CN112276957B - Smooth transition method and system for straight line segment and circular arc - Google Patents
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Abstract
The invention provides a smooth transition method of a straight line segment and a circular arc. The transition curve obtained according to the method has two-step geometric continuity with the connection part of the straight line section and the circular arc, in addition, the transition curve is cut from one section on the plane elastic line, the curvature of the transition curve is ensured to be changed smoothly, and the cut curve section can not have the conditions of local closure and sharp points.
Description
Technical Field
The invention relates to the technical field of robot path planning, in particular to a smooth transition method and system for straight line segments and circular arcs.
Background
In industrial robot programming, the motion path of the tool center point TCP has two basic forms: and the straight line segment and the circular arc correspond to the linear motion command and the circular arc motion command respectively. Through the combination and sequential execution of motion instructions, complex end tool motions may be achieved. If the TCP (tool center point) moves according to the path defined by the instruction strictly, in order to ensure that the acceleration does not jump, the robot should stop at the end point of each path, and the robot is frequently started and stopped for a task needing to execute a plurality of movement instructions. Some applications (e.g., handling, loading, unloading) may desire that the robot maintain motion between commands and allow TCP to deviate from a specified path within a certain range. For such a requirement, it is typical to add parameters to the instruction and define an intersection region, in which a transition curve smoothly connecting adjacent path segments is constructed. The constructed transition curve should meet two requirements: firstly, the connection part of the transition curve and the front and rear path sections has two-step and more than two-step geometric continuity so as to ensure that the acceleration does not jump; secondly, the curvature change is gentle, and the phenomena of local closure, sharp points and the like are avoided.
Regardless of the order, the connections between the basic paths are divided into three cases: straight line segments and straight line segments, straight line segments and arcs, arcs and arcs. Accordingly, transitions between the basic paths are also classified into three cases. The transition of the straight line segment and the straight line segment is relatively simple, and in this case, the construction method of the transition curve comprises using a Bezier curve or performing vector superposition on the front path and the rear path. However, for the transition between a straight line segment and a circular arc, or between a circular arc and a circular arc, although the transition curve constructed by the above method has two-step geometric continuity at the joint, phenomena such as local closure and sharp point may occur.
For example, in the method, the system and the robot for transitioning the spatial trajectory of the industrial robot disclosed in application No. 201811627820.8, a 5-order bezier curve is adopted to construct a transition curve, the constructed transition curve has two-order geometric continuity with a front path and a rear path at a joint, but the transition curve may have local closure and cusp phenomena, so that the local change of curvature is severe, and the passing speed of the robot is affected.
Disclosure of Invention
The invention aims to provide a smooth transition method for straight line segments and circular arcs.
The invention solves the technical problems through the following technical means:
a smooth transition method for straight line segment and circular arc assumes that the known straight line segment is P0As a starting point, P1Is an end point, and is at P1Point and arc connected by P0To P1Is given by a direction vector of0，P0And P1The distance between is d; arc of a circle with P1As a starting point, P2As an end point, the center of the circle is O, the radius is R, and the circular arc is at a starting point P1Has a tangent vector of t1Normal vector is n1The central angle of the arc is theta; t is t0And n1Has an included angle of alpha, t0And t1The included angle of the angle is beta;
the method comprises the following specific steps: given a radius R of the fusion zonecThe intersection region is represented by P1As a circle center, RcA region of radius; constructing a transition curve in the blending region, and marking the starting point of the transition curve as PsEnd point is marked as Pe；PsOn a straight line segment, PeOn a circular arc, PsTo P1Is recorded as ds，PeTo P1Is recorded as de；
With O as the origin, t0In the x-axis direction, t0Rotated by an angle of beta to t1Is taken as a counterclockwise direction, a plane rectangular coordinate system XOY is established and taken as a reference coordinate system, and under the reference coordinate system, P is taken assThe coordinates are (x)s，ys)，PeThe coordinates are (x)e，ye)；
The transition curve is cut from a section on the plane elastic line, and under a reference coordinate system, the parameter equation has the following form,
in the formula, phi is a parameter of a parameter equation and ranges from-pi/2 to phi1And there is a flow of water and a gas,
u (phi), v (phi) is expressed as
Wherein E (phi, m) is the second type of elliptic integral, F (phi, m) is the first type of elliptic integral, m is the parameter of the elliptic integral, and the value range is more than 0 and less than m and less than 1; i.e. ic，jcIs a unit vector perpendicular to each other, related to m, and expressed by
When p is phi is 0, the coordinate of the corresponding point on the transition curve;
determining m, phi1And p is as follows:
according to the values of alpha and beta, the two situations are processed, wherein alpha is larger than 90 degrees or alpha is 90 degrees, beta is 0 degrees, alpha is smaller than 90 degrees or alpha is 90 degrees, and beta is 180 degrees, and the other situation is two;
for case one, the end point P of the transition curve is first determinedeCalculating P according to the following formulaeTo P1Distance d ofe，
From deTo obtain OPeArgument theta ofeComprises the following steps:
thus PeCoordinate (x)e，ye) Is composed of
(xe，ye)＝(R cosθe，R sinθe) (7)
Then find m and phi1，φ1The following relationship exists with respect to m,
by numerical means, in the interval [ sin2(θe/4+π/8)，sin2(θe/2+π/4)]Above solve the equation for m:
in the formula (9), E (m) is a second type complete elliptic integral, K (m) is a first type complete elliptic integral, and according to the formula (8), two sides of the equal sign of the formula (9) are only related to m; after m is solved, phi can be obtained from the formula (8)1；
Finally, p is obtained, and the p calculation formula obtained from the formula (2) is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (10)
For case two, first determine when φ1At pi, the starting point P of the transition curvesTo P1Distance d ofsAnd PeTo P1Distance d ofeThe specific method comprises the following steps: using numerical algorithms, in the interval [0, 0.1 ]]The following equation for mu is solved internally,
after solving for μ, dsAnd deIs determined by
Then, P is calculated according to the following formulasTo P1Maximum allowable distance d ofmax，
According to dmaxAnd ds，deThe difference in relative size is handled in two cases, the first case, dmax≥dsAnd d ismax≥deAt this time have
m＝u，φ1＝π (14)
PeCoordinate (x)e，ye) Is composed of
Represented by the formula (2), p is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (16)
Second case, dmax＜dsOr dmax＜deFor this case, the transition curve starting point P is first fixedsSo that ds＝dmaxSolving the parameter equation of the transition curve and verifying the end point PeTo P1Distance d ofeWhether or not it is not more than dmaxIf d ise≤dmaxThe equation of the transition curve is the solved parameter equation, if ds＞dmaxThen the end point P of the transition curve is fixedeSo that de＝dmaxAnd solving the parameter equation of the transition curve again.
Further, the starting point P of the transition curve is fixedsSo that ds＝dmaxThe process of solving the transition curve parameter equation is as follows: from ds＝dmaxTo obtain PsHas the coordinates of
(xs，ys)＝(-R sinβ-dmax，R cosβ) (17)
φ1M has the following relation
Wherein
c1＝dmax/R+sinβ，c2＝2-cosβ (19)
Using numerical methods, in intervals
The following equation for m is solved internally,
according to the formula (18), the equal sign sides of the formula (20) are only related to m; after m is obtained, phi can be obtained from the formula (18)1And then is composed ofFixed PsM and phi already obtained1From equation (2), it can be determined
p＝(xs，ys)-R|cosφ1|(u(-π/2)ic+v(-π/2)jc) (21)。
Further, the end point P of the transition curve is fixedeSo that de＝RcThe process of solving the parameter equation of the transition curve is as follows: from de＝dmaxTo obtain OPeArgument theta ofeIs composed of
Then P iseHas the coordinates of
(xe，ye)＝(R cosθe，R sinθe) (23)
At this moment phi1M has the following relation
By numerical means, in the interval [ sin2(θe/4-π/8)，sin2(θe/2-π/4)]The following equation for m is solved above,
according to the formula (24), the equal sign sides of the formula (25) are only related to m; after m is obtained, phi can be obtained from the formula (10)1And then from the fixed PeM and phi already obtained1From equation (2), it can be determined
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (26)。
The transition curve obtained according to the method has two-step geometric continuity with the connection part of the straight line section and the circular arc, in addition, the transition curve is cut from one section on the plane elastic line, the curvature of the transition curve is ensured to be changed smoothly, and the cut curve section can not have the conditions of local closure and sharp points.
The present invention also provides a smooth transition system for straight line segments and arcs, assuming that the straight line segment is known as P0As a starting point, P1Is an end point, and is at P1Point and arc connected by P0To P1Is given by a direction vector of0，P0And P1The distance between is d; arc of a circle with P1As a starting point, P2As an end point, the center of the circle is O, the radius is R, and the circular arc is at a starting point P1Has a tangent vector of t1Normal vector is n1The central angle of the arc is theta; t is t0And n1Has an included angle of alpha, t0And t1The included angle of the angle is beta;
the system comprises
A transition curve construction module for setting the radius R of the blend zonecThe intersection region is represented by P1As a circle center, RcA region of radius; constructing a transition curve in the blending region, and marking the starting point of the transition curve as PsEnd point is marked as Pe；PsOn a straight line segment, PeOn a circular arc, PsTo P1Is recorded as ds，PeTo P1Is recorded as de；
With O as the origin, t0In the x-axis direction, t0Rotated by an angle of beta to t1Is taken as a counterclockwise direction, a plane rectangular coordinate system XOY is established and taken as a reference coordinate system, and under the reference coordinate system, P is taken assThe coordinates are (x)s，ys)，PeThe coordinates are (x)e，ye)；
The transition curve is cut from a section on the plane elastic line, and under a reference coordinate system, the parameter equation has the following form,
wherein phi is a parameterThe parameters of the equation range from-pi/2 to phi1And there is a flow of water and a gas,
u (phi), v (phi) is expressed as
Wherein E (phi, m) is the second type of elliptic integral, F (phi, m) is the first type of elliptic integral, m is the parameter of the elliptic integral, and the value range is more than 0 and less than m and less than 1; i.e. ic，jcIs a unit vector perpendicular to each other, related to m, and expressed by
When p is phi is 0, the coordinate of the corresponding point on the transition curve;
a parameter value calculation module for determining m, phi1And p is as follows:
according to the values of alpha and beta, the two situations are processed, wherein alpha is larger than 90 degrees or alpha is 90 degrees, beta is 0 degrees, alpha is smaller than 90 degrees or alpha is 90 degrees, and beta is 180 degrees, and the other situation is two;
for case one, the end point P of the transition curve is first determinedeCalculating P according to the following formulaeTo P1Distance d ofe，
From deTo obtain OPeArgument theta ofeComprises the following steps:
thus PeCoordinate (x)e，ye) Is composed of
(xe，ye)＝(R cosθe，R sinθe) (7)
Then find m and phi1，φ1The following relationship exists with respect to m,
by numerical means, in the interval [ sin2(θe/4+π/8)，sin2(θe/2+π/4)]Above solve the equation for m:
in the formula (9), E (m) is a second type complete elliptic integral, K (m) is a first type complete elliptic integral, and according to the formula (8), two sides of the equal sign of the formula (9) are only related to m; after m is solved, phi can be obtained from the formula (8)1；
Finally, p is obtained, and the p calculation formula obtained from the formula (2) is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (10)
For case two, first determine when φ1At pi, the starting point P of the transition curvesTo P1Distance d ofsAnd PeTo P1Distance d ofeThe specific method comprises the following steps: using numerical algorithms, in the interval [0, 0.1 ]]The following equation for mu is solved internally,
after solving for μ, dsAnd deIs determined by
Then, P is calculated according to the following formulasTo P1Maximum allowable distance d ofmax，
According to dmaxAnd dsc，decThe difference in relative size is handled in two cases, the first case, dmax≥dsAnd d ismax≥deAt this time have
m＝u，φ1＝π (14)
PeCoordinate (x)e，ye) Is composed of
Represented by the formula (2), p is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (16)
Second case, dmax＜dsOr dmax＜deFor this case, the transition curve starting point P is first fixedsSo that ds＝dmaxSolving the parameter equation of the transition curve and verifying the end point PeTo P1Distance d ofeWhether or not it is not more than dmaxIf d ise≤dmaxThe equation of the transition curve is the solved parameter equation, if ds＞dmaxThen the end point P of the transition curve is fixedeSo that de＝dmaxAnd solving the parameter equation of the transition curve again.
Further, the starting point P of the transition curve is fixedsSo that ds＝dmaxThe process of solving the transition curve parameter equation is as follows: from ds＝dmaxTo obtain PsHas the coordinates of
(xs，ys)＝(-R sinβ-dmax，R cosβ) (17)
φ1M has the following relation
Wherein
c1＝dmax/R+sinβ，c2＝2-cosβ (19)
Using numerical methods, in intervals
The following equation for m is solved internally,
according to the formula (18), the equal sign sides of the formula (20) are only related to m; after m is obtained, phi can be obtained from the formula (18)1And then from the fixed PsM and phi already obtained1From equation (2), it can be determined
p＝(xs，ys)-R|cosφ1|(u(-π/2)ic+v(-π/2)jc) (21)。
Further, the end point P of the transition curve is fixedeSo that de＝RcThe process of solving the parameter equation of the transition curve is as follows: from de＝dmaxTo obtain OPeArgument theta ofeIs composed of
Then P iseHas the coordinates of
(xe，ye)＝(R cosθe，R sinθe) (23)
At this timeφ1M has the following relation
By numerical means, in the interval [ sin2(θe/4-π/8)，sin2(θe/2-π/4)]The following equation for m is solved above,
according to the formula (24), the equal sign sides of the formula (25) are only related to m; after m is obtained, phi can be obtained from the formula (10)1And then from the fixed PeM and phi already obtained1From equation (2), it can be determined
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (26)。
The invention also provides a straight line segment and circular arc smooth transition processing device, which comprises at least one processor and at least one memory connected with the processor in a communication way, wherein: the memory stores program instructions executable by the processor, which when called by the processor are capable of performing the methods described above.
The present invention also provides a computer-readable storage medium storing computer instructions that cause the computer to perform the above-described method.
The invention has the advantages that:
the parameter equation (1) of the transition curve and the equations (8), (18) and (24) in the solving process ensure that the transition curve has equal tangent and curvature at the joint of the straight line segment and the circular arc, so that the transition curve has two-order geometric continuity with the front curve and the rear curve at the joint; in addition, if the parameter value range is not limited, the parameter equation (1) represents a plane elastic line, the transition curve is actually a section of curve intercepted from the plane elastic line, and the plane elastic line is an energy minimum curve, so that the curvature of the transition curve is ensured to change smoothly, and the situations of closure and sharp points can not occur in the intercepted curve section.
Drawings
FIG. 1 is a schematic view of a transition curve in example 2 of the present invention;
FIG. 2 is a schematic view of a transition curve in example 3 of the present invention;
fig. 3 is a schematic diagram of a transition curve in embodiment 4 of the present invention.
Fig. 4 is a schematic diagram of a transition curve in embodiment 5 of the present invention.
Fig. 5 is a schematic diagram of a transition curve in embodiment 6 of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention, and it is obvious that the described embodiments are some embodiments of the present invention, but not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Example 1
A smooth transition method for straight line segment and circular arc, known as P0As a starting point, P1Is an end point, and is at P1Point and arc connected by P0To P1Is given by a direction vector of0，P0And P1The distance between is d; arc of a circle with P1As a starting point, P2As an end point, the center of the circle is O, the radius is R, and the circular arc is at a starting point P1Has a tangent vector of t1Normal vector is n1The central angle of the arc is theta; t is t0And n1Has an included angle of alpha, t0And t1The included angle of (b) is beta.
Given a radius R of the fusion zonecThe intersection region is represented by P1As a circle center, RcIs the area of the radius. Constructing a transition curve in the blending region, and marking the starting point of the transition curve as PsEnd point is marked as Pe。PsOn a straight line segment, PeOn a circular arc, PsTo P1Is recorded as ds，PeTo P1Is recorded as de。
With O as the origin, t0In the x-axis direction, t0Rotated by an angle of beta to t1Is taken as a counterclockwise direction, a plane rectangular coordinate system XOY is established and taken as a reference coordinate system, and under the reference coordinate system, P is taken assThe coordinates are (x)s，ys)，PeThe coordinates are (x)e，ye)。
The transition curve is cut from a section on the plane elastic line, and under a reference coordinate system, the parameter equation has the following form,
in the formula, phi is a parameter of a parameter equation and ranges from-pi/2 to phi1And there is a flow of water and a gas,
u (phi), v (phi) is expressed as
Wherein E (phi, m) is the second type of elliptic integral, F (phi, m) is the first type of elliptic integral, m is the parameter of the elliptic integral, and the value range is more than 0 and less than m and less than 1; i.e. ic，jcIs a unit vector perpendicular to each other, related to m, and expressed by
When p is equal to 0, the coordinate of the corresponding point on the transition curve.
If it can be confirmedDetermine m, phi1And p, the parametric equation for the transition curve will be fully determined, and the transition curve will be determined accordingly. M, phi below1And a method and a process for determining p.
According to the values of α and β, the two cases are handled, where α > 90 °, or α ═ 90 °, β ═ 0 °, or α < 90 °, or α ═ 90 °, or β ═ 180 °, or two.
For case one, the end point P of the transition curve is first determinedeCalculating P according to the following formulaeTo P1Distance d ofe，
From deTo obtain OPeArgument theta ofeComprises the following steps:
thus PeCoordinate (x)e，ye) Is composed of
(xe，ye)＝(R cosθe，R sinθe) (7)
Then find m and phi1，φ1The following relationship exists with respect to m,
by numerical means, e.g. dichotomy, in the interval [ sin2(θe/4+π/8)，sin2(θe/2+π/4)]Above solve the equation for m:
in the formula (9), E (m) is a second kind of complete elliptic integral, K (m) is a first kind of complete elliptic integral, and the equations (8) and (9) are equal in signEdges are only associated with m; after m is solved, phi can be obtained from the formula (8)1。
Finally, p is obtained, and the p calculation formula obtained from the formula (2) is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (10)
For case two, first determine when φ1At pi, the starting point P of the transition curvesTo P1Distance d ofsAnd PeTo P1Distance d ofeThe specific method comprises the following steps: using numerical algorithms, e.g. dichotomy, in the interval [0, 0.1]The following equation for mu is solved internally,
after solving for μ, dsAnd deIs determined by
Then, P is calculated according to the following formulasTo P1Maximum allowable distance d ofmax，
According to dmaxAnd ds，deThe difference in relative size is handled in two cases, the first case, dmax≥dsAnd d ismax≥deAt this time have
m＝u，φ1＝π (14)
PeCoordinate (x)e，ye) Is composed of
Represented by the formula (2), p is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (16)
Second case, dmax＜dsOr dmax＜deFor this case, the transition curve starting point P is first fixedsSo that ds＝dmaxSolving the parameter equation of the transition curve and verifying the end point PeTo P1Distance d ofeWhether or not it is not more than dmaxIf d ise≤dmaxThe equation of the transition curve is the solved parameter equation, if ds＞dmaxThen the end point P of the transition curve is fixedeSo that de＝dmaxAnd solving the parameter equation of the transition curve again.
Further, the starting point P of the transition curve is fixedsSo that ds＝dmaxThe process of solving the transition curve parameter equation is as follows: from ds＝dmaxTo obtain PsHas the coordinates of
(xs，ys)＝(-R sinβ-dmax，R cosβ) (17)
φ1M has the following relation
Wherein
c1＝dmax/R+sinβ，c2＝2-cosβ (19)
Using numerical methods, e.g. dichotomy, in intervals
The following equation for m is solved internally,
according to the formula (18), the equal sign sides of the formula (20) are only related to m; after m is obtained, phi can be obtained from the formula (18)1And then from the fixed PsM and phi already obtained1From equation (2), it can be determined
p＝(xs，ys)-R|cosφ1|(u(-π/2)ic+v(-π/2)jc) (21)
Further, the end point P of the transition curve is fixedeSo that de＝RcThe process of solving the parameter equation of the transition curve is as follows: from de＝dmaxTo obtain OPeArgument theta ofeIs composed of
Then P iseHas the coordinates of
(xe，ye)＝(R cosθe，R sinθe) (23)
At this moment phi1M has the following relation
By numerical means, e.g. dichotomy, in the interval [ sin2(θe/4-π/8)，sin2(θe/2-π/4)]The following equation for m is solved above,
according to the formula (24), the equal sign sides of the formula (25) are only related to m; after m is obtained, phi can be obtained from the formula (10)1And then from the fixed PeM and phi already obtained1From equation (2), it can be determined
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (26)
The transition curve obtained by the method of the embodiment has two-step geometric continuity with the straight line segment and the circular arc at the joint, and in addition, the transition curve is cut from one segment on the plane elastic line, so that the curvature of the transition curve is ensured to be changed smoothly, and the cut curve segment is free from the conditions of local closure and sharp points.
The embodiment also provides a system for smooth transition of a straight line segment and a circular arc, which comprises a transition curve construction module and a given blending area radius RcThe intersection region is represented by P1As a circle center, RcA region of radius; constructing a transition curve in the blending region, and marking the starting point of the transition curve as PsEnd point is marked as Pe；PsOn a straight line segment, PeOn a circular arc, PsTo P1Is recorded as ds，PeTo P1Is recorded as de；
With O as the origin, t0In the x-axis direction, t0Rotated by an angle of beta to t1Is taken as a counterclockwise direction, a plane rectangular coordinate system XOY is established and taken as a reference coordinate system, and under the reference coordinate system, P is taken assThe coordinates are (x)s，ys)，PeThe coordinates are (x)e，ye)；
The transition curve is cut from a section on the plane elastic line, and under a reference coordinate system, the parameter equation has the following form,
in the formula, phi is a parameter of a parameter equation and ranges from-pi/2 to phi1And there is a flow of water and a gas,
u (phi), v (phi) is expressed as
Wherein E (phi, m) is the second type of elliptic integral, F (phi, m) is the first type of elliptic integral, m is the parameter of the elliptic integral, and the value range is more than 0 and less than m and less than 1; i.e. ic，jcIs a unit vector perpendicular to each other, related to m, and expressed by
When p is phi is 0, the coordinate of the corresponding point on the transition curve;
a parameter value calculation module for determining m, phi1And p is as follows:
according to the values of alpha and beta, the two situations are processed, wherein alpha is larger than 90 degrees or alpha is 90 degrees, beta is 0 degrees, alpha is smaller than 90 degrees or alpha is 90 degrees, and beta is 180 degrees, and the other situation is two;
for case one, the end point P of the transition curve is first determinedeCalculating P according to the following formulaeTo P1Distance d ofe，
From deTo obtain OPeArgument theta ofeComprises the following steps:
thus PeCoordinate (x)e，ye) Is composed of
(xe，ye)＝(R cosθe，R sinθe) (7)
Then find m and phi1，φ1The following relationship exists with respect to m,
by numerical means, in the interval [ sin2(θe/4+π/8)，sin2(θe/2+π/4)]Above solve the equation for m:
in the formula (9), E (m) is a second type complete elliptic integral, K (m) is a first type complete elliptic integral, and according to the formula (8), two sides of the equal sign of the formula (9) are only related to m; after m is solved, phi can be obtained from the formula (8)1；
Finally, p is obtained, and the p calculation formula obtained from the formula (2) is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (10)
For case two, first determine when φ1At pi, the starting point P of the transition curvesTo P1Distance d ofsAnd PeTo P1Distance d ofeThe specific method comprises the following steps: using numerical algorithms, in the interval [0, 0.1 ]]The following equation for mu is solved internally,
after solving for μ, dsAnd deIs determined by
Then, P is calculated according to the following formulasTo P1Maximum allowable distance d ofmax，
According to dmaxAnd ds，deThe difference in relative size is handled in two cases, the first case, dmax≥dsAnd d ismax≥deAt this time have
m＝u，φ1＝π (14)
PeCoordinate (x)e，ye) Is composed of
Represented by the formula (2), p is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (16)
Second case, dmax＜dsOr dmax＜deFor this case, the transition curve starting point P is first fixedsSo that ds＝dmaxSolving the parameter equation of the transition curve and verifying the end point PeTo P1Distance d ofeWhether or not it is not more than dmaxIf d ise≤dmaxThe equation of the transition curve is the solved parameter equation, if ds＞dmaxThen the end point P of the transition curve is fixedeSo that de＝dmaxAnd solving the parameter equation of the transition curve again.
Further, the starting point P of the transition curve is fixedsSo that ds＝dmaxThe process of solving the transition curve parameter equation is as follows: from ds＝dmaxTo obtain PsHas the coordinates of
(xs，ys)＝(-R sinβ-dmax，R cosβ) (17)
φ1M has the following relation
Wherein
c1＝dmax/R+sinβ，c2＝2-cosβ (19)
Using numerical methods, in intervals
The following equation for m is solved internally,
according to the formula (18), the equal sign sides of the formula (20) are only related to m; after m is obtained, phi can be obtained from the formula (18)1And then from the fixed PsM and phi already obtained1From equation (2), it can be determined
p＝(xs，ys)-R|cosφ1|(u(-π/2)ic+v(-π/2)jc) (21)。
Further, the end point P of the transition curve is fixedeSo that de＝RcThe process of solving the parameter equation of the transition curve is as follows: from de＝dmaxTo obtain OPeArgument theta ofeIs composed of
Then P iseHas the coordinates of
(xe，ye)＝(R cosθe，R sinθe) (23)
At this moment phi1M has the following relation
By numerical means, in the interval [ sin2(θe/4-π/8)，sin2(θe/2-π/4)]The following equation for m is solved above,
according to the formula (24), the equal sign sides of the formula (25) are only related to m; after m is obtained, phi can be obtained from the formula (10)1And then from the fixed PeM and phi already obtained1From equation (2), it can be determined
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (26)。
Example 2
As shown in fig. 1, the present embodiment takes a specific input corresponding to case one as an example to describe the process of determining the transition curve in detail.
Given d 2.5, R1, θ 180, α 120, β 30, Rc＝1.2。
From α > 90 °, case one can be determined, and according to the case one processing method, d is obtained from equation (5)eθ was obtained from formula (6) as 1.2e(x) 0.2398 from formula (7)e，ye) (0.97138, 0.23751) using dichotomy, in the interval
The equation for m determined by equation (9) is solved above to obtain m as 0.29288, and then phi is obtained by equation (8)10.64952, and finally determining p (0.92593, -0.32098) from (10). m, phi1And p are determined, the parametric equation of the transition curve can be obtained by the formula (1). Fig. 1 is a schematic representation of the resulting transition curve.
Example 3
As shown in fig. 2, the present embodiment gives another example corresponding to case one.
Given d-4, R-1, θ -240, α -90, β -0, Rc＝1.8。
The case one can be determined from α being 90 ° and β being 0 °, a parametric equation of the transition curve is obtained according to the processing method of the case one, and fig. 2 is a schematic diagram of the obtained transition curve.
Example 4
As shown in fig. 3, the present embodiment takes a specific input corresponding to the second case as an example to describe the process of determining the transition curve in detail.
Given d 3.5, R2, θ 180, α 30, β 60, Rc＝1.7。
The case two can be judged by alpha < 90 degrees, and according to the processing method of the case two, a dichotomy is used in the interval [0, 0.1 ]]The equation expressed by equation (11) is internally solved, μ is obtained as 0.020663, and d is obtained from equation (12)s＝1.34701，deWhen d is obtained from formula (13) after 1.48127max1.7. Due to dmax≥dsAnd d ismax≥deIn accordance with the first case, m is 0.020663, phi is obtained from equation (14) according to the processing flow of the first case1Pi, is represented by formula (15) to give (x)e，ye) (-0.56902, 1.9173), and finally p (-2.17967, 1.09448) from formula (16). m, phi1And p are determined, the parametric equation of the transition curve can be obtained by the formula (1). Fig. 3 is a schematic diagram of the resulting transition curve.
Example 5
As shown in fig. 4, the present embodiment takes another specific input corresponding to the second case as an example to describe the process of determining the transition curve in detail.
Given d 2, R1, θ 180, α 60, β 150, Rc＝1。
The case two can be judged by alpha < 90 degrees, and according to the processing method of the case two, a dichotomy is used in the interval [0, 0.1 ]]The equation expressed by equation (11) is internally solved, μ is obtained as 0.08406, and d is obtained from equation (12)s＝1.99423，deWhen d is obtained from formula (13) after 1.6988max1. Due to dmax＜dsCorresponding to the second case. According to the second case of the process, the starting point P of the transition curve is first fixedsSo that ds＝dmax(x) is obtained from the formula (17)s，ys) Formula (19) yields c 1-1, c 2-2.86603, using the dichotomy method in the interval (-1.5, -0.86603)
The equation for m determined by equation (20) is solved internally to obtain m as 0.179253, and phi is obtained by equation (18)1And (3) p (-1.03924, -0.71694) from formula (21) is 1.93861. According to the obtained m, phi1And P, obtaining an end point P from the formulae (1) and (2)eCoordinate (x) ofe，ye)＝(-0.9933，-0.11557)，P1Coordinate (x) of1，y1)＝(R cos(β+π/2)，R sin(β+π/2))＝(-0.5，-0.866025)，PeAnd P1A distance d betweene0.898067. Due to de≤dmaxThe equation of the transition curve is the solved parameter equation. Fig. 4 shows a schematic representation of the resulting transition curve.
Example 6
As shown in fig. 5, the present embodiment takes a specific input as an example, and two methods are respectively used: the 5 th-order Bezier curve transition and the method provided by the invention are carried out, and the results are compared.
Given d 2, R1, θ 30 °, α 160 °, β 70 °, Rc＝0.17。
The transition is performed by using a 5-order Bezier curve and the method of the present invention, and FIG. 5 is a schematic diagram of the transition curve obtained by using the two methods, in which the 5-order Bezier curve has a local closure phenomenon.
The above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.
Claims (8)
1. A smooth transition method of a straight line segment and a circular arc is characterized in that the straight line segment is assumed to be known as P0As a starting point, P1Is an end point, and is at P1Point and arc connected by P0To P1Is given by a direction vector of0，P0And P1The distance between is d; arc of a circle with P1As a starting point, P2As an end point, the center of the circle is O, the radius is R, and the circular arc is at a starting point P1Has a tangent vector of t1Normal vector is n1The central angle of the arc is theta; t is t0And n1Has an included angle of alpha, t0And t1The included angle of the angle is beta;
the method comprises the following specific steps: given a radius R of the fusion zonecThe intersection region is represented by P1As a circle center, RcA region of radius; constructing a transition curve in the blending region, and marking the starting point of the transition curve as PsEnd point is marked as Pe；PsOn a straight line segment, PeOn a circular arc, PsTo P1Is recorded as ds，PeTo P1Is recorded as de；
With O as the origin, t0In the x-axis direction, t0Rotated by an angle of beta to t1Is taken as a counterclockwise direction, a plane rectangular coordinate system XOY is established and taken as a reference coordinate system, and under the reference coordinate system, P is taken assThe coordinates are (x)s，ys)，PeThe coordinates are (x)e，ye)；
The transition curve is cut from a section on the plane elastic line, and under a reference coordinate system, the parameter equation has the following form,
in the formula, phi is a parameter of a parameter equation and ranges from-pi/2 to phi1And there is a flow of water and a gas,
u (phi), v (phi) is expressed as
Wherein E (phi, m) is the second type of elliptic integral, F (phi, m) is the first type of elliptic integral, m is the parameter of the elliptic integral, and the value range is more than 0 and less than m and less than 1; i.e. ic，jcIs a unit vector perpendicular to each other, related to m, and expressed by
When p is phi is 0, the coordinate of the corresponding point on the transition curve;
determining m, phi1And p is as follows:
according to the values of alpha and beta, the two situations are processed, wherein alpha is larger than 90 degrees or alpha is 90 degrees, beta is 0 degrees, alpha is smaller than 90 degrees or alpha is 90 degrees, and beta is 180 degrees, and the other situation is two;
for case one, the end point P of the transition curve is first determinedeCalculating P according to the following formulaeTo P1Distance d ofe，
From deTo obtain OPeArgument theta ofeComprises the following steps:
thus PeCoordinate (x)e，ye) Is composed of
(xe，ye)＝(R cosθe，R sinθe)(7)
Then find m and phi1，φ1The following relationship exists with respect to m,
by numerical means, in the interval [ sin2(θe/4+π/8)，sin2(θe/2+π/4)]Above solve the equation for m:
in the formula (9), E (m) is a second type complete elliptic integral, K (m) is a first type complete elliptic integral, and according to the formula (8), two sides of the equal sign of the formula (9) are only related to m; after m is solved, phi can be obtained from the formula (8)1；
Finally, p is obtained, and the p calculation formula obtained from the formula (2) is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (10)
For case two, first determine when φ1At pi, the starting point P of the transition curvesTo P1Distance d ofsAnd PeTo P1Distance d ofeThe specific method comprises the following steps: using numerical algorithms, in the interval [0, 0.1 ]]The following equation for mu is solved internally,
after solving for μ, dsAnd deIs determined by
Then according toCalculation of P by the following equationsTo P1Maximum allowable distance d ofmax，
According to dmaxAnd ds，deThe difference in relative size is handled in two cases, the first case, dmax≥dsAnd d ismax≥deAt this time have
m＝u，φ1＝π (14)
PeCoordinate (x)e，ye) Is composed of
Represented by the formula (2), p is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (16)
Second case, dmax＜dsOr dmax＜deFor this case, the transition curve starting point P is first fixedsSo that ds＝dmaxSolving the parameter equation of the transition curve and verifying the end point PeTo P1Distance d ofeWhether or not it is not more than dmaxIf d ise≤dmaxThe equation of the transition curve is the solved parameter equation, if ds＞dmaxThen the end point P of the transition curve is fixedeSo that de＝dmaxAnd solving the parameter equation of the transition curve again.
2. The method for smooth transition of straight line segment and circular arc as claimed in claim 1, wherein the starting point P of the transition curve is fixedsSo that ds＝dmaxThe process of solving the transition curve parameter equation is as follows: from ds＝dmaxTo obtain PsHas the coordinates of
(xs，ys)＝(-R sinβ-dmax，R cosβ)(17)
φ1M has the following relation
Wherein
c1＝dmax/R+sinβ，c2＝2-cosβ (19)
Using numerical methods, in intervals
The following equation for m is solved internally,
according to the formula (18), the equal sign sides of the formula (20) are only related to m; after m is obtained, phi can be obtained from the formula (18)1And then from the fixed PsM and phi already obtained1From equation (2), it can be determined
p＝(xs，ys)-R|cosφ1|(u(-π/2)ic+v(-π/2)jc) (21)。
3. A smooth transition method of straight line segment and circular arc as claimed in claim 1, characterized by fixing transition curve end point PeSo that de＝RcThe process of solving the parameter equation of the transition curve is as follows: from de＝dmaxTo obtain OPeArgument theta ofeIs composed of
Then P iseHas the coordinates of
(xe，ye)＝(R cosθe，R sinθe)(23)
At this moment phi1M has the following relation
By numerical means, in the interval [ sin2(θe/4-π/8)，sin2(θe/2-π/4)]The following equation for m is solved above,
according to the formula (24), the equal sign sides of the formula (25) are only related to m; after m is obtained, phi can be obtained from the formula (10)1And then from the fixed PeM and phi already obtained1From equation (2), it can be determined
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (26)。
4. A smooth transition system for straight line segments and arcs, characterized by assuming a known straight line segment as P0As a starting point, P1Is an end point, and is at P1Point and arc connected by P0To P1Is given by a direction vector of0，P0And P1The distance between is d; arc of a circle with P1As a starting point, P2As an end point, the center of the circle is O, the radius is R, and the circular arc is at a starting point P1Has a tangent vector of t1Normal vector is n1The central angle of the arc is theta; t is t0And n1Has an included angle of alpha, t0And t1The included angle of the angle is beta;
the system comprises
A transition curve construction module for setting the radius R of the blend zonecThe intersection region is represented by P1As a circle center, RcA region of radius; constructing a transition curve in the blending region, and marking the starting point of the transition curve as PsEnd point is marked as Pe；PsOn a straight line segment, PeOn a circular arc, PsTo P1Is recorded as ds，PeTo P1Is recorded as de；
With O as the origin, t0In the x-axis direction, t0Rotated by an angle of beta to t1Is taken as a counterclockwise direction, a plane rectangular coordinate system XOY is established and taken as a reference coordinate system, and under the reference coordinate system, P is taken assThe coordinates are (x)s，ys)，PeThe coordinates are (x)e，ye)；
The transition curve is cut from a section on the plane elastic line, and under a reference coordinate system, the parameter equation has the following form,
in the formula, phi is a parameter of a parameter equation and ranges from-pi/2 to phi1And there is a flow of water and a gas,
u (phi), v (phi) is expressed as
Wherein E (phi, m) is the second type of elliptic integral, F (phi, m) is the first type of elliptic integral, m is the parameter of the elliptic integral, and the value range is more than 0 and less than m and less than 1; i.e. ic，jcIs a unit vector perpendicular to each other, related to m, and expressed by
When p is phi is 0, the coordinate of the corresponding point on the transition curve;
a parameter value calculation module for determining m, phi1And p is as follows:
according to the values of alpha and beta, the two situations are processed, wherein alpha is larger than 90 degrees or alpha is 90 degrees, beta is 0 degrees, alpha is smaller than 90 degrees or alpha is 90 degrees, and beta is 180 degrees, and the other situation is two;
for case one, the end point P of the transition curve is first determinedeCalculating P according to the following formulaeTo P1Distance d ofe，
From deTo obtain OPeArgument theta ofeComprises the following steps:
thus PeCoordinate (x)e，ye) Is composed of
(xe，ye)＝(R cosθe，R sinθe)(7)
Then find m and phi1，φ1The following relationship exists with respect to m,
by numerical means, in the interval [ sin2(θe/4+π/8)，sin2(θe/2+π/4)]Above solve the equation for m:
in the formula (9), E (m) is a second type complete elliptic integral, K (m) is a first type complete elliptic integral, and according to the formula (8), two sides of the equal sign of the formula (9) are only related to m; after m is solved, phi can be obtained from the formula (8)1；
Finally, p is obtained, and the p calculation formula obtained from the formula (2) is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (10)
For case two, first determine when φ1At pi, the starting point P of the transition curvesTo P1Distance d ofsAnd PeTo P1Distance d ofeThe specific method comprises the following steps: using numerical algorithms, in the interval [0, 0.1 ]]The following equation for mu is solved internally,
after solving for μ, dsAnd deIs determined by
Then, P is calculated according to the following formulasTo P1Maximum allowable distance d ofmax，
According to dmaxAnd ds，deThe difference in relative size is handled in two cases, the first case, dmax≥dsAnd d ismax≥deAt this time have
m＝u，φ1＝π (14)
PeCoordinate (x)e，ye) Is composed of
Represented by the formula (2), p is
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (16)
Second case, dmax＜dsOr dmax＜deFor this case, the transition curve starting point P is first fixedsSo that ds＝dmaxSolving the parameter equation of the transition curve and verifying the end point PeTo P1Distance d ofeWhether or not it is not more than dmaxIf d ise≤dmaxThe equation of the transition curve is the solved parameter equation, if ds＞dmaxThen the end point P of the transition curve is fixedeSo that de＝dmaxAnd solving the parameter equation of the transition curve again.
5. A smooth transition system of straight line segments and arcs as claimed in claim 4, wherein the starting point P of the transition curve is fixedsSo that ds＝dmaxThe process of solving the transition curve parameter equation is as follows: from ds＝dmaxTo obtain PsHas the coordinates of
(xs，ys)＝(-R sinβ-dmax，R cosβ)(17)
φ1M has the following relation
Wherein
c1＝dmax/R+sinβ，c2＝2-cosβ (19)
Using numerical methods, in intervals
The following equation for m is solved internally,
according to the formula (18), the equal sign sides of the formula (20) are only related to m; after m is obtained, phi can be obtained from the formula (18)1And then from the fixed PsM and phi already obtained1From equation (2), it can be determined
p＝(xs，ys)-R|cosφ1|(u(-π/2)ic+v(-π/2)jc) (21)。
6. -a smooth transition system of straight line segments and circular arcs in accordance with claim 4, characterized by the fixed transition curve end point PeSo that de＝RcThe process of solving the parameter equation of the transition curve is as follows: from de＝dmaxTo obtain OPeArgument theta ofeIs composed of
Then P iseHas the coordinates of
(xe，ye)＝(R cosθe，R sinθe)(23)
At this moment phi1M has the following relation
By numerical means, in the interval [ sin2(θe/4-π/8)，sin2(θe/2-π/4)]The following equation for m is solved above,
according to the formula (24), the equal sign sides of the formula (25) are only related to m; after m is obtained, phi can be obtained from the formula (10)1And then from the fixed PeM and phi already obtained1From equation (2), it can be determined
p＝(xe，ye)-R|cosφ1|(u(φ1)ic+v(φ1)jc) (26)。
7. A straight line segment and circular arc smooth transition processing device comprising at least one processor and at least one memory communicatively coupled to the processor, wherein: the memory stores program instructions executable by the processor, the processor invoking the program instructions to perform the method of any of claims 1 to 3.
8. A computer-readable storage medium storing computer instructions for causing a computer to perform the method of any one of claims 1 to 3.
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