CN110788874A - Manipulator for controlling clamping force to measure distance between clamping points - Google Patents

Abstract

The invention discloses a manipulator for controlling the clamping force and measuring the distance between clamping points, which is used for industrial robots. The device consists of clamping arm, coding screw, frame, driver and numerical control. The gripper arm includes a movable arm and a fixed arm with force and deformation sensing functions. The coding screw is composed of a micrometer screw and a tri-state encoder, and the tri-state encoder includes a toothed disc and a cantilever beam sensor. The clamping arm and the coded screw rod form a force-deformation-displacement composite sensing mechanism, which is installed on the frame. The CNC controls the clamping arm to clamp the object through the driver and the coded screw. The composite sensing mechanism interacts with the CNC to transmit measurement and control signals. The CNC controls the clamping force and gives the distance between the clamping points.

Description

Manipulator for controlling gripping force measurement gripping point spacing

technical field

This design is a manipulator that controls the gripping force and measures the distance between gripping points and is used in industrial robots.

Background technique

The manipulator is the front-end working mechanism of various robots. It is generally installed on the robotic arm of the robot. Its basic function is to pick and place, that is, to grab, transfer, and release objects. During the pick-and-place process, it is necessary to control the gripping force of the manipulator on the object. To configure manipulators for robots including a large number of automated production, processing, and testing equipment in the industrial field, in addition to the pick-and-place function, other functions, such as measurement, identification, assembly, adjustment, etc., need to be added according to different tasks. In the machinery industry, length dimension measurement is the most common measurement, but so far, there is still a lack of manipulators capable of dimension measurement. When designing such a manipulator, the key problem to be solved is: how to control the clamping force effectively, and measure the dimensional data conveniently and quickly with sufficient accuracy.

SUMMARY OF THE INVENTION

The purpose of this design is to provide an industrial robot with a manipulator that controls the clamping force and measures the distance between the clamping points. It is used to pick and place objects such as mechanical parts, and to control the clamping force. Spacing obtains dimensional data such as object length, thickness, diameter, and other data such as deformation. The manipulator consists of a guide rail seat, a sliding frame, a transmission shaft, a coding screw, a driver, a clamping arm and a numerical controller.

The structure of the guide seat includes a rectangular thick plate, a parallel male guide and a coupling. The parallel male rails are located on the front and rear sides above the thick plate, and are used to cooperate with the sliding frame. The coupling is fixed under the thick plate and is used to connect the arm of the robot.

The structure of the sliding frame includes a rectangular bottom plate, parallel female guide rails located at the front and rear edges of the bottom plate, a protrusion with a threaded through hole located at the bottom left of the bottom plate, a left bearing support plate located above the bottom plate and a left bearing support plate located above the bottom plate. The right bearing support plate near the right side and the U-shaped guide limit groove between the two bearing support plates on the upper surface of the bottom plate. A left bearing is embedded on the left bearing support plate, and a right bearing is embedded on the right bearing support plate, and the two are in a coaxial position. The axis of the threaded through hole on the protruding platform, the axis of the guiding and limiting groove, and the axes of the two bearings are all located in the longitudinal symmetry plane of the parallel female guide rail and are parallel to the axis of the parallel female guide rail. The sliding frame and the guide rail seat are installed together through the cooperation of the male guide rail and the female guide rail.

The structure of the transmission shaft includes a threaded rod and a connecting shaft on the left side of the threaded rod. The threaded rod is installed on the sliding frame through the cooperation with the threaded through hole on the protruding table, and the connecting shaft extends to the left side of the protruding table.

The coding screw consists of a micrometer screw and a three-state encoder. The micrometer screw is a threaded stepped shaft, and its structure is divided into three sections from left to right: I-II, II-III, and III-IV. Sections I-II and III-IV are optical axes, and sections II-III are threaded shafts. , the outer diameter of the thread is larger than the diameter of the optical axis of the I-II segment and the optical axis of the III-IV segment. The micrometer screw rod is installed on the sliding frame through the cooperation between the optical axis of the I-II section and the left bearing and the cooperation of the optical axis of the III-IV section and the right bearing, and the threaded shaft of the II-III section is located on the left bearing support plate and the right bearing. Between the support plates, the optical axis of the I-II segment extends to the left side of the left bearing support plate, and the optical axis of the III-IV segment extends to the right side of the right bearing support plate. The left and right end faces of the II-III threaded shaft are respectively matched with the right end face of the left bearing and the left end face of the right bearing. These two matching pairs prevent the micrometer screw from moving left and right. The three-state encoder consists of a sensor bracket, a toothed disc, an upper left cantilever beam sensor, an upper right cantilever beam sensor, a lower left cantilever beam sensor, and a lower right cantilever beam sensor. The sensor bracket is generally a rectangular frame, which is fixed on the upper right of the bottom plate of the sliding frame, and surrounds the toothed disk in the middle. A rectangular through hole on the side and a threaded hole in vertical communication with the rectangular through hole. The toothed disc is an equal-thickness disc with circular arc teeth distributed around it, and is coaxially fixed on the optical axis of the III-IV segment of the micrometer screw. The number of arc teeth is an integer multiple of 4, such as 180, 500. The shape and size of the four cantilever beam sensors are the same. The elastic body used is a constant-section elastic beam or a variable-section elastic beam. Side inner wall, right inner wall, left inner wall and lower inner wall. The four elastic beams are respectively machined with left triangular protruding edge, upper triangular protruding edge, right triangular protruding edge and lower triangular protruding edge near the free end towards the side of the toothed disc, and are respectively attached with a single shaft along the beam axis near the root. Resistance strain gauges [R ₅ , R ₆ ], [R ₇ , R ₈ ], [R ₉ , R ₁₀ ] and [R ₁₁ , R ₁₂ ]. Each of the four elastic beams has a certain amount of pre-deformation, and the elastic pressure generated by the pre-deformation keeps the tops of the four triangular ridges in contact with the circular arc teeth around the toothed disc respectively. The specific position of the contact point is determined according to the following conditions:

a. The longitudinal symmetry line of the toothed plate just passes through the center of the arc tooth just above and the arc tooth directly below, and the horizontal line of symmetry of the tooth plate just passes through the center of the leftmost arc tooth and the rightmost arc tooth.

b. At this time, the left triangular protruding edge is located on the horizontal symmetry line of the toothed disc, and is in contact with the vertex of the leftmost circular arc tooth. The right triangular protruding edge is located on the horizontal symmetry line of the toothed disc and the upper side of the rightmost circular arc tooth, and is aligned with the valley bottom between two adjacent circular arc teeth. Both the upper triangular protruding rib and the lower triangular protruding rib are located on the right side of the longitudinal symmetry line of the toothed disc, respectively contacting the right side of the directly above circular arc tooth and the right side of the directly lower circular arc tooth. The distance from the left triangular ridge to the longitudinal symmetry line of the chainring is represented by h _max , the distance from the right triangular ridge to the longitudinal symmetry line of the chainring is represented by h _min , and the contact point between the upper triangular ridge and the arc tooth just above is at the level of the chainring The distance of the symmetry line is equal to the distance from the contact point of the lower triangular protruding edge and the arc tooth directly below to the horizontal symmetry line of the chainring, both of which are represented by h _mid . There is a relationship between h _mid and h _min and h _max expressed by formula (1):

h _min , h _mid and h _max are collectively referred to as feature heights, where h _min is the minimum feature height, h _mid is the average feature height, and h _max is the maximum feature height.

The driver includes a first driver and a second driver, both of which can use a stepper motor with a coupling. The first driver is installed on the upper left of the guide rail seat, and is fixedly connected with the connecting shaft of the transmission shaft through the coupling. The second driver is installed on the left side of the left bearing support plate, and is fixedly connected with the optical axis of the I-II section of the micrometer screw through the coupling.

The gripping arm consists of a movable arm and a fixed arm, both of which are cantilevered elastic beam sensors with grippers, and the elastic beams used in the two sensors are of the same material, shape and size. The elastic beam of the movable arm is fixedly connected with a shaft sleeve to form an integrated structure, a transmission nut is embedded in the middle of the shaft sleeve, and a limit guide rod is embedded at the bottom. The movable arm is installed on the sliding frame through the cooperation between the transmission nut and the micrometer screw rod and the cooperation between the limit guide rod and the U-shaped guide limit groove. The cooperation of the limit guide rod and the U-shaped guide limit groove prevents the movable arm from rotating around the axis of the micrometer screw. When the micrometer screw rotates, the movable arm is driven to move along the axis of the micrometer screw. The matching pair of the transmission nut and the micrometer screw must take anti-backlash measures to meet two requirements: one is that theoretically, when the micrometer screw changes the direction of rotation, it can drive the transmission nut to move in the opposite direction without lag; the other is the transmission nut. The rotational degrees of freedom of the axis in the xy plane are zero. The elastic beam of the fixed arm is fixedly connected with the right bearing support plate to form an integrated structure. The directions of the movable arm and the fixed arm are vertically upward, and the two are in symmetrical positions. The elastic beam of the movable arm is attached with uniaxial resistance strain gauges R ₂ and R ₁ on the left and right sides of the bc section along the beam axis direction, and the elastic beam of the fixed arm is attached along the beam axis direction on the left and right sides of the bc section. Uniaxial resistance strain gauges R ₃ and R ₄ , resistance strain gauges R ₁ , R ₂ , R ₃ , and R ₄ form a full-bridge measurement circuit. The elastic beam of the clamping arm has the following two basic structural forms:

a) Fishhook-shaped folded beam, its structure includes a long arm ac section, a turning section ad section and a short arm de section, the cross sections of the three sections are all rectangular, and the section c is the fixed end of the folded beam. The section size of section ad is much larger than that of section ac and section de, and this section can be regarded as a rigid node of polybeam. The structure of the ac segment of the long arm is divided into two segments ab and bc from top to bottom, and the height H of the ab segment is greater than the height h ₁ of the bc segment. The de section of the short arm is generally a beam of equal section.

b) Straight beam, its structure is divided into two sections a-b and b-c from top to bottom, the height H of the a-b section is greater than the height h of the b-c section, the cross-sections of the a-b and b-c sections are rectangular, and the beam is fixed at the section c. end, the section a is the free end of the beam.

The gripper of the gripping arm has three types of surface contact, line contact and point contact, which are selected according to the requirements of use. The gripper of the fishhook-shaped folding beam is arranged at the section e of the de section of the short arm, for example, the left gripping surface on the movable arm and the right gripping surface on the fixed arm in Fig. 1 constitute a pair of surface-contact gripping surfaces. device. The distance between the two clamping surfaces is represented by s. The clamps of the straight beam are arranged on the left and right sides of the free end a, such as the first clamping thimble and the second clamping thimble in Figure 4(a), which constitute a pair of point-contact clamps; The third clamping thimble and the fourth clamping thimble constitute a pair of point-contact type clamps that are supported outward. Fishhook polybeams can be designed according to the condition of zero rotation of the gripper. This condition means: when the action line of the clamping force coincides with the intersection of the longitudinal symmetry plane of the beam and the cross section e, the rotation angle of the cross section e and the clamp in the longitudinal symmetry plane is zero. For example, it is assumed that the width of each segment of the polybeam is equal, which is indicated by b in Figure 1. The heights of the bc segment and the de segment are equal, that is, h ₁ =h ₂ =h. The height H of segment ab is ≥ 5h, so this segment can be regarded as a rigid body. The length L of the ac segment is 5 times the length l ₁ of the bc segment, that is, L=5l ₁ . According to the bending theory, under the condition of small linear elastic deformation, the relation can be obtained:

In formula (2), l ₂ is the length of de segment. As long as L, l ₁ and l ₂ satisfy equation (2), the gripper can maintain zero rotation angle.

表示。第一驱动器和第二驱动器分别接入驱动器控制电路。The numerical control device is a microcomputer measurement and control system with a strain signal acquisition-conditioning circuit, a driver control circuit and a measurement software, and is generally integrated with the robot control system. A full-bridge measurement circuit composed of resistance strain gauges R ₁ , R ₂ , R ₃ and R ₄ is connected with the strain signal acquisition-conditioning circuit. The resistance strain gauges [R ₅ , R ₆ ], [R ₇ , R ₈ ], [R ₉ , R ₁₀ ], [R ₁₁ , R ₁₂ ] are respectively connected to the strain signal acquisition-conditioning circuit in a half-bridge manner, and the numerical control The measured strain readings for these four half-bridge measurement circuits are given by

express. The first driver and the second driver are respectively connected to the driver control circuit.

In this design, the coded screw is a relatively independent component that is used to track and measure the displacement of the movable arm.

The coded lead screw works as follows:

均发生连续周期性变化，变化周期用T表示，T也表示齿盘上相邻两个圆弧齿的齿顶间距。齿盘每转过一个齿，即一个周期T，应变读数

分别完成一次循环。跟踪观察

的变化，当

达到最小值ε_rmin时，停止齿盘的转动，调节数控器上电阻应变计[R₅,R₆]所在电桥的平衡电路，使之达到平衡状态，即

重复前述动作，依次在

取得最小值ε_rmin时，调节电阻应变计[R₇,R₈]、[R₉,R₁₀]和[R₁₁,R₁₂]所在电桥的平衡电路，使

按以上方法完成四个半桥测量电路零位调整后，再转动齿盘，则应变读数

均在最小值0和一个最大值ε_rmax之间循环变化，最小值0对应于左三角形突棱或上三角形突棱或右三角形突棱或下三角形突棱处在正对相邻两圆弧齿之间的谷底位置，即对应最于小特征高度h_min，最大值ε_rmax对应于四个三角形突棱之一处在与圆弧齿顶点接触的位置，即对应于最大特征高度h_max。以上调整三态编码器测量电路零位的方法，称为零位四步调整法。1) Zero position adjustment of the three-state encoder measurement circuit: use the numerical controller to control the rotation of the second drive, the micrometer screw and the toothed disc rotate accordingly, and the strain reading

All have continuous periodic changes, the change period is represented by T, and T also represents the tooth tip spacing of two adjacent circular arc teeth on the toothed disc. Every time the toothed disc rotates one tooth, that is, one cycle T, the strain reading

Complete one cycle separately. Follow up

changes, when

When the minimum value ε _rmin is reached, stop the rotation of the toothed disc, and adjust the balance circuit of the bridge where the resistance strain gauges [R ₅ , R ₆ ] on the numerical control device are located to reach a balanced state, that is,

Repeat the above actions, in turn

When the minimum value ε _rmin is obtained, adjust the balance circuit of the bridge where the resistance strain gauges [R ₇ , R ₈ ], [R ₉ , R ₁₀ ] and [R ₁₁ , R ₁₂ ] are located so that

After completing the zero position adjustment of the four half-bridge measurement circuits according to the above method, and then rotating the toothed plate, the strain reading

Both are cyclically changed between a minimum value of 0 and a maximum value of ε _rmax . The minimum value of 0 corresponds to the left triangular protruding edge or the upper triangular protruding edge or the right triangular protruding edge or the lower triangular protruding edge facing the adjacent two circular arc teeth. The valley bottom position between , that is, corresponds to the minimum characteristic height h _min , and the maximum value ε _rmax corresponds to the position where one of the four triangular ridges is in contact with the apex of the arc tooth, that is, corresponds to the maximum characteristic height h _max . The above method of adjusting the zero position of the three-state encoder measurement circuit is called the zero position four-step adjustment method.

的最大值ε_rmax、最小值0和平均值0.5ε_rmax。数字1与最大特征高度h_max对应，定义为满值。数字0与最小特征高度h_min对应，定义为零值。数字1/2与平均特征高度h_mid对应，定义为中值。满值、零值和中值共同定义为应变读数的三态编码值，简称三态值。齿盘旋转时，三态值0、1/2、1按周期T循环变化。三态值的循环变化，用于确定齿盘的旋转状态，即齿盘的旋转方向和旋转角度。三态值总共有四种不同的组合，如表1所示：2) Determine the relationship between the strain reading and the rotation state of the toothed disc: After completing the zero position adjustment of the three-state encoder measurement circuit, it is specified that the numbers 1, 0 and 1/2 are used to represent the strain readings respectively.

The maximum value ε _rmax , the minimum value 0 and the average value _{0.5ε rmax} . The number 1 corresponds to the maximum feature height _hmax , which is defined as the full value. The number 0 corresponds to the minimum feature height _hmin and is defined as a zero value. The number 1/2 corresponds to the mean feature height _hmid , defined as the median. The full, zero, and median values are collectively defined as the tri-state coded value of the strain reading, or tri-state value for short. When the toothed disc rotates, the tri-state values 0, 1/2, and 1 change cyclically according to the period T. The cyclic change of the three-state value is used to determine the rotation state of the chainring, that is, the rotation direction and rotation angle of the chainring. There are a total of four different combinations of tristate values, as shown in Table 1:

的三态值组合Table 1 Strain readings

Tri-state value combination of

Table 2 Changes of tri-state values 0, 1/2, and 1 in each period T during the clockwise rotation of the chainring.

In the table, "↑" indicates that the tri-state value increases, and "↓" indicates that the tri-state value decreases

Table 3 Changes of tri-state values 0, 1/2, and 1 in each period T during the counterclockwise rotation of the chainring.

In the table, "↑" indicates that the tri-state value increases, and "↓" indicates that the tri-state value decreases

的三态值0、1/2、1分别发生不同的升降变化。编号①至⑧的八行数据，两两互异，均具有唯一性，其中每一行数据都唯一地表示齿盘的一个特定旋转状态。例如，编号③的一行数据表示且只表示齿盘沿顺时针方向转过一个周期T内的第三个1/4周期，即0.5T至0.75T。编号⑤的一行数据表示且只表示齿盘沿逆时针方向转过一个周期T内的第一个1/4周期，即0至0.25T。连续变化的应变读数与三态值配合，监测齿盘的旋转状态。Choose a tri-state value combination from Table 1 as the starting point to determine the rotation state of the chainring. For the sake of clarity, choose the tri-state value combination 1, then every time the chainring rotates a tooth in the clockwise direction, the tri-state value A cycle of one period T is completed as shown in Table 2. Each time the toothed disc rotates one tooth in the counterclockwise direction, the tri-state value completes a cycle of one period T as shown in Table 3. In Tables 2 and 3, period T is divided into four 1/4 sub-periods, and within each 1/4 sub-period, four strain readings

The three-state values of 0, 1/2, and 1 have different up and down changes respectively. The eight rows of data numbered ① to ⑧ are different from each other and are unique, and each row of data uniquely represents a specific rotation state of the chainring. For example, a line of data numbered ③ represents and only represents that the toothed disc rotates clockwise in the third 1/4 cycle within one cycle T, that is, 0.5T to 0.75T. The row of data numbered ⑤ represents and only represents that the toothed disc rotates counterclockwise in the first 1/4 cycle of one cycle T, that is, 0 to 0.25T. Continuously varying strain readings are paired with tri-state values to monitor the rotational state of the chainrings.

3) Measure the displacement of the movable arm: use the numerical controller to control the rotation of the micrometer screw, adjust the movable arm to a specified position or any position on the micrometer screw, and record the position as the movable arm. Displace the origin, and record the position of the chainring at this time as the zero position of the chainring. The toothed plate starts to rotate from the zero position of the toothed plate, and the movable arm is displaced from the displacement origin. S is used to represent the displacement of the movable arm relative to the displacement origin, which is called the displacement of the origin, and S is calculated by formula (3):

In formula (3), t represents the lead of the micrometer screw (3), N _c represents the number of teeth of the toothed disc, nz _{, s} represents the number of teeth that the toothed disc has rotated clockwise since the zero position of the toothed disc, n _{z ,n} represents the number of teeth that the chainring has rotated counterclockwise since the zero position of the chainring, and n _z,s and n _z,n are always positive. n _z represents the difference between n _z,s and n _z,n , which is defined as the effective number of rotating teeth. n _z,s , n _z,n and n _z are also referred to as sprocket rotation parameters. n _z and S are the number of generations. When the chainring rotates clockwise, the movable arm moves to the right, and the signs of n _z and S are both "-". When the chainring rotates counterclockwise, the movable arm moves to the left, and the signs of n _z and S are both "+".

The method of using the manipulator of this design includes the following steps:

1) Measure the line connection. The full-bridge measurement circuit composed of resistance strain gauges R ₁ , R ₂ , R ₃ , and R ₄ is connected to the numerical controller. sensory measurement. When it is used to measure F, the strain reading measured by the numerical control device is expressed by ε _rf ; when used to measure s, the strain reading measured by the numerical control device is expressed by ε _rd .

然后以

为标定数，用力F与应变读数ε_rf的函数关系，即式(4)计算力F：2) Calibration of the force measuring system. Generally, a standard load cell or a standard force measuring ring is used as the force value standard device, and the force value standard device is clamped by the gripper on the clamping arm to calibrate the force measuring system. Taking the surface contact type gripper shown in Figure 1 as an example, when calibrating, use the left grip surface on the movable arm and the right grip surface on the fixed arm to clamp the standard load cell or the standard force measuring ring, and the numerical control Control the movement of the movable arm, and apply a set of standard forces F ₁ , F ₂ , ..., F _N to the movable arm and the fixed arm, (F ₁ <F ₂ <..., < F _N , N is a positive integer not less than 2 , the value range is determined according to the needs, such as 2≤N≤10), and the numerical controller records the strain readings corresponding to F ₁ , F ₂ , ..., F _N at the same time

then with

is the calibration number, the functional relationship between the force F and the strain reading ε _rf , that is, the formula (4) calculates the force F:

In formula (4), A ₁ and B ₁ are constants, which are calculated according to formulas (5) and (6) respectively:

表示与力F_i对应的应变读数，即标定数

标定中，弹性梁在标准力F_N作用下产生的最大应力不得超过所用材料的比例极限。In equations (5) and (6), N represents the ordinal number of the standard force, F _i represents the force value of the standard force with different ordinal numbers,

Represents the strain reading corresponding to the force F _i , i.e. the calibration number

During calibration, the maximum stress generated by the elastic beam under the standard force F _N shall not exceed the proportional limit of the material used.

一般在5με～20με范围内取值，例如取

第二步，取标准块规d₀，用数控器控制可动臂移动，由第一夹持顶针和第二夹持顶针夹持d₀，当数控器测得的应变读数时，可动臂停止移动，将可动臂的当前位置记作位移原点，将齿盘的当前位置记作齿盘零位；第三步，用标准块规d₁，d₂…，d_n依次替换d₀，重复完成第二部动作，记下相应的应变读数

第四步，以

为标定数，按长度s与应变读数ε_rd的函数关系，即式(7)，计算s：3) Calibration of length measuring system. For the calibration of the length measuring (grip point spacing) system, the appropriate length standard must be selected according to the type of the holder. The standard diameter gauge and the standard thickness gauge are the most commonly used length standards. The standard diameter gauge is a set of standard cylinders with different diameters for line contact grippers and surface contact grippers; the standard thickness gauge is a set of standard thickness block gauges with different thicknesses, mainly used for point contact gripping It can also be used for line contact grippers and surface contact grippers. _The thickness value _of each standard cylinder or standard block _{gauge is} represented by d ₀ , d ₁ , d _{2 .} The number is generally 2≤n≤10 (for example, n=7). d ₀ , d ₁ , d ₂ , ..., _dn are also used to denote the corresponding standard cylinder or standard block gauge. Here, the calibration method will be described by taking the point contact type gripper shown in FIG. 4( a ) as an example. Using the standard thickness gauge as the standard, the thickness value of the standard block gauge should meet the conditions: when the deflection of the elastic beam of the clamping arm at the section e is λ=d _n -d ₀ , the maximum stress value of the beam does not exceed the material used ratio limit. The calibration is performed in four steps: In the first step, an initial value is preset for the strain reading ε _rd

Generally, the value is in the range of 5με～20με, for example, take

In the second step, take the standard block gauge d ₀ , use the numerical controller to control the movement of the movable arm, and clamp d ₀ by the first clamping thimble and the second clamping thimble, when the numerical controller measures the strain reading When the movable arm stops moving, the current position of the movable arm is recorded as the displacement origin, and the current position of the toothed plate _is recorded as the zero position of the toothed plate; in the third step, use the standard block gauge d ₁ , d ₂ … Substitute d ₀ in turn, repeat the second action, and record the corresponding strain reading

The fourth step is to

is the calibration number, according to the functional relationship between the length s and the strain reading ε _rd , that is, formula (7), calculate s:

In formula (7), A ₂ and B ₂ are constants, which are calculated by formulas (8) and (9), respectively:

表示与d_i对应的应变读数，即标定数 In formulas (8) and (9), n is the number of standard thickness gauges, d _i is the thickness value of standard thickness gauges with different thicknesses,

Indicates the strain reading corresponding to d _i , that is, the calibration number

Equations (4) and (9) are both derived by the linear fitting method.

时，停止第一驱动器的转动；第三步，控制第二驱动器转动，使可动臂向右移动，右夹面与物体接触，当应变读数

时，停止第二驱动器的转动；第四步，控制两驱动器同时转动，使可动臂和固定臂同步相向移动，当应变读数ε_rd取得

范围的任何一值ε_rdx时，记下ε_rdx、对应于ε_rdx的应变读数ε_rf以及赖原点位移S；第五步，将ε_rf代入式(4)，将ε_rdx和S代入公式(10)：4) Clamping and measuring operations. The operation process will be described using the surface contact type gripper shown in Fig. 1 as an example. For simplicity, let the object be a rigid body with a fixed position. The operation procedure can be divided into five steps: the first step is to adjust the position of the manipulator and the distance s between the two clamping surfaces by the numerical controller, so that s is greater than the length m of the object, and the object enters between the two clamping surfaces; the second step is to control the first The drive rotates, so that the fixed arm moves to the left with the sliding frame, and the right clamping surface is in contact with the object. When the strain reading

When , stop the rotation of the first driver; in the third step, control the rotation of the second driver, so that the movable arm moves to the right, and the right clamping surface is in contact with the object, when the strain reading

When , stop the rotation of the second driver; in the fourth step, control the two drivers to rotate at the same time, so that the movable arm and the fixed arm move synchronously towards each other, when the strain reading _εrd is obtained

At any value of ε _rdx in the range, write down ε _rdx , the strain reading ε _rf corresponding to ε _rdx , and the displacement S of the origin; the fifth step, substitute ε _rf into equation (4), and substitute ε _rdx and S into the formula ( 10):

Equation (4) gives the clamping force F of the object, and Equation (10) gives the length m of the object.

做为测量起点；第二步，用数控器控制夹持器夹持弹簧，使应变读数

记下

和与之对应的赖原点位移S^*；第三步，增大夹持力F，跟踪记录F、应变读数ε_rd和赖原点位移S，按式(11)计算弹簧的变形量ν：Based on the measurement of the clamping point distance s, the deformation of deformed objects such as springs and rubber can be measured. Taking a compression spring as an example, the measurement procedure is divided into three steps: The first step is to set a strain reading

As the starting point of measurement; in the second step, use the CNC to control the gripper to clamp the spring to make the strain reading

Write down

and the corresponding origin displacement S ^* ; in the third step, increase the clamping force F, track and record F, the strain reading ε _rd and the origin displacement S, and calculate the spring deformation ν according to formula (11):

Using the measurement software, the force-deformation relationship curve, that is, the F-ν curve, can be drawn.

Compared with the existing manipulator, this design has the following characteristics:

1. Use the combination mechanism of the clamping arm and the coding screw to control the clamping force and measure the distance between the clamping points. The core part of the manipulator of this design is a force-deformation-displacement composite sensing mechanism composed of a clamping arm and a coded screw rod. closed-loop control and measurement. The measured F value directly represents the force on the object. The measured s value, for a rigid object or structure, directly represents the thickness, outer diameter, inner diameter, hole distance, spacing and other length dimensions; for elastic elements such as springs and easily deformable objects such as rubber and plant fruits, s can be used to calculate The amount of deformation that needs to be measured is obtained, and the relationship between force and deformation is given by the software.

2. The clamping is flexible. The design uses the elastic force of the clamping arm to clamp the object, and the change of the clamping force is controlled by the numerical controller, so the clamping method is flexible.

3. The measurement is flexible. When the designed manipulator clamps a rigid object, if the toothed plate is controlled to rotate at a small angle, so that the strain reading ε _rd changes continuously, it can be seen that although ε _rd and the displacement S of the origin point are both changing, formula (10) gives The resulting length value m remains unchanged. This phenomenon is called "flexible equal difference output", which is a unique property of the force-deformation-displacement composite sensing mechanism composed of the clamping arm and the coding screw, and is also the principle basis for measuring the distance between the clamping points in this design. Changing ε _rd within a certain range, the measured length m remains unchanged, and different values of ε _rd correspond to different clamping forces. Therefore, the algorithm design of the measurement software can make use of the flexible equidistant output properties according to different operation objects and measurement requirements.

4. The design range of the clamping force and its measurement and control accuracy is wide. For the manipulator designed in this design, the value range of the clamping force and the measurement resolution of the clamping force are mainly determined by the structural form, size and material properties of the elastic beam used in the clamping arm, and the design flexibility is very large. The value range is 0 to 10N, and the gripping arm with the resolution less than 0.1N can also be designed with the force value range of 0 to 5000N and the gripping arm with the resolution less than 1N.

5. The design range of clamping point spacing adjustment is wide. For the manipulator designed in this design, the adjustment range of the distance between the clamping points mainly depends on the movement of the movable arm, and the theoretical minimum distance is zero. Therefore, selecting the appropriate length of micrometer screw for the clamping arm-encoding screw combination mechanism can obtain the required adjustment range of the clamping point spacing, such as 0-20 mm, 0-100 mm, 200-500 mm.

6. The resolution of the distance measurement between the clamping points can reach the medium level of the length measurement in the machinery industry. For the manipulator designed in this design, the measurement resolution of the distance between the clamping points is determined by the two factors of the clamping arm and the coding screw. The displacement resolution of the clamping arm and the displacement resolution of the coding screw are basically the same in design range, ranging from 0.1 μm to 10 μm.

Description of drawings

Fig. 1 is the front view of the structural sketch of this design device;

Fig. 2 is the right side view of the device structure diagram;

Figure 3 is a schematic diagram of a full-bridge measurement circuit of a dual cantilever beam sensor;

Fig. 4 is the measurement circuit of the three-state encoder, in which (a) half-bridge diagram of resistance strain gage [R ₅ , R ₆ ], (b) half-bridge diagram of resistance strain gage [R 7 , R 8 ], (c) resistance strain gage [R ₇ , R ₈ ] half-bridge diagram Half-bridge diagram of strain gage [R ₉ , R ₁₀ ], (d) half-bridge diagram of resistance strain gage [R ₁₁ , R ₁₂ ];

Fig. 5 is a schematic diagram of the construction principle of a three-state encoder;

Fig. 6 is the schematic diagram of the clamping arm in the form of a straight beam;

In the picture: 1. Guide rail seat, 2. Coupling, 3. Male guide rail, 4. Transmission shaft, 5. Protruding table, 6. First driver, 7. Second driver, 8. Left bearing support plate, 9. Micrometer screw, 10. Left bearing, 11. U-shaped guide limit groove, 12. Transmission nut, 13. Limit guide rod, 14. Right bearing, 15. Right bearing support plate, 16. Movable arm, 17 .Fixed arm, 18. Left clamp surface, 19. Right clamp surface, 20. Sensor bracket, 21. Upper right cantilever beam sensor, 22. Gear plate, 23. Arc tooth, 24. Left lower cantilever beam sensor, 25. Right lower Cantilever beam sensor, 26. Sliding frame, 27. Female guide rail, 28. Left upper cantilever beam sensor, 29. Left triangular ridge, 30. Upper triangular ridge, 31. Right triangular ridge, 32. Lower triangular ridge, 33. Rectangular through hole, 34. Threaded hole, 35. Set screw, 36. First clamping thimble, 37. Second clamping thimble, 38. Third clamping thimble, 39. Fourth clamping thimble, 40 .1st clamping blade, 41. 2nd clamping blade, 42. 3rd clamping blade, 43. 4th clamping blade, TE. Tri-state encoder, W. Object.

Detailed ways

The design will be further described below in conjunction with the accompanying drawings.

Referring to Figures 1 to 6, this design is a manipulator that controls the clamping force and measures the distance between the clamping points. The manipulator consists of a guide rail seat 1, a sliding frame 26, a transmission shaft 4, a coding screw, a driver, a clamping arm and a numerical controller.

The structure of the guide rail seat 1 includes a rectangular thick base plate, a coupling 2 and a parallel male guide rail 3 . The coupling 2 is fixed under the thick base plate and is used to connect the robot arm. The parallel male guide rails 3 are located on the front and rear sides of the upper surface of the thick substrate.

The structure of the sliding frame 26 includes a rectangular bottom plate, parallel female guide rails 27 located at the front and rear edges of the bottom plate, a protrusion 5 with threaded through holes located at the bottom left of the bottom plate, and a left bearing support plate 8 located on the left side of the bottom plate. , The right bearing support plate 15 located above the bottom plate and close to the right side, and the U-shaped guide limiting groove 11 located on the upper surface of the bottom plate. The left bearing support plate 8 is embedded with the left bearing 10 , and the right bearing support plate 15 is embedded with the right bearing 14 . The left bearing 10 is in a coaxial position with the right bearing 14 . The axis of the threaded through hole on the boss 5 , the axis of the guide limit groove 11 , and the axes of the left bearing 10 and the right bearing 14 are all located in the longitudinal symmetry plane of the parallel female guide rail 27 and are parallel to the axis of the parallel female guide rail 27 . The sliding frame 26 and the guide rail seat 1 are installed together by the cooperation of the male guide rail 3 and the female guide rail 27 .

The structure of the transmission shaft 4 includes a threaded rod and a connecting shaft on the left side of the threaded rod.

The encoder screw consists of a micrometer screw 9 and a three-state encoder TE.

The micrometer screw 9 is a stepped shaft with a thread, and its structure is divided into three sections from left to right: I-II, II-III, and III-IV. Sections I-II and III-IV are optical axes, and sections II-III are threads. The outer diameter of the thread is larger than the diameter of the optical axis of the I-II section and the optical axis of the III-IV section. The micrometer screw 9 is installed on the sliding frame 26 through the cooperation between the optical axis of the I-II section and the left bearing 10 and the cooperation of the optical axis of the III-IV section and the right bearing 14, and the threaded shaft of the II-III section is located on the left bearing support. Between the plate 8 and the right bearing support plate 15 , the optical axis of the I-II section extends to the left side of the left bearing support plate 8 , and the optical axis of the III-IV section extends to the right side of the right bearing support plate 15 . The left and right end faces of the II-III threaded shaft are respectively matched with the right end face of the left bearing 10 and the left end face of the right bearing 14. These two mating pairs prevent the micrometer screw 9 from moving left and right.

The three-state encoder TE consists of a sensor bracket 20 , a toothed disc 22 , an upper left cantilever beam sensor 28 , an upper right cantilever beam sensor 21 , a lower left cantilever beam sensor 24 and a lower right cantilever beam sensor 25 . The sensor bracket 20 can adopt a rectangular frame, and four sides of the frame are sequentially processed with a rectangular through hole 33 whose axis is parallel to the plane of the frame and perpendicular to the side, and a rectangular through hole 33 which is in vertical communication with the rectangular through hole 33 at the position close to one side corner. Threaded hole 34. The toothed disc 22 is a disc of equal thickness with circular arc teeth 23 evenly distributed around the periphery, and the circular arc teeth 23 can be made by inlaying steel balls or by cutting technology. The number of circular arc teeth 23 is an integer multiple of 4, for example, 180, 500. The shape and size of the four cantilever beam sensors are the same, and the elastic body used is a constant-section elastic beam or a variable-section elastic beam. The screw 35 is screwed into the threaded hole 34 to generate the pressing force, and is fixed on the upper inner wall, the right inner wall, the left inner wall and the lower inner wall of the sensor bracket 20 . The four elastic beams are respectively attached with uniaxial resistance strain gauges [R ₅ , R ₆ ], [R ₉ , R ₁₀ ], [R ₁₁ , R ₁₂ ] and [R ₇ , R ₈ along the beam axis direction near the roots. ]. The four elastic beams are respectively machined with a left triangular protruding rib 29 , an upper triangular protruding rib 30 , a lower triangular protruding rib 32 , and a right triangular protruding rib 31 at the side near the free end facing the toothed plate 22 . The four elastic beams have a certain amount of pre-deformation, and the elastic pressure generated by the pre-deformation keeps the tops of the four triangular ridges in contact with the arc teeth 23 respectively. The specific position of the contact point is determined according to the following conditions:

a. Set the longitudinal symmetry line of the toothed plate 22 to pass through the center of the arc tooth directly above and the arc tooth directly below, while the horizontal line of symmetry of the tooth plate 22 passes through the center of the leftmost arc tooth and the rightmost arc tooth.

b. At this time, the left triangular protruding rib 29 is located on the horizontal symmetry line of the toothed disc 23 and is in contact with the apex of the leftmost circular arc tooth. The right triangular protruding rib 31 is located on the horizontal symmetry line of the toothed plate 22 and the upper side of the rightmost circular arc tooth, facing the valley bottom between two adjacent circular arc teeth 23 . Both the upper triangular protruding rib 30 and the lower triangular protruding rib 32 are located on the right side of the longitudinal symmetry line of the toothed plate 23, and are in contact with the right side of the directly above circular arc tooth and the right side of the directly below circular arc tooth respectively. The distance from the left triangular ridge 29 to the longitudinal symmetry line of the chainring 22 is represented by h _max , the distance from the right triangular ridge 31 to the longitudinal symmetry line of the chainring 23 is represented by h _min , and the upper triangular ridge 30 is in contact with the circular arc teeth directly above. The distance from the point to the horizontal symmetry line of the chainring 22 is the same as the distance from the contact point of the lower triangular protruding rib 32 and the arc tooth directly below to the horizontal symmetry line of the chainring 22, and both are represented by h _mid . There is a relationship between h _mid and h _min and h _max expressed by formula (1):

h _min , h _mid and h _max are collectively referred to as feature heights, where h _min is the minimum feature height, h _mid is the average feature height, and h _max is the maximum feature height.

The toothed disc 22 is coaxially fixed on the optical axis of the III-IV section of the micrometer screw 9 . The sensor bracket 20 is fixed on the upper right of the bottom plate of the sliding frame 26 and surrounds the toothed plate 22 in the middle.

The driver includes a first driver 6 and a second driver 7, both of which can use a stepping motor. The first driver 6 is installed on the upper left of the guide rail seat 1 and is fixedly connected with the connecting shaft of the transmission shaft 4 . The second driver 7 is installed on the left side of the left bearing support plate 8 and is fixedly connected with the optical axis of the section I-II of the micrometer screw 9 .

The clamping arm is composed of a movable arm 16 and a fixed arm 17, both of which are cantilevered elastic beam sensors with clamps, and the two pairs of elastic beams of the sensor are of the same material, shape and size. The elastic beam of the movable arm 16 is fixedly connected with a shaft sleeve to form an integral structure, the middle part of the shaft sleeve is embedded with a transmission nut 12, and the bottom part is embedded with a limit guide rod 13. The movable arm 16 is installed on the sliding frame 26 through the cooperation between the transmission nut 12 and the micrometer screw 9 and the cooperation between the limit guide rod 13 and the U-shaped guide limit groove 11 . The cooperation of the limit guide rod 13 and the U-shaped guide limit groove 11 prevents the movable arm 16 from rotating around the axis of the micrometer screw 9 . When the micrometer screw 9 rotates, the movable arm 16 is driven along the micrometer screw 9 . move in the direction of the axis. The transmission nut 12 and the micrometer screw 9 must be matched with backlash-elimination measures to meet two requirements: First, it can be considered theoretically that when the micrometer screw 9 changes the direction of rotation, it can drive the transmission nut 12 to move in the opposite direction without lag; Second, the rotational freedom of the axis of the transmission nut 12 in the x-y plane is zero. The elastic beam of the fixed arm 17 is fixedly connected with the right bearing support plate 15 to form an integrated structure. The elastic beam of the movable arm 16 and the elastic beam of the fixed arm 17 are vertically upward, and the two are in a symmetrical position. The elastic beam of the clamping arm has two basic structural forms:

a) Fishhook-shaped folded beam, its structure includes a long arm ac section, a turning section ad section and a short arm de section, the cross-section of the three sections are all rectangles, and the section c is the fixed end of the folded beam; the section size of the ad section It is much larger than the section size of the ac and de sections, so this section can be regarded as a rigid node of the polybeam. The structure of the ac segment of the long arm is divided into two segments ab and bc from top to bottom, and the height H of the ab segment is greater than the height h ₁ of the bc segment. The de section of the short arm is a beam of equal section. The heights of the long arm section ac, the turning section ad section and the short arm section de may be equal or unequal.

b) Straight beam, its structure is divided into two sections a-b and b-c from top to bottom, the height H of the a-b section is greater than the height h of the b-c section, the cross-sections of the a-b and b-c sections are rectangular, and the beam is fixed at the section c. end, the section a is the free end of the beam.

The elastic beam of the movable arm 16 is attached with uniaxial resistance strain gauges R ₂ and R ₁ on the left and right sides of the bc section along the beam axis direction, and the elastic beam of the fixed arm 17 is on the left and right sides of the bc section along the beam axis direction The uniaxial resistance strain gauges R ₃ and R ₄ are attached, and the resistance strain gauges R ₁ , R ₂ , R ₃ , and R ₄ form a full-bridge measurement circuit.

The gripper of the gripping arm has three types of surface contact, line contact and point contact, which are selected according to the requirements of use. The gripper of the fishhook-shaped folding beam is arranged at the section e of the short arm de section, for example, the left clamping surface 18 on the movable arm 16 and the right clamping surface 19 on the fixed arm 17 in FIG. 1 constitute a pair of surfaces. Contact type gripper. The distance between the left clamping surface 18 and the right clamping surface 19 is denoted by s. The clamps of the straight beam are arranged on the left and right sides of the free end a, such as the first clamping thimble 36 on the movable arm 16 and the second clamping thimble 37 on the fixed arm 17 in FIG. 6(a), The two form a pair of point-contact clamps; the third clamp thimble 38 on the movable arm 16 and the fourth clamp thimble 39 on the fixed arm 17 form a pair of point-contact clamps supported outward. When the action line of the clamping force of the fishhook-shaped folded beam coincides with the intersection of the longitudinal symmetry plane of the beam (xy plane in Figure 1) and the cross-section e, the structural dimensions of each part of the beam can be coordinated to make the cross-section e. The angle of rotation is equal to zero, thus keeping the gripper at zero angle of rotation. For example, suppose that the width b of each segment of the polybeam is equal, as shown in Figure 1; the height of segment bc and segment de are equal, that is, h ₁ =h ₂ =h; the height of segment ab is greater than or equal to 5h, so this segment can be regarded as rigid body. The length L of the ac segment is 5 times the length l ₁ of the bc segment, that is, L=5l ₁ . According to the bending theory, under the condition of small linear elastic deformation, the relation can be obtained:

In formula (2), l ₂ is the length of de segment. As long as L, l ₁ and l ₂ satisfy formula (2), the corner of the cross section e is equal to zero, and the left and right clip surfaces 18 and 19 in FIG. 1 can maintain zero corners.

The numerical control device (not shown in the figure) is a microcomputer system with a strain signal acquisition-conditioning circuit, a driver control circuit and measurement software, and the measurement software can be written according to the following methods and ideas. A full-bridge measurement circuit composed of resistance strain gauges R ₁ , R ₂ , R ₃ and R ₄ is connected with the strain signal acquisition-conditioning circuit. The resistance strain gauges [R ₅ , R ₆ ], [R ₇ , R ₈ ], [R ₉ , R ₁₀ ], [R ₁₁ , R ₁₂ ] are respectively connected to the strain signal acquisition-conditioning circuit in a half-bridge manner, and the numerical control The measured strain readings for these four half-bridge measurement circuits are given by express. The first driver 6 and the second driver 7 are respectively connected to the driver control circuit. The numerical controller is generally integrated with the control system of the robot, and can also be set separately.

In this design, the coded screw is a component with independent functions that can track and measure the displacement of the movable arm 16 .

The coded lead screw works as follows:

均发生连续周期性变化，变化周期用T表示。齿盘22每转过一个齿，即一个周期T，

分别完成一次循环。跟踪观察

的变化，当

达到最小值ε_rmin时，停止齿盘22的转动，调节数控器中电阻应变计[R₅,R₆]所在电桥的平衡电路，使重复前述动作，依次在

取得最小值ε_rmin时，调节电阻应变计[R₇,R₈]、[R₉,R₁₀]和[R₁₁,R₁₂]所在电桥的平衡电路，使

完成四个半桥电路零位调整后，再转动齿盘22，则

均在最小值0和一个最大值ε_rmax之间循环变化，最小值0对应于左三角形突棱29或上三角形突棱30或右三角形突棱31或下三角形突棱32处在正对相邻两圆弧齿23之间的谷底位置，即对应最于小特征高度h_min，最大值ε_rmax对应于四个三角形突棱之一处在与圆弧齿23顶点接触的位置，即对应于最大特征高度h_max。调整三态编码器TE测量电路零位的方法，称为零位四步调整法。1) Zero position adjustment of the three-state encoder measurement circuit: use the numerical controller to control the second driver 7 to drive the encoder screw to rotate, and the strain reading

There are continuous periodic changes, and the change period is represented by T. Each time the toothed disc 22 rotates one tooth, that is, one cycle T,

Complete one cycle separately. Follow up

changes, when

When reaching the minimum value ε _rmin , stop the rotation of the toothed plate 22, and adjust the balance circuit of the bridge where the resistance strain gauges [R ₅ , R ₆ ] in the numerical control device are located to make Repeat the above actions, in turn

When the minimum value ε _rmin is obtained, adjust the balance circuit of the bridge where the resistance strain gauges [R ₇ , R ₈ ], [R ₉ , R ₁₀ ] and [R ₁₁ , R ₁₂ ] are located so that

After completing the zero position adjustment of the four half-bridge circuits, turn the toothed plate 22, then

Both vary cyclically between a minimum value of 0 and a maximum value of ε _rmax . The minimum value of 0 corresponds to the left triangular ridge 29 or the upper triangular ridge 30 or the right triangular ridge 31 or the lower triangular ridge 32. The valley bottom position between the two arc teeth 23 corresponds to the minimum characteristic height h _min , and the maximum value ε _rmax corresponds to the position where one of the four triangular protruding edges is in contact with the apex of the arc teeth 23 , that is, corresponds to the maximum value ε rmax . Feature height h _max . The method of adjusting the zero position of the three-state encoder TE measurement circuit is called the zero position four-step adjustment method.

的最大值ε_rmax、最小值0和平均值0.5ε_rmax。数字1与最大特征高度h_max对应，定义为满值。数字0与最小特征高度h_min对应，定义为零值。数字1/2与平均特征高度h_mid对应，定义为中值。满值、零值和中值共同定义为应变读数

的三态编码值，简称三态值。齿盘22旋转时，三态值0、1/2、1按周期T循环变化。三态值的循环变化，用于确定齿盘22的旋转状态，即齿盘22的旋转方向和旋转角度。三态值总共有四种不同的组合，如表1所示：2) Determine the relationship between the strain reading and the rotation state of the toothed disc: After completing the zero position adjustment of the TE measurement circuit of the three-state encoder, it is specified that the numbers 1, 0 and 1/2 are used to represent the strain readings respectively.

The maximum value ε _rmax , the minimum value 0 and the average value _{0.5ε rmax} . The number 1 corresponds to the maximum feature height _hmax , which is defined as the full value. The number 0 corresponds to the minimum feature height _hmin and is defined as a zero value. The number 1/2 corresponds to the mean feature height _hmid , defined as the median. The full, zero, and median values are collectively defined as the strain reading

The three-state encoded value of , referred to as the three-state value. When the toothed disc 22 rotates, the tri-state values 0, 1/2, and 1 change cyclically according to the period T. The cyclic change of the tri-state value is used to determine the rotation state of the chainring 22 , that is, the rotation direction and the rotation angle of the chainring 22 . There are a total of four different combinations of tristate values, as shown in Table 1:

的三态值组合Table 1 Strain readings

Tri-state value combination of

Table 2 Changes of the tri-state values 0, 1/2, and 1 in each period T during the clockwise rotation of the chainring 22 .

In the table, "↑" indicates that the tri-state value increases, and "↓" indicates that the tri-state value decreases

Table 3 Changes of the tri-state values 0, 1/2, and 1 in each period T during the counterclockwise rotation of the toothed disc 22 .

In the table, "↑" indicates that the tri-state value increases, and "↓" indicates that the tri-state value decreases

的三态值0、1/2、1分别发生不同的升降变化。编号①至⑧的八行数据，两两互异，均具有唯一性，其中每一行数据都唯一地表示齿盘的一个特定旋转状态。例如，编号③的一行数据表示且只表示齿盘沿顺时针方向转过一个周期T内的第三个1/4周期，即0.5T至0.75T。编号⑤的一行数据表示且只表示齿盘沿逆时针方向转过一个周期T内的第一个1/4周期，即0至0.25T。连续变化的应变读数与三态值配合，监测齿盘的旋转状态。Choose a tri-state value combination from Table 1 as the starting point to determine the rotation state of the chainring. For the sake of clarity, choose the tri-state value combination 1, then every time the chainring rotates a tooth in the clockwise direction, the tri-state value A cycle of one period T is completed as shown in Table 2. Each time the toothed disc rotates one tooth in the counterclockwise direction, the tri-state value completes a cycle of one period T as shown in Table 3. In Tables 2 and 3, period T is divided into four 1/4 sub-periods, and within each 1/4 sub-period, four strain readings

The three-state values of 0, 1/2, and 1 have different up and down changes respectively. The eight rows of data numbered ① to ⑧ are different from each other and are unique, and each row of data uniquely represents a specific rotation state of the chainring. For example, a line of data numbered ③ represents and only represents that the toothed disc rotates clockwise in the third 1/4 cycle within one cycle T, that is, 0.5T to 0.75T. The row of data numbered ⑤ represents and only represents that the toothed disc rotates counterclockwise in the first 1/4 cycle of one cycle T, that is, 0 to 0.25T. Continuously varying strain readings are paired with tri-state values to monitor the rotational state of the chainrings.

3) Measure the displacement of the movable arm: use the numerical controller to control the rotation of the micrometer screw 9, adjust the movable arm 16 to a specified position or any position on the micrometer screw 9, and record the position as the movable The displacement origin of the arm 16, the position of the chainring 22 at this time is recorded as the chainring zero position. When the chainring 22 starts to rotate from the zero position of the chainring, the movable arm 16 is displaced from the displacement origin. Let S represent the displacement of the movable arm 16 relative to the displacement origin, and define it as the displacement at the origin, and calculate S with the formula (3):

In formula (3), t represents the lead of the micrometer screw 9, N _c represents the number of teeth of the toothed disc 22, n _{z, s} represents the number of teeth that the toothed disc 22 has rotated clockwise since the zero position of the toothed disc, n _{z ,n} represents the total number of teeth that the chainring 22 has rotated counterclockwise since the zero position of the chainring, and nz _,s and nz _,n are always positive values. n _z represents the difference between n _z,s and n _z,n , which is defined as the effective number of rotating teeth. n _z,s , n _z,n and n _z are also referred to as sprocket rotation parameters. n _z and S are the number of generations. When the toothed plate 22 rotates clockwise, the movable arm 16 moves to the right, and the signs of n _z and S are both "-". When the toothed plate 22 rotates counterclockwise, the movable arm 16 moves to the left, and the signs of n _z and S are both "+".

The manipulator designed in this design is used as follows:

1) Measure the line connection. A full-bridge measuring circuit composed of resistance strain gauges R ₁ , R ₂ , R ₃ , and R ₄ is connected to the numerical controller. The circuit is used for both force sensing measurement and length sensing measurement. When used for measuring force, the strain reading measured by the numerical control device is expressed by ε _rf ; when used for measuring length, the strain reading measured by the numerical control device is expressed by ε _rd .

然后以

为标定数，用力F与应变读数ε_rf的函数关系，即式(4)计算力F：2) Calibration of the force measuring system. Generally, a standard load cell or a standard force measuring ring is used as the force value standard device, and the force value standard device is clamped by the gripper on the clamping arm to calibrate the force measuring system. Taking the surface contact type gripper shown in FIG. 1 as an example, when calibrating, use the left gripping surface 18 on the movable arm 16 and the right gripping surface 19 on the fixed arm 17 to grip the standard load cell or the standard force measuring ring. , the movement of the movable arm ₁₆ is controlled by the numerical controller, and _a set of standard forces _{F 1} , _{F 2} , . It is a positive integer not less than 2, and the value range is determined according to needs, for example, 2≤N≤10), and the numerical controller records the strain reading corresponding to F ₁ , F ₂ , ..., F _N

then with

is the calibration number, the functional relationship between the force F and the strain reading ε _rf , that is, the formula (4) calculates the force F:

In formula (4), A ₁ and B ₁ are constants, which are calculated according to formulas (5) and (6) respectively:

表示与力F_i对应的应变读数，即标定数

标定中，弹性梁在标准力F_N作用下产生的最大应力不得超过所用材料的比例极限。In equations (5) and (6), N represents the ordinal number of the standard force, F _i represents the force value of the standard force with different ordinal numbers,

Represents the strain reading corresponding to the force F _i , i.e. the calibration number

During calibration, the maximum stress generated by the elastic beam under the standard force F _N shall not exceed the proportional limit of the material used.

一般在5με～20με范围内取值，例如取第二步，取标准块规d₀，用数控器控制可动臂16移动，由第一夹持顶针36和第二夹持顶针37夹持d₀，当应变读数

时，可动臂16停止移动，将可动臂16的当前位置记作位移原点，将齿盘22的当前位置记作齿盘零位；第三步，用标准块规d₁，d₂…，d_n依次替换d₀，重复完成第二步动作，记下相应的应变读数

第四步，以

为标定数，按长度s与应变读数ε_rd的函数关系，即式(7)计算长度s：3) Calibration of length measuring system. First select a length standard according to the type of holder, standard diameter gauge and standard thickness gauge are commonly used length standards. A standard diameter gauge is a set of standard cylinders of different diameters for use with line contact grippers and face contact grippers. Standard thickness gauges are a set of standard thickness block gauges of different thicknesses for point contact grippers, line contact grippers and surface contact grippers. _The thickness value _of each standard cylinder or standard block _{gauge is} represented by d ₀ , d ₁ , d _{2 .} The number of gauges generally takes 2≤n≤10 (for example, n=7). d ₀ , d ₁ , d ₂ , ..., _dn also represent the corresponding standard cylinder or standard block gauge. The calibration method will be described by taking the point contact type gripper shown in Fig. 6(a) as an example. The standard thickness gauge is used as the standard, and the thickness value of the standard block gauge should meet the conditions: when the deflection of the elastic beam of the clamping arm at the section e is λ=d _n -d ₀ , the maximum stress of the beam does not exceed the proportion of the material used limit. The calibration is performed in four steps: In the first step, an initial value is preset for the strain reading ε _rd

Generally, the value is in the range of 5με～20με, for example, take In the second step, take the standard block gauge d ₀ , use the numerical controller to control the movement of the movable arm 16 , and clamp d ₀ by the first clamping thimble 36 and the second clamping thimble 37 , when the strain reading

, the movable arm 16 stops moving, the current position of the movable arm 16 is recorded as the displacement origin, and the current position of the toothed plate 22 is recorded as the toothed plate zero position; in the third step, use the standard block gauge d ₁ , d ₂ . . . , d _n replaces d ₀ in turn, repeat the second step, and record the corresponding strain reading

The fourth step is to

is the calibration number, and the length s is calculated according to the functional relationship between the length s and the strain reading ε _rd , that is, formula (7):

In formula (7), A ₂ and B ₂ are constants, which are calculated by formulas (8) and (9), respectively:

表示与d_i对应的应变读数，即标定数

In formulas (8) and (9), n is the number of standard thickness gauges, d _i is the thickness value of standard thickness gauges with different thicknesses,

Indicates the strain reading corresponding to d _i , that is, the calibration number

The above equations (4) and (9) are both derived by the linear fitting method. When calibrating the force measuring system and the length measuring system, it is possible to design a force standard device positioning device and a length standard device positioning device to facilitate the correct installation of these two standard devices.

时，停止第一驱动器6的转动；第三步，控制第二驱动器7转动，使可动臂16向右移动，右夹面19与物体W接触，当应变读数

时，停止第二驱动器7的转动；第四步，控制第一驱动器6和第二驱动器7同时转动，使可动臂16和固定臂17同步相向移动，当应变读数ε_rd取得

范围的任何一值ε_rdx时，记下ε_rdx、对应于ε_rdx的应变读数ε_rf和赖原点位移S；第五步，将ε_rf代入式(4)，将ε_rdx和S代入公式(10)：4) Clamping and measuring operations. The manipulator is connected with the robot through the coupling 2, and performs the clamping and measuring operation under the control of the numerical controller. The operation process will be described by taking the surface contact type gripper shown in Fig. 1 as an example. For the sake of simplicity, let the object W be a rigid body and its position is fixed. The operation process is divided into five steps: the first step is to adjust the position of the manipulator and the distance s between the two clamping surfaces, so that s is greater than the length m of the object W, and the object W enters between the two clamping surfaces; the second step is to control the first driver 6 Rotate, so that the fixed arm 17 moves to the left with the sliding frame 26, and the right clamping surface 19 is in contact with the object W, when the strain reading

When the rotation of the first driver 6 is stopped; the third step is to control the rotation of the second driver 7 to make the movable arm 16 move to the right, and the right clamping surface 19 is in contact with the object W. When the strain reading

, stop the rotation of the second driver 7; in the fourth step, control the first driver 6 and the second driver 7 to rotate at the same time, so that the movable arm 16 and the fixed arm 17 move synchronously towards each other, when the strain reading ε _rd is obtained

At any value of ε _rdx in the range, write down ε _rdx , the strain reading ε _rf corresponding to ε _rdx and the displacement S of the origin; the fifth step, substitute ε _rf into equation (4), and substitute ε _rdx and S into the formula ( 10):

Equation (4) gives the clamping force F of the object W, and Equation (10) gives the length m of the object W.

例如设

做为测量起点；第二步，用数控器控制夹持器夹持弹簧，使应变读数记下

和与之对应的赖原点位移S^*；第三步，增大夹持力F，跟踪记录F、应变读数ε_rd和赖原点位移S，按式(11)计算弹簧的变形量ν：For deformed objects such as springs and rubber, the amount of deformation can be measured. In the case of a compression spring, a surface contact type holder is used. The measurement procedure is divided into three steps: The first step is to set a strain reading

For example, let

As the starting point of measurement; in the second step, use the CNC to control the gripper to clamp the spring to make the strain reading Write down

and the corresponding origin displacement S ^* ; in the third step, increase the clamping force F, track and record F, the strain reading ε _rd and the origin displacement S, and calculate the spring deformation ν according to formula (11):

Using the measurement software, the force-deformation relationship curve, that is, the F-ν curve, can be drawn.

做为测量起点。测量时，控制夹持器夹物体W，先使应变读数

记下

和与之对应的赖原点位移S^*，然后增大夹持力F，跟踪记录F、ε_rd和赖原点位移S，按式(12)计算压入深度δ：For polymer material products, plant fruits, animal organs and other objects, the indentation deformation can be measured. Suppose the object W is a spherical body or a regular hexahedron, and use the clamping thimble as shown in Figure 6(a) to measure the depth of the clamping thimble pressed into the object W. Set a strain reading first For example, let

as a starting point for measurement. When measuring, control the gripper to clamp the object W, and make the strain reading first

Write down

and the corresponding displacement S ^* at the origin, then increase the clamping force F, track and record F, ε _rd and displacement S at the origin, and calculate the indentation depth δ according to formula (12):

Using the measurement software, the force-indentation depth relationship curve, that is, the F-delta curve, can be drawn. F, ν, δ, and S can be used as control parameters, so as to realize the control of the manipulator in different ways.

Claims (1)

1. The manipulator for controlling the clamping force to measure the distance between the clamping points is characterized in that the manipulator is composed of a guide rail seat, a sliding frame, a transmission shaft, a coding screw, a driver, a clamping arm and a numerical control device; The structure of the guide rail seat includes a base plate, a parallel male guide rail on the top of the base plate, and a coupling under the base plate for connecting the robot arm; The structure of the sliding frame includes a bottom plate, a parallel female guide rail located under the bottom plate, a protrusion with a threaded through hole located at the bottom left of the bottom plate, a left bearing support plate and a right bearing respectively embedded with bearings on the left and right sides above the bottom plate. The support plate, the U-shaped guide limit slot on the upper surface of the base plate; the two bearings are in a coaxial position; the axis of the threaded through hole on the protruding table, the axis of the guide limit slot, and the axes of the two bearings are all located in parallel female guide rails in the longitudinal symmetry plane of the frame, and parallel to the axis of the parallel female guide rail; the sliding frame and the guide rail seat are installed together through the cooperation of the male guide rail and the female guide rail; The structure of the transmission shaft includes a threaded section and a connecting shaft on the left side of the threaded section, which is installed on the sliding frame through the cooperation between the threaded section and the threaded through hole on the protruding table, and the connecting shaft extends to the left side of the protruding table; The coding screw consists of a micrometer screw and a three-state encoder; The micrometer screw is a threaded stepped shaft, and its structure is divided into three sections from left to right: I-II, II-III, and III-IV. Sections I-II and III-IV are optical axes, and sections II-III are threaded shafts. , the outer diameter of the thread is larger than the diameter of the optical axis of the I-II section and the optical axis of the III-IV section; the micrometer screw rod is installed on the bearing through the cooperation of the optical axis of the I-II section and the III-IV section and the bearings on the two bearing support plates. On the sliding frame, the threaded shafts of sections II-III are located between the two bearing support plates, the optical shafts of sections I-II extend to the left side of the left bearing support plate, and the optical shafts of sections III-IV extend to the right bearing support plate the right side; The three-state encoder consists of a toothed disc, a sensor bracket and four cantilever beam sensors; the toothed disc is a disc with circular arc teeth evenly distributed around the periphery, which is coaxially fixed on the III-IV optical axis of the micrometer screw. The number of teeth of the chainring is an integer multiple of 4; the sensor bracket is a rectangular frame, which is fixed on the upper right of the bottom plate of the sliding frame, and surrounds the chainring in the middle; the four cantilever beam sensors face the side of the chainring near the free end The left triangular protruding edge, the upper triangular protruding edge, the right triangular protruding edge and the lower triangular protruding edge are processed respectively, and the resistance strain gauges [R ₅ , R ₆ ], [R ₇ , R ₈ ], [R ₉ , R ₁₀ ] and [R ₁₁ , R ₁₂ ]; the four cantilever beam sensors are respectively fixed on the inner walls of the four sides of the sensor bracket, and all have a certain amount of pre-deformation. The elastic pressure generated by the pre-deformation makes the The ridge tops of the four triangular ridges are in contact with the circular arc teeth around the toothed disc respectively, and the specific position of the contact point is determined according to the following conditions: a. Set the longitudinal symmetry line of the toothed plate to pass through the center of the arc tooth just above and the arc tooth directly below, and the horizontal line of symmetry of the tooth plate passes through the center of the leftmost arc tooth and the rightmost arc tooth; b. At this time, the left triangular protruding edge is located on the horizontal symmetry line of the toothed disc and is in contact with the apex of the leftmost circular arc tooth; the right triangular protruding edge is located on the horizontal symmetry line of the toothed disc and the upper side of the rightmost circular arc tooth, facing the The valley bottom between two adjacent circular arc teeth; the upper triangular protruding edge and the lower triangular protruding edge are both located on the right side of the longitudinal symmetry line of the toothed disc, and are in contact with the right side of the circular arc tooth just above and the right side of the circular arc tooth just below. ;The distance from the left triangular ridge to the longitudinal symmetry line of the chainring is represented by h _max , the distance from the right triangular ridge to the longitudinal symmetry line of the chainring is represented by h _min , the contact point between the upper triangular ridge and the arc tooth directly above the chainring The distance of the horizontal symmetry line and the distance between the lower triangular protruding edge and the contact point of the circular arc tooth directly below and the horizontal symmetry line of the chainring are all expressed by h _mid ; there is a relationship between h _mid and h _min and h _max expressed by formula (1). :

h _min , h _mid and h _max are collectively referred to as feature heights, where h _min is the minimum feature height, h _mid is the average feature height, and h _max is the maximum feature height; The driver includes a first driver and a second driver; the first driver is installed on the upper left of the guide rail seat and is connected with the connecting shaft of the transmission shaft; the second driver is installed on the left side of the left bearing support plate, and is connected with the micrometer screw rod The I-II section optical axis connection; The clamping arm is composed of a movable arm and a fixed arm, both of which are cantilevered elastic beam sensors with clamps; the elastic beam of the movable arm is fixedly connected to a shaft sleeve to form an integral structure, and the middle of the shaft sleeve is embedded There is a transmission nut, and a limit guide rod is embedded at the bottom; the movable arm is installed on the sliding frame through the cooperation between the transmission nut and the micrometer screw and the cooperation between the limit guide rod and the U-shaped guide limit groove; It cannot rotate around the axis of the micrometer screw. When the micrometer screw rotates, it drives the movable arm to move along the axis of the micrometer screw; the elastic beam of the fixed arm and the right bearing support plate are fixedly connected to form an integral structure; the movable arm It is vertically upward with the fixed arm, and the two are in a symmetrical position; the elastic beam of the movable arm is attached with resistance strain gauges R ₂ and R ₁ on the left and right sides of the movable arm along the beam axis direction, respectively. The elastic beam of the fixed arm Resistance strain gauges R ₃ and R ₄ are respectively attached to the left and right sides near the root along the beam axis. The resistance strain gauges R ₁ , R ₂ , R ₃ , and R ₄ form a full-bridge measurement circuit; the elasticity of the clamping arm Beams have the following two basic structural forms: a) Fishhook-shaped polybeam, its structure includes a long arm ac section, a turning section ad section and a short arm de section, the cross sections of the three sections are all rectangular, the section c is the fixed end of the polybeam, and the ad section is the polybeam The section d is the fixed end of the short arm de section; the structure of the long arm ac section is divided into two sections ab and bc from top to bottom, the height H of the ab section is greater than the height h ₁ of the bc section; the short arm de section is a beam of equal section; b) Straight beam, its structure is divided into two sections a-b and b-c from top to bottom, the height H of the a-b section is greater than the height h of the b-c section, the section c is the fixed end of the beam, and the section a is the free end of the beam; The gripper of the gripping arm has three types: surface contact, line contact and point contact; the gripper of the fishhook-shaped polybeam is set at the section e of the d-e section of the short arm, and meets the zero-angle condition; the zero-angle condition is Refers to: when the action line of the clamping force coincides with the intersection of the longitudinal symmetry plane of the folded beam and the cross section e, the corner of the cross section e and the corner of the gripper are both zero; the gripper of the straight beam is set at the free end the left and right sides of a; The numerical control device is a microcomputer measurement and control system with strain signal acquisition-conditioning circuit, driver control circuit and measurement software; The coded screw is used to measure the displacement of the movable arm, and its measurement method is as follows: 1) Zero position adjustment of circuit: control the rotation of the second drive with the numerical controller, the micrometer screw and toothed disc rotate accordingly, and the strain reading of the four half-bridge measuring circuits of the three-state encoder measured by the numerical controller

There are continuous periodic changes, and the change period is represented by T; every time the toothed disc rotates one tooth, that is, a period T,

Complete a cycle separately; follow up changes, when

When the minimum value ε _rmin is reached, stop the rotation of the toothed disc, and adjust the balance circuit of the bridge where the resistance strain gauges [R ₅ , R ₆ ] on the numerical control device are located to reach a balanced state, that is,

Repeat the above actions, in turn

When the minimum value ε _rmin is obtained, adjust the balance circuit of the bridge where the resistance strain gauges [R ₇ , R ₈ ], [R ₉ , R ₁₀ ] and [R ₁₁ , R ₁₂ ] are located so that

After completing the zero position adjustment of the four half-bridge measurement circuits according to the above method, and then rotating the toothed plate, the

Both cyclically change between a minimum value of 0 and a maximum value of ε _rmax ; the minimum value of 0 corresponds to the position of the valley bottom between the triangular protruding edge and the two adjacent circular arc teeth, that is, corresponding to the minimum characteristic height h _min ; The maximum value ε _rmax corresponds to the position where the triangular protruding edge is in contact with the apex of the arc tooth, that is, corresponds to the maximum characteristic height h _max ; the above method of adjusting the zero position of the three-state encoder measurement circuit is called the zero position four-step adjustment method ; 2) Determine the relationship between the strain reading and the rotation state of the toothed disc: After completing the zero position adjustment of the three-state encoder measurement circuit, it is specified that the numbers 1, 0 and 1/2 are used to represent the strain readings respectively.

The maximum value ε _rmax , the minimum value 0 and the average value _{0.5ε rmax} ; the number 1 corresponds to the maximum characteristic height h _max and is defined as the full value; the number 0 corresponds to the minimum characteristic height h _min and is defined as zero value; the number 1/2 Corresponding to the average characteristic height h _mid , defined as the median value; the full value, zero value and the median value are jointly defined as the strain reading

The tri-state code value of , referred to as tri-state value; when the toothed disc rotates, the tri-state values 0, 1/2, 1 change cyclically according to the period T; the cyclic change of the tri-state value is used to determine the rotational state of the toothed disc, that is, the tooth The rotation direction and rotation angle of the disk; 3) Measure the displacement of the movable arm: use the numerical controller to adjust the movable arm to a specified position or any position on the micrometer screw, record the position as the displacement origin of the movable arm, and use the gear plate at this time. The position of the chain is recorded as the zero position of the toothed plate; the toothed plate is rotated from the zero position of the toothed plate, and the movable arm is displaced from the displacement origin; S is used to represent the displacement of the movable arm relative to the displacement origin, and is defined as Origin displacement, calculate S using formula (3):

In formula (3), t represents the lead of the micrometer screw, N _c represents the number of teeth of the toothed disc, nz _{, s} represents the number of teeth that the toothed disc has rotated clockwise since the zero position of the toothed disc, and n _{z, n} represents The total number of teeth rotated counterclockwise from the zero position of the chainring, n _z,s and n _z,n are always positive values; n _z represents the difference between n _z,s and n _z,n , which is defined as the effective number of rotating teeth ; n _z,s , n _z,n and n _z are also called the rotation parameters of the toothed disc; n _z and S are the generation quantities; when the toothed disc rotates clockwise, the movable arm moves to the right, and the symbols of n _z and S are both is "-"; when the toothed plate rotates counterclockwise, the movable arm moves to the left, and the signs of n _z and S are both "+"; The method of using the manipulator of this design includes the following steps: 1) Connection of measurement lines; connect the full-bridge measurement circuit composed of resistance strain gauges R ₁ , R ₂ , R ₃ , and R ₄ to the numerical controller. This circuit is used for both force sensing measurement and clamping point spacing. When measuring the force, the strain reading measured by the numerical control device is expressed by ε _rf ; when measuring the distance between the clamping points, the strain reading measured by the numerical control device is expressed by ε _rd ; 2) Calibration of the force measuring system; the force value standard device is used to calibrate the force measuring system; the clamping arm clamping force value standard device is controlled by the numerical controller, and a set of standard forces F ₁ , F ₂ , ..., are applied to the clamping arm. F _N , (F ₁ <F ₂ <..., <F _N , N is a positive integer not less than 2), write down the strain readings corresponding to F ₁ , F ₂ ,..., F _N

by

is the calibration number, using the functional relationship between the force F and the strain reading ε _rf , that is, formula (4), calculate F:

In formula (4), A ₁ and B ₁ are constants, which are calculated according to formulas (5) and (6) respectively:

In equations (5) and (6), N represents the ordinal number of the standard force, F _i represents the force value of the standard force with different ordinal numbers,

Represents the strain reading corresponding to the force F _i , i.e. the calibration number

3) Calibration of the length measuring system; the length measuring system is calibrated with a length standard; the standard includes a standard diameter gauge and a standard thickness gauge, the former is a set of standard cylinders with different diameters, and the latter is a set of standard thickness blocks with different thicknesses The standard length value of the standard device is represented by d ₀ , d ₁ , d ₂ , ..., dn in turn, d ₀ <d ₁ <d ₂ , ..., <d _n , _n represents the number of standard cylinders or standard block gauges ;d ₀ , d ₁ , d ₂ , ..., _dn also represent standard cylinders or standard block gauges; the calibration procedure is divided into four steps: the first step is to preset an initial value for the strain reading ε _rd

Take the value in the range of 5με～20με; in the second step, take d ₀ from the standard device, control the movable arm to clamp d ₀ , when the strain reading

When the movable arm stops moving, the current position of the movable arm is recorded as the displacement origin, and the current position of the toothed plate is recorded as the zero position of the toothed plate; in the third step, replace d with d ₁ , d ₂ ..., d _n in turn ₀ , repeat the second step, record the corresponding strain reading

The fourth step is to

is the calibration number, calculate the length according to (7), that is, the clamping point spacing s:

In formula (7), A ₂ and B ₂ are constants, which are calculated by formulas (8) and (9), respectively:

In formulas (8) and (9), n represents the number of standard cylinders or standard thickness block gauges, d _i represents different standard length values,

Indicates the calibration number corresponding to d _i 4) Clamping measurement operation; measuring the length of rigid objects, the operation process is divided into five steps: the first step, the second driver is controlled by the numerical control to adjust the gap between the grippers, and the object enters the clamping space; the second step, control The first drive rotates so that the fixed arm moves to the left with the sliding frame to contact the object, when the strain reading

When the rotation of the first driver is stopped; in the third step, the rotation of the second driver is controlled to make the movable arm move to the right and contact the object. When the strain reading

When , stop the rotation of the second driver; in the fourth step, control the first driver and the second driver to rotate at the same time, so that the movable arm and the fixed arm move toward each other, when the strain reading ε _rd is obtained

When the specified value of the range or any value ε _rdx , write down ε _rdx , the strain reading ε _rf corresponding to ε _rdx , and the displacement S at the origin; the fifth step, substitute ε _rf into formula (4) to calculate the clamping force F, Substitute ε _rdx and S into equation (10) to calculate the length m of the object: The deformation of the deformed object is measured by measuring the distance s between the clamping points.

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