direction and the curvature of the starting point and the end point of the addition operation, and a three-dimensional clothoid segment on the multiple spatial points and the next three-dimensional The conditional expression number of the conditional expression of the position, tangent direction, normal direction and curvature of the clothoid segment connecting the two, and the seven parameters a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b of the three-dimensional clothoid segment _2. The unknowns of h are consistent; by making the conditional formula consistent with the unknowns, the seven parameters a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , h of the three-dimensional clothoid segment are calculated.

The eighth invention is an industrial product designed by the method for designing an industrial product according to any one of inventions 1 to 7.

The ninth invention is a program for making a computer use a three-dimensional curve (referred to as a three-dimensional rondo) given by the curve length or the quadratic equation of the curve length variable using the inclination angle and the deflection angle in the tangential direction, respectively, in order to design the shape of an industrial product. line), which plays a role as a means of designing the shape of industrial products.

The tenth invention is a computer-readable recording medium on which is recorded a program for making the computer use the inclination angle and the deflection angle in the tangential direction as the curve length or the curve length variable, respectively, in order to design the shape of an industrial product. The three-dimensional curve given by the quadratic formula (called the three-dimensional clothoid curve) plays a role as a means of designing the shape of industrial products.

B. Numerical control method using clothoid curve

Hereinafter, the invention using the numerical control method of the clothoid curve described in claims 11 to 27 will be described.

The 11th invention, in order to solve the above-mentioned problem, provides a kind of numerical control method, wherein, adopts the three-dimensional curve (referred to as three-dimensional clothoid curve) that the inclination angle and the deflection angle of the tangential direction are respectively given by the quadratic formula of the curve length or the curve length variable ), which represents the tool trajectory or the contour shape of the workpiece, and the movement of the tool is controlled by the three-dimensional curve.

A twelfth invention is the numerical control method according to the eleventh invention, wherein the three-dimensional convolution is defined by the following equation.

[Formula 3]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^{\mathrm{the\; s}}\mathrm{uds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S\mathrm{wxya},$ 0≤s≤h, $0\&le;S=\frac{\mathrm{the\; s}}{h}\&le;1$

(1)

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}$

(2)

α＝a ₀ +a ₁ S+a ₂ S ² (3)

β=b ₀ +b ₁ S+b ₂ S ² (4)

here,

[Formula 4]

$\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Respectively represent the position vector of the point on the three-dimensional clothoid curve and its initial value.

Let s be the length of the curve from the start point, and let its total length (the length from the start point to the end point) be h. Let S denote the value of dividing s by h. S is a value of a dimensionless quantity, which is referred to as a curve length variable.

i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively.

u is a unit vector representing the tangential direction of the curve on the point P, and is given by Equation (2). E ^kβ and E ^jα are rotation matrices, and represent the rotation of the angle β of the k-axis system and the rotation of the angle α of the j-axis system, respectively. The former is called yaw rotation, and the latter is called pitch rotation. Equation (2) indicates that the tangent vector u is obtained by first rotating the unit vector in the i-axis by only α on the j-axis, and then only rotating by β on the k-axis.

a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ are constants.

The thirteenth invention is a numerical control device in which a tool locus or a workpiece is expressed using a three-dimensional curve (referred to as a three-dimensional clothoid curve) whose inclination angle and deflection angle in the tangential direction are respectively given by the curve length or the quadratic equation of the curve length variable. The outline shape of the 3D curve controls the movement of the tool.

The 14th invention is a program for making the computer use a three-dimensional curve (referred to as a three-dimensional rondo) given by the curve length or the quadratic equation of the curve length variable, respectively, using the inclination angle and deflection angle in the tangential direction for numerically controlling the movement of the tool. line), it functions as a means of expressing the tool trajectory or the contour shape of the workpiece.

The fifteenth invention is a computer-readable recording medium on which is recorded a program or a calculation result obtained by the program for numerically controlling the motion of the tool, making the computer use the inclination and deflection in the tangential direction The angle is a three-dimensional curve (called a three-dimensional clothoid curve) given by the curve length or the quadratic formula of the curve length variable, and it functions as a means of expressing the tool trajectory or the contour shape of the workpiece.

In order to solve the above problems, the sixteenth invention provides a numerical control method in which a three-dimensional curve (three-dimensional clothoid segment) whose inclination angle and deflection angle in the tangential direction are respectively given by the curve length or the quadratic equation of the curve length variable is used, and interpolated Complement between the point columns arbitrarily given in the three-dimensional coordinates, and control the movement of the tool through the three-dimensional clothoid segment.

The seventeenth invention is a numerical control method in which a plurality of three-dimensional curves (three-dimensional clothoid segments) whose inclination angle and deflection angle in the tangential direction are respectively given by the curve length or the quadratic equation of the curve length variable are connected, and through the plurality of A 3D clothoid segment controls the motion of the tool.

The eighteenth invention is the numerical control method according to the sixteenth or seventeenth invention, wherein the three-dimensional clothoid curve is defined by the following equation.

[Formula 5]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^{\mathrm{the\; s}}\mathrm{uds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S\mathrm{wxya},$ 0≤s≤h, $0\&le;S=\frac{\mathrm{the\; s}}{h}\&le;1$

(1)

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}$

(2)

α＝a ₀ +a ₁ S+a ₂ S ² (3)

β=b ₀ +b ₁ S+b ₂ S ² (4)

here,

[Formula 6]

$\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Respectively represent the position vector of the point on the three-dimensional clothoid curve and its initial value.

Let s be the length of the curve from the start point, and let its total length (the length from the start point to the end point) be h. Let S denote the value of dividing s by h. S is a value of a dimensionless quantity, which is referred to as a curve length variable.

i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively.

u is a unit vector representing the tangential direction of the curve on the point P, and is given by Equation (2). E ^kβ and E ^jα are rotation matrices, and represent the rotation of the angle β of the k-axis system and the rotation of the angle α of the j-axis system, respectively. The former is called yaw rotation, and the latter is called pitch rotation. Equation (2) indicates that the tangent vector u is obtained by first rotating the unit vector in the i-axis by only α on the j-axis, and then only rotating by β on the k-axis.

a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ are constants.

The nineteenth invention, the numerical control method as described in the eighteenth invention, is characterized in that: on the joint of a three-dimensional clothoid segment and the next three-dimensional clothoid segment, the position and tangent direction (and curvature according to the situation) of the two are continuous In this way, the seven parameters a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , and h of the three-dimensional clothoid segment are calculated.

The 20th invention is a numerical control device in which a point arbitrarily given in three-dimensional coordinates is interpolated using a three-dimensional clothoid segment whose inclination angle and deflection angle in the tangential direction are respectively given by the curve length or the quadratic equation of the curve length variable Between columns, the movement of the tool is controlled by the three-dimensional clothoid segment.

The 21st invention is a program for causing the computer to interpolate the three-dimensional clothoid segment using the inclination angle and the deflection angle in the tangential direction respectively given by the quadratic expression of the curve length or the curve length variable for numerically controlling the movement of the tool. It functions as a means between arbitrarily given point sequences in three-dimensional coordinates.

The 22nd invention is a computer-readable recording medium on which a program or a calculation result obtained by the program is recorded, the program is used for numerically controlling the movement of the tool, and the computer is used as the inclination angle and the deflection in the tangential direction. The three-dimensional clothoid segments whose angles are given by the curve length or the quadratic formula of the curve length variable, respectively, function by means of interpolating between point sequences arbitrarily given in the three-dimensional coordinates.

The 23rd invention is a numerical control method, wherein: a three-dimensional curve (referred to as a three-dimensional clothoid curve) whose inclination angle and deflection angle in the tangential direction are respectively given by the curve length or the quadratic equation of the curve length variable is used to express the tool path or the workpiece The shape of the contour; specify the motion of the tool moving along the three-dimensional curve; calculate the moving position of the tool per unit time according to the specified motion. Here, motion refers to positional information that changes as a function of time.

A twenty-fourth invention is the numerical control method according to the twenty-third invention, wherein the three-dimensional clothoid curve is defined by the following formula.

[Formula 7]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^{\mathrm{the\; s}}\mathrm{uds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S\mathrm{wxya},$ 0≤s≤h, $0\&le;S=\frac{\mathrm{the\; s}}{h}\&le;1$

(1)

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}$

(2)

α＝a ₀ +a ₁ S+a ₂ S ² (3)

β=b ₀ +b ₁ S+b ₂ S ² (4)

here,

[Formula 8]

$\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Respectively represent the position vector of the point on the three-dimensional clothoid curve and its initial value.

Let s be the length of the curve from the start point, and let its total length (the length from the start point to the end point) be h. Let S denote the value of dividing s by h. S is a value of a dimensionless quantity, which is referred to as a curve length variable.

i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively.

u is a unit vector representing the tangential direction of the curve on the point P, and is given by Equation (2). E ^kβ and E ^jα are rotation matrices, and represent the rotation of the angle β of the k-axis system and the rotation of the angle α of the j-axis system, respectively. The former is called yaw rotation, and the latter is called pitch rotation. Equation (2) indicates that the tangent vector u is obtained by first rotating the unit vector in the i-axis by only α on the j-axis, and then only rotating by β on the k-axis. a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ are constants.

The twenty-fifth invention is a numerical control device, wherein: a three-dimensional curve (referred to as a three-dimensional clothoid curve) whose inclination angle and deflection angle in the tangential direction are respectively given by the curve length or the quadratic equation of the curve length variable is used to express the tool path or the workpiece The shape of the contour; specify the motion of the tool moving along the three-dimensional curve; calculate the moving position of the tool per unit time according to the specified motion. Here, motion refers to positional information that changes as a function of time.

The twenty-sixth invention is a program for causing a computer to function as a means for numerically controlling the motion of a tool by using a three-dimensional method in which the inclination angle and deflection angle in the tangential direction are respectively given by the curve length or the quadratic formula of the curve length variable. Cloth curve (referred to as a three-dimensional clothoid curve), a means of expressing the trajectory of a tool or the contour shape of a workpiece; a means of specifying the motion of a tool moving along the three-dimensional curve; calculating the movement position of the tool per unit time according to the specified motion s method. Here, motion refers to positional information that changes as a function of time.

The twenty-seventh invention is a computer-readable recording medium on which is recorded a program for causing a computer to function as means for numerically controlling the movement of a tool, or a calculation result obtained by the program by using The inclination angle and deflection angle in the tangential direction are respectively given by the curve length or the quadratic formula of the curve length variable. The three-dimensional loop curve (called three-dimensional clothoid curve) is a means of expressing the contour shape of the tool trajectory or workpiece; specifying along the three-dimensional A means of moving the tool in a curve; a means of calculating the moving position of the tool per unit time according to the specified motion. Here, motion refers to positional information that changes as a function of time.

A. Design method of industrial products using clothoid curve

According to the inventions described in the first to tenth inventions, by using the three-dimensional clothoid curve, it is possible to design a motion track that makes the motion of the machine element smooth. If the track can be designed in this way, mechanically reasonable motion can be realized, and a machine with less functional degradation or track damage caused by motion errors can be manufactured.

Especially for screw devices, it is possible to provide a universal method of generating the spatial curves necessary for designing the circulation paths of the rolling elements. In the case that the rolling element moves along the space curve of the cyclic path with acceleration and deceleration, it is possible to design a design in which the constraint force changes smoothly. According to this feature, since the rolling elements move smoothly and smoothly, the power transmission efficiency of the screw device can be improved, and the occurrence of excessive frictional force or inertial force can be suppressed. Accordingly, damage to components can be prevented, and a highly reliable screw device can be realized.

In addition, the application can control the characteristics of the curvature change graph, which can increase the application in the industrial field. For example, this widely-used curve design method can be effectively applied to the design of a pattern shape that requires aesthetics.

B. Numerical control method using clothoid curve

According to the inventions described in the 11th to 27th inventions, since the main variable of the curve is the curve length or the curve length variable, and the inclination angle and the deflection angle of the tangential direction are given according to the quadratic formula of the curve length or the curve length variable respectively, it can be It is guaranteed that the normal direction obtained by the first differential and the curvature obtained by the second differential are continuous with respect to the length of the curve or the variable of the length of the curve. In other words, in a clothoid curve, the normal direction and the curvature are connected. Therefore, a smooth curve with good properties can be obtained, and it becomes possible to realize a numerical control method of a mechanically reasonable speed change.

Description of drawings

FIG. 1 is a diagram representing a two-dimensional clothoid curve on xy coordinates.

FIG. 2 is a diagram showing the shape of a typical two-dimensional clothoid curve.

FIG. 3 is a diagram showing definitions of inclination α and deflection β of a three-dimensional clothoid curve.

Fig. 4 is a diagram showing the shape of a typical three-dimensional clothoid curve.

FIG. 5 is a graph showing the amount of change of the unit normal vector.

Fig. 6 is a diagram showing two two-dimensional or three-dimensional clothoid curves with the same size and shape but opposite directions.

Fig. 7 is a diagram showing division of a three-dimensional clothoid curve.

Fig. 8 is a diagram showing conditions of G2 ^- continuous interpolation.

FIG. 9 is a diagram showing the concept of a contact plane.

FIG. 10 is a diagram showing a schematic flow of a method of convolution interpolation.

FIG. 11 is a diagram showing a schematic flow of a convolution interpolation method satisfying the condition of ^G2 continuity.

Fig. 12 is a diagram showing three-dimensional convolutional interpolation of points P ₁ , P ₂ , P ₃ .

Figure 13 is a diagram showing 3D Discrete Clothoid Splines for r=4.

Fig. 14 is a diagram illustrating 3D Discrete Clothoid Splines.

Fig. 15 is a perspective view of a three-dimensional clothoid curve generated by interpolation.

Fig. 16 is a curvature change graph in which the moving distance from the starting point is taken on the horizontal axis and the curvature is taken on the vertical axis.

Fig. 17 is a diagram showing a schematic flow of three-dimensional convolution interpolation in which values are controlled at both ends.

Fig. 18 is a schematic diagram showing three-dimensional convolution interpolation in which values are controlled at both ends.

Fig. 19 is a graph showing the results of actually performing interpolation.

FIG. 20 is a graph showing the relationship between the moving distance from the starting point of each curve and the curvature.

Fig. 21 is a diagram showing control of values at intermediate points.

Fig. 22 is a diagram showing a schematic flow of an interpolation method using a three-dimensional convolution that controls each value at a start point and an end point.

Figure 23 is a diagram showing 3D Discrete Clothoid Splines for r=4.

Fig. 24 is a diagram showing generated polygons.

Fig. 25 is a diagram showing three-dimensional convolutional interpolation of points P ₁ , P ₂ , P ₃ .

Fig. 26 is a diagram showing generated curves and polygons.

Figure 27 is an illustration of insertion points.

Fig. 28 is a diagram representing a segmented three-dimensional clothoid curve.

Fig. 29 is a diagram showing the generated curves.

FIG. 30 is a graph showing the relationship between the moving distance s from the starting point of each curve and the curvature κ.

Fig. 31 is a diagram showing a ball screw in which the reverser is a nut and other reverser methods.

Fig. 32 is a diagram showing a nut in which the reverser is a ball screw integrated with the nut.

Fig. 33A is a perspective view of the nut with the ball circulation groove seen.

33B is a perspective view of the nut in a state where the loaded ball rolling groove is seen.

Fig. 34 is a diagram showing a state where a nut is assembled on a screw shaft.

Fig. 35 is a developed view of a circulation path of a conventional ball screw.

FIG. 36 is a graph showing the curvature of the circulation path of a conventional ball screw.

Figure 37 is a diagram showing the track of the center of the ball.

Fig. 38 is a diagram showing a coordinate system.

Fig. 39 is a diagram showing a coordinate system viewed from the z-axis.

FIG. 40 is a diagram representing a curve plotting a locus of the center of a ball moving along a screw groove.

FIG. 41 is a graph showing curves _C0 and _C1 viewed from the y-axis.

FIG. 42 is a graph showing curves _C0 and _C1 around the point _Ps viewed from the z-axis.

Figure 43 is an illustration of the insertion point _P2 .

FIG. 44 is a diagram showing the generated regression path and curve _C0 .

Fig. 45 is a graph showing the relationship between the moving distance from the point _Pe and the curvature.

Fig. 46 is a diagram representing a two-dimensional clothoid curve on x, y coordinates.

Fig. 47 is a graph showing a two-dimensional clothoid curve.

Fig. 48 is a diagram showing definitions of α and β of a three-dimensional clothoid curve.

Fig. 49 is a diagram representing a graph of a typical three-dimensional clothoid curve.

Fig. 50 is a diagram showing conditions of ^G2 continuous interpolation.

Fig. 51 is a diagram showing the concept of a contact plane.

Fig. 52 is a diagram showing a schematic flow of the convolution interpolation method.

FIG. 53 is a diagram showing a schematic flow of a convolution interpolation method satisfying the condition of ^G2 continuity.

Fig. 54 is a diagram showing three-dimensional convolutional interpolation of points P ₁ , P ₂ , and P ₃ .

Figure 55 is a diagram showing 3D Discrete Clothoid Splines for r=4.

Figure 56 is a diagram illustrating 3D Discrete Clothoid Splines.

Fig. 57 is a perspective view of a three-dimensional clothoid curve generated by interpolation.

Fig. 58 is a curve graph showing the movement distance from the starting point on the horizontal axis and the curvature change graph on the vertical axis.

Fig. 59 is a diagram showing a schematic flow of three-dimensional convolution interpolation in which values are controlled at both ends.

Fig. 60 is a schematic diagram showing three-dimensional convolutional interpolation in which values are controlled at both ends.

Fig. 61 is a graph showing the result of actually performing interpolation.

Fig. 62 is a graph showing the relationship between the moving distance from the starting point of each curve and the curvature.

Fig. 63 is a diagram showing control of values at intermediate points.

Fig. 64 is a diagram showing a schematic flow of an interpolation method using three-dimensional convolution that controls values at the start point and end point.

Figure 65 is a diagram showing 3D Discrete Clothoid Splines for r=4.

Fig. 66 is a diagram showing generated polygons.

Fig. 67 is a diagram showing three-dimensional convolutional interpolation of points P ₁ , P ₂ , and P ₃ .

Fig. 68 is a diagram showing generated curves and polygons.

Figure 69 is an illustration of an insertion point.

Fig. 70 is a diagram representing a segmented three-dimensional clothoid curve.

Fig. 71 is a diagram showing a generated curve.

FIG. 72 is a graph showing the relationship between the moving distance s from the starting point of each curve and the curvature κ.

Fig. 73 is a flowchart showing a numerical control method.

Fig. 74 is a comparison diagram showing conventional spline curves.

Detailed ways

A. Design method of industrial products using clothoid curve

Hereinafter, regarding the embodiment of the invention of the design method of an industrial product using a clothoid curve, it is divided into 1. the definition and characteristics of a three-dimensional clothoid curve, 2. the interpolation method using a three-dimensional clothoid curve, and 3. the use of a three-dimensional clothoid interpolation method to design as The method of the return path of the ball screw of the screw device, and 4. The numerical control method using three-dimensional convolution interpolation will be described in order.

1. Definition and characteristics of three-dimensional clothoid curve

(1-1) The basic method of three-dimensional rotation

Clothoid curve, also known as Cornu's spiral, is a curve whose curvature changes proportionally to the length of the curve.

The two-dimensional clothoid curve proposed by the inventor is a type of plane curve (two-dimensional curve), and is represented by the following equation on the xy coordinates shown in FIG. 1 .

[Formula 9]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^{\mathrm{the\; s}}{e}^{\mathrm{j\&phi;}}\mathrm{ds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S{e}^{\mathrm{j\&phi;}}\mathrm{wxya},$ 0≤s≤h, $0\&le;S=\frac{\mathrm{the\; s}}{h}\&le;1$

(1-1)

φ＝c ₀ +c ₁ s+c ₂ s ² ＝φ ₀ +φ _v S+φ _u S ² (1-2)

here,

[Formula 10]

P=x+jy, $j=\sqrt{-1}---(1-3)$

is the position vector representing the point on the curve,

[Formula 11]

P ₀ =x ₀ +jy ₀ (1-4)

is its initial value (the position vector of the starting point).

[Formula 12]

e ^jφ = cos φ + jsin φ (1-5)

is a position vector (vector with a length of 1) representing the tangent direction of the curve, and this direction Φ is measured counterclockwise from the original line (x-axis direction). If this unit vector is multiplied by a small length ds integral, the point P on the curve can be obtained.

Let the length from the start point of the curve measured along the curve be s, and let the total length (length from the start point to the end point) be h. Let S denote the value of dividing s by h. S is a value of a dimensionless quantity, which is referred to as a curve length variable.

The characteristic of the clothoid curve, as shown in the formula (1-2), is that the tangent direction angle Φ is expressed by the quadratic formula of the curve length s or the curve variable S. c ₀ , c ₁ , c ₂ or Φ _o , Φ _v , Φ _u are the coefficients of the quadratic equation, and these numbers and the total length h of the curve are called the parameters of the convolution. Figure 2 shows the shape of a general clothoid curve.

For the relationship above three-dimensional expansion, create an expression for a three-dimensional clothoid curve. The formula for giving a three-dimensional clothoid curve was not known in the past, so the inventors first derived the formula.

The three-dimensional clothoid curve is defined by the following equation.

[Formula 13]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^{\mathrm{the\; s}}\mathrm{uds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S\mathrm{wxya},$ 0≤s≤h, $0\&le;S=\frac{\mathrm{the\; s}}{h}\&le;1$

(1-6)

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}$

(1-7)

α＝a ₀ +a ₁ S+a ₂ S ² (1-8)

β=b ₀ +b ₁ S+b ₂ S ² (1-9),

here,

[Formula 14]

$\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

Respectively represent the position vector of the point on the three-dimensional clothoid curve and its initial value. i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively.

u is a unit vector representing the tangent direction of the curve on point P, and is given by equation (1-7). In formula (1-7), E ^kβ and E ^jα are rotation matrices, as shown in Figure 3, which represent the rotation of the angle β in the k-axis (z-axis) system and the angle α in the j-axis (y-axis) system, respectively. rotate. The former is called yaw rotation, and the latter is called pitch rotation. Equation (1-7) indicates that the tangent vector u is obtained by first rotating the unit vector in the direction of the i-axis (x-axis) by α in the j-axis (y-axis) system, and then by only rotating β in the k-axis (z-axis) system .

That is, in two dimensions, the unit vector e ^jΦ representing the tangential direction of the curve is obtained from the inclination angle Φ from the x-axis. In three dimensions, the tangent vector u of the curve can be obtained from the inclination angle α and the deflection angle β. If the inclination angle α is 0, a two-dimensional clothoid curve rolled up in the xy plane can be obtained, and if the deflection angle β is 0, a two-dimensional clothoid curve rolled up in the xz plane can be obtained. If the integral is multiplied by the small length ds in the tangential direction vector u, a three-dimensional clothoid curve can be obtained.

In the three-dimensional clothoid curve, the inclination angle α and the deflection angle β of the tangent vector are shown in formula (1-8) and formula (1-9) respectively, which can be given by the quadratic formula of the curve length variable S. In this way, the change in the direction of the tangent can be freely chosen and also given continuity in its change.

As shown in the above formula, the three-dimensional clothoid curve is defined as "a curve in which the inclination angle α and the deflection angle β in the tangential direction are respectively expressed by the quadratic equation of the curve length variable".

A three-dimensional clothoid curve starting from P ₀ , given by

[Formula 15]

a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , h (1-11)

These 7 parameters are determined. The six variables of a ₀ to b ₂ have angle units and represent the shape of the clothoid curve. In contrast, h has units of length, representing the size of the clothoid curve. A typical example of a three-dimensional clothoid curve is a spiral curve shown in FIG. 4 .

(1-2) Fresney frame and curvature on three-dimensional clothoid

When there is an arbitrary ternary curve, it is specified that t is used as a parameter and expressed as R(t). In particular, when the moving distance s from the starting point is used as a parameter, it is represented by R(s).

If the absolute value of the relative position vector dR(s) of two points on the curve having a degree difference of ds is regarded as the linear element ds, the relationship of the following formula (2-1) exists between ds and dt. To simplify the differentiation of R using the parameter t, dots are attached to the letters.

[Formula 16]

$\mathrm{<span\; class="notranslate">\; ds</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; |</span>\frac{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; dt</span>}}<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; =</span>\sqrt{\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Since the unit tangent vector u(t) normalizes the line element vector dR(t) of the curve, it can be expressed by formula (2-2) if referring to formula (2-1).

[Formula 17]

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; ds</span>}}<span\; class="notranslate">\; =</span>\frac{\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}{<span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

Next, the change amount du of the unit tangent vector is considered. Fig. 5 shows the amount of change of the unit normal vector. Since the tangent direction does not change when it is a straight line, it is du(t)={0, 0, 0}, but it is not like this when it is a curve, the change of the unit tangent vector du at the position of the separation distance ds and the tangent vector u Orthogonal. This can also be clarified from the fact that if the relation u·u=1 is differentiated, the orthogonal relation u·du=0 can be obtained. What normalizes the change amount du of the unit tangent vector is the unit main normal vector n(t). That is, the unit main normal vector n(t) is represented by formula (2-3)

[Formula 18]

$\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{<span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

The normal direction is positive when the person faces the tangent direction to the left. More precisely, in the plane made by the vector du and the unit tangent vector u(t), the direction rotated 90 degrees counterclockwise from the unit tangent vector u(t) is defined as the unit principal normal vector n(t ) in the positive direction.

Also, the slave normal vector b(t) is a vector orthogonal to both the unit tangent vector u(t) and the unit main normal vector n(t), and is defined by Equation (2-4).

[Formula 19]

b(t)=u(t)×n(t) (2-4)

Define the defined unit tangent vector u(t), unit principal normal vector n(t), and normal vector b(t) as three vector groups {u(t), n(t), b(t)} , is called the Frenet Frame on the position R(t) of the curve.

Next, the curvature κ which is the rate at which the unit tangent vector bends along the line element of the curve will be described. The curvature in three dimensions is defined by formula (2-5).

[Formula 20]

$\mathrm{<span\; class="notranslate">\; \&kappa;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}{{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Regarding the basic quantity on the three-dimensional curve defined above, it is described by using the curve length variable S as a parameter in the three-dimensional clothoid curve.

When considering any three-dimensional clothoid curve P(S), the unit tangent vector u(S) can be expressed by formula (2-2) and formula (2-6).

[Formula 21]

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

Furthermore, the unit tangent vector u(S) can also be represented by the following equation (2-7) if the definition equations (1-7), (1-8), and (1-9) of the three-dimensional clothoid curve are considered. In this specification, these representations are mainly used.

[Formula 22]

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

Use formula (2-8) to express the curve length variable S1 order differential according to the unit tangent vector u(S) of the three-dimensional clothoid curve, and use formula (2-9) to express its magnitude.

[Formula 23]

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right\}$

(2-8)

$<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; =</span>\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

Next, consider the unit principal normal vector n(S). Since the normal vector of the three-dimensional curve is expressed by formula (2-3), the normal vector of the three-dimensional clothoid curve is expressed by formula (2-10).

[Formula 24]

$\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}$

$<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}}\left\{\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right\}$

(2-10)

Regarding the secondary normal vector b(S), it is stipulated that the unit tangent vector u(S) from the formula (2-7) and the unit main normal vector n(S) from the formula (2-10) are defined by the formula (2-4) Find out.

[Formula 25]

b(S)=u(S)×n(S) (2-11)

Finally, regarding the curvature, if the form (2-5) is changed, it can be expressed by formula (2-12).

[Formula 26]

$\mathrm{<span\; class="notranslate">\; \&kappa;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}{{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; =</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}$

(2-12)

From the above, the Fresnel frame and curvature κ at each point on the three-dimensional clothoid curve can be obtained from the curve length variable S.

(1-3) Generation of three-dimensional clothoid curves facing opposite

Consider generating a three-dimensional clothoid curve shown in Figure 6 that has the same size and shape as a certain three-dimensional clothoid curve but with the opposite orientation.

Assume that there is a start point P _s and an end point _Pe , the three-dimensional clothoid curve parameters, and a three-dimensional clothoid curve C ₁ determined by seven values including h, a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ . At this time, the tangential rotation angles α ₁ and β ₁ are represented by the following formulas (2-13) (2-14).

[Formula 27]

α ₁ =a ₀ +a ₁ S+a ₂ S ² (2-13)

β ₁ =b ₀ +b ₁ S+b ₂ S ² (2-14)

In generating a three-dimensional clothoid curve C ₂ with the same size and shape as the three-dimensional clothoid curve but with the opposite orientation, if the starting point is set as P' _s and P' _e is used as the end point, P' _s =P _e , P' _e = P _s . First consider the curve length h, but if the size is considered to be the same, the curve lengths are equal in the curves C ₁ and C ₂ . Next, if the tangent line t on the three-dimensional clothoid curve _C2 is considered to be oriented opposite to the tangent line t on the three-dimensional clothoid curve _C1 with the same coordinates in general, the tangent rotation angles _α1 , _β1 of the curve _C1 and the curve _C2 There is the following relationship between the tangential rotation angles α ₂ and β ₂ .

[Formula 28]

α ₂ (S)=α ₁ (1-S)+π (2-15)

β ₂ (S) = β ₁ (1-S) (2-16)

When these formulas are arranged, they are represented by the following formulas (2-7) (2-18).

[Formula 29]

α ₂ (S)=(a ₀ +a ₁ +a ₂ +π)-(a ₁ +2a ₂ )S+a ₂ S ² (2-17)

β ₂ (S)=(b ₀ +b ₁ +b ₂ )-(b ₁ +2b ₂ )S+b ₂ S ² (2-18)

Since the remaining parameters are thus determined, the convolution parameters h', a' ₀ , a' ₁ , a' ₂ , b' ₀ , b' ₁ , b' ₂ of the curve C ₂ can be used by using the parameters of the curve C ₁ Formula (2-19) represents.

[Formula 30]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&pi;</span>}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; h</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 19</span>}<span\; class="notranslate">\; )</span>$

Using this relational expression, it is possible to generate three-dimensional clothoid curves having the same size and shape but opposite directions.

(1-4) Segmentation of three-dimensional clothoid curve

Assume that there is a start point P ₁ and an end point P ₂ , the three-dimensional clothoid curve parameters, and a three-dimensional clothoid curve C ₀ determined by seven values including h, a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ . At this time, as shown in Figure 7, the curve length variable in the application is the point P _m of S=S _d to divide the three-dimensional clothoid curve C ₀ connecting the points P ₁ and P ₂ , and consider the method of dividing it into curves C ₁ and C ₂ .

Consider the curve C ₁ starting from the point P ₁ among the divided curves. If the curve length h is considered, it can be seen from the definition of the three-dimensional clothoid curve that the curve length _{h 1} of the curve C 1 is equal to S _d times the curve length _{h 0 of} the curve C 0 . In addition, if the curve length variable of the curve C ₀ when representing a point on the curve C ₁ is S ₀ , and the curve length variable of the curve C ₁ is S ₁ , the following relationship is established between them.

[Formula 31]

S ₁ =S _d S ₀ (2-20)

That is, it is found that the following relationship exists between the tangential rotation angles α ₀ , β ₀ of the curve C ₀ and the tangential rotation angles α ₁ , β ₁ of the curve C ₁ .

[Formula 32]

α ₁ (S ₁ )=α ₀ (S _d S ₀ )

(2-21)

β ₁ (S ₁ )=β ₀ (S _d S ₀ )

When these formulas are rearranged, they are represented by the following formula (2-22).

[Formula 33]

α ₁ (S)＝a ₀ +a ₁ S _d S+a ₂ S _d ² S ²

(2-22)

β ₁ (S)=b ₀ +b ₁ S _d S+b ₂ S _d ² S ²

Since the direction of the tangent line is determined by _this , the parameters of the convolution h', a' ₀ , a' ₁ , a' ₂ , b' ₀ , b' ₁ , and b' ₂ of the curve C 1 adopt the parameters of the curve C ₀ , and the formula can be used (2-23) said.

[Formula 34]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="S\; d\; b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">S</figure-callout></span><figure-callout\; id="2"\; label="S\; d\; b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}& \\ {}_{}^{}{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; wxya</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; twenty\; three</span>}<span\; class="notranslate">\; )</span>$

Next, consider the curve C ₂ starting from the split point P _m . The curve C ₂ can be generated by combining the method of generating the curves described in 1-3 with the same size and shape but facing opposite directions, and the method used to generate the curve C ₁ .

First, a curve having the same size and shape as the curve C ₀ but facing in the opposite direction is set as the curve C' ₀ . The dividing point P _m is represented by P _m =C' ₀ (1-S _d ) on this curve. Here, if it is considered that the curve C' ₀ is divided by the point P _m , the curve C' ₂ starting from P ₂ in the divided curve becomes a curve with the same size and shape as the curve C ₂ but facing opposite. Since the curve C'2 can be generated by using the method described in 1-3 and the method used for the curve _C1 , here, if the method described in 1-3 is additionally used for the curve _C'2 , _it can be Curve _C2 is generated.

The convolution parameters h″, a″ ₀ , a″ ₁ , a″ ₂ , b″ ₀ , b″ ₁ , b″ ₂ of _the curve C 2 are expressed by the following formula (2-24) using the parameters of the curve C ₀ .

[Formula 35]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; \}</span>\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; \}</span>\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; twenty\; four</span>}<span\; class="notranslate">\; )</span>$

From the above, the curve can be divided into curves C ₁ and C ₂ by using the point P _m on the three-dimensional clothoid curve C ₀ whose curve variable is S=S _d .

(1-5) Characteristics of three-dimensional clothoid curves

(a) Continuity of the curve

In a clothoid curve (a clothoid curve represented by the same parameter), since the inclination angle and the deflection angle of the tangential direction are given according to the quadratic formula of the curve length or the curve length variable S, respectively, with respect to the curve length variable S, it can be guaranteed that 1 The normal direction obtained by the sub-differentiation, and the curvature obtained by the second-order differentiation are continuous. In other words, in a clothoid curve, the direction of the normal and the curvature are continuous. Therefore, smooth, well-characterized curves can be obtained. Even in the case of connecting two clothoid curves, it is possible to create a smooth connected curve by selecting parameters so that the tangent, normal, and curvature are continuous at their joints. Call it the clothoid group.

(b) Applicability

Since the tangent direction of the curve can be shared by two angles (inclination angle and deflection angle), three-dimensional curves that meet various conditions can be made arbitrarily, can be used for various purposes, and can provide a wide range of spatial curves necessary for the design of industrial products. method of occurrence. In the case that the object moves along the space curve with acceleration and deceleration, it is possible to design a smooth change of the constraint force. In addition, since changes in curvature can be appropriately designed with respect to the length of the curve, it can be effectively used in various industrial fields such as artistic curve design that requires aesthetics.

(c) Integration with geometric curves

Geometric curves such as straight lines, circular arcs, and spiral curves can be created by setting some of the convolution parameters to 0, or setting a specific functional relationship between several parameters. These curves are a type of clothoid curve and can be represented in a clothoid format.

Furthermore, since two-dimensional convolutions can be made by generally setting either of α or β to 0, resources already obtained for two-dimensional convolutions can be applied.

That is, by appropriately setting α or β, individual curves such as circular arcs and straight lines can also be expressed including known two-dimensional convolutions. Since the three-dimensional clothoid equation of the same form can be used for such individual curves, calculation procedures can be simplified.

(d) Inferred goodness

In conventional interpolation methods such as spline interpolation, when formulating a free curve, it is difficult to separate its overall form or local form. It is easy to grasp the overall image.

In addition, at the middle end expressed as a clothoid curve, values such as line length, tangent direction, and curvature are known, and recalculation is not required unlike conventional interpolation methods. That is, corresponding to the parameter S of the curve, the tangent, or normal, and curvature of the curve are directly obtained as shown in equations (1-7), (2-10) and (2-12).

(e) Ease of motion control

The principal variable of the curve is the length s or normalized length S, and the equation of the curve is given by the equation of nature relative to this length. Therefore, by determining the length s as a function of time t, motion characteristics such as acceleration and deceleration can be given arbitrarily, and by adopting a motion curve with good characteristics conventionally used for cams, etc., it is possible to increase the speed of machining operations. Since the length s can be given as a value in the actually existing Cartesian space, and the velocity and acceleration can be obtained with respect to the tangential direction, it is not necessary to combine the values given for each axis like the conventional interpolation method. In addition, since the calculation of the curvature is easy, it is also easy to obtain the centrifugal acceleration during exercise, and control in accordance with the trajectory of the exercise can be performed.

2. Using the interpolation method of the three-dimensional clothoid curve

(2-1) Mathematical conditions for smooth connection

In one three-dimensional clothoid curve, the shape expression of the curve has a limit. Here, a plurality of three-dimensional clothoid curves (three-dimensional clothoid segments) are connected for the main purpose of controlling the motion of a tool using numerical control, and the shape of an industrial product is designed through the plurality of three-dimensional clothoid curves. Hereinafter, the interpolation method using a three-dimensional clothoid curve will be referred to as three-dimensional clothoid interpolation. Hereinafter, the entire group of curves generated by interpolation is referred to as a three-dimensional clothoid curve, and the unit curves constituting it are referred to as a three-dimensional clothoid segment.

Fluidly connecting two three-dimensional clothoid segments at their endpoints is defined as continuously connecting endpoint positions, tangents, and curvatures. Using the above definition formula, this condition is described as follows. The first 3 formulas represent the continuity of the position, the next 2 formulas represent the continuity of the tangent line, the next 1 formula represents the coincidence of the normal line, and the last formula represents the continuity of the curvature.

[Formula 36]

Px _i (1) = Px _i+1 (0)

Py _i (1) = Py _i+1 (0)

Pz _i (1) = Pz _i+1 (0)

α _i (1) = α _i+1 (0)

β _i (1)=β _i+1 (0) (3-1)

tan γ _i (1) = tan γ _i+1 (0)

κ _i (1) = κ _{i + 1} (0)

This is to satisfy the condition that the tangent vector and the normal vector are continuous, and the curvature and α, β are continuous at the connection point, and sometimes the condition is too strict. Therefore, it is also possible to simply satisfy the conditions by changing the conditions as shown below.

[Formula 37]

Px _i (1) = Px _i+1 (0)

Py _i (1) = Py _i+1 (0)

Pz _i (1) = Pz _i+1 (0)

cos[α _i (1)-α _i+1 (0)]=1

(3-2)

cos[β _i (1)-β _i+1 (0)]=1

tan γ _i (1) = tan γ _i+1 (0)

κ _i (1) = κ _{i + 1} (0)

Here, in addition,

[Formula 38]

cos[α _i (1)-α _i+1 (0)]=1

If the above relationship is also taken into account,

[Formula 39]

tan γ _i (1) = tan γ _i+1 (0)

It is replaced by the following condition.

[Formula 40]

tan γ _i (1) = tan γ _i+1 (0)

$\frac{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}$

∵α′ _i (1)β′ _i+1 (0)=α′ _i+1 (0)β′ _i (1)

As a result, it was found that the objective can be achieved if the following conditions are satisfied.

[Formula 41]

Px _i (1) = Px _i+1 (0)

Py _i (1) = Py _i+1 (0)

Pz _i (1) = Pz _i+1 (0)

cos[α _i (1)-α _i+1 (0)]=1

cos[β _i (1)-β _i+1 (0)]=1 (3-3)

${\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

κ _i (1) = κ _{i + 1} (0)

In the formula (3-3), the first formula 3 represents the continuity of the position, the next formula 2 represents the continuity of the tangent line, the formula 1 below represents the coincidence of the normal line, and the last formula represents the continuity of the curvature. To perform ^G2 continuous interpolation, two three-dimensional clothoid curves are required to satisfy the seven conditional expressions of Equation (3-3) at their endpoints.

Supplementary about G ² continuous (G is the prefix of Geometry). Fig. 8 shows the conditions of ^G2 continuous interpolation.

The so-called G ⁰ continuity means that the two three-dimensional clothoid curves have the same endpoint position, the so-called G ¹ continuity means that the tangent direction is consistent, and the so-called G ² continuity means that the contact plane (normal line) and curvature are consistent. In Table 1 below, the C ⁰ -C ² continuum used for the comparative spline curve and the G ⁰ -G ² continuum used in the clothoid curve of the present invention are compared.

Table 1

C⁰: position G⁰: position C¹: first order differential coefficient G¹: Tangent direction C²: quadratic differential coefficient G²: contact plane (normal), curvature

When considering the continuity of two three-dimensional clothoid curves, the interpolation conditions become stricter as C ⁰ →C ¹ →C ² and G ⁰ →G ¹ →G ² are reached. In C ¹ continuity, the size and direction of tangents must be consistent, but in G ¹ continuity, only the direction of tangents may be consistent. When connecting tangent lines smoothly with two three-dimensional clothoid curves, it is preferable to make a conditional expression continuously using ^G1 . For example, if the spline curve is made into a conditional expression continuously with C ¹ , the condition is too strict due to the addition of the condition that the sizes of the tangent lines that have no relationship in geometry are consistent. If ^G1 is used to make the conditional expression continuously, it has the advantage of being able to freely set the size of the primary differential coefficient.

Make the contact plane (normal) consistent in ^G2 continuation. The so-called contact planes, as shown in FIG. 9 , refer to the planes S1 and S2 partially containing the curve C. FIG. 9 shows an example in which the point P is continuous in the tangential direction, but the contact planes S1 and S2 are discontinuous. When considering the continuity of a three-dimensional curve, the consistency of the contact plane must be considered after the consistency of the tangent direction. When discussing the curvature, it does not mean that the contact planes are not consistent, but the curvature needs to be consistent after the contact planes are consistent. Using two three-dimensional curves to make the coordinates, tangent direction, contact plane (normal line) and curvature consistent can meet the condition of G ² continuity.

(2-2) Specific calculation order

There are the following two calculation sequences.

(a) Given the parameters h, α, and β of the curve, a three-dimensional clothoid curve is generated, and at its endpoint, the parameters of the next three-dimensional clothoid curve are determined in a manner that satisfies formula (3-3). In this way, three-dimensional clothoid curves connected smoothly one by one can be generated. According to this calculation procedure, the curve parameters can be easily calculated, which is called a sequential solution. According to this method, curves of various shapes can be easily generated, but the connection points through which the curves pass cannot be clearly specified.

(b) The three-dimensional clothoid curves can be connected so that the point groups specified in advance become the connection points of the curves. Here, a short clothoid curve (clothoid segment) is created for each section of each discretely given point sequence. In this case, the calculation sequence to determine the curve parameters in a manner that satisfies formula (3-3) is more complex than (a), which is repeated convergence calculation. Since the curve parameters are determined inversely from the connection conditions, this calculation order is called an inverse solution.

Regarding the inverse solution of (b) above, the calculation method will be described in detail. The computational problem to be solved is formulated as follows.

Unknown parameter: Curve parameter

Constraints: Formula (3-3) or a part thereof

Depending on the required problem, varying the number of constraints can be used as unknown parameters to set the corresponding number of curve parameters. For example, in the case where the continuity of the curvature is not required, some curve parameters can be freely operated. Or, when the curvature is continuous and the tangent direction is specified, it is necessary to increase the number of three-dimensional clothoid curves used for interpolation through segmentation and increase the corresponding unknown curve parameters.

In order to make the above repeated convergence calculations converge stably, it is necessary to work hard on the calculations. In order to avoid calculation divergence and speed up convergence, an effective method for unknown parameters is to set better initial values. Therefore, an effective method is to generate a more simple interpolation curve that satisfies the given constraints such as connection points, such as a linear spline curve, etc., and calculate the curve parameters of the three-dimensional clothoid curve from its curve shape as a repeated convergence calculation the initial value of .

Alternatively, the method of sequentially increasing the conditional expressions without satisfying the constraint conditions that should be satisfied is also effective as a method for stably obtaining a solution. For example, the sequence in which the curves occur is divided into the following three STEPs, which are performed sequentially. After interpolating so that the position information and the tangent direction coincide in the first step, interpolate so that the normal direction coincides as the second step, and interpolate so that the curvature coincides in the third step. Fig. 10 shows a schematic flow of this method. The necessary three-dimensional clothoid equations and their definitions for tangents, normals, or curvatures are shown.

(2-3) Example using interpolation method of three-dimensional clothoid curve

(a) Flow of imputation method

An embodiment of a method for smoothly interpolating between a given point sequence by using a three-dimensional clothoid curve is described in detail.

As the basic flow of three-dimensional convolutional interpolation, each parameter of the three-dimensional cycloidal segment connecting the points of the interpolation object is used as an unknown, and the points of the interpolation object are strictly passed through, and the satisfaction is obtained by Newton-Raphson's method ^. Continuous conditional solutions generate curves. FIG. 11 is a diagram summarizing the outline of this flow. The so-called G ² continuity means that the two three-dimensional clothoid curves have the same position, tangent direction, normal direction and curvature at their endpoints.

(b) Conditions for ^G2 continuous interpolation

In the three-dimensional convolutional interpolation, specific conditions are considered regarding the condition that the point of the interpolation object passes closely and becomes ^G2 continuous.

Now, simply having 3 points P ₁ ={Px ₁ , Py ₁ , Pz ₁ }, P ₂ ={Px ₂ , Py ₂ , Pz ₂ } and P ₃ ={Px ₃ , Py ₃ , Pz ₃ }, Consider interpolating the point with a 3D clothoid segment. Fig. 12 shows three-dimensional convolutional interpolation of points _P1 , _P2 and _P3 . If the curve between connection points P ₁ and P ₂ is set as curve C ₁ , and the curve between connection points P ₂ and P ₃ is set as curve C ₂ , in this case, the unknown is the parameter of curve C ₁ a0 ₁ , a1 ₁ , a2 ₁ , b0 ₁ , b1 ₁ , b2 ₁ , h ₁ , and 14 parameters a0 ₂ , a1 _{2 , a2 2} , b0 ₂ , b1 ₂ , b2 ₂ , and _{h 2 of} the curve C ₂ . In addition, the subscripts of characters appearing in the description below correspond to the subscripts of the respective curves.

Next, consider the point that passes the interpolation object strictly and achieves the condition of _G2 continuity. First, the condition of passing the point of the interpolation object strictly at the starting point is considered from the definition of the three-dimensional clothoid curve, because it must be satisfied when the starting point is given, so there is no interpolation condition. Next, at the connection point P ₁ , three cases are established regarding the position, two cases are established regarding the tangent vector, two cases are established regarding the magnitude and direction of the conditional expression of curvature continuity, and a total of seven cases are established. Also about the end point, at point P ₂ is 3 in terms of position. The total number of conditional expressions obtained from the above is 10. However, since there are only 10 conditional expressions with respect to 14 unknowns in this way, the solution to the unknowns cannot be obtained. Therefore, in this study, the tangent vectors of the two endpoints are given, and two conditions are added for each of the two endpoints, so that the number of conditional expressions and unknowns are equal. In addition, if the tangent direction at the starting point is determined, since a0 ₁ and b0 ₁ can be obtained from the definition formula, they need not be treated as unknowns. Hereinafter, each condition is considered.

First, considering the positional conditions, the following three expressions (4-1), (4-2), and (4-3) are established. (Hereafter, the natural number i<3 is specified.)

[Formula 42]

${\mathrm{<span\; class="notranslate">\; Px</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Px</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}$

(4-1)

${\mathrm{<span\; class="notranslate">\; Python</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Python</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}$

(4-2)

${\mathrm{<span\; class="notranslate">\; Pz</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Pz</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}$

(4-3)

Next, if the tangential direction is considered, two equations (4-4) and (4-5) are established.

[Formula 43]

cos(a0 _i +a1 _i +a2 _i -a0 _i+1 )＝1 (4-4)

cos(b0 _i +b1 _i +b2 _i -b0 _i+1 )＝1 (4-5)

Regarding the magnitude of the curvature κ, the following formula (4-6) is established

[Formula 44]

κ _i (1)-κ _i+1 (0) = 0 (4-6)

Finally consider the normal direction vector n. The normal vector n of the three-dimensional clothoid curve is represented by formula (2-10).

Here, similar to the determination of the tangent vector u of the three-dimensional clothoid curve, the normal vector n is considered using the clothoid curve. For the initial tangent direction (1, 0, 0), a constant γ is used, and (0, cosγ, -sinγ)i is used to represent the initial normal direction. If it is convolved in the same way as the tangent, the normal n is shown in formula (4-7).

[Formula 45]

$\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\end{array}\right\}$

$<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right\}$

(4-7)

Comparing formulas (2-10) and (4-7), it can be seen that sinγ and cosγ correspond to formula (4-8).

[Formula 46]

$\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}}$

$\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

That is, it can be known from formula (4-8) that to achieve the continuity of the normals of the connection points in the three-dimensional convolution interpolation, as long as tanγ is continuous.

[Formula 47]

$\mathrm{<span\; class="notranslate">\; the\; tan</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

That is, the condition for knowing that the normal line is continuous is Equation (4-10).

[Formula 48]

tan γ _i (1) = tan γ _i+1 (0) (4-10)

Here, in addition, as

[Formula 49]

cos[α _i (1)-α _i+1 (0)]=1 (4-11)

Also under consideration, the conditional expression (4-10) can be replaced by the following conditional expression. That is, the condition for the continuity of the normal line is Equation (4-12).

[numeral formula 50]

α′ _i (1)β′ _i+1 (0)=α′ _i+1 (0)β′ _i (1) (4-12)

To sum up, it is known that the condition of passing the interpolation object point strictly and becoming _G2 continuous is as formula (4-13) at the tangent point. Also, some of these conditions can be selected even at the start point and the end point.

[Formula 51]

Px _i (1) = Px _i+1 (0)

Py _i (1) = Py _i+1 (0)

Pz _i (1) = Pz _i+1 (0)

cos[α _i (1)-α _i+1 (0)]=1

cos[β _i (1)-β _i+1 (0)]=1

${\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

κ _i (1) = κ _{i + 1} (0) (4-13)

From the above, we know that for the 12 unknowns a1 ₁ , a2 ₁ , b1 ₁ , b2 ₁ , h ₁ , a0 ₂ , a1 ₂ , a2 ₂ , b0 ₂ , b1 ₂ , b2 ₂ and h ₂ , the conditional expression holds Remember 12. (The tangential rotation angle of point P ₃ is set to α ₃ , β ₃ .)

[Formula 52]

Px ₁ (1) = Px ₂ (0)

Py ₁ (1) = Py ₂ (0)

Pz ₁ (1) = Pz ₂ (0)

cos[α ₁ (1)-α ₂ (0)]=1

cos[β ₁ (1)-β ₂ (0)]=1

α′ ₁ (1)β′ ₂ (0)=α′ ₂ (0)β′ ₁ (1)

κ ₁ (1) = κ ₂ (0)

Px ₂ (1) = Px ₃ (0)

Py ₂ (1) = Py ₃ (0)

Pz ₂ (1) = Pz ₃ (0)

cos[α ₂ (1)-α ₃ ]=1

cos[β ₂ (1)-β ₃ ]=1 (4-14)

In this way, since 12 equations are established for 12 unknowns, they can be solved. This can be explained by Newton Raphson's method, and the solution can be obtained.

In addition, generally, when interpolation of n point columns is considered, the conditional expression only needs to expand the above-mentioned natural number i to i<n. Then there is the question of the number of unknowns and conditional expressions.

For example, when there are n-1 point columns, it is considered that N unknown numbers and N relational expressions are established. Here, if it is assumed that one more point is added, seven unknowns will be added, including the three-dimensional clothoid segment P _n-1 , the cycloidal parameters a0 _n , a1 _n , a2 n , _{b0 n} , b1 _{n ,} b2 _n and h _n . On the one hand, the conditional expression, since the connection point increases by 1, so at the point P _n-1 , 3 are added in terms of position, 2 are added in terms of tangent vector, and 2 are added in terms of the magnitude and direction of the conditional expression with continuous curvature , adding 7 in total.

Since it is known that when n=3, there are 12 unknowns and relational expressions, so when n≥3, the unknowns are 7(n-2)+5, and the established formula is also 7(n-2)+ 5. In this way, since the unknown number is equal to the number of conditions related to it, the same method as that of 3 points can be used for n free point columns. As a solution method, the Newton-Raphson method using the relationship of the equations (4-15) and (4-16) established between the unknown number and the conditional expression is used. (Set the condition to be F, the unknown to be u, and the error Jacobian to be j.)

[Formula 53]

ΔF＝[J]Δu (4-15)

Δu＝[J] ^-1 ΔF (4-16)

From the above, it is possible to perform three-dimensional convolutional interpolation that closely passes through the interpolation target points for n point columns and achieves ^G2 continuity.

(C) Determination of initial value

In the Newton-Raphson method, appropriate initial values need to be given when starting the search for a solution. The initial value can be given in any way, but only one example of the way of giving the initial value will be described here.

In interpolation, first, it is necessary to determine the initial value of each unknown from the point sequence, but in this study, the interpolation target point sequence of the single shape of the polygon Q generated in Li et al.'s 3D Discrete Clothoid Splines has 4 For the vertices, their initial values are calculated from the polygon Q and determined. 3DDiscrete Clothoid Splines, strictly by interpolating object points, has the property that the curvature changes smoothly with respect to the moving distance from the starting point. In this specification, the initial value used for three-dimensional convolution interpolation is determined by calculation from here by making a polygon Q of 3D Discrete Clothoid Splines with r=4 as shown in FIG. 13 .

Next, 3D Discrete Clothoid Splines are supplemented. First, as shown in Figure 14, a polygon P with the point row of the interpolation object as the vertex is made, and the same number r of new vertices are inserted between the vertices of P, and it is made as $P\&Subset;Q$ The polygon Q. Here, if the number of vertices of P is set to n, there are rn vertices when the polygon Q is closed, and r(n-1)+1 vertices when the polygon Q is open. Subscripts are defined hereafter as consecutive numbers from the starting point, and each vertex is represented by qi. In addition, at each vertex, a normal vector b is determined as a direction, and a vector k having a curvature κ is determined as a magnitude.

At this time, when the equation (4-17) in which the vertices are equidistant from each other is satisfied, the curvature is closest to the condition proportional to the moving distance from the starting point (the function of the equation (4-18) is minimized When) polygon Q, called 3D Discrete Clothoid Splines.

[Formula 54]

|q _i-1 q _i |＝|q _i+1 q _i |, $(q_i\&NotElement;P)---(4-17)$

i＝{0...n-1}，Δ²k_i＝k_i-1-2k_i+k_i+1

i={0...n-1}, Δ ² k _i =k _i-1 -2k _i +k _i+1

(4-18)

In 3D Discrete Clothoid Splines, the Fresnel frame of each vertex has been calculated. Therefore, the parameters a ₀ and b ₀ are obtained from the unit tangential direction vector t. The tangential direction vector t is known when the polygon Q is obtained, and the tangential direction rotation angles α, β of the vertices of the polygon Q are obtained from this t and the tangent equation of the three-dimensional clothoid curve. From this, the initial values of a ₀ and b ₀ of each curve are obtained. Also, on the 3D clothoid segment from the start point, this value is given.

[Formula 55]

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 19</span>}<span\; class="notranslate">\; )</span>$

Here, regarding the 3D Discrete Clothoid Splines, if the vertices are arranged equidistantly, at point q _4i+1 in FIG. 13 , it can be approximated that the curve length variable S is 1/4. Also at point q _4(i+1)-1 , it can be approximated that the curve length variable S is 3/4. When these are considered together with the formula of α of the three-dimensional clothoid curve, the following formula (4-20) is established.

[Formula 56]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 20</span>}<span\; class="notranslate">\; )</span>$

This formula becomes a two-dimensional simultaneous equation with unknowns a1 _4i and a2 _4i , which is solved and used as the initial values of parameters a ₁ and a ₂ . It is likewise possible to determine initial values for the parameters b ₁ , b ₂ .

The remaining unknown is the curve length h, but its initial value can be calculated from the formula of the curvature of the three-dimensional clothoid curve. The curvature of the three-dimensional clothoid curve can be expressed by formula (4-21).

[Formula 57]

$\mathrm{<span\; class="notranslate">\; \&kappa;</span>}<span\; class="notranslate">\; =</span>\frac{\sqrt{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; twenty\; one</span>}<span\; class="notranslate">\; )</span>$

If this formula is changed to formula (4-22), the initial value of h can be determined.

[Formula 58]

${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\sqrt{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}{\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}}{{\mathrm{<span\; class="notranslate">\; \&kappa;</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}}$

(4-22)

Using the above method, initial values can be determined for the seven three-dimensional convolution parameters. Using this determined initial value, under the condition of achieving ^G2 continuity described in (b), the approximate value of the parameters of each curve is obtained by the Newton-Raphson method. A three-dimensional clothoid segment is generated from the parameters thus obtained, and a three-dimensional clothoid curve is used to interpolate between point columns.

(d) Imputation example

As an example of actually interpolating a point sequence using the method described above, three-dimensional convolutional interpolation (0.0, 0.0, 0.0), (2.0, 2.0, 2.0), (4.0, 0.0, 1.0) and (5.0, 0.0, 2.0) are given ) Examples of these 4 points. A perspective view of a three-dimensional clothoid curve generated by interpolation is shown in FIG. 15 . The solid line in Fig. 15 is a three-dimensional clothoid curve, and the straight line of dotted line, dot-dash line, and two-dot-dash line is that the size of each point on the curve is log (radius of curvature+natural logarithm e), and the direction is normal The radius of curvature variation pattern along the line.

In addition, Table 2 shows the parameters of each curve, and Table 3 shows the coordinates of each tangent point, tangent line, normal line, and curvature deflection. As can be seen from these tables, a three-dimensional clothoid curve that is ^G2 continuous is generated at each tangent point. In addition, FIG. 16 is a curvature change graph in which the moving distance from the starting point is taken on the horizontal axis and the curvature is taken on the vertical axis.

Table 2

The mode of each 3D clothoid segment

Curve 1 (Curvature Radius Variation Mode Dashed Line)

α＝-0.657549-1.05303S+1.84584S ²

β=1.03297+1.29172S- ^2.55118S2

h＝3.82679

_P0 = (0.0, 0.0, 0.0)

Curve 2 (curvature radius change mode dot-dash line)

α＝0.135559+2.18537S- ^2.69871S2

β=-0.226655-3.15603S+3.03298S ²

h＝3.16932

_P0 = (2.0, 2.0, 2.0)

Curve 3 (curvature radius change mode two dotted line)

α＝-0.377569-1.45922S+0.984945S ²

β=-0.349942+1.32198S- ^0.873267S2

h＝1.43987

_P0 = (4.0, 0.0, 1.0)

table 3

Coordinates, tangents, normals, curvature skews at each tangent point

The connection point of curve 1 and curve 2

Coord: (1.16×10 ^-5 , 2.00×10 ^-6 , 3.82×10 ^-6 )

Tvector: (7.59×10 ^-5 , 1.50×10 ^-5 , 2.95×10 ^-4 )

Nvector: (2.93×10 ^-4 , 9.19×10 ^-5 , -7.57×10 ^-6 )

Curvature: 3.06×10 ^-7

The connection point of curve 2 and curve 3

Coord: (-4.33×10 ^-6 , -1.64×10 ^-6 , 1.11×10 ^-5 )

Tvector: (2.06×10 ^-6 , 2.33×10 ^-4 , 1.97×10 ^-4 )

Nvector: (3.30×10 ^-4 , 1.19×10 ^-5 , -3.23×10 ^-5 )

Curvature: 5.96×10 ^-6

(2-4) ^G2 continuous three-dimensional convolution interpolation considering the control of each value at both ends

(a) Imputation conditions and unknowns

As described in (2-3), when the curve is turned on, when there are n points to be interpolated, the three-dimensional convolution interpolation point sequence is performed using n-1 curves. If strictly passing through each point, for each three-dimensional clothoid segment, since there are 7 unknowns such as a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , h, etc., the total number of unknowns is 7(n-1) . On the other hand, regarding the conditional expression, since there are 7 each of coordinates, tangents, normals, and curvatures and 3 coordinates on the end point for n-2 connection points, all of them are 7(n-2)+ 3. In the method (2-3), four conditions are added by giving the tangent vectors at the start point and the end point, and the conditional expression is made to be opposite to the number of unknowns.

Here, if the tangent, normal, and curvature at the start point and end point are controlled, and the interpolation is achieved in a ^G2 continuous manner, the condition is compared with that of controlling the tangent at both ends. In addition, at the start point and end point, in terms of normal line and curvature Add 2 each, and add 4 in total. Thus, all the conditional expressions reach 7n-3. In this case, the Newton-Raphson method cannot be used to solve the problem because the number of unknowns is less than the condition. Therefore, what method needs to be used to increase the unknown.

Therefore, here, the unknown number and the number of the conditional expression are equalized by reinserting interpolation target points. For example, if there are more than one of four unknowns, two new points are inserted, and two of the coordinates of each point are treated as unknowns.

In this case, since two connection points are added, 14 of each of seven coordinates, tangents, normals, and curvatures are added for each connection point condition. On the other hand, since two three-dimensional clothoid segments are added to the unknowns, a total of seven of a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , and h is added to the total of 14 unknowns. Since the number of points contained in the point column is n+2 at this time, if considered as a whole, the number of unknowns reaches 7(n+1), and the number of conditional expressions reaches 7(n+1)+4. Here, assuming that two of the coordinates of the newly inserted point are treated as unknowns, the number of unknowns increases by four. Therefore, the number of unknowns and conditional expressions reaches 7(n+2)-3, and the solution of the unknowns can be obtained. In this way, by inserting new points, it is possible to perform interpolation that strictly passes through each given point, is ^G2 continuous, and controls the tangent, normal, and curvature of both ends.

Also, consider the general situation. When interpolating n point columns, the number of points inserted when controlling m items at both ends and the number of coordinates handled as unknowns at this point are considered. It is also described above, but when the curve is turned on, n-1 curves are used to interpolate the point sequence. If each point is strictly passed, since there are 7 unknowns of a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , h for each three-dimensional clothoid segment, there are 7(n-1) unknowns as a whole. On the one hand, regarding the conditional expression, since there are n-2 connection points, there are 7 each of coordinates, tangents, normals, and curvatures, and 3 coordinates on the end point, so all of them are 7(n-2)+3 , the number of conditional expressions is less than 4. That is, there are four or more items to be controlled at both ends. Hereinafter, in the description, m is a natural number greater than or equal to 4, k is a natural number greater than or equal to 2, and a method of making the conditional expression equal to the number of the unknown when inserting a new point will be described.

(i) When m=2k

When both ends are combined to control m=2k items, there are 7(n-1) unknowns and 7(n-1)-4+2k overall conditional expressions. At this time, there are 2k-4 excess conditional expressions. Now, if we consider reinserting k-2 points, since the three-dimensional clothoid segment increases by k-2 and the connection points increase by k-2, there are 7 (n+k-3) unknowns and 7 ( n+k-3)-4+2k pieces. Here, it is also assumed that two of the values of the coordinates of each newly inserted point (for example, x and y) are handled as unknowns, and the total number of unknowns is 7(n+k-3)+2(k-2). The conditional expression The overall number is 7(n+k-3)+2(k-2), and the number of unknowns and conditional expressions are equal.

(ii) When m=2k+1

When both ends are combined to control m=2k+1 items, the total number of unknowns is 7(n-1), and the total number of conditional expressions is 7(n-1)+2k-3. At this time, there are 2k-3 excess conditional expressions. Now, if you consider reinserting k-1 points, since the three-dimensional clothoid segment increases by k-1, and the connection points increase by k-1, the total number of unknowns is 7(n+k-2), and the total number of conditional expressions is 7( n+k-2)-3+2k pieces. Here, it is also assumed that two of the values of the coordinates of each newly inserted point (for example, x and y) are treated as unknowns, and there are 7(n+k-2)+2(k-2) unknowns as a whole. The conditional expression The overall number is 7(n+k-2)+2k-3, and the number of conditional expressions is one more. Therefore, at one of the points to be inserted when m=2k+1, only one of the coordinate values is handled as an unknown. In this way, the overall number of unknowns is 7(n+k-2)+2(k-2), the overall number of conditional expressions is 7(n+k-2)+2(k-2), and the number of unknowns and conditional expressions The numbers are equal.

In the method described above, even in various situations such as when controlling the tangent line, normal line, and curvature other than the tangent line, normal line, and curvature, such as the tangent line angle of rotation α, by adjusting the unknown number in the coordinates of the inserted point by comparing with the added condition In this case, the unknown number and the number of the conditional expression can also be compared, and the values at both ends can be controlled theoretically. In addition, regarding control items, unknowns, and conditional expressions, Table 4 lists the summed numbers

Table 4

The number of control items and unknowns and conditional expressions at both ends in interpolation of n points

^* k: natural number above 2

(b) method

The interpolation method using the three-dimensional convolution that controls each value at the start point and the end point is performed in the following flow as shown in Fig. 17 and Fig. 18 .

Step1) Using only 4 of the conditions to be controlled, perform interpolation that strictly passes through the interpolation object points, and ^G2 continuous interpolation, to generate a curve.

Step2) Insert a new point on the generated curve, adjust the conditional expression and the number of unknowns.

Step3) Using the curve parameters of Step1 as initial values, use the Newton-Raphson method to obtain the approximate values of the parameters of each curve that meet the objective conditions.

Hereinafter, supplementary explanations are given for each Step. First in Step1, as long as the tangent direction is controlled, the method (2-3) can be used to generate the curve. Also, even when the tangential direction is not controlled, the same initial values as in the method (2-3) are used as initial values when obtaining the parameters of the curve.

Next, insert a new point in Step2, and adjust the conditional expression and the number of unknowns. At this time, the number of points to be newly inserted is one or less among the interpolation target points as much as possible. In addition, as the interpolation point, an intermediate point between the three-dimensional clothoid segments generated in Step 1 between the interpolation objects is inserted. In addition, the inserted points should be inserted sequentially from both ends. That is, the first inserted point is between the start point and its adjacent point, and between the end point and its adjacent point.

Finally about Step3, but the initial values for the Newton-Raphschen method performed in Step3 need to be re-determined. Therefore, for a curve to which a new point is inserted, the curve is divided by the method of dividing a three-dimensional clothoid curve as described in (1-4), and determined from each value of the generated curve. For the curve without point insertion, directly adopt the value of the curve generated in Step1. Above, the initial value of each parameter of the curve in Step3 is determined. Using this initial value, a three-dimensional clothoid curve is generated from parameters obtained by the Newton-Raphson method, and interpolation between point rows is performed using a three-dimensional clothoid curve satisfying the objective condition.

(C) Interpolation example

An example of performing three-dimensional convolution interpolation is shown in which the tangent, normal, and curvature at both ends are actually controlled using the conditions in Table 5. Consecutive numbers are allocated to interpolation target points that should pass strictly to form P ₁ , P ₂ , and P ₃ .

Table 5 Conditions for each point of the interpolation object and the start point and end point

coordinate unit tangent vector principal normal vector curvature P₁ (0,0,0) (Cos(θ), Sin(θ), 0) (-Sin(θ), Cos(θ), 0) 0.2 P₂ (4, -4, -4) — — — P₃ (8, -4, -5) (1,0,0) (0, -1, 0) 0.2

*θ＝－(π/6)

Fig. 19 shows the results of actually performing interpolation under these conditions. A solid line represents a three-dimensional clothoid curve, and a dotted line, a one-dot chain line, a two-dot chain line, and a three-dot chain line represent changes in the radius of curvature of each curve. In addition, FIG. 20 is a graph showing the relationship between the movement distance and the curvature from the starting point of each curve corresponding to the line type of the curve in FIG. 19 . It can be seen from the figure that the generated curve meets the conditions given in Table 6.

Differences between the values given in Table 6 and the tangent, normal, and curvature of the start point and end point of the generated curve

(d) Control of values at intermediate points

Using the method of (b), continue to control the values at both ends, and perform ^G2 continuous interpolation. Here, it is considered to control the value not at both ends but at an intermediate point.

For example, in the case of interpolating a point sequence as shown in FIG. 21, it is considered to control the tangent and the normal at the intermediate point _Pc . However, the value at the intermediate point cannot be controlled in the aforementioned method. So here by splitting the point column into 2, the value at the middle point is controlled.

That is, the point sequence is not interpolated at one go, but is interpolated by dividing it into a curve _C1 and a curve _C2 with the middle point _Pc in between. In this case, since the point P _c corresponds to an end point, the value can be controlled by adopting the method (b).

As a result of separating the points with values to be controlled, controlling the values at both ends, and performing interpolation, as long as the generated curves are connected, it is theoretically possible to perform three-dimensional control of tangent, normal, and curvature at each point. Convolutional interpolation.

(2-5) Three-dimensional convolutional interpolation to control the tangent, normal, and curvature of the two ends

(a) Flow of the method

The interpolation method using the three-dimensional convolution that controls each value at the start point and the end point can be performed in the following flow shown in FIG. 22 . Hereinafter, description will be made along this flow.

(b-1) Give the point of the interpolation object

In this example, 3 points {0.0, 0.0, 0.0}, {5.0, 5.0, 10.0}, {10.0, 10.0, 5.0} of the three-dimensional space are given. Table 7 summarizes the conditions of tangent, normal, curvature, etc. given at other points.

Table 7 Conditions for each point of the interpolation object and the start point and end point

coordinate unit tangent vector principal normal vector curvature P₁ (0.0, 0.0, 0.0) {0.0, 1.0, 0.0} {1.0, 0.0, 0.0} 0.1 P₂ (5.0, 5.0, 10.0) — — — P₃ (10.0, 10.0, 5.0) {1.0, 0.0, 0.0} {0.0, -1.0, 0.0} 0.1

(b-2) Generation of 3DDCS with r=4

In the Newton-Raphson method, appropriate initial values need to be given when starting the search for a solution. Here, preparations for finding the initial value are performed. 3D Discrete Clothoid Splines, which is the first research, has the property that the curvature changes smoothly with respect to the moving distance from the starting point by strictly interpolating the target point. Therefore, in this study, the polygon Q of 3D Discrete Clothoid Splines with r=4 as shown in Figure 23 is made, and the initial value for three-dimensional convolution interpolation is determined from here by calculation. In addition, FIG. 24 shows the polygon actually generated by this point sequence, and the coordinates of the vertices are listed in Table 8.

Table 8 Vertex coordinates of the generated polygon

Vertex coordinates P₁ {0.0, 0.0, 0.0} {0.4677, 0.4677, 3.1228} {0.9354, 0.9354, 6.2456} {2.3029, 2.3029, 9.4966} P₂ {5.0, 5.0, 10.0} {6.7095, 6.7095, 9.9244} {8.0655, 8.0655, 8.4732} {9.0327, 9.0327, 6.7366} P₃ {10.0, 10.0, 5.0}

(b-3) Determination of initial value

To use the Newton-Raphson method to solve, it is necessary to determine the initial value of each unknown. In this method, the approximate value of each unknown is obtained using the polygon Q generated in (b-2), and the value is determined. In 3D Discrete Clothoid Splines, the Fresnel frame of each vertex has been calculated. Therefore, the parameters a ₀ and b ₀ are obtained from the unit tangent direction vector t of the polygon Q generated in (b-2). The tangential direction vector t is known when the polygon Q is obtained, and the tangential direction clothoid angles α, β of the vertices of the polygon Q are obtained from this t and the tangent equation of the three-dimensional clothoid curve. From this, the initial values of a ₀ and b ₀ of each curve are obtained. Also, on the 3D clothoid segment from the start point, the value is given.

[Formula 59]

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}$

Here, regarding the 3D Discrete Clothoid Splines, if considering that the vertices are arranged equidistantly, at the point q _4i+1 in Fig. 23, it can be approximated that the curve length variable S is 1/4. Also at point q _4(i+1)-1 , it can be approximated that the curve length variable S is 3/4. When these are considered together with the expression of α of the three-dimensional clothoid curve, the following expression is established.

[Formula 60]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}\right.$

This formula becomes a two-dimensional simultaneous equation with unknowns a1 _4i and a2 _4i , which is solved and used as the initial values of parameters a ₁ and a ₂ . It is likewise possible to determine initial values for the parameters b ₁ , b ₂ .

The remaining unknown is the curve length h, but its initial value can be calculated from the formula of the curvature of the three-dimensional clothoid curve. The curvature of the three-dimensional clothoid curve can be expressed as follows.

[Formula 61]

$\mathrm{<span\; class="notranslate">\; \&kappa;</span>}<span\; class="notranslate">\; =</span>\frac{\sqrt{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}}{\mathrm{<span\; class="notranslate">\; h</span>}}$

If this formula is changed to the following formula, the initial value of h can be determined.

[Formula 62]

${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\sqrt{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}{\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}}{{\mathrm{<span\; class="notranslate">\; \&kappa;</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}}$

Using the above method, it is possible to determine initial values for the seven three-dimensional convolution parameters.

Table 9 shows the initial values actually obtained by this method.

Table 9 Initial value

(b-4) Pass through each point strictly, ^G2 continuous three-dimensional convolution interpolation

Using the initial values determined by (b-3), under the condition of achieving ^G2 continuity, use the Newton-Raphson method to obtain approximate values of the parameters of each curve. A three-dimensional clothoid segment is generated from the parameters thus obtained, and a three-dimensional clothoid curve is used to interpolate between point columns.

Here, in the 3-point three-dimensional convolutional interpolation, specific conditions are considered regarding the condition that the interpolation target points are passed strictly and ^G2 continuity is achieved. Fig. 25 shows three-dimensional convolutional interpolation of points P ₁ , P ₂ , and P ₃ . If the curve between connection points P ₁ and P ₂ is taken as curve C ₁ , and the curve between connection points P ₂ and P ₃ is taken as curve C ₂ , since a0 ₁ and b0 ₁ are known, the unknown is curve C ₁ parameters a1 ₁ , a2 ₁ , b1 ₁ , b2 ₁ , h ₁ , and 12 parameters a0 ₂ , a1 2 , a2 ₂ , _{b0 2} , b1 ₂ , b2 ₂ , h ₂ of curve C ₂ . The subscripts of the text appearing in the description later correspond to the subscripts of each curve, as a function of the curve length variable S, such as Pxi _, Py _i , Pzi _, α _i , β _i , ni _, and κ _i , indicating that each Coordinates on the curve, tangent turning angle α, β, normal, curvature.

First of all, the condition of strictly passing the interpolation object point at point _P1 , if considered from the definition of the three-dimensional clothoid curve, must be achieved when the starting point is given. Also, regarding the tangential direction, since it is already given as a known value, the condition on the point _P1 is not particularly specified.

Next, consider point _P2 . Point P ₂ is the connection point between the curves, to achieve G ² continuity requires position, tangent, normal, and curvature continuity. That is, the conditions to be established at the point _P2 are as follows.

[Formula 63]

Px ₁ (1) = Px ₂ (0)

Py ₁ (1) = Py ₂ (0)

Pz ₁ (1) = Pz ₂ (0)

cos[α ₁ (1)-α ₂ (0)]=1

cos[β ₁ (1)-β ₂ (0)]=1

n ₁ (1)·n ₂ (0)=1

κ ₁ (1) = κ ₂ (0)

Finally consider point _P3 . The point _P3 is the end point, and since the conditions to be satisfied are only the position and the tangent, the following five conditions are satisfied. Here, α ₃ , β ₃ are considered to be tangential turning angles α, β that determine the tangent vector at a given end point.

[Formula 64]

Px ₂ (1) = Px ₃

Py ₂ (1) = Py ₃

Pz ₂ (1) = Pz ₃

cos[α ₂ (1)-α ₃ ]=1

cos[β ₂ (1)-β ₃ ]=1

From the above, we know that for the 12 unknowns a1 ₁ , a2 ₁ , b1 ₁ , b2 ₁ , h ₁ , a0 ₂ , a1 ₂ , a2 ₂ , b0 ₂ , b1 ₂ , b2 ₂ , h ₂ , the conditional expression holds Remember 12. The conditional expression for induction is as follows.

[Formula 65]

Px ₁ (1) = Px ₂ (0)

Py ₁ (1) = Py ₂ (0)

Pz ₁ (1) = Pz ₂ (0)

cos[α ₁ (1)-α ₂ (0)]=1

cos[β ₁ (1)-β ₂ (0)]=1

n ₁ ·n ₂ =1

κ ₁ (1) = κ ₂ (0)

Px ₂ (1) = Px ₃

Py ₂ (1) = Py ₃

Pz ₂ (1) = Pz ₃

cos[α ₂ (1)-α ₃ ]=1

cos[β ₂ (1)-β ₃ ]=1

Since 12 equations are established for 12 unknowns, they can be solved. Use the Newton-Raphson method to solve this equation and find the solution. Table 10 lists the initial values and solutions.

Table 10 Initial value and solution

(b-5) Curve generation

Fig. 26 shows both the curve generated based on the parameters obtained in (b-4) and the polygon generated in (b-2). The curve of the solid line is the curve C ₁ , and the curve of the broken line is the curve C ₂ . At this stage, a ^G2 continuous three-dimensional clothoid curve is formed in which the tangential direction is controlled at the start point and the end point.

(b-6) Conditional expressions and unknowns

Here, it is additionally considered that the normal and curvature on the starting point _P1 and the ending point _P3 are also determined as the values given in Table 7. To control the normal line and curvature at the start point and end point, two additional conditions on the start point and end point are required. However, in a state where four conditions are added, a solution satisfying the conditions cannot be found from the viewpoint of the relationship with the unknowns. Therefore, in order to make the unknown number and the number of the conditional expression correspond, as shown in FIG. 27 , point DP ₁ is reinserted at the position of curve length variable S=0.5 of curve C ₁ . Furthermore, for the curve C ₂ , the point DP ₂ is also reinserted at the position of the curve length variable S=0.5.

In this case, let the curve connecting point P ₁ and point DP ₁ be curve C' ₁ , let the curve connecting point DP ₁ and point P ₂ be curve C' ₂ , and let the curve connecting point P ₂ and point DP ₂ be curve C' ₃ , let the curve connecting the point DP ₂ and the point P ₃ be the curve C' ₄ . The subscripts of the text appearing in the explanation later correspond to each curve name, for example, as a function of the curve length variable S, such as Px _c , Py _c , Pz _c , α _c , β _c , n _c , κ _c , which represent the curve C Coordinates on , tangent turning angle α, β, normal, curvature. In addition, at the start point and end point, at the start point such as Px _s , Py _s , Pz _s , α _s , β _s , n _s , κ _s , at the end point such as Px _e , Py _e , Pz _e , α _e , β _e , n _e and K _e represent coordinates, tangential turning angles α, β, normal, and curvature.

Conditions established at each point will be described below.

[Formula 66]

Point P ₁ : tangent, normal, curvature: 4

cos[α _C′1 (0)-α _s ]=1

cos[β _C′1 (0)-β _s ]=1

n _C′1 (0) n _s =1

κ _C′1 (0)=κ _s

Point DP ₁ : position, tangent, normal, curvature: 7

Px _C'1 (1) = Px _C'2 (0)

Py _C'1 (1) = Py _C'2 (0)

Pz _C'1 (1) = Pz _C'2 (0)

cos[α _C′1 (1)-α _C′2 (0)]=1

cos[β _C′1 (1)-β _C′2 (0)]=1

n _C′1 (1)·n _C′2 (0)=1

κ _C'1 (1) = κ _C'2 (0)

Point P ₂ : position, tangent, normal, curvature: 7

Px _C'2 (1) = Px _C'3 (0)

Py _C'2 (1) = Py _C'3 (0)

Pz _C'2 (1) = Pz _C'3 (0)

cos[α _C′2 (1)-α _C′3 (0)]=1

cos[β _C′2 (1)-β _C′3 (0)]=1

n _C′2 (1)·n _C′3 (0)=1

κ _C′2 (1)=κ _C′3 (0)

Point DP ₂ : position, tangent, normal, curvature: 7

Px _C'3 (1) = Px _C'4 (0)

Py _C'3 (1) = Py _C'4 (0)

Pz _C'3 (1) = Pz _C'4 (0)

cos[α _C′3 (1)-α _C′4 (0)]=1

cos[β _C′3 (1)-β _C′4 (0)]=1

n _C′3 (1)·n _C′4 (0)=1

κ _C′3 (1)=κ _C′4 (0)

Point P ₃ : position, tangent, normal, curvature: 7

Px _C'4 (1) = Px _e

Py _C'4 (1) = Py _e

Pz _C'4 (1) = Pz _e

cos[α _C′4 (1)-α _e ]=1

cos[β _C′4 (1)-β _e ]=1

n _C′4 (1)·n _e =1

κ _C'4 (1) = κ _e

There are 32 conditional expressions that should all be satisfied as described above. Here, each curve has seven convolution parameters of a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , and h, and since there are four curves, there are 28 unknowns. However, in this way, since the unknown number and the number of the conditional expression are not equal, the solution cannot be obtained. Therefore, the y and z coordinates of the reinserted two points DP ₁ and DP ₂ are treated as unknowns, and four unknowns are added. By doing this, there are 32 unknowns and conditional expressions, and a solution can be obtained.

(b-7) Determination of initial value

In order to obtain a solution satisfying the conditional expression established in (b-6), the Newton-Raphson method is used, but the initial value of the unknown is determined in order to increase the convergence rate. As a method, four three-dimensional clothoids are created by dividing the three-dimensional clothoids generated in (b-5) before and after the newly inserted point as shown in FIG. 28, and their clothoid parameters are given.

Regarding the division method of _the curve, if the method of dividing the curve C ₁ into the curve C' ₁ and the curve C' ₂ is explained, the convolution parameters h', a' ₀ , a' ₁ , a' ₂ , b' of the curve C' 1 ₀ , b' ₁ , and b' ₂ are represented by the following formula using the parameters of the curve C ₁ . Here Sd is the curve length variable at the split point, which is 0.5 here.

[Formula 67]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="S\; d\; b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">S</figure-callout></span><figure-callout\; id="2"\; label="S\; d\; b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}& \\ {}_{}^{}{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; wxya</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}\end{array}\right.$

Consider next the curve C' ₂ starting from the split point DP ₁ . First of all, if the curve with the same size and shape as the curve C ₁ but facing the opposite direction is taken as the curve C” ₁ , the convolution parameters of the curve h”, a” ₀ , a” ₁ , a” ₂ , b” ₀ , b” ₁ , b” ₂ , the parameters of the curve using curve C ₁ are represented by the following formula.

[Formula 68]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&pi;</span>}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; h</span>}\end{array}\right.$

On this curve, the dividing point DP ₁ is represented by DP ₁ =C” ₁ (1-S _d ). Here, if considering dividing the curve C” ₁ at the point DP ₁ , the point P ₂ in the divided curve is taken as The curve C" ₂ at the starting point becomes a curve having the same size and shape as the curve C" ₂ but facing in the opposite direction. _{The curve C" 2 can be generated by the method of generating the curve C'1. Here, the curve C 2 can be} generated by generating a curve having the same size and shape but facing opposite to the curve C" ₂ .

Using the above method, the curve C ₁ can be divided into C' 1 and C _{' 2 at} the point DP ₁ where the curve length variable S=0.5 on the three-dimensional clothoid curve C ₁ . Using the same method, the curve C ₂ can also be divided into C' ₃ and C _{' 4 at} the point DP ₂ on the curve C 2 where the curve length variable S=0.5.

Table 11 lists the parameters of the 4 curves segmented by this method. The parameters of this curve are used for the initial values of the Newton-Raphson method used to obtain a solution satisfying the conditional expression established in b-4.

Table 11 Parameters of the curve generated by segmentation

(b-8) Calculate the convolution parameter satisfying the condition

Based on the initial value determined in (b-7), a solution satisfying the conditional expression established in (b-6) is found by the Newton-Raphson method. Table 12 shows the calculated parameters of each curve. In addition, Table 13 shows the difference between the given value and the tangent, normal, and curvature of the start point and end point of the generated curve.

Table 12 Parameters of the generated curves

Differences between the values given in Table 13 and the tangent, normal, and curvature of the start point and end point of the generated curve

(b-9) Curve generation

Fig. 29 shows a graph generated from the parameters obtained in (b-8). The solid line represents the three-dimensional clothoid curve, and the dotted line, one-dot-dash line, two-dot-dash line, and three-dot-dash line indicate that the direction of each curve is in the direction of the main normal, and the size is the radius, which satisfies the natural logarithm, and takes the logarithmic radius of curvature change model. In addition, FIG. 30 is a graph showing the relationship between the moving distance s from the starting point of each curve corresponding to the line type in FIG. 29 and the curvature κ. It can be seen from the figure that the generated curve meets the conditions given in Table 12.

In the above, an example of generating a curve using the three-dimensional convolutional interpolation method in which the tangent, the normal, and the curvature are controlled at both ends has been described.

3. The design method of the return path of the ball screw using three-dimensional convolution interpolation

As an application example in the mechanical design of the three-dimensional clothoid curve, the design of the return path of the ball screw of the inverter system is performed.

(3-1) Explanation of the ball screw of the inverter type

31 to 35 show the ball screw of the inverter type. The reverser constitutes the return path of the balls rotating along the lead screw groove. The inverter is formed separately from the nut and fixed to the nut, or formed integrally with the nut. Figure 31 shows the manner in which the reverser is formed separately from the nut.

Hereinafter, a ball screw in which the inverter and the nut are integrated will be described. Fig. 32 shows the nut 1 of the ball screw in which the reverser is integrated with the nut. On the inner peripheral surface of the nut 1, a loaded ball rolling groove 2 is formed as a helical loaded rolling element rolling groove with less than one turn. The loaded ball rolling groove 2 has a lead corresponding to the ball rolling groove of the screw shaft described later. The ball rolling groove 3 as a return path connects one end and the other end of the loaded rolling groove, and has a lead in the direction opposite to that of the loaded ball rolling groove 2 . These loaded ball rolling grooves 2 and ball rolling grooves 3 form a roll groove 4 . 33A is a perspective view of the nut 1 with the ball circulation groove 3 seen, and FIG. 33B is a perspective view of the nut 1 with the loaded ball rolling groove 2 seen.

Fig. 34 shows the state where the nut 1 is assembled on the screw shaft.

On the outer peripheral surface of the screw shaft 5, a ball rolling groove 6 is formed as a spiral rolling element rolling groove having a predetermined lead. The loaded ball rolling groove 2 of the nut 1 is opposed to the ball rolling groove 6 of the screw shaft 5 . Between the loaded ball rolling groove 2 and the ball circulation groove 3 of the nut 1 and the ball rolling groove 6 of the screw shaft 5, a plurality of balls are arranged as a plurality of rotatable rolling elements. As the nut 1 rotates relative to the rotating shaft 5 , a plurality of balls rotate while receiving a load between the loaded ball rolling groove 2 of the nut 1 and the ball rolling groove 6 of the screw shaft 5 .

The ball circulation groove 3 of the nut 1 shown in FIG. 32 corresponds to the reverser shown in FIG. 31 . The ball circulation groove 3 is such that the balls rolling along the load ball rolling groove 2 of the rotating shaft 5 go around the circumference of the screw shaft 5 and return to the original load ball rolling groove. Thread 7 of shaft 5.

The circulation path of the conventional model was created by separating from the center of the screw shaft 5 to such an extent that the thread and the ball were not touched along the path when the screw shaft was wound around the developed view of Fig. 35 . As seen from the curvature change, the path is a curvature discontinuity. Therefore, three-dimensional convolutional interpolation is used to redesign the circular path on the path with continuous curvature.

Figure 37 shows the orbit of the center of the ball. For the ball's recirculation path to be G2 ^- continuous as a whole, it needs to be ^G2- continuous at the point where the ball moves on the return path. Therefore, in the design of the regression path, it is considered necessary to control the tangent, normal, and curvature at both ends of the regression path.

(3-2) In the following, an example of designing a return path of a ball screw with an inverter system using a three-dimensional clothoid curve will be described.

(a-1) Screw shaft and balls

Table 14 lists the dimensions of the screw shaft and balls used in this design.

Table 14 Dimensions of screw shafts and balls

Screw shaft outer diameter (mm) 28.0 Ball Center Diameter (mm) 28.0 Thread inner diameter (mm) 24.825 Spacing(mm) 5.6 Ball diameter(mm) 3.175

(a-2) Symmetry and coordinates

The return path of the ball screw of the inverter type needs to be axisymmetric in view of its application. Therefore, the coordinate system used in this design is described.

First, as shown in Figure 38, take the z-axis as the direction of the screw axis. The solid line in FIG. 38 is the trajectory drawn by the center of the ball when the ball is moved along the screw groove. Also, let the point entering the return path be the point P _s , let the point returning from the return path to the screw groove be the point P _e , and let the midpoint between the point P _s and the point P _e be the point P _m . As shown in Fig. 39, if P _s and point P _e are viewed from the projection diagram to the xy plane, an equilateral triangle is drawn from the origin 0, P _s and point P _e , but the ∠P _s of the two equilateral triangle is taken The direction of the vertical bisector of OP _e is the y-axis direction. In addition, in consideration of symmetry, it is prescribed that the y-axis passes through the point P _m . The directions of the respective axes are as shown in FIGS. 38 and 39 . In this way, the regression path is designed in such a way that the coordinate system is axisymmetric.

In actual design, the coordinates of each point are determined with θ=15°. Table 15 lists the coordinates, tangents, normals, and curvatures thus determined.

Table 15 Coordinates, tangents, normals, and curvatures of each point

coordinate tangent normal curvature Click P_s {-3.6088, -13.5249, 2.5563} {0.96397, -025829, 0.063533} {0.25881, 0.96592, 0.0} 0.071428 Click P_e {3.6088, -13.5249, -2.5563} {0.96397, 0.25829, 0.063533} {-0.25881, 0.96592, 0.0} 0.071428 Click P_m {0.0, -13.5249, 0.0} — — —

(a-3) Constraints

Constraints in the design of the return path of the ball screw of the inverter system are studied. First, the curve drawn with the trajectory of the center of the ball moving along the screw groove at point _Ps and point _Pe must be _G2 continuous.

Next, if considering the height of the example ball, since the center of the ball passes through a certain point on the y-axis as long as the return path is considered to be y-axis symmetric, this point is taken as point _Ph (see Figures 38 and 39). At this time, for the ball to pass over the thread teeth, the absolute value of the y coordinate of the point P _h needs to satisfy at least:

(The absolute value of the y coordinate of the point P _h ) ≥ (screw shaft outer diameter + ball diameter)/2 Therefore, in this design, it is:

(The absolute value of the y-coordinate of point P _h ) ≥ (screw shaft outer diameter + ball diameter × 1.2)/2 In addition, considering that the y-axis is symmetrical, the normal direction needs to be {0, 1, 0}, and the tangent direction has only a degree of freedom of rotation around it.

Satisfying the above conditions, use the three-dimensional clothoid curve to generate a y-axis symmetric regression path. In fact, in addition to this, the interference to the screw shaft must also be considered, but the regression path of the interference design can be checked. When there is interference, the initial value of the interpolation can be changed, or the interpolation object point can be increased, or redesigned. path to resolve.

(a-4) To avoid interference

Interference with the screw shaft is likely to occur on the side of the return path, and since the path is created by free interpolation, interference is likely to occur. It is required that the return path leave the screw shaft and return to the original position over the thread teeth, but to avoid interference, it is best to leave the screw shaft to a certain extent and return to the original position over the thread teeth. As a method of generating this regression path, there are methods of adding interpolation target points to avoid interference, and methods of manually creating the first curve that enters the regression path and forcibly separating it from the screw shaft. Among them, in this design, the first root curve entering the return path is manually generated, and the method of forcibly separating from the screw shaft is adopted.

Here, the first curve C ₁ entering the regression path starting from the point P _s will be described. Variables as the curve length variable S, such as Px ₁ , (S), Py ₁ , (S), Pz ₁ (S), α ₁ (S), β ₁ (S), n ₁ (S), κ ₁ ( s), indicating the coordinates on the curve _C1 , the tangential turning angles α, β, normal, and curvature. Furthermore, at point P _s ·point Ph _h , at point P _s such as Px _s , Py _s , Pz _s , α _s , β _s , n _s , κ _s , at point P _h such as Px _h , Py _h , Pz _h , α _h , β _h , n _h , κ _h , represent coordinates, tangent turning angles α, β, normal, and curvature. The curve drawn with the trajectory of the center of the ball moving along the screw groove is a condition of ^G2 continuity, and the following expression is established at the point _Ps .

[Formula 69]

Point P _s : tangent, normal, curvature: 4

cos[α ₁ (0)-α _s ]=1

cos[β ₁ (0)-β _s ]=1

n ₁ (0) n _s =1

κ ₁ (0)=κ _s

In addition, the curve drawn by the trajectory of the center of the ball moving along the screw groove is represented by a three-dimensional clothoid curve, but from the point shown in _FIG . . Here, the pitch of the thread is defined as pit, the outer shape of the screw shaft is defined as R, and the pitch angle of the thread is defined as α ₀ .

[numeral formula 70]

α ₀ (S)=-α ₀

β ₀ (S)=β _e +2πS

${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="h"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">h</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\sqrt{{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="pit"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">pit</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;R</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="h"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">h</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; wxya</span>}$

In the formula of the curve C ₀ , the point P _s is expressed as P _s =P ₀ (11/12). Now, if a curve C ₁ that continues from the point P _s to the curve C ₀ to G ² at the point P _s is generated, a curve having the following parameters can be forcibly separated from the screw shaft.

[Equation 71]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 60</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 11</span>}}{\mathrm{<span\; class="notranslate">\; 6</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 15</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 11</span>}}{\mathrm{<span\; class="notranslate">\; 6</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="h"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">h</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}{\mathrm{<span\; class="notranslate">\; 60</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; wxya</span>}\end{array}\right.$

For example, as the curve C ₁ satisfying this condition, a three-dimensional clothoid curve having the parameters of Table 16 is generated.

Table 16 Parameters of curve C ₁

At this time, if the values of the tangent, normal, and curvature of the curve _C0 and the curve _C1 on the point _Ps are compared, Table 17 is formed, and it is judged that ^G2 is continuous.

Table 17 Offset of tangent, normal and curvature at point P _s

In addition, this curve, as judged from FIGS. 41 and 42 , only forms a shape separated from the screw shaft. Therefore, as for the first curve C ₁ entering the regression path starting from the point P _s , the curve of this parameter is used.

(a-5) Conditions and unknowns of three-dimensional convolutional interpolation

Adding the conditions described in (a-3), under the condition of achieving ^G2 continuity, use the Newton-Raphson method to obtain the approximate values of the parameters of each curve. Here, since the curve C ₁ starting from the point P _s has already been generated, the design of the route between the end point P ₁ of the curve C ₁ and the point _Ph will be described later in the description. The subscript of the text appearing in the description corresponds to the subscript of each curve, as a function of the curve length variable S, such as _Pxi , (S)Py _i , (S)Pz _i (S), α _i (S), β _i (S), _ni (S), and κ _i (S) represent coordinates on each curve, tangential turning angles α, β, normal line, and curvature. In addition, at the point P _h , coordinates, tangential turning angles α, β, normal, and curvature are expressed as Px _h , Py _h , Pz _h , α _h , β _h , n _h , and h _h .

In the design of the route, since the two points to be passed strictly are the point _P1 and the point _Ph , this is a three-dimensional convolutional interpolation of the two interpolated points. Here, if the interpolation conditions at both ends are considered, since the number of conditional expressions is two more than the number of unknowns, in order to perform ^G2 continuous three-dimensional convolutional interpolation, as shown in Figure 43, it is determined at point _P1 and point P Insert point P ₂ between _h . Also, let the curve connecting the point P ₁ and the point _Ph be the curve C ₂ , and let the curve connecting the point P ₂ and the point _Pe be the curve C ₃ .

The interpolation conditions at each point will be described below.

[Equation 72]

Point P ₁ : tangent, normal, curvature: 4

cos[α ₂ (0)-α ₁ (1)]=1

cos[β ₂ (0)-β ₁ (1)]=1

n ₂ (0)·n ₁ (1)=1

κ ₂ (0) = κ ₁ (1)

Point P ₂ : position, tangent, normal, curvature: 7

Px ₃ (1) = Px ₂ (0)

Py ₃ (1) = Py ₂ (0)

Pz ₃ (1) = Pz ₂ (0)

cos[α ₃ (1)-α ₂ (0)]=1

cos[β ₃ (1)-β ₂ (0)]=1

n ₃ (1)·n ₂ (0)=1

κ ₃ (1) = κ ₂ (0)

Point _Ph : position, β, normal: 5

Px ₃ (1) = Px _h

Py ₃ (1) = Py _h

Pz ₃ (1)=Pz _h

cos[β ₃ (1)]=1

n ₃ (1)·{0, 1, 0}=1

The above shows that there are 16 conditional expressions that should all be established. Here, each curve has 7 convolution parameters such as a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , and h, and since there are 2 curves, there are 14 unknowns. However, in this way, since the unknown number and the number of the conditional expression are not equal, the solution cannot be obtained. Therefore, the y and z coordinates of the reinserted 2 points P ₂ are treated as unknowns, adding 2 unknowns. By doing this, there are 16 unknowns and conditional expressions, and a solution can be obtained. In addition, although it is not carried out in this design example, as long as the unknown number and the number of the conditional expression give a point that should be strictly passed on the way, and ^G2 continuity is achieved before and after this point, it is usually established, so even at point P A solution can also be obtained by adding interpolation target points between ₁ and the point _Ph .

(a-6) Calculate the convolution parameters satisfying the conditions

A solution satisfying the conditional expression established in (a-5) is found by the Newton-Raphson method. The interpolation method and the generation method of the initial value follow the method of three-dimensional convolution interpolation. Table 18 lists the calculated parameters of each curve, and Table 19 lists the offsets of coordinates, tangents, normals, and curvatures on the drawn connection points.

Table 18 Parameters of the generated curves

Table 19 Coordinates, tangents, normals, and curvature offsets at each tangent point

(a-7) Path generation

Using the parameters obtained from (a-5) and (a-6), it is possible to design a route from the point P _s to the point _Ph . In addition, because the path from point P _s to point P _e is y-axis symmetric, it is the same as the path generated by retaking the coordinate system and considering point P _e as point P _s , so these can also be generated by the same curve generate.

Fig. 44 shows a route generated by the above method. The solid line is the center orbit of the ball on the screw shaft, that is, the curve C ₀ , and the three curves from the point P _s to the point P _n are the curves C ₁ , C ₂ , and C ₃ . In addition, the three curves of the dashed-two dotted line, the dashed-dotted line, and the dotted line from the point _{P n} to the point Pe are curves C ₃ , C ₂ , and C ₁ , and curves symmetrical to the y-axis, respectively.

FIG. 45 is a graph showing the relationship between the movement distance and the curvature κ along the circular path counterclockwise from the point P _e viewed from the positive direction of the z-axis. The line type of the graph corresponds to the line type of the graph in FIG. 44 .

Using the above method, the three-dimensional clothoid curve is used to design the circulation path of the reverser type ball screw. In addition, the method of using the three-dimensional clothoid curve to design the circulation path is of course not limited to the reverser type ball screw, and is also applicable to the so-called return tube type ball screw that uses the tube to form the return path, or the ball screw that is installed on the end surface of the nut. The ball screw is called an end clearance type ball screw that picks up the ball from the ball rolling groove of the screw shaft, passes through the nut, and returns to the ball rolling groove of the screw shaft from the end clearance on the opposite side.

However, when the program for realizing the design method of the present invention is executed by a computer, the program is stored in an auxiliary storage device such as a hard disk drive of the computer, and loaded into the main memory for execution. In addition, such a program may be stored on a portable recording medium such as a CD-ROM for sale, or may be stored in a recording device of a computer connected via a network, and may be transmitted to other computers via a network.

B. Numerical control method using clothoid curve

In the following, 1. the definition and characteristics of the three-dimensional clothoid curve, 2. the interpolation method using the three-dimensional clothoid curve, and 3. the numerical control method using the three-dimensional clothoid interpolation method, sequentially describe the embodiment of the invention using the numerical control method using the clothoid curve .

1. Definition and characteristics of three-dimensional clothoid curve

(1) The basic formula of three-dimensional convolution

Clothoid curve, also known as Cornu's spiral, is a curve whose curvature changes proportionally to the length of the curve. The conventionally known two-dimensional clothoid curve is a type of plane curve (two-dimensional curve), and is represented by the following equation on the xy coordinates shown in FIG. 46 .

[Equation 73]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^{\mathrm{the\; s}}{e}^{\mathrm{j\&phi;}}\mathrm{ds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S{e}^{\mathrm{j\&phi;}}\mathrm{wxya},$ 0≤s≤h, $0\&le;S=\frac{\mathrm{the\; s}}{h}\&le;1$

(1)

φ＝c ₀ +c ₁ s+c ₂ s ² ＝φ ₀ +φ _v S+φ _u S ² (2)

here,

[Equation 74]

P=x+jy, $j=\sqrt{-1}---\left(3\right)$

is the position vector representing the point on the curve,

[Equation 75]

P ₀ ＝x ₀ +jy ₀ (4)

is its initial value (the position vector of the starting point).

[Equation 76]

e ^jφ = cosφ+jsinφ (5)

is a position vector (vector with a length of 1) representing the tangent direction of the curve, and this direction Φ is measured counterclockwise from the original line (x-axis direction). If this unit vector is multiplied by a small length ds integral, the point P on the curve can be obtained.

Let the length from the start point of the curve measured along the curve be s, and let the total length (length from the start point to the end point) be h. Let S denote the value of dividing s by h. S is a value of a dimensionless quantity, which is referred to as a curve length variable.

The characteristic of the clothoid curve, as shown in the formula (2), is that the tangent direction angle Φ is expressed by the quadratic formula of the curve length s or the curve variable S. c ₀ , c ₁ , c ₂ or Φ _o , Φ _v , Φ _u are the coefficients of the quadratic equation, and these numbers and the total length h of the curve are called the parameters of the convolution. Fig. 47 shows the shape of a general clothoid curve.

For the relationship above three-dimensional expansion, create an expression for a three-dimensional clothoid curve. The formula for giving a three-dimensional clothoid curve was not known in the past, so the inventors first derived the formula.

The three-dimensional clothoid curve is defined by the following equation.

[Formula 77]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^{\mathrm{the\; s}}\mathrm{uds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S\mathrm{wxya},$ 0≤s≤h, $0\&le;S=\frac{\mathrm{the\; s}}{h}\&le;1$

(6)

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}$

(7)

α＝a ₀ +a ₁ S+a ₂ S ² (8)

β=b ₀ +b ₁ S+b ₂ S ² (9)

here,

[Equation 78]

$\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="P"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">P</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

Respectively represent the position vector of the point on the three-dimensional clothoid curve and its initial value. i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively.

u is a unit vector representing the tangential direction of the curve on the point P, and is given by Equation (7). In Equation (7), E ^kβ and E ^jα are rotation matrices, and as shown in FIG. 48 , represent the rotation of the angle β on the k-axis (z-axis) system and the rotation of the angle α on the j-axis (y-axis) system, respectively. The former is called yaw rotation, and the latter is called pitch rotation. Equation (7) indicates that the tangent vector u is obtained by first rotating the unit vector in the direction of the i-axis (x-axis) by only α in the j-axis (y-axis) system, and then by only rotating β in the k-axis (z-axis) system.

That is, in two dimensions, the unit vector e ^jΦ representing the tangential direction of the curve is obtained from the inclination angle Φ from the x-axis. In three dimensions, the tangent vector u of the curve is obtained from the inclination angle α and the deflection angle β. If the inclination angle α is 0, a two-dimensional clothoid curve rolled up in the xy plane can be obtained, and if the deflection angle β is 0, a two-dimensional clothoid curve rolled up in the xz plane can be obtained. If the tangential direction vector u is multiplied by the small long ds integral, a three-dimensional clothoid curve can be obtained.

In the three-dimensional clothoid curve, the inclination angle α and the deflection angle β of the tangent vector, as shown in equations (8) and (9) respectively, can be given by the quadratic equation of the curve length variable S. In this way, the change in the direction of the tangent can be freely chosen and also given continuity in its change.

As shown in the above formula, the three-dimensional clothoid curve is defined as "a curve in which the inclination angle and deflection angle in the tangential direction are respectively expressed by the quadratic equation of the curve length variable".

A three-dimensional clothoid curve starting from P ₀ , given by

[Equation 79]

a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , h (11)

These 7 parameters are determined. The six variables of a ₀ to b ₂ have angle units and represent the shape of the clothoid curve segment. In contrast, h has units of length, representing the size of a clothoid segment.

A typical example of a three-dimensional clothoid curve is a spiral curve shown in FIG. 49 .

(2) Activity frame

In formula (7), if instead of the basic tangent direction vector i, the basic coordinates [i, j, k] are substituted, the next moving frame (moving frame) E is obtained.

[numeral formula 80]

$\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; u</span>}& \mathrm{<span\; class="notranslate">\; v</span>}& \mathrm{<span\; class="notranslate">\; w</span>}\end{array}\right]<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; i</span>}& \mathrm{<span\; class="notranslate">\; j</span>}& \mathrm{<span\; class="notranslate">\; k</span>}\end{array}\right]<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}$

$<span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\\ \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$

Here, v and w are unit vectors included in a plane perpendicular to the tangent of the curve, are orthogonal to each other, and are also orthogonal to the tangential direction unit vector u. The group of these three unit vectors (three-point fluorescent group) is a frame (coordinate system, frame) that acts with the moving point P, and is called a moving frame.

Since the moving frame can be obtained by the above formula, the calculation of the principal normal and the subnormal can be easily performed, and the shape analysis of the curve can be easily performed.

In addition, the posture of the tool point of the robot can be obtained by using E, and the position and posture of the object grasped by the robot handle can be obtained.

If the initial value and final value of E are taken as E ₀ and E ₁ respectively, it is:

[Formula 81]

${\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="E.\; kb"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">E.</figure-callout></span><figure-callout\; id="0"\; label="E.\; kb"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">\; </figure-callout>}}^{{\mathrm{<span\; class="notranslate">\; </span>}}_{}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{{\mathrm{<span\; class="notranslate">\; ja</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 14</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; )</span>$

(3) Scroll

By taking into account the active frame, a third rotation "roll" can be handled. A roll is a rotation around a tangential direction. The presence of rolling does not affect the shape of the 3D gyro itself, but affects the active frame induced by the 3D gyro. Through the twisted metal thin abacus beads, it can freely rotate around the wire, but it does not change the shape of the metal thin.

When considering roll rotation, the active frame is expressed as follows.

[Formula 82]

E＝E ^kβ E ^jα E ^iγ I＝E ^kβ E ^jα E ^iγ (16)

The roll angle γ can be expressed as a function of S.

[Equation 83]

γ＝c ₀ +c ₁ S+c ₂ S ² (17)

(4) The geometric properties of the three-dimensional clothoid curve

(a) The normal of the three-dimensional clothoid curve

It is known that the normal vector of a three-dimensional curve is expressed by the following formula using the tangent direction u.

[Formula 84]

$\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span>}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$

Here, an appropriate amount of primary differentiation from the tangent line of the three-dimensional clothoid curve in Equation (7) is as follows.

[Formula 85]

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right\}$

$<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; =</span>\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}$

  (19)

That is, an appropriate amount of normal to the three-dimensional clothoid curve is expressed in the following form using S.

[Formula 86]

$\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}$

$<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}}\left\{\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right\}$

  (20)

(b) Normals to 3D clothoid curves with rotation

Here, similar to the determination of the tangent u in (7), the normal n is also considered. For the initial tangent direction (1, 0, 0), it is assumed that a constant γ is used, and the initial normal direction is represented by (0, cosγ, -sinγ). When it is rotated in the same way as the tangent, the normal n is expressed as follows.

[Formula 87]

$\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\end{array}\right\}$

$<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right\}$

(twenty one)

Comparing the formulas of (20) and (21), it is known that sinγ and cosγ correspond to the following.

[Formula 88]

$\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}}$

$\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; two</span>}<span\; class="notranslate">\; )</span>$

(c) Normal line continuity at connection points in 3D convolution interpolation

In order to achieve the continuity of the normal at the connection point in the three-dimensional convolution interpolation, it can be obtained from formula (22), as long as

[Formula 89]

$\mathrm{<span\; class="notranslate">\; the\; tan</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; three</span>}<span\; class="notranslate">\; )</span>$

It is continuous.

(d) The curvature of the three-dimensional clothoid curve

The curvature of the three-dimensional clothoid curve is represented by the following formula.

[Number 90]

$\mathrm{<span\; class="notranslate">\; \&kappa;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; four</span>}<span\; class="notranslate">\; )</span>$

From formula (19), the curvature is expressed as:

[Formula 91]

$\mathrm{<span\; class="notranslate">\; \&kappa;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\sqrt{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 25</span>}<span\; class="notranslate">\; )</span>$

(5) Characteristics of three-dimensional clothoid curves

(a) Continuity of the curve

In a clothoid curve segment (clothoid curve represented by the same parameter), since the inclination angle and deflection angle in the tangential direction are given according to the quadratic formula of the curve length or the curve length variable S, respectively, the curve length variable S can be guaranteed to be 1 The normal direction obtained by the sub-differentiation, and the curvature obtained by the second-order differentiation are continuous. In other words, within a clothoid segment, the normal direction and curvature are continuous. Therefore, smooth, well-characterized curves can be obtained. Even in the case of connecting two clothoid curves, it is possible to create a smooth connected curve by selecting parameters so that the tangent, normal, and curvature are continuous at their joints. Call it a convolutional spline.

(b) Applicability

Since the tangent direction of the curve can be divided by two angles (inclination angle and deflection angle), it is possible to create arbitrarily three-dimensional curves that meet various conditions, and can be used for various purposes.

(c) Integration with geometric curves

Geometric curves such as straight lines, circular arcs, and spiral curves can be made by setting some of the convolution parameters to 0, or setting a specific functional relationship between several parameters. These curves are a type of clothoid curve and can be represented in a clothoid format. Therefore, it is not necessary to perform processing by changing the above-mentioned format by straight lines, arcs, free curves, etc. like conventional NC, and calculation and control can be performed using the same format.

Furthermore, since two-dimensional convolutions can be made by generally setting either of α or β to 0, resources already obtained for two-dimensional convolutions can be applied.

That is, by appropriately setting α or β, individual curves such as circular arcs and straight lines can also be expressed including known two-dimensional convolutions. Since the three-dimensional clothoid equation of the same form can be used for such individual curves, calculation procedures can be simplified.

(d) Inferred goodness

In conventional interpolation methods such as spline interpolation, when formulating a free curve, it is difficult to separate its overall form or local form. It is easy to grasp the overall image.

In addition, at the middle end expressed as a clothoid curve, the values of the line length, tangential direction, curvature, etc. are known, and there is no need for recalculation like the conventional interpolation method. That is, corresponding to the parameter S of the curve, the tangent, or normal, and curvature of the curve are directly obtained as shown in equations (7), (20) and (26). This is a very effective feature for the numerical control method described later. In this way, calculation time can be greatly shortened, resources such as memory can be saved, and real-time interpolation calculations can be performed.

In NC processing, the minimum curvature radius of the tool path is an important issue. In spline interpolation, etc., it needs complicated calculations to solve it. The value of is known, so it is advantageous in the selection of the tool diameter, etc.

(e) Ease of motion control

The principal variable of the curve is the length s or normalized length S, and the equation of the curve is given by the equation of nature relative to this length. Therefore, by determining the length s as a function of time t, motion characteristics such as acceleration and deceleration can be given arbitrarily, and by adopting a motion curve with good characteristics conventionally used for cams, etc., it is possible to increase the speed of machining operations. Since the length s can be given as a value in the actually existing Cartesian space, and the velocity and acceleration can be obtained with respect to the tangential direction, it is not necessary to combine the values given for each axis like the conventional interpolation method. In addition, since the calculation of the curvature is easy, it is also easy to obtain the centrifugal acceleration during exercise, and control in accordance with the trajectory of the exercise can be performed.

(6) The generation of the curve and the properties of each parameter

According to the definition, the influence of each parameter of the three-dimensional clothoid curve on the curve is as follows. By giving each parameter, a three-dimensional clothoid curve can be generated as shown in FIG. 49 .

Table 20 summarizes the properties of the parameters of the three-dimensional clothoid curve.

Table 20

parameter mean P₀ Parallel Shifting 3D Clothoid Curve h Determine Size of 3D Clothoid Curve a₀, b₀ Rotate 3D Clothoid Curve a₁, a₂, b₁, b₂ Determining the Shape of a 3D Clothoid Curve

2. Using the interpolation method of the three-dimensional clothoid curve

(1) Mathematical conditions for smooth connection

In one three-dimensional clothoid curve, the shape expression of the curve has a limit. Here, for the main purpose of controlling the motion of the tool using numerical control, a plurality of three-dimensional clothoid curves (three-dimensional clothoid segments) are connected, and the motion of the tool is controlled by the plurality of three-dimensional clothoid segments.

Fluidly connecting two three-dimensional clothoid curves at their endpoints is defined as continuously connecting endpoint positions, tangents, and curvatures. Using the above definition formula, this condition is described as follows. The first 3 formulas represent the continuity of the position, the next 2 formulas represent the continuity of the tangent line, the next 1 formula represents the coincidence of the normal line, and the last formula represents the continuity of the curvature.

[Number 92]

Px _i (1) = Px _i+1 (0)

Py _i (1) = Py _i+1 (0)

Pz _i (1) = Pz _i+1 (0)

α _i (1) = α _i+1 (0)

β _i (1) = β _{i + 1} (0) (26)

tan γ _i (1) = tan γ _i+1 (0)

κ _i (1) = κ _{i + 1} (0)

This is to satisfy the condition that the tangent vector and the normal vector are continuous, and the curvature and α, β are continuous at the connection point, and sometimes the condition is too strict. Therefore, it is also possible to simply satisfy the conditions by changing the conditions as shown below.

[Formula 93]

Px _i (1) = Px _i+1 (0)

Py _i (1) = Py _i+1 (0)

Pz _i (1) = Pz _i+1 (0)

cos[α _i (1)-α _i+1 (0)]=1

cos[β _i (1)-β _i+1 (0)]=1 (27)

tan γ _i (1) = tan γ _i+1 (0)

κ _i (1) = κ _{i + 1} (0)

Here, in addition,

[Number 94]

cos[α _i (1)-α _i+1 (0)]＝1

If the above relationship is also taken into account,

[Formula 95]

tan γ _i (1) = tan γ _i+1 (0)

It is replaced by the following condition.

[Number 96]

tan γ _i (1) = tan γ _i+1 (0)

$\frac{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}$

∵α′ _i (1)β′ _i+1 (0)=α′ _i+1 (0)β′ _i (1)

As a result, it was found that the object can be achieved if the following conditions are satisfied.

[Number 97]

Px _i (1) = Px _i+1 (0)

Py _i (1) = Py _i+1 (0)

Pz _i (1) = Pz _i+1 (0)

cos[α _i (1)-α _i+1 (0)]=1

cos[β _i (1)-β _i+1 (0)]=1 (28)

${\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

κ _i (1) = κ _{i + 1} (0)

In Equation (28), the first 3 equations represent the continuity of position, the next 2 equations represent the continuity of tangent lines, the next 1 equation represents the coincidence of normal lines, and the last equation represents the continuity of curvature. To perform G ² continuous interpolation, two three-dimensional clothoid curves need to satisfy the seven conditional expressions of Equation (28) at their endpoints.

Supplementary about G ² continuous (G is the prefix of Geometry). Fig. 50 shows the conditions of ^G2 consecutive interpolation.

The so-called G ⁰ continuity means that the two three-dimensional clothoid curves have the same endpoint position, the so-called G ¹ continuity means that the tangent direction is consistent, and the so-called G ² continuity means that the contact plane (normal line) and curvature are consistent. In Table 21 below, the C ⁰ -C ² continuum used for the comparative spline curve and the G ⁰ -G ² continuum used in the clothoid curve of the present invention are compared.

Table 21

C⁰: position G⁰: position C¹: first order differential coefficient G¹: Tangent direction C²: quadratic differential coefficient G²: contact plane (normal), curvature

When considering the continuity of two three-dimensional clothoid curves, the interpolation conditions become stricter as C ⁰ →C ¹ →C ² and G ⁰ →G ¹ →G ² are reached. In C ¹ continuity, the size and direction of tangents must be consistent, but in G ¹ continuity, only the direction of tangents may be consistent. When connecting tangent lines smoothly with two three-dimensional clothoid curves, it is preferable to make a conditional expression using ^G1 continuity. For example, if the spline curve is made into a conditional expression continuously with C ¹ , the condition is too strict due to the addition of the condition that the sizes of the tangent lines that have no relationship in geometry are consistent. If G ¹ is used to make the conditional expression continuously, it has the advantage of freely setting the size of the primary differential coefficient.

Make the contact plane (normal) consistent in ^G2 continuation. The so-called contact planes, as shown in FIG. 51 , refer to the planes S1 and S2 partially including the curve C. FIG. 51 shows an example in which the point P is continuous in the tangential direction, but the contact planes S1 and S2 are discontinuous. When considering the continuity of a three-dimensional curve, the consistency of the contact plane must be considered after the consistency of the tangent direction. When discussing the curvature, it does not mean that the contact planes are not consistent, but the curvature needs to be consistent after the contact planes are consistent. Use two three-dimensional curves to make the coordinates, tangent direction, contact plane (normal line) and curvature consistent to meet the condition of ^G2 continuity.

(2) Specific calculation order

There are the following two calculation sequences.

(a) Given the parameters h, α, and β of the curve, a three-dimensional clothoid curve is generated, and at its endpoint, the parameters of the next three-dimensional clothoid curve are determined in a manner that satisfies formula (28). In this way, three-dimensional clothoid curves connected smoothly one by one can be generated. According to this calculation procedure, the curve parameters can be easily calculated, which is called a sequential solution. According to this method, curves of various shapes can be easily generated, but the connection points through which the curves pass cannot be clearly specified.

(b) The three-dimensional clothoid curves can be connected so that the point groups specified in advance become the connection points of the curves. Here, a short clothoid curve (clothoid segment) is created for each section of each discretely given point sequence. In this case, the calculation sequence to determine the curve parameters in a manner that satisfies formula (28) is more complicated than that in (a), which is repeated convergence calculation. Since the curve parameters are determined inversely from the connection conditions, this calculation order is called an inverse solution.

Regarding the inverse solution of (b) above, the calculation method will be described in detail. The computational problem to be solved is formulated as follows.

Unknown parameter: Curve parameter

Constraints: Formula (28) or a part thereof

Depending on the required problem, varying the number of constraints can be used as unknown parameters to set the corresponding number of curve parameters. For example, in the case where the continuity of the curvature is not required, some curve parameters can be freely operated. Or, when the curvature is continuous and the tangent direction is specified, it is necessary to increase the number of three-dimensional clothoid curves used for interpolation through segmentation and increase the corresponding unknown curve parameters.

In order to make the above repeated convergence calculations converge stably, it is necessary to work hard on the calculations. In order to avoid calculation divergence and speed up convergence, an effective method for unknown parameters is to set better initial values. Therefore, an effective method is to generate a more simple interpolation curve that satisfies the given constraints such as connection points, such as a linear spline curve, etc., and calculate the curve parameters of the three-dimensional clothoid curve from its curve shape as a repeated convergence calculation initial value of .

Alternatively, it is also effective as a method of stably obtaining a solution by sequentially increasing the conditional expressions without satisfying the constraint conditions that should be satisfied all at once. For example, the sequence in which the curves occur is divided into the following three STEPs, which are performed sequentially. After interpolating so that the position information coincides with the tangent direction as the 1st STEP, interpolate so as to match the normal direction as the 2nd STEP, and interpolate so as to match the curvature in the 3rd STEP. Fig. 52 shows a schematic flow of this method. The necessary three-dimensional clothoid equations and their definitions for tangents, normals, or curvatures are shown.

(3) The embodiment that adopts the interpolation method of three-dimensional clothoid curve

(a) Flow of imputation method

An embodiment of a method for smoothly interpolating between a given point sequence by using a three-dimensional clothoid curve is described in detail. Hereinafter, the interpolation method using a three-dimensional clothoid curve is referred to as three-dimensional clothoid interpolation. A group of curves generated by interpolation is collectively called a three-dimensional clothoid curve, and unit curves constituting it are called a three-dimensional clothoid segment.

As the basic flow of three-dimensional convolutional interpolation, each parameter of the three-dimensional cycloidal segment connecting the points of the interpolation object is used as the unknown, and the points of the interpolation object are strictly passed through, and the satisfying ^G2 Continuous conditional solutions generate curves. FIG. 53 is a diagram summarizing the outline of this flow. The so-called G ² continuity means that the two three-dimensional clothoid curves have the same position, tangent direction, normal direction and curvature at their endpoints.

(b) Conditions for ^G2 continuous interpolation

In the three-dimensional convolutional interpolation, specific conditions are considered regarding the condition that the point of the interpolation object passes closely and becomes ^G2 continuous.

Now, simply having 3 points P ₁ ={Px ₁ , Py ₁ , Pz ₁ }, P ₂ ={Px ₂ , Py ₂ , Pz ₂ } and P ₃ ={Px ₃ , Py ₃ , Pz ₃ }, Consider interpolating the point with a 3D clothoid segment. Fig. 54 shows three-dimensional convolutional interpolation of points _P1 , _P2 and _P3 . If the curve between connection points P ₁ and P ₂ is set as curve C ₁ , and the curve between connection points P ₂ and P ₃ is set as curve C ₂ , in this case, the unknown is the parameter of curve C ₁ a0 ₁ , a1 ₁ , a2 ₁ , b0 ₁ , b1 ₁ , b2 ₁ , h ₁ , 14 parameters of curve C ₂ , such as a0 ₂ , a1 ₂ , a2 ₂ , b0 ₂ , b1 ₂ , b2 ₂ , h ₂ . In addition, the subscripts of characters appearing in the description below correspond to the subscripts of the respective curves.

Next, consider the point that passes the interpolation object strictly and achieves the condition of _G2 continuity. First of all, the condition of strictly passing the point of the interpolation object at the starting point is considered from the definition of the three-dimensional clothoid curve, because it must be satisfied when the starting point is given, so there is no interpolation condition. Next, at the connection point P ₁ , three cases are established regarding the position, two cases are established regarding the tangent vector, two cases are established regarding the magnitude and direction of the conditional expression of curvature continuity, and a total of seven cases are established. Also about the end point, at point P ₂ is 3 in terms of position. The total number of conditional expressions obtained from the above is 10. However, since there are only 10 conditional expressions for 14 unknowns in this way, the solution to the unknowns cannot be obtained. Therefore, in this study, the tangent vectors of the two ends are given, and two conditions are added at each end point, so that the number of conditional expressions and unknowns are equal. In addition, if the tangent direction at the starting point is determined, since a0 ₁ and b0 ₁ can be obtained from the definition formula, they need not be treated as unknowns. Hereinafter, each condition is considered.

First, considering the condition of the position, the following three expressions are established from the expressions (1-1), (1-2), and (1-3). (Hereafter, the natural number i<3 is specified.)

[Number 98]

${\mathrm{<span\; class="notranslate">\; Px</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Px</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}$

(1-1)

${\mathrm{<span\; class="notranslate">\; Python</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Python</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}$

(1-2)

${\mathrm{<span\; class="notranslate">\; Pz</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Pz</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}$

(1-3)

Next, if the tangential direction is considered, two equations (1-4) and (1-5) are established.

[Number 99]

cos(a0 _i +a1 _i +a2 _i -a0 _i+1 )＝1 (1-4)

cos(b0 _i +b1 _i +b2 _i -b0 _i+1 )＝1 (1-5)

Regarding the magnitude of the curvature κ, the following formula (1-6) is established.

[numeral formula 100]

κ _i (1)-κ _i+1 (0) = 0 (1-6)

Finally consider the normal direction vector n. The normal vector n of the three-dimensional clothoid curve is expressed by formula (21).

Here, as in the determination of the tangent vector u of the three-dimensional clothoid curve, rotation is also used, and the normal vector n is considered. For the initial tangent direction (1, 0, 0), it is assumed that a constant γ is used, and the initial normal direction is represented by (0, cosγ, -sinγ). When it is rotated in the same way as the tangent, the normal n is expressed by Equation (1-7).

[Number 101]

$\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\end{array}\right\}$

$<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\end{array}\right\}$

(1-7)

Comparing formulas (21) and (1-7), we know that sinγ and cosγ correspond to formula (1-8).

[Formula 102]

$\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}}$

$\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

That is, it is known from formula (1-8) that in order to achieve the continuity of the normal at the connection point in the three-dimensional convolutional interpolation, only tanγ is continuous.

[Formula 103]

$\mathrm{<span\; class="notranslate">\; the\; tan</span>}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

That is, the condition for knowing that the normal line is continuous is Equation (1-10).

[Formula 104]

tan γ _i (1) = tan γ _i+1 (0) (1-10)

Here, additionally, if the

[Formula 105]

cos[α _i (1)-α _i+1 (0)]=1 (1-11)

Taking this into consideration, conditional expression (1-10) can be replaced by the following conditional expression (1-12). That is, the condition for the continuity of the normal line is Equation (1-12).

[Formula 106]

α′ _i (1)β′ _i+1 (0)=α′ _i+1 (0)β′ _i (1) (1-12)

To sum up, it is known that by interpolating the object points strictly and reaching the condition of ^G2 continuity, the connection point is formula (1-13). Also, some of these conditions can be selected even at the start point and end point.

[Formula 107]

Px _i (1) = Px _i+1 (0)

Py _i (1) = Py _i+1 (0)

Pz _i (1) = Pz _i+1 (0)

cos[α _i (1)-α _i+1 (0)]=1

cos[β _i (1)-β _i+1 (0)]=1 (1-13)

${\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

κ _i (1) = κ _{i + 1} (0)

From the above, we know that for 12 unknowns a1 ₁ , a2 ₁ , b1 ₁ , b2 ₁ , h ₁ , a0 ₂ , a1 2 , a2 ₂ , b0 ₂ , _{b1 2 ,} b2 ₂ , h _{2 ,} etc., the conditional expression holds Remember 12. (The tangential rotation angle at point P ₃ is defined as α ₃ , β ₃ .)

[Formula 108]

Px ₁ (1) = Px ₂ (0)

Py ₁ (1) = Py ₂ (0)

Pz ₁ (1) = Pz ₂ (0)

cos[α ₁ (1)-α ₂ (0)]=1

cos[β ₁ (1)-β ₂ (0)]=1

α′ ₁ (1)β′ ₂ (0)=α′ ₂ (0)β′ ₁ (1)

κ ₁ (1) = κ ₂ (0)

(1-14)

Px ₂ (1) = Px ₃ (0)

Py ₂ (1) = Py ₃ (0)

Pz ₂ (1) = Pz ₃ (0)

cos[α ₂ (1)-α ₃ ]=1

cos[β ₂ (1)-β ₃ ]=1

In this way, since 12 equations are established for 12 unknowns, they can be solved. This can be explained by Newton Raphson's method, and the solution can be obtained.

In addition, generally, when interpolation of n point columns is considered, the conditional expression only needs to expand the above-mentioned natural number i to i<n. Then there is the question of the number of unknowns and conditional expressions.

For example, when there are n-1 point columns, it is considered that N unknown numbers and N relational expressions are established. Here, if it is assumed that one more point is added, the unknowns will be increased by seven convolution parameters a0 _n , a1 _n , a2 n , b0 _n , b1 _n , _{b2 n and} h _n of the three-dimensional clothoid segment P _n-1 , P _n . On the one hand, the conditional expression, since the connection point increases by 1, so at the point P _n-1 , 3 are added in terms of position, 2 are added in terms of tangent vector, and 2 are added in terms of the magnitude and direction of the conditional expression with continuous curvature , adding 7 in total.

Since it is known that when n=3, there are 12 unknowns and relational expressions, so when n≥3, the unknowns are 7(n-2)+5, and the formula for this treatment is also 7(n-2)+ 5. In this way, since the unknown number is equal to the number of conditions related to it, the same method as that of 3 points can be used for n free point columns. As a solution method, the Newton-Raphson method using the relationship of the equations (1-15) and (1-16) established between the unknown number and the conditional expression is used. (The condition is set to F, the unknown is set to u, and the error Jacobian matrix is set to j.)

[Formula 109]

ΔF＝[J]Δu (1-15)

Δu＝[J] ^-1 ΔF (1-16)

From the above, it is possible to perform three-dimensional convolutional interpolation that closely passes through the interpolation target points for n point sequences, and ^G2 continuous.

(c) Determination of initial value

In the Newton-Raphson method, appropriate initial values need to be given when starting the search for a solution. The initial value can be given in any way, but only one example of the way of giving the initial value will be described here.

3D Discrete Clothoid Splines, a research conducted earlier, has the property that the curvature changes smoothly with respect to the moving distance from the starting point by strictly interpolating the target point. Therefore, in this study, the initial value for three-dimensional convolutional interpolation is determined by calculation from here by making the polygon Q of 3D DiscreteClothoid Splines with r=4 as shown in Figure 55.

Next, 3D Discrete Clothoid Splines are supplemented. First, as shown in Fig. 56, a polygon P with the point row of the interpolation object as a vertex is made, and the same number r of new vertices are inserted between each vertex of P, and it is made as $P\&Subset;Q$ The polygon Q. Here, if the number of vertices of P is set to n, there are rn vertices when the polygon Q is closed, and r(n-1)+1 vertices when the polygon Q is open. Subscripts are defined hereafter as consecutive numbers from the starting point, and each vertex is represented by qi. In addition, at each vertex, a normal vector b is determined as a direction, and a vector k having a curvature κ is determined as a magnitude.

At this time, when the equation (1-17) in which the vertices are equidistant from each other is satisfied, the curvature is closest to the condition proportional to the moving distance from the starting point (the function of equation (1-18) is minimized When) polygon Q, called 3D Discrete Clothoid Splines.

[Formula 110]

|q _i-1 q _i |＝|q _i+1 q _i |, $(q_i\&NotElement;P)---(1-17)$

 i＝{0...n-1}，Δ²k_i＝k_i-1-2k_i+k_i+1

i={0...n-1}, Δ ² k _i =k _i-1 -2k _i +k _i+1

(1-18)

In 3D Discrete Clothoid Splines, the Fresnel frame of each vertex has been calculated. Therefore, the parameters a ₀ and b ₀ are obtained from the unit tangential direction vector t. The tangential direction vector t is already known when the polygon Q is obtained, and the tangential direction rotation angles α, β of the vertices of the polygon Q are obtained by using this t and the tangent equation of the three-dimensional clothoid curve. From this, the initial values of a ₀ and b ₀ of each curve are obtained. In addition, its value is given on the three-dimensional clothoid segment from the starting point.

[Equation 111]

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 19</span>}<span\; class="notranslate">\; )</span>$

Here, regarding the 3D Discrete Clothoid Splines, if the vertices are arranged equidistantly, at point q _4i+1 in FIG. 55 , it can be approximated that the curve length variable S is 1/4. Also at point q _4(i+1)-1 , it can be approximated that the curve length variable S is 3/4. When these are considered together with the formula of α of the three-dimensional clothoid curve, the following formula (1-20) is established.

[Formula 112]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 20</span>}<span\; class="notranslate">\; )</span>$

This formula becomes a two-dimensional simultaneous equation with unknowns a1 _4i and a2 _4i , which is solved and used as the initial values of parameters a ₁ and a ₂ . It is likewise possible to determine initial values for the parameters b ₁ , b ₂ .

The remaining unknown is the curve length h, but this initial value can be calculated from the expression of the curvature of the three-dimensional clothoid curve. The curvature of the three-dimensional clothoid curve can be expressed by formula (1-21).

[Formula 113]

$\mathrm{<span\; class="notranslate">\; \&kappa;</span>}<span\; class="notranslate">\; =</span>\frac{\sqrt{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; twenty\; one</span>}<span\; class="notranslate">\; )</span>$

If this formula is changed to formula (1-22), the initial value of h can be determined.

[Formula 114]

${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\sqrt{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}{\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}}{{\mathrm{<span\; class="notranslate">\; \&kappa;</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}}$

(1-22)

Using the above method, it is possible to determine initial values for the seven three-dimensional convolution parameters. Using this determined initial value, under the condition of achieving ^G2 continuity described in (b), the approximate value of the parameters of each curve is obtained by the Newton-Raphson method. A three-dimensional clothoid segment is generated from the parameters thus obtained, and a three-dimensional clothoid curve is used to interpolate between point columns.

(b) Imputation example

As an example of actually interpolating a point sequence using the method described above, three-dimensional convolutional interpolation (0.0, 0.0, 0.0), (2.0, 2.0, 2.0), (4.0, 0.0, 1.0) and (5.0, 0.0, 2.0) are given ) Examples of these 4 points. A perspective view of a three-dimensional clothoid curve generated by interpolation is shown in FIG. 57 . The solid line in Fig. 57 is a three-dimensional clothoid curve, and the straight line of dotted line, one-dot-dash line, and two-dot-dash line is that the size of each point on the curve is log (radius of curvature+natural logarithm e), and the direction is normal The radius of curvature variation pattern along the line.

In addition, Table 22 shows the parameters of each curve, and Table 23 shows the coordinates of each tangent point, tangent line, normal line, and curvature deflection. As can be seen from these tables, a three-dimensional clothoid curve that is ^G2 continuous is generated at each tangent point. In addition, FIG. 58 is a curvature change graph in which the movement distance from the starting point is taken on the horizontal axis and the curvature is taken on the vertical axis.

Table 22

  Parameters of each 3D clothoid segment

Curve 1 (Curvature Radius Variation Mode Dashed Line)

α＝-0.657549-1.05303S+1.84584S ²

β=1.03297+1.29172S- ^2.55118S2

h＝3.82679

_P0 = (0.0, 0.0, 0.0)

Curve 2 (curvature radius change mode dot-dash line)

α＝0.135559+2.18537S- ^2.69871S2

β=-0.226655-3.15603S+3.03298S ²

h＝3.16932

^P0 = (2.0, 2.0, 2.0)

Curve 3 (curvature radius change mode two dotted line)

α＝-0.377569-1.45922S+0.984945S ²

β=-0.349942+1.32198S- ^0.873267S2

h＝1.43987

_P0 = (4.0, 0.0, 1.0)

Table 23

Coordinates, tangents, normals, curvature skew at each tangent point

The connection point of curve 1 and curve 2

Coord: (1.16×10 ^-5 , 2.00×10 ^-6 , 3.82×10 ^-6 )

Tvector: (7.59×10 ^-5 , 1.50×10 ^-5 , 2.95×10 ^-4 )

Nvector: (2.93×10 ^-4 , 9.19×10 ^-5 , -7.57×10 ^-6 )

Curvature: 3.06×10 ^-7

The connection point of curve 2 and curve 3

Coord: (-4.33×10 ^-6 , -1.64×10 ^-6 , 1.11×10 ^-5 )

Tvector: (2.06×10 ^-6 , 2.33×10 ^-4 , 1.97×10 ^-4 )

Nvector: (3.30×10 ^-4 , 1.19×10 ^-5 , -3.23×10 ^-5 )

Curvature: 5.96×10 ^-6

(4) ^G2 continuous three-dimensional convolution interpolation considering the control of each value at both ends

(a) Imputation conditions and unknowns

As described in (3), when the curve is turned on, when there are n points to be interpolated, the three-dimensional convolution interpolation point sequence is performed using n-1 curves. If strictly passing through each point, for each three-dimensional clothoid segment, since there are 7 unknowns such as a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , h, etc., the total number of unknowns is 7(n-1) . On the other hand, regarding the conditional expression, since there are 7 each of coordinates, tangents, normals, and curvatures and 3 coordinates on the end point for n-2 connection points, all of them are 7(n-2)+ 3. In the method (3), four conditions are added by giving the tangent vector on the start point and the end point, and the conditional expression and the number of unknowns are compared.

Here, if the tangent, normal, and curvature at the start point and end point are controlled, and the interpolation is achieved in a ^G2 continuous manner, the condition is that the normal line and curvature at the start point, end point, and the Increase by 2, a total of 4. Thus, all the conditional expressions reach 7n-3. In this case, the Newton-Raphson method cannot be used to solve the problem because the number of unknowns is less than the condition. Therefore, what method needs to be used to increase the unknown.

Therefore, here, the unknown number and the number of the conditional expression are equalized by reinserting interpolation target points. For example, if there are more than four unknowns, two new points are inserted, and two of the coordinates of each point are treated as unknowns.

In this case, since two connection points are added, 14 of each of seven coordinates, tangents, normals, and curvatures are added for each connection point condition. On the other hand, since two three-dimensional clothoid segments are added to the unknowns, a total of seven of a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , and h is added to the total of 14 unknowns. Since the number of points contained in the point column is n+2 at this time, if considered as a whole, the number of unknowns reaches 7(n+1), and the number of conditional expressions reaches 7(n+1)+4. Here, assuming that two of the coordinates of the newly inserted point are treated as unknowns, the number of unknowns increases by four. Therefore, there are 7(n+2)-3 unknowns and conditional expressions, and the solution to the unknowns can be obtained. In this way, by inserting new points, it is possible to perform interpolation that strictly passes through each given point, is ^G2 continuous, and controls the tangent, normal, and curvature of both ends.

In addition, the general case is also considered. When interpolating n point columns, the number of points inserted when controlling m items at both ends and the number of coordinates handled as unknowns at this point are considered. It is also described above, but when the curve is turned on, n-1 curves are used to interpolate the point sequence. Since if the points are strictly passed through, for each three-dimensional clothoid segment, there are 7 unknowns such as a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , h, etc., so the total number of unknowns is 7(n-1) indivual. On the one hand, regarding the conditional expression, since there are n-2 connecting points, there are 7 each of coordinates, tangents, normals, and curvatures, and 3 coordinates on the end point, so all of them are 7(n-2)+3 There are fewer conditional expressions, 4. That is, there are four or more items to be controlled at both ends. Hereinafter, in the description, m is a natural number greater than or equal to 4, k is a natural number greater than or equal to 2, and a method of making the conditional expression equal to the number of the unknown when reinserting a point will be described.

(i) When m=2k

When both ends are combined to control m=2k items, the total number of unknowns is 7(n-1), and the total number of conditional expressions is 7(n-1)-4+2k. At this time, there are 2k-4 excess conditional expressions. Now, if we consider reinserting k-2 points, since the three-dimensional clothoid segment increases by k-2 and the connection points increase by k-2, there are 7 (n+k-3) unknowns and 7 ( n+k-3)-4+2k pieces. Here, it is also assumed that two of the values of the coordinates of each newly inserted point (for example, x and y) are treated as unknowns, and the total number of unknowns is 7(n+k-3)+2(k-2). The conditional expression The overall number is 7(n+k-3)+2(k-2), and the number of unknowns and conditional expressions are equal.

(ii) When m=2k+1

When both ends are combined to control m=2k+1 items, the total number of unknowns is 7(n-1), and the total number of conditional expressions is 7(n-1)+2k-3. At this time, there are 2k-3 excess conditional expressions. Now, if we consider reinserting k-1 points, since the three-dimensional clothoid segment increases by k-1, and the connection points increase by k-1, the total number of unknowns is 7(n+k-2), and the total number of conditional expressions is 7( n+k-2)-3+2k pieces. Here, it is also assumed that two of the values of the coordinates of each newly inserted point (for example, x and y) are handled as unknowns, and the total number of unknowns is 7(n+k-2)+2(k-2). The conditional expression The overall number is 7(n+k-2)+2k-3), and the number of conditional expressions is one more. Therefore, at one of the points to be inserted when m=2k+1, only one of the coordinate values is handled as an unknown. In this way, the overall number of unknowns is 7(n+k-2)+2(k-2), the overall number of conditional expressions is 7(n+k-2)+2(k-2), and the number of unknowns and conditional expressions The numbers are equal.

In the method described above, even when adjusting the unknown number in the coordinates of the point to be inserted by adding together the number of the added condition, and controlling the tangent line, normal line, and curvature other than, for example, the tangent line angle of rotation α, etc. In various cases, the unknown number and the number of the conditional expression can also be compared, and the values at both ends can be controlled theoretically. In addition, regarding control items, unknowns, and conditional expressions, Table 24 lists the summed numbers

Table 24

Control items at both ends of n-point interpolation, unknowns, and numbers of conditional expressions

^* k: natural number above 2

(b) method

The interpolation method using the three-dimensional convolution that controls each value at the start point and the end point is performed in the following flow as shown in Fig. 59 and Fig. 60 .

Step1) Using only 4 of the conditions to be controlled, perform interpolation that strictly passes through the interpolation object points, and ^G2 continuous interpolation, to generate a curve.

Step2) Insert a new point on the generated curve, adjust the conditional expression and the number of unknowns.

Step3) Using the curve parameters of Step1 as initial values, use the Newton-Raphson method to obtain the approximate values of the parameters of each curve that meet the objective conditions.

Hereinafter, supplementary explanations are given for each Step. First, in Step1, as long as the tangent direction is controlled, the method (3) can be used to generate the curve. Also, even when the tangential direction is not controlled, the same initial values as those in the method (3) are used as initial values when obtaining the parameters of the curve.

Next, insert a new point in Step2, and adjust the conditional expression and the number of unknowns. At this time, the number of points to be newly inserted is one or less among the interpolation target points as much as possible. Also, as the interpolation point, an intermediate point between the three-dimensional clothoid segments generated in Step 1 between the interpolation objects is inserted. In addition, the inserted points should be inserted sequentially from both ends. That is, the first inserted point is between the start point and its adjacent point, and between the end point and its adjacent point.

Finally about Step3, but the initial values for the Newton-Raphschen method performed in Step3 need to be re-determined. Therefore, for a curve to which a new point is inserted, the curve is divided by the method of dividing a three-dimensional clothoid curve as described in (1-4), and determined from each value of the generated curve. For the curve without point insertion, directly adopt the value of the curve generated in Step1. Above, the initial value of each parameter of the curve in Step3 is determined. Using this initial value, a three-dimensional clothoid curve is generated from parameters obtained by the Newton-Raphson method, and interpolation between point rows is performed using a three-dimensional clothoid curve satisfying the objective condition.

(C) Interpolation example

An example of performing three-dimensional convolution interpolation is shown in which the tangent, normal, and curvature at both ends are actually controlled using the conditions in Table 25. Consecutive numbers are allocated to interpolation target points that should pass strictly to form P ₁ , P ₂ , and P ₃ .

Table 25 Conditions for each point of the interpolation object and the start point and end point

coordinate unit tangent vector principal normal vector curvature P₁ (0,0,0) (Cos(θ), Sin(θ), 0) (-Sin(θ), Cos(θ), 0) 0.2 P₂ (4, -4, -4) — — — P₃ (8, -4, -5) (1,0,0) (0, -1, 0) 0.2

*θ＝-(π/6)

Fig. 61 shows the results of actually performing interpolation under these conditions. A solid line represents a three-dimensional clothoid curve, and a dotted line, a one-dot chain line, a two-dot chain line, and a three-dot chain line represent changes in the radius of curvature of each curve. In addition, FIG. 62 is a graph showing the relationship between the movement distance and the curvature from the starting point of each curve corresponding to the line type of the curve in FIG. 61 . It can be seen from the figure that the generated curve meets the conditions given in Table 26.

Differences between the values given in Table 26 and the tangent, normal, and curvature of the start point and end point of the generated curve

(d) Control of values at intermediate points

Using the method of (b), continue to control the values at both ends, and perform ^G2 continuous interpolation. Here, it is considered to control the value not at both ends but at an intermediate point.

For example, in the case of interpolating a point sequence as shown in FIG. 63, it is considered to control the tangent and the normal at the intermediate point _Pc . However, the value at the intermediate point cannot be controlled in the aforementioned method. So here by splitting the point column into 2, the value at the middle point is controlled.

That is, the point sequence is not interpolated at one go, but is interpolated by dividing it into a curve _C1 and a curve _C2 with the middle point _Pc in between. In this case, since the point P _c corresponds to an end point, the value can be controlled by adopting the method (b).

As a result of separating the points with values to be controlled, controlling the values at both ends, and performing interpolation, as long as the generated curves are connected, it is theoretically possible to perform three-dimensional control of tangent, normal, and curvature at each point. Convolutional interpolation.

(5) Three-dimensional convolutional interpolation to control the tangent, normal, and curvature of the two ends

(a) Flow of the method

The interpolation method using the three-dimensional convolution that controls each value at the start point and the end point can be performed in the following flow shown in FIG. 64 . Hereinafter, description will be made along this flow.

(b-1) Give the point of the interpolation object

In this example, 3 points {0.0, 0.0, 0.0}, {5.0, 5.0, 10.0}, {10.0, 10.0, 5.0} of the three-dimensional space are given. Table 27 summarizes the conditions for tangent, normal, curvature, etc. given at other points.

Table 27 Conditions for each point of the interpolation object and the start point and end point

coordinate unit tangent vector principal normal vector curvature P₁ (0.0, 0.0, 0.0) {0.0, 1.0, 0.0} {1.0, 0.0, 0.0} 0.1 P₂ (5.0, 5.0, 10.0) — — — P₃ (10.0, 10.0, 5.0) {1.0, 0.0, 0.0} {0.0, -1.0, 0.0} 0.1

(b-2) Generation of 3DDCS with r=4

In the Newton-Raphson method, appropriate initial values need to be given when starting the search for a solution. Here, preparations for finding the initial value are performed. 3D Discrete Clothoid Splines, which is the first research, has the property that the curvature changes smoothly with respect to the moving distance from the starting point by strictly interpolating the target point. Therefore, in this study, the polygon Q of 3D Discrete Clothoid Splines with r=4 as shown in Figure 65 is made, and the initial value for three-dimensional convolution interpolation is determined from here by calculation. In addition, FIG. 66 shows a polygon actually generated by this point sequence, and the coordinates of the vertices are listed in Table 28.

Table 28 Vertex coordinates of the generated polygon

Vertex coordinates P₁ {0.0, 0.0, 0.0} {0.4677, 0.4677, 3.1228} {0.9354, 0.9354, 6.2456} {2.3029, 2.3029, 9.4966} P₂ {5.0, 5.0, 10.0} {6.7095, 6.7095, 9.9244} {8.0655, 8.0655, 8.4732} {9.0327, 9.0327, 6.7366} P₃ {10.0, 10.0, 5.0}

(b-3) Determination of initial value

To use the Newton-Raphson method to solve, it is necessary to determine the initial value of each unknown. In this method, the approximate value of each unknown is obtained using the polygon Q generated in (b-2), and the value is determined. In 3D Discrete Clothoid Splines, the Fresnel frame of each vertex has been calculated. Therefore, the parameters a ₀ and b ₀ are obtained from the unit tangent direction vector t of the polygon Q generated in (b-2). The tangential direction vector t is known when the polygon Q is obtained, and the tangential direction clothoid angles α, β of the vertices of the polygon Q are obtained from this t and the tangent equation of the three-dimensional clothoid curve. From this, the initial values of a ₀ and b ₀ of each curve are obtained. Also, on the 3D clothoid segment from the start point, the value is given.

[Formula 115]

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}$

Here, regarding 3D Discrete Clothoid Splines, if considering that vertices are arranged equidistantly, at point q _4i+1 in Fig. 65, it can be approximated that the curve length variable S is 1/4. Also at point q _4(i+1)-1 , it can be approximated that the curve length variable S is 3/4. When these are considered together with the expression of α of the three-dimensional clothoid curve, the following expression is established.

[Formula 116]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}\right.$

This formula becomes a two-dimensional simultaneous equation with unknowns a1 _4i and a2 _4i , which is solved and used as the initial values of parameters a ₁ and a ₂ . It is likewise possible to determine initial values for the parameters b ₁ , b ₂ .

The remaining unknown is the curve length h, but its initial value can be calculated from the formula of the curvature of the three-dimensional clothoid curve. The curvature of the three-dimensional clothoid curve can be expressed as follows.

[Formula 117]

$\mathrm{<span\; class="notranslate">\; \&kappa;</span>}<span\; class="notranslate">\; =</span>\frac{\sqrt{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&prime;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; \&prime;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cos"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">cos</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}}{\mathrm{<span\; class="notranslate">\; h</span>}}$

If this formula is changed to the following formula, the initial value of h can be determined.

[Formula 118]

${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\sqrt{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}{\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}}{{\mathrm{<span\; class="notranslate">\; \&kappa;</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}}$

Using the above method, it is possible to determine initial values for the seven three-dimensional convolution parameters.

Table 29 shows the initial values actually obtained by this method.

  Table 29 Initial value

(b-4) Pass through each point strictly, ^G2 continuous three-dimensional convolution interpolation

Using the initial values determined by (b-3), under the condition of achieving ^G2 continuity, use the Newton-Raphson method to obtain approximate values of the parameters of each curve. A three-dimensional clothoid segment is generated from the parameters thus obtained, and a three-dimensional clothoid curve is used to interpolate between point columns.

Here, in the 3-point three-dimensional convolutional interpolation, specific conditions are considered regarding the condition that the interpolation target points are passed strictly and ^G2 continuity is achieved. Fig. 67 shows three-dimensional convolutional interpolation of points P ₁ , P ₂ , and P ₃ . If the curve between connection points P ₁ and P ₂ is taken as curve C ₁ , and the curve between connection points P ₂ and P ₃ is taken as curve C ₂ , since a0 ₁ and b0 ₁ are known, the unknown is curve C ₁ There _are 12 parameters a1 ₁ , a2 ₁ , b1 ₁ , b2 ₁ , h ₁ for curve C 2 , a0 ₂ , a1 ₂ , a2 ₂ , b0 ₂ , b1 ₂ , b2 ₂ , h ₂ . The subscripts of the text appearing in the description later correspond to the subscripts of each curve, as a function of the curve length variable S, such as Pxi _, Py _i , Pzi _, α _i , β _i , ni _, and κ _i , indicating that each Coordinates on the curve, tangent turning angle α, β, normal, curvature.

First of all, the condition of strictly passing the interpolation object point at point _P1 , if considered from the definition of the three-dimensional clothoid curve, must be achieved when the starting point is given. Also, regarding the tangential direction, since it is already given as a known value, the condition on the point _P1 is not particularly specified.

Next, consider point _P2 . Point P ₂ is the connection point between the curves, to achieve G ² continuity requires position, tangent, normal, and curvature continuity. That is, the conditions to be established at the point _P2 are as follows.

[Formula 119]

Px ₁ (1) = Px ₂ (0)

Py ₁ (1) = Py ₂ (0)

Pz ₁ (1) = Pz ₂ (0)

cos[α ₁ (1)-α ₂ (0)]=1

cos[β ₁ (1)-β ₂ (0)]=1

n ₁ (1)·n ₂ (0)=1

κ ₁ (1) = κ ₂ (0)

Finally consider point _P3 . The point _P3 is the end point, and since the conditions to be satisfied are only the position and the tangent, the following five conditions are satisfied. Here, α ₃ , β ₃ are considered to be tangential turning angles α, β that determine the tangent vector at a given end point.

[Number 120]

Px ₂ (1) = Px ₃

Py ₂ (1) = Py ₃

Pz ₂ (1) = Pz ₃

cos[α ₂ (1)-α ₃ ]=1

cos[β ₂ (1)-β ₃ ]=1

From the above, we know that for 12 unknowns a1 ₁ , a2 ₁ , b1 ₁ , b2 ₁ , h ₁ , a0 ₂ , a1 2 , a2 ₂ , b0 ₂ , _{b1 2 ,} b2 ₂ , h _{2 ,} etc., the conditional expression holds Remember 12. The conditional expression for induction is as follows.

[Formula 121]

Px ₁ (1) = Px ₂ (0)

Py ₁ (1) = Py ₂ (0)

Pz ₁ (1) = Pz ₂ (0)

cos[α ₁ (1)-α ₂ (0)]=1

cos[β ₁ (1)-β ₂ (0)]=1

n ₁ ·n ₂ =1

κ ₁ (1) = κ ₂ (0)

Px ₂ (1) = Px ₃

Py ₂ (1) = Py ₃

Pz ₂ (1) = Pz ₃

cos[α ₂ (1)-α ₃ ]=1

cos[β ₂ (1)-β ₃ ]=1

In this way, since 12 equations are established for 12 unknowns, they can be solved. Use the Newton-Raphson method to solve this equation and find the solution. Table 30 lists the initial values and solutions.

Table 30 Initial value and solution

(b-5) Curve generation

Fig. 68 shows both the curve generated based on the parameters obtained in (b-4) and the polygon generated in (b-2). The curve of the solid line is the curve C ₁ , and the curve of the broken line is the curve C ₂ . At this stage, a ^G2 continuous three-dimensional clothoid curve is formed in which the tangential direction is controlled at the start point and the end point.

(b-6) Conditional expressions and unknowns

Here, it is additionally considered that the normal line and curvature on the starting point _P1 and the ending point _P3 are also determined as the values given in Table 27. To control the normal line and curvature at the start point and end point, two additional conditions on the start point and end point are required. However, in a state where four conditions are added, a solution satisfying the conditions cannot be found from the viewpoint of the relationship with the unknowns. Therefore, in order to make the unknown number and the number of the conditional expression correspond, as shown in FIG. 69 , point DP ₁ is reinserted at the position of curve length variable S=0.5 of curve C ₁ . Furthermore, for the curve C ₂ , the point DP ₂ is also reinserted at the position of the curve length variable S=0.5.

In this case, let the curve connecting point P ₁ and point DP ₁ be curve C' ₁ , let the curve connecting point DP ₁ and point P ₂ be curve C' ₂ , and let the curve connecting point P ₂ and point DP ₂ be curve C' ₃ , let the curve connecting the point DP ₂ and the point P ₃ be the curve C' ₄ . The subscripts of the text that will appear in the description later correspond to the names of the curves, for example, as a function of the curve length variable S, such as Px _c , Py _c , Pz _c , α _c , β _c , n _c , κ _c , representing the curve C Coordinates on , tangent turning angle α, β, normal, curvature. In addition, at the start point and end point, at the start point such as Px _s , Py _s , Pz _s , α _s , β _s , n _s , κ _s , at the end point such as Px _e , Py _e , Pz _e , α _e , β _e , n _e , κ _e , indicating coordinates, tangent turning angle α, β, normal, curvature

Conditions established at each point will be described below.

[Formula 122]

Point P ₁ : tangent, normal, curvature: 4

cos[α _C′1 (0)-α _s ]=1

cos[β _C′1 (0)-β _s ]=1

n _C′1 (0) n _s =1

κ _C′1 (0)=κ _s

Point DP ₁ : position, tangent, normal, curvature: 7

Px _C'1 (1) = Px _C'2 (0)

Py _C'1 (1) = Py _C'2 (0)

Pz _C'1 (1) = Pz _C'2 (0)

cos[α _C′1 (1)-α _C′2 (0)]=1

cos[β _C′1 (1)-β _C′2 (0)]=1

n _C′1 (1)·n _C′2 (0)=1

κ _C'1 (1) = κ _C'2 (0)

Point P ₂ : position, tangent, normal, curvature: 7

Px _C'2 (1) = Px _C'3 (0)

Py _C'2 (1) = Py _C'3 (0)

Pz _C'2 (1) = Pz _C'3 (0)

cos[α _C′2 (1)-α _C′3 (0)]=1

cos[β _C′2 (1)-β _C′3 (0)]=1

n _C′2 (1)·n _C′3 (0)=1

κ _C′2 (1)=κ _C′3 (0)

Point DP ₂ : position, tangent, normal, curvature: 7

Px _C'3 (1) = Px _C'4 (0)

Py _C'3 (1) = Py _C'4 (0)

Pz _C'3 (1) = Pz _C'4 (0)

cos[α _C′3 (1)-α _C′4 (0)]=1

cos[β _C′3 (1)-β _C′4 (0)]=1

n _C′3 (1)·n _C′4 (0)=1

κ _C′3 (1)=κ _C′4 (0)

Point P ₃ : position, tangent, normal, curvature: 7

Px _C'4 (1) = Px _e

Py _C'4 (1) = Py _e

Pz _C'4 (1) = Pz _e

cos[α _C′4 (1)-α _e ]=1

cos[β _C′4 (1)-β _e ]=1

n _C′4 (1)·n _e =1

κ _C'4 (1) = κ _e

There are 32 conditional expressions that should all be satisfied as described above. Here, each curve has seven convolution parameters of a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , and h, and since there are four curves, there are 28 unknowns. However, in this way, since the unknown number and the number of the conditional expression are not equal, the solution cannot be obtained. Therefore, the y and z coordinates of the reinserted two points DP ₁ and DP ₂ are treated as unknowns, and four unknowns are added. By doing this, there are 32 unknowns and conditional expressions, and a solution can be obtained.

(b-7) Determination of initial value 2

In order to find a solution that satisfies the conditional expression established in (b-6), the Newton-Raphson method is used, but the initial value of the unknown is determined in order to increase the convergence rate. As a method, four three-dimensional clothoids are created by dividing the three-dimensional clothoids generated in (b-5) before and after the newly inserted point as shown in FIG. 70, and their clothoid parameters are given.

Regarding the division method of the curve, if the method of dividing the curve C ₁ into the curve C' ₁ and the curve C' _{2 is} explained, the convolution parameters h', a' ₀ , a' ₁ , a' ₂ , b' of the curve C' 1 ₀ , b' ₁ , and b' ₂ are represented by the following formula using the parameters of the curve C ₁ . Here S _d is the curve length variable at the split point, which is 0.5 here.

[Formula 123]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="S\; d\; b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">S</figure-callout></span><figure-callout\; id="2"\; label="S\; d\; b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}& \\ {}_{}^{}{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; \&prime;</span>{<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; wxya</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}\end{array}\right.$

Consider next the curve C' ₂ starting from the split point DP ₁ . First of all, if the curve with the same size and shape as the curve C ₁ but facing the opposite direction is taken as the curve C” ₁ , the convolution parameters of the curve h”, a” ₀ , a” ₁ , a” ₂ , b” ₀ , b” ₁ , b” ₂ , the parameters of the curve using curve C ₁ are represented by the following formula.

[Formula 124]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&pi;</span>}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; h</span>}\end{array}\right.$

On this curve, the dividing point DP ₁ is represented by DP ₁ =C” ₁ (1-S _d ). Here, if considering dividing the curve C” ₁ at the point DP ₁ , the point P ₂ in the divided curve is taken as The curve C" ₂ at the starting point becomes a curve having the same size and shape as the curve C" ₂ but facing in the opposite direction. _{The curve C" 2 can be generated by the method of generating the curve C'1. Here, the curve C 2 can be} generated by generating a curve having the same size and shape but facing opposite to the curve C" ₂ .

The convolution parameters h", a" ₀ , a" ₁ , a" ₂ , b" ₀ , b" ₁ , and b" ₂ of this curve C 2 are represented by the following formula using the parameters of the curve C _{0 .}

[Number 125]

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; \}</span>\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="b"\; filenames="C200580006053E01121.png,C200580006053E01131.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; \}</span>\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="b"\; filenames="C200580006053E01142.png,C200580006053E01162.png"\; state="\{\{state\}\}">b</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>\end{array}\right.$

Using the above method, the curve C ₁ can be divided into C' 1 and _{C 2 at} the point DP ₁ where the curve length variable S=0.5 on the three-dimensional clothoid curve C ₁ . Using the same method, the curve C ₂ can also be divided into C' ₃ and C _{' 4 at} the point DP ₂ on the curve C 2 where the curve length variable S=0.5.

Table 31 lists the parameters of the 4 curves segmented by this method. The parameters of this curve are used as initial values of the Newton-Raphson method used to obtain a solution satisfying the conditional expression established in b-6.

Table 31 Parameters of the curve generated by segmentation

(b-8) Calculate the convolution parameter satisfying the condition

Based on the initial value determined in (b-7), a solution satisfying the conditional expression established in (b-6) is found by the Newton-Raphson method. Table 32 shows the calculated parameters of each curve. In addition, Table 33 lists the difference between the given value and the tangent, normal, and curvature of the start point and end point of the generated curve.

Table 32 Parameters of the generated curves

Differences between the values given in Table 33 and the tangent, normal, and curvature of the start point and end point of the generated curve

(b-9) Curve generation

Fig. 71 shows a graph generated from the parameters obtained in (b-8). The solid line represents the three-dimensional clothoid curve, and the dotted line, one-dot-dash line, two-dot-dash line, and three-dot-dash line indicate that the direction of each curve is in the direction of the main normal, and the size is the radius, which satisfies the natural logarithm, and takes the logarithmic radius of curvature change model. In addition, FIG. 72 is a graph showing the relationship between the moving distance s from the starting point of each curve corresponding to the line type in FIG. 71 and the curvature κ. It can be seen from the figure that the generated curve meets the conditions given in Table 33.

In the above, an example of generating a curve using the three-dimensional convolutional interpolation method in which the tangent, the normal, and the curvature are controlled at both ends has been described.

3. Adopt the numerical control method of three-dimensional convolution interpolation

The three-dimensional convolutional interpolation curve described above is effectively used to generate numerical control information for motion control of a tool of a machine tool or other moving objects. It is characterized in that the speed can be easily controlled and the speed can be changed smoothly.

(1) Numerical control method using three-dimensional convolution interpolation

The numerical control method using the three-dimensional convolution interpolation curve is constituted by the following procedure shown in FIG. 73 .

(a) Design of tool movement trajectory (Fig. 73, S1)

Using the method described in the previous section, determine the three-dimensional convolution interpolation curve that meets the conditions. When working with a tool such as a robot, the representative point (tool center point) of the tool can be considered and moved along a continuous trajectory curve (including a straight line) drawn in a plane or space in time. The position of the tool point is represented by coordinates (x, y, z), and the posture of the work point is represented by, for example, a rotation angle with respect to the x, y, and z axes. No matter how complicated the work is, the trajectory of the tool points will not be intermittent, but continuously connected. The first stage of motion control consists in designing the shape of this trajectory in terms of a three-dimensional clothoid curve.

(b) Matching of motion curves (Fig. 73, S2)

According to the request from the numerical control, along the three-dimensional convolution interpolation curve, the distribution of the moving speed of the control target point on the curve is specified. That is, the second stage of motion control is to determine the velocity and acceleration of the tool point acting along the designed trajectory. It is determined by what time function the tool point acts on the trajectory, or by determining the velocity and acceleration of the tool point. The velocity and acceleration of the tool point may be determined relative to time, and may be determined according to the shape of the trajectory. Generally, it is determined relative to the time, but for example, when processing a curved surface, it is required to move at a high speed on a flat part and move at a low speed on a curved part, so the speed is determined according to the shape of the trajectory.

In this embodiment, for example, a curve with good characteristics used in a cam mechanism is used. Constitutes a continuous position and posture curve group defined in Cartesian space (actual space), but applies a motion curve to each of these curves, and specifies the acceleration. The so-called Cartesian space is a three-dimensional coordinate system created using three axes of x, y, and z that are perpendicular to each other at the origin, and can express not only the position of the tool point but also the posture.

(c) Time division (FIG. 73, S3) and calculation of the position and orientation of the tool using the Cartesian coordinate system (FIG. 73, S3)

Here, the movement position and posture of the tool point are calculated at the movement speed specified by the control object every unit time for calculating the numerical control information. Since the trajectory and motion are determined, the position and orientation of the tool point can be given as a function of time t. Thus, when the time t is given at minute time intervals, the displacement of the tool point with respect to each time point can be obtained. The calculation of (c) is specifically performed as follows. At the current point, position information or values of tangent, curvature, etc. are known. As long as the specified moving speed is multiplied by the unit time, the length of the moving curve in the unit time can be known, and the parameter of the moving curve length can be calculated from this. From this moved curve length parameter, it is possible to calculate positional information on the moved point, values of tangents, curves, and the like.

Through the above steps, the position and orientation of the tool point with respect to time t on the Cartesian coordinate system (actual existence space) can be calculated. As a variable, it is (x, y, z, γ, μ, ν, θ) in three dimensions. However, (γ, μ, ν, θ) represents the pose E with an equivalent rotation, where (γ, μ, ν) represents the axis of the equivalent rotation, and θ represents the rotation angle.

In addition, according to the request from the numerical control, an offset point that is offset by a predetermined size in the normal direction along the three-dimensional convolution interpolation curve is obtained, and this is used as the tool (the trajectory of the tool center). This calculation is facilitated by obtaining the normal direction.

(d) Inverse mechanism solution (Fig. 73, S5)

Next, the rotation angles of each axis necessary to give the above-mentioned position and orientation of the tool point are obtained. This process is generally called inverse kinematics. For example, if it is a 6-axis robot, since there are 6 joints, the position and posture of the tool point are determined by how many degrees to rotate the shoulder joint, arm joint, elbow joint, wrist joint, etc. This is called the inverse mechanism solution. The inverse mechanism solution, on the contrary, obtains the rotation angles θ1 to θ6 in the axial space from the position and orientation in space. The transmission mechanism of each axis is not limited to a rotary motor, and sometimes a linear motor or other direct-motion transmission mechanism may be used. However, at this time, the calculation of the electronic transmission device that converts the minimum actual displacement into the input pulse number of the linear motor is required. . The inverse mechanism solution is unique to each type of mechanism such as robots, and solutions must be prepared individually for various robots.

(e) Calculation of the motor displacement of each axis using the axis coordinate system (Figure 73, S6)

The inverse mechanism solution is obtained for each time-divided tool point, and it is integerized as the displacement pulse of each axis motor (including the direct drive mechanism). In the case of non-pulse control, the minimum resolution unit (resolution) of displacement of each axis is used, and it is obtained as integerized data corresponding to the number of pulses.

The above (a) and (b) are the order of preparation and are only carried out once. (c) to (e) are performed every specified unit time until the target time or the target condition is satisfied.

Also can carry out above-mentioned all calculations in the numerical control device, or utilize another computer to calculate and set (a) and (b), also can send this curve parameter to the numerical control device, carry out in the numerical control device (c ) and (e) calculations.

(2) NC device and CNC device

Hereinafter, a case of using an independent numerical control device (NC device) and a case of using a CNC device in which a computer having a program function and an NC device are integrated will be described.

(a) When using an independent NC device

In a conventional general NC machine, the hardware is divided into two devices, namely, a program device for programming and creating NC data, and an NC device for operating the machine device using the NC data. On the other hand, in recent CNC machines, a computer for programming is installed in the NC device to form an integrated device.

First, in the former case where an independent device is used, a numerical control method using three-dimensional convolution is proposed. In this case, it is stipulated to use the convolution parameters in the handover of the convolution data, and the format of the convolution is defined in the G code. This, for example, is shown below.

G ^***

A0, A1, A2, B0, B1, B2, H

Here, G ^*** indicates the number of the G code. A0～H represent seven parameters of the three-dimensional clothoid segment. Before executing this code, the tool comes to the position of _P0 . Using this parameter in the NC device, the runtime calculates the instantaneous tool position or the difference between the tool positions. This operation is called "sequential solution". The reason for performing the sequential solution on the NC device side is to prevent an increase in the amount of data, and therefore some kind of calculation needs to be performed on the NC device. By expressing the clothoid with G code, it is possible to incorporate the clothoid curve into the existing NC device.

(b) CNC method

Next, a CNC device in which a computer and an NC device function as a program are integrated will be described. In this case, it is not a problem which part of the hardware is used for the calculation of the convolution. In addition, the volume of data or the speed of delivery is also being addressed.

Generally, this program includes a process of determining parameters of convolutions suitable for each condition. This is called an "inverse solution". The inverse solution, for example, also includes, given several discrete point sequences, the procedure for computing a smooth curve passing through these points exactly (free point sequence interpolation). In addition, a program (so-called CAM) for determining the tool trajectory required for machining is often included.

(3) Features of the numerical control method using three-dimensional convolution interpolation

The numerical control method using three-dimensional convolution interpolation has the following advantages.

(a) As described above, since the curve is expressed with the curve length from the reference point as an independent parameter, numerical control information corresponding to the designated moving speed can be generated. On other curves such as spline curves expressed by independent parameters that have nothing to do with the length of the curve, even if the moved point is calculated, it is difficult to calculate the value of the independent parameter corresponding to the point, and it is difficult to generate a curve corresponding to the specified moving speed. Numeric control information.

To explain this in detail, consider a case where a tool is moved at a certain linear velocity from a point R ₀ on a trajectory represented by a spline curve R(t) as shown in FIG. 74 . When calculating the target point of the tool at regular time intervals, the tool movement amount ΔS after the elapse of unit time can be known. However, since the independent variable t is not a parameter related to time or the length of the curve, the change amount Δt of the independent variable cannot be obtained immediately. Find out. When solving the formula of R ₀ +ΔS=R(t0+Δt), since the target point cannot be calculated unless Δt is obtained, this calculation must be repeated at regular time intervals.

(b) On the three-dimensional clothoid curve, it is expected that the variation of the curvature with respect to the curve length is approximately constant, and the corresponding numerical control information is expected to be mechanically reasonable control information from the viewpoint of motion control. In general spline interpolation, etc., it is difficult to predict and control changes in curvature.

(c) Three-dimensional clothoid curves include, as special cases, straight lines, circular arcs, spiral curves, etc., and can express numerical control information for various curves in detail without incorporating individual curve expressions.

(d) The three-dimensional clothoid curve is a natural equation that does not depend on the establishment method of coordinate axes. In a conventional NC device that expresses curves on the x, y, and z axes, for example, when machining a workpiece at an inclination, machining may be easy or difficult depending on how the workpiece is mounted. In the three-dimensional clothoid curve, since the curve is given according to the line length, even when machining an inclined surface, as long as a trajectory is made on the inclined surface, it can be processed in the same way as when machining a horizontal plane.

In addition, when the three-dimensional curves (called three-dimensional clothoid curves) of the inclination angle and the deflection angle in the tangential direction are given by the quadratic formula of the curve length or curve length variable of the present invention, the contour of the tool track or the workpiece is executed by the computer. In the case of a program for a shape, the program is stored in an auxiliary storage device such as a hard disk drive of a computer, loaded into the main memory, and executed. In addition, such a program may be stored on a portable recording medium such as a CD-ROM for sale, or may be stored in a recording device of a computer connected via a network, and may be transmitted to other computers via a network. In addition, the calculation results (curve parameters of the three-dimensional clothoid curve, displacement pulses of the motors of each axis, etc.) obtained by such a program can also be stored in a portable recording medium such as a CD-ROM for sale.

According to the three-dimensional clothoid curve of the present invention, it is possible to provide a widely used generation method for spatial curves required for the design and production of industrial products. In the case that the object moves along the space curve with acceleration and deceleration, the constraint force can be designed smoothly. Its characteristics can be widely used in the design method of the motion track of the mechanical element with mass. This time, as an application example of the design, the design method of the return path of the ball screw is explained, but other than that, it can also be used for the track of the jet inertial power unit that travels rapidly along the track that curves up, down, left, and right, for example. Design methods, linear guides, etc. In addition, since changes in curvature can be appropriately designed with respect to the length of the curve, it can be effectively used in various industrial fields such as aesthetically crafted curve design.

Claims (15)

1. A design method of an industrial product, characterized in that: the shape of the industrial product is designed by a three-dimensional clothoid curve defined by the following formula, and the inclination α and the deflection angle β of the tangential direction of the curve are determined by the curve length s or the curve length variable The quadratic form of S gives, The three-dimensional clothoid curve is defined as follows, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\mathrm{<span\; class="notranslate">\; uds</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; h</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; S</span>}}\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ α＝a ₀ +a ₁ S+a ₂ S ² (3) β=b ₀ +b ₁ S+b ₂ S ² (4) here, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ Represent the position vector of the point on the three-dimensional clothoid curve and its initial value, respectively, Set the length of the curve from the starting point as s, and set its total length, that is, the length from the starting point to the end point, as h, and use S to represent the value of dividing s by h, and S is a dimensionless value, which is called the curve length variable , i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively, u is the unit vector representing the tangent direction of the curve on the point P, given by formula (2), E ^kβ and E ^jα are rotation matrices, which respectively represent the rotation of the angle β of the k-axis system and the angle α of the j-axis system Rotation, the former is called deflection rotation, and the latter is called tilt rotation. Equation (2) means that by first making the unit vector in the i-axis rotate only α in the j-axis system, and then only rotate β in the k-axis system, Get the tangent vector u, a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ are constants. 2. The design method of industrial products as claimed in claim 1, characterized in that: The article of manufacture is a machine comprising a mechanism for moving a mechanical element having a mass, The trajectory of the movement of the mechanical element is designed using the three-dimensional clothoid curve. 3. The design method of industrial products as claimed in claim 2, characterized in that: said machine is a screw device comprising a mechanism for moving rolling bodies as elements of said machine; The screw device includes: a screw shaft having a helical rolling element raceway groove on the outer peripheral surface; a loaded rolling element raceway groove opposite to the rolling element raceway groove on the inner peripheral surface; one end of the load rolling element raceway groove and the nut of the return path at the other end; and arranged between the rolling element raceway groove of the screw shaft and the load rolling element raceway groove of the nut and multiple rolling elements on the return path, The regression path of the screw device is designed by using the three-dimensional clothoid curve. 4. The method for designing an industrial product according to any one of claims 1 to 3, characterized in that: a plurality of spatial points are specified in the three-dimensional coordinates, and by interpolating these spatial points using the three-dimensional clothoid curve, the designed The shape of the industrial product described. 5. The design method of an industrial product as claimed in claim 4, characterized in that: at said plurality of spatial points, a three-dimensional clothoid segment and the next three-dimensional clothoid used as a unit curve constituting a curve group generated by interpolation Clothoid segment, the position, tangent direction, normal direction and curvature of two three-dimensional clothoid segments are connected, and the seven parameters a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ of the three-dimensional clothoid segment are calculated , h. 6. The design method of industrial products as claimed in claim 5, characterized in that: Specify the tangent direction, normal direction and curvature of the starting point and the ending point in the plurality of spatial points, Re-insert interpolation object points between the pre-specified space points, thereby, The sum of the number of conditional expressions of the tangent direction, normal direction and curvature of the starting point and the end point and a three-dimensional clothoid segment and the next three-dimensional clothoid segment on the plurality of space points and the interpolation object point The number of conditional expressions obtained by adding the number of conditional expressions whose position, tangent direction, normal direction and curvature are continuous, and the seven parameters a ₀ , a ₁ , a ₂ of the three-dimensional clothoid segment , b ₀ , b ₁ , b ₂ , and h have the same number of unknowns; By matching the number of conditional expressions with the number of unknowns, the seven parameters a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , and h of the three-dimensional clothoid segment are calculated. 7. An industrial product, designed by using the design method for an industrial product as claimed in any one of claims 1-6. 8. A numerical control method, wherein, a three-dimensional clothoid curve defined by the following formula is used to represent the tool trajectory or the contour shape of the workpiece, the movement of the tool is controlled by the three-dimensional clothoid curve, the inclination angle α and the deflection angle of the tangential direction of the curve β is given by the quadratic expression of the curve length s or the curve length variable S, and the three-dimensional clothoid curve is defined as follows, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\mathrm{<span\; class="notranslate">\; uds</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; h</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; S</span>}}\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ α＝a ₀ +a ₁ S+a ₂ S ² (3) β=b ₀ +b ₁ S+b ₂ S ² (4) here, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ Represent the position vector of the point on the three-dimensional clothoid curve and its initial value, respectively, Set the length of the curve from the starting point as s, and set its total length, that is, the length from the starting point to the end point, as h, and use S to represent the value of dividing s by h, and S is a dimensionless value, which is called the curve length variable , i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively, u is the unit vector representing the tangent direction of the curve on the point P, given by formula (2), E ^kβ and E ^jα are rotation matrices, which respectively represent the rotation of the angle β of the k-axis system and the angle α of the j-axis system Rotation, the former is called deflection rotation, and the latter is called tilt rotation. Equation (2) means that by first making the unit vector in the i-axis rotate only α in the j-axis system, and then only rotate β in the k-axis system, Get the tangent vector u, a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ are constants. 9. The numerical control method as claimed in claim 8, characterized in that: Using the three-dimensional clothoid curve to express the contour shape of the tool trajectory or workpiece; specifying the motion of the tool moving along said three-dimensional clothoid curve; According to the specified movement, calculate the movement position of the tool per unit time, Here, motion refers to positional information that changes as a function of time. 10. A numerical control device, wherein a three-dimensional clothoid curve defined by the following formula is used to express the tool trajectory or the contour shape of the workpiece, the movement of the tool is controlled by the three-dimensional clothoid curve, the inclination angle α and the deflection angle of the tangential direction of the curve β is given by the curve length s or quadratic in the curve length S variable, The three-dimensional clothoid curve is defined as follows, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\mathrm{<span\; class="notranslate">\; uds</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; h</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; S</span>}}\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ α＝a ₀ +a ₁ S+a ₂ S ² (3) β=b ₀ +b ₁ S+b ₂ S ² (4) here, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ Represent the position vector of the point on the three-dimensional clothoid curve and its initial value, respectively, Set the length of the curve from the starting point as s, and set its total length, that is, the length from the starting point to the end point, as h, and use S to represent the value of dividing s by h, and S is a dimensionless value, which is called the curve length variable , i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively, u is the unit vector representing the tangent direction of the curve on the point P, given by formula (2), E ^kβ and E ^jα are rotation matrices, which respectively represent the rotation of the angle β of the k-axis system and the angle α of the j-axis system Rotation, the former is called deflection rotation, and the latter is called tilt rotation. Equation (2) means that by first making the unit vector in the i-axis rotate only α in the j-axis system, and then only rotate β in the k-axis system, Get the tangent vector u, a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ are constants. 11. The numerical control device as claimed in claim 10, characterized in that: Using the three-dimensional clothoid curve to express the contour shape of the tool trajectory or workpiece; specifying the motion of the tool moving along said three-dimensional clothoid curve; According to the specified movement, calculate the movement position of the tool per unit time, Here, motion refers to positional information that changes as a function of time. 12. A numerical control method, wherein, using a three-dimensional clothoid segment defined by the following formula to interpolate between point columns arbitrarily given in three-dimensional coordinates, the movement of the tool is controlled by the three-dimensional clothoid segment, and the tangent direction of the curve The inclination angle α and the deflection angle β are given by the quadratic formula of the curve length s or the curve length variable S, and the three-dimensional clothoid segment is defined by the following formula, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\mathrm{<span\; class="notranslate">\; uds</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; h</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; S</span>}}\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}$ (2) α＝a ₀ +a ₁ S+a ₂ S ² (3) β=b ₀ +b ₁ S+b ₂ S ² (4) here, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ represent the position vector and its initial value of the point on the three-dimensional clothoid segment respectively, Set the length of the curve from the starting point as s, and set its total length, that is, the length from the starting point to the end point, as h, and use S to represent the value of dividing s by h, and S is a dimensionless value, which is called the curve length variable , i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively, u is the unit vector representing the tangent direction of the curve on the point P, given by formula (2), E ^kβ and E ^jα are rotation matrices, which respectively represent the rotation of the angle β of the k-axis system and the angle α of the j-axis system Rotation, the former is called deflection rotation, and the latter is called tilt rotation. Equation (2) means that by first making the unit vector in the i-axis rotate only α in the j-axis system, and then only rotate β in the k-axis system, Get the tangent vector u, a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ are constants. 13. A numerical control method, wherein a plurality of three-dimensional clothoid segments defined by the following formula are connected, and the movement of the tool is controlled by the plurality of three-dimensional clothoid segments. The quadratic formula of s or the curve length variable S is given, and the described three-dimensional clothoid segment is defined by the following formula, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\mathrm{<span\; class="notranslate">\; uds</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; h</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; S</span>}}\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ α＝a ₀ +a ₁ S+a ₂ S ² (3) β=b ₀ +b ₁ S+b ₂ S ² (4) here, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ represent the position vector and its initial value of the point on the three-dimensional clothoid segment respectively, Set the length of the curve from the starting point as s, and set its total length, that is, the length from the starting point to the end point, as h, and use S to represent the value of dividing s by h, and S is a dimensionless value, which is called the curve length variable , i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively, u is the unit vector representing the tangent direction of the curve on the point P, given by formula (2), E ^kβ and E ^jα are rotation matrices, which respectively represent the rotation of the angle β of the k-axis system and the angle α of the j-axis system Rotation, the former is called deflection rotation, and the latter is called tilt rotation. Equation (2) means that by first making the unit vector in the i-axis rotate only α in the j-axis system, and then only rotate β in the k-axis system, Get the tangent vector u, a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ are constants. 14. The numerical control method as claimed in claim 12 or 13, characterized in that: on the joint of a three-dimensional clothoid segment and the next three-dimensional clothoid segment, with the positions of the two, the tangent direction and the curvature continuous mode according to the situation, Calculate the seven parameters a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , b ₂ , and h of the three-dimensional clothoid segment. 15. A numerical control device, wherein a three-dimensional clothoid segment defined by the following formula is used to interpolate between point columns arbitrarily given in three-dimensional coordinates, and the movement of the tool is controlled by the three-dimensional clothoid segment, and the tangent direction of the curve The inclination angle α and deflection angle β are given by the quadratic formula of the curve length s or the curve length variable S, The three-dimensional clothoid segment is defined by the following formula, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\mathrm{<span\; class="notranslate">\; uds</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; h</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; S</span>}}\mathrm{<span\; class="notranslate">\; wxya</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; k\&beta;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}^{\mathrm{<span\; class="notranslate">\; j\&alpha;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right]\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right\}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ \mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ α＝a ₀ +a ₁ S+a ₂ S ² (3) β=b ₀ +b ₁ S+b ₂ S ² (4) here, $\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}\\ \mathrm{<span\; class="notranslate">\; z</span>}\end{array}\right\}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right\}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ represent the position vector and its initial value of the point on the three-dimensional clothoid segment respectively, Set the length of the curve from the starting point as s, and set its total length, that is, the length from the starting point to the end point, as h, and use S to represent the value of dividing s by h, and S is a dimensionless value, which is called the curve length variable , i, j, and k are unit vectors in the x-axis, y-axis, and z-axis directions, respectively, u is the unit vector representing the tangent direction of the curve on the point P, given by formula (2), E ^kβ and E ^jα are rotation matrices, which respectively represent the rotation of the angle β of the k-axis system and the angle α of the j-axis system Rotation, the former is called deflection rotation, and the latter is called tilt rotation. Equation (2) means that by first making the unit vector in the i-axis rotate only α in the j-axis system, and then only rotate β in the k-axis system, Get the tangent vector u, a ₀ , a ₁ , a ₂ , b ₀ , b ₁ , and b ₂ are constants.

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