Parameter calibration method of parallel structure micromanipulation robot based on microscopic vision

Abstract

The invention provides a parallel structure micro-operation robot parameter calibration method based on microscopic vision, and belongs to the technical field of robot parameter calibration, the method mainly uses a microscopic camera to amplify the motion of an end effector of a micro-operation robot, obtains the position of the end effector through image processing, and determines the position of the end effector according to a kinematics analysis result. Kinematics parameters are obtained through calculation, and then the motion precision of the micro-operation robot is measured. The method mainly comprises the following steps of parallel structure robot modeling and kinematics analysis, camera correction, proportional scale determination, robot orthogonality correction, parameter calibration, repeated positioning precision measurement, translation precision measurement and rotation precision measurement. According to the method, the problem of kinematics parameter calibration of the parallel structure micro-operation robot is solved, calibration of the kinematics parameters under the high-precision requirement is achieved, and the kinematics precision of the micro-operation robot is improved.

Description

Technical Field

[0001] The invention relates to the technical field of robot parameter calibration, and in particular to a parameter calibration method of a parallel structure micro-manipulation robot based on microscopic vision. Background Art

[0002] Micro-nano technology is one of the most important technological fields in this century. It will enable humans to enter the field of micro-nano scale operations, such as biological cell operations, assembly of micro-electromechanical system components, docking of communication optical fibers, microelectronic integrated circuit manufacturing, microsurgery, etc., thus greatly changing human production and life, and profoundly affecting the development of civil and national defense science and technology. The key to human beings' operation of the microscopic world is to develop a micro-manipulation robot system that can process and assemble tiny objects at the microscopic scale.

[0003] For a robot system, a kinematic model containing parameters needs to be established after the design is completed. Since there are often errors in the assembly of the robot system, the kinematic parameters need to be calibrated, that is, the position error is obtained by measuring the actual position of the end effector and the theoretical position calculated by the kinematic model, and the parameters are corrected by the algorithm to make the position error as small as possible. Through parameter calibration, the motion accuracy of the robot can be greatly improved. The micro-manipulation robot system has extremely high requirements for motion accuracy, so it has high requirements for measurement accuracy. The optical measuring instruments commonly used for general robot calibration are not accurate enough and cannot be used. The laser rangefinder with higher accuracy is large in size and heavy in weight for the micro-manipulation robot, and cannot be installed on the end effector, nor can it be used for parameter calibration of the micro-manipulation robot.

[0004] Since conventional calibration methods cannot calibrate the parameters of a micro-manipulation robot, and microscopic vision can magnify the movement of the end point without contacting the robot, microscopic vision can be used as a measurement method to calibrate the parameters of the micro-manipulation robot. Therefore, for a parallel structure micro-manipulation robot, the present invention proposes a robot kinematic parameter calibration method based on microscopic vision. Using this method, the kinematic parameters of the micro-manipulation robot can be successfully calibrated and the robot's motion accuracy can be improved. Summary of the invention

[0005] In order to overcome the limitations of calibration of optical trackers and laser trackers, the present invention uses a visual method to calibrate parameters of a parallel structure micro-manipulation robot, determines the parameters in the motion model, and improves the motion accuracy.

[0006] The present invention adopts the following technical solutions to solve the above problems:

[0007] A parameter calibration method for a parallel structure micro-manipulation robot based on microscopic vision, the method comprising the following steps:

[0008] S1: Modeling and kinematic analysis of the parallel structure robot to be calibrated, and constructing a kinematic model;

[0009] S2: designing a calibration method based on the kinematic model described in S1, setting a micromanipulator and a camera, and using the micromanipulator as a standard reference object to calibrate the camera to correct the pitch, roll and yaw angles of the camera;

[0010] S3: Fix the camera calibrated in S2, place a robot with a micrometer on the end effector in front of the camera, adjust the distance between the robot and the camera until the scale on the micrometer is clearly imaged in the field of view, and save the image to determine the corresponding relationship between the real length and the image pixels;

[0011] S4: Move the end effector to obtain a picture of the end effector under the camera, process the image to obtain data, calculate the kinematic parameters of the robot, and then measure the repeatability, translation and rotation accuracy of the robot.

[0012] Furthermore, in S1, the design of the parallel structure robot is a planar five-bar structure, which includes two parallel linear motors, two rotary joints and an end effector. When the displacements of the two linear motors are both zero, the end effector is located at the initial position; when the displacements of the two linear motors are equal, the end effector translates; when the displacements of the two linear motors are unequal, the end effector rotates. This structure can realize two degrees of freedom movement of the end effector, i.e., one translation movement and one rotation movement, through two linear motors.

[0013] Further, in S1, the displacement of the linear motor located on the upper side is L1, the displacement of the linear motor located on the lower side is L2, the distance between the two linear motor motion axes is d, and the distance between the end effector and the rotation axis of the rotary joint parallel to the rod direction is d m , the distance between the end effector and the rotation axis of the rotary joint perpendicular to the rod direction is L tool , the relationship between the translation distance L, the rotation angle θ and L1, L2 is:

[0014] L=L1=L2

[0015]

[0016] A s (x s ,y s ) is the initial position of the end point of the end effector, A f (x f ,y f) is the final position of the end point of the end effector. From the analysis, we can know that the position coordinates of the end point movement (x f ,y f ) and the relationship between L1 and L2 is:

[0017]

[0018] Furthermore, in S2, in order to solve the problem of defocusing caused by the non-parallelism between the camera's viewing plane and the robot's motion plane, which results in large measurement errors, a standard xyz three-degree-of-freedom electric micromanipulator is used as a standard reference object, and its y-axis and z-axis motion directions are set as orthogonal horizontal and vertical directions to calibrate the camera and correct its pitch, roll and yaw angles. After the correction is completed, it is considered that the camera's viewing plane is parallel to the yz plane of the robot's base coordinate system.

[0019] Furthermore, in S4, the image processing and accuracy measurement specifically include the following steps:

[0020] S401: Correct the orthogonality of the robot: Due to the error in the assembly of the robot, the motion axes of the robot are likely to be non-orthogonal, so the orthogonality between the axes needs to be corrected under the microscope lens;

[0021] S402: Parameter calibration: Move the robot's motion axis to make the displacement difference ΔL of two parallel linear motors equal to different values, save the image of the end effector, and obtain the coordinates of the motion point through image processing for calculating kinematic parameters;

[0022] S403: Measuring repeated positioning accuracy: moving the robot's motion axis to make the end effector repeatedly reach a certain position, repeating n times, saving the image, and calculating the repeated positioning error;

[0023] S404: measuring translation accuracy: making the linear motors performing translation motion translate the same distance each time, for a total of n times, and saving images for translation accuracy measurement;

[0024] S405: Measure the rotation accuracy: Use the kinematic parameters calculated in S402 to calculate the theoretical rotation angle at each ΔL, and subtract it from the actual rotation angle to obtain the rotation accuracy.

[0025] Furthermore, in S401, in order to solve the problem of large measurement errors caused by errors in robot assembly resulting in non-orthogonality between the robot's motion axes, the orthogonality between the axes is corrected under the microscope lens, and the robot's motion axes are moved, and the orthogonality between the robot's motion axes is corrected based on the vertical and horizontal directions in the camera's field of view.

[0026] Further, in S402, the image is subjected to straight line detection based on Hough transform to obtain the two end point coordinates M (x m ,y m )、N(x n ,y n ), image processing can obtain the coordinates A of the end point of the micrometer before and after movement s (x s ,y s ), A f (x f ,y f ), then the kinematic parameters d, d m and L tool The expression is:

[0027]

[0028] in,

[0029]

[0030] The beneficial effects of the present invention are:

[0031] 1. The present invention proposes a parameter calibration method based on microscopic vision, which determines the parameters in the kinematic model and improves the accuracy of parameter calibration.

[0032] 2. The present invention proposes a method for calibrating parameters of a micro-manipulation robot, which solves the problem that optical trackers and laser trackers cannot be used for calibrating parameters of a micro-manipulation robot.

[0033] 3. The present invention proposes a method for parameter calibration of a parallel structure robot, which solves the problem of parameter calibration of a parallel structure robot and improves the movement accuracy of the robot. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] In order to more clearly illustrate the specific embodiments of the present invention, the drawings required for use in the description of the specific embodiments will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0035] Figure 1 It is a flow chart of the parameter calibration method of the present invention;

[0036] Figure 2 A simplified model diagram of the micro-manipulation robot targeted by the present invention;

[0037] Figure 3 It is a schematic structural diagram of the micro-manipulation robot targeted by the present invention;

[0038] Figure 4 This is a schematic diagram of a planar five-bar structure according to the present invention;

[0039] Figure 5 The image of the field of view of the end effector (i.e., micrometer) used in the present invention under a microscope;

[0040] Figure 6 It is a diagram of the equipment installation for the camera correction step involved in the present invention;

[0041] Figure 7 This is an equipment installation diagram for calibrating the robot involved in the present invention.

[0042] In the figure: 1. Linear motor 1; 2. Linear motor 2; 3. Linear motor 3; 4. Linear motor 4; 5. Linear motor 5; 6. Slide rail; 7. Rotary joint 1; 8. Rotary joint 2; 9. Rotary joint 3; 10. Rotary joint 4; 11. Base; 12. End effector (micrometer); 13. Upper linear motor; 14. Lower linear motor; 15. Upper rotary joint; 16. Lower rotary joint; 17. Upper rail of slide rail; 18. Lower rail of slide rail; 19. Base; 20. End effector (micrometer); 21. Microscope lens; 22. Lens mounting bracket; 23. XYZ three-degree-of-freedom electric micromanipulator; 24. Computer; 25. Industrial camera. DETAILED DESCRIPTION

[0043] In order to enable those skilled in the art to better understand the technical solution of the present invention, the present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0044] like Figure 1 As shown, the present invention proposes a parameter calibration method for a parallel structure micro-manipulation robot based on microscopic vision, which includes eight steps: parallel structure robot modeling and kinematic analysis, camera correction, scale determination, robot orthogonality correction, parameter calibration, measurement of repeat positioning accuracy, measurement of translation accuracy and measurement of rotation accuracy.

[0045] Example 1

[0046] like Figure 2 , Figure 3As shown, the simplified model diagram of the micro-manipulation robot targeted by the present invention includes 5 linear motors and 4 rotary joints. The linear motors include linear motor 1 and linear motor 3, linear motor 2 and linear motor 4, which are corresponding to each other up and down, and linear motor 5, which is arranged at the end. The above-mentioned linear motors are active moving pairs, and the rotary joints include rotary joint 1 7 and rotary joint 3 9, rotary joint 2 8 and rotary joint 4 10, which are opposite to each other up and down, that is, passive rotary pairs, and also include slide rails 6, that is, passive moving pairs, 11 is a fixed base, 12 is an end effector, and in this embodiment, the end effector 12 is a micrometer; the micro-manipulation robot targeted by the present invention uses 5 linear motors, including two groups of planar five-bar structures, each group can realize one translation motion and one rotation motion in the plane, and the two groups of structures are orthogonal to each other, so this structure contains five degrees of freedom. This parameter calibration method includes the following detailed steps:

[0047] S1: Parallel structure robot modeling and kinematic analysis

[0048] like Figure 4 As shown, the planar five-bar structure consists of an upper linear motor 13, a lower linear motor 14, an upper rotary joint 15, a lower rotary joint 16, an upper rail 17, and a lower rail 18 of a slide rail. The upper linear motor 13 and the lower linear motor 14 are installed in parallel on the upper and lower surfaces of a base 19. The upper linear motor 13 is connected to the upper rotary joint 15, and the lower linear motor 14 is connected to the lower rotary joint 16. The upper rotary joint 15 is connected to the upper rail 17 of the slide rail, and the lower rotary joint 16 is connected to the lower rail 18 of the slide rail. The end effector 20 is assembled on the upper rail 17 of the slide rail.

[0049] The displacement of the upper linear motor 13 is set to L1, the displacement of the lower linear motor 14 is set to L2, the spacing between the motion axes of the upper linear motor 13 and the lower linear motor 14 is set to d, and the distance between the end effector 20 and the rotation axis of the upper rotary joint 15 parallel to the connecting rod direction is set to d m , the distance between the end effector 20 and the rotation axis of the upper rotation joint 15 perpendicular to the connecting rod direction is set to L tool When L1=L2=0, the end effector 20 is located at the initial position; when L1=L2≠0, the end effector 20 translates a distance L; when L1≠L2, the end effector 20 rotates an angle θ.

[0050] The relationship between the translation distance L, the rotation angle θ, L1, and L2 is:

[0051] L=L1=L2

[0052]

[0053] like Figure 5As shown in the figure, in the field of view image of the end effector (i.e., micrometer) under the microscope, the horizontal right direction is the positive x direction, and the vertical downward direction is the positive y direction. s (x s ,y s ) is the initial position of the end point of the end effector, A f (x f ,y f ) is the final position of the end point of the end effector, M(x m ,y m )、N(x n ,y n ) are two points on the horizontal line of the micrometer.

[0054] Combine the following Figure 5 The principle of the parameter calibration method of the parallel structure micro-manipulation robot based on vision is introduced: when the end effector 20 is in the initial position, the angle between the straight line MN and the horizontal direction is 0°; when the end effector 20 rotates, the angle between the straight line MN and the horizontal direction is θ, and

[0055]

[0056] From the analysis, we can know that the position coordinates (x f ,y f ) and the relationship between L1 and L2 is:

[0057] x f =-L1+d m -d m cosθ-L tool sinθ-x s

[0058] y f =-L tool +(L tool cosθ-d m sinθ)-y s

[0059] Combining the above equations, we can get the kinematic parameters d, d m and L tool The expression is:

[0060]

[0061] Therefore, it is only necessary to obtain the initial position coordinates (x s ,y s ), the final position coordinates (x f ,y f ) and the coordinates of two points on the micrometer horizontal line (x m ,ym )、(x n ,y n ), substituting into the above formula, we can get the kinematic parameters d, d m and L tool .

[0062] S2: Correction Camera

[0063] Figure 6 The device installation diagram of the camera correction step involved in the present invention, in which 21 is a microscope lens, 22 is a lens mounting bracket, 23 is an xyz three-degree-of-freedom electric micromanipulator, 24 is a computer, and 25 is an industrial camera. This embodiment uses a Siskiyou MX7600 electric micromanipulator as a reference. Figure 6 Introduce the steps to calibrate the camera.

[0064] Using the xyz three-degree-of-freedom electric micromanipulator 23 as a reference object, it is assumed that its three motion axes are completely orthogonal, such as Figure 6 Place the xyz three-degree-of-freedom electric micromanipulator 23 in front of the microscope lens 21, select a point on the end effector of the electric micromanipulator as the object point, adjust the x-axis motor to make the object point clearly imaged, and then adjust the camera pitch and deflection angles to make the entire end effector plane clearly imaged. Move the end effector of the electric micromanipulator along the y-axis and z-axis directions respectively, and observe whether the object point moves horizontally or vertically in the field of view. If not, adjust the roll of the camera until the object point moves horizontally or vertically in the field of view. After the correction is completed, it is considered that the camera viewing plane is parallel to the yz plane of the robot base coordinate system.

[0065] S3: Determine the scale

[0066] like Figure 7 As shown, the end effector of the micromanipulator robot is placed in front of the microscope lens 21. The end effector of the robot is a micrometer. The linear motors of the micromanipulator robot are placed in the initial position, and the linear motor 2 and the linear motor 4 are adjusted until the scale on the micrometer is clearly imaged, and a clear picture of the scale is saved; it is known that the spacing between two adjacent scale lines on the micrometer is 0.1mm, and the pixel spacing between the two scale lines is obtained by image processing, thereby determining the ratio between the real length and the image pixel, which is used to convert the pixel distance into the actual distance and substitute it into the formula in subsequent calculations.

[0067] S401: Correcting robot orthogonality

[0068] Due to the error during robot assembly, the movement axes of the robot may not be completely orthogonal, so the orthogonality of the robot needs to be corrected. Take a point on the micrometer as the object point, move the linear motor 5, and observe whether the object point moves vertically in the field of view. If not, adjust the position of the linear motor 1 and the linear motor 3 until the object point moves vertically in the field of view. At this time, it is considered that the movement direction of the linear motor 5 and the movement direction of the linear motor 1 and the linear motor 3 are orthogonal to each other. Record the displacement of the linear motor 1 and the linear motor 3 at this time, and use it as the initial position of the linear motor 1 and the linear motor 3 in the subsequent steps.

[0069] S402: Parameter calibration

[0070] Assume that the displacements of linear motor 1 and linear motor 3 are L2 and L1 respectively, place the two motors in their initial positions, save the image of the micrometer under the camera at this time, and obtain the coordinates A of the end point of the micrometer after processing the image. s (x s ,y s ); Set the motor displacement to make ΔL i =L 2i -L 1i = ±0.1, ±0.2, ±0.3, ±0.4, ±0.5, save the image of the micrometer under the camera, and after processing the image, you can get M i (x mi ,y mi ), N i (x ni ,y ni ), A fi (x fi ,y fi ), according to the calibration principle:

[0071]

[0072] in,

[0073]

[0074] The calibration result is:

[0075]

[0076] S403: Measuring repeatability

[0077] Take a point on the micrometer as the object point, and move the linear motor 1, linear motor 3, and linear motor 5 to move the object point to a fixed position A. At this time, the displacement of the linear motor 1, linear motor 3, and linear motor 5 is recorded as (a, b, c); then move the object point to any random position. Then move the displacement of the linear motor 1, linear motor 3, and linear motor 5 to (a, b, c), save the image of the object point at this time, and repeat this operation 10 times; the position coordinates of the object point each time are obtained through image processing as (x i ,y i ), then the mean of the 10 position coordinates for:

[0078]

[0079] Then the deviation e i for:

[0080]

[0081] Then the repeat positioning accuracy RP is:

[0082]

[0083] S404: Measuring translation accuracy

[0084] Move linear motor 1 and linear motor 3 simultaneously by 5um each time and save the image of the object point at this time. Move 10 times in total. Get the position of the object point through image processing and calculate the change Δx of the position of the object point before and after each movement. i , then the error k of position change each time moving i for:

[0085] k i =|Δx i -5|

[0086] The translation accuracy AP1 of linear motor 1 and linear motor 3 is:

[0087]

[0088] Make the linear motor 5 move 5um each time, and the rest of the operations are the same as above, and the translation accuracy AP2 of the linear motor 5 is obtained.

[0089] S405: Measuring rotation accuracy

[0090] The calibration results are obtained in S402 Then each ΔL i Theoretical rotation angle θ i for:

[0091]

[0092] The actual rotation angle σ i for:

[0093]

[0094] Angle error r i for:

[0095] r i =|θ i -σ i |

[0096] Then the rotation precision SP is:

[0097]

[0098] The calibration results are shown in Table 1 below:

[0099] Table 1 Calibration results

[0100] Kinematic parameters Average value / mm Standard deviation / mm d 25.44 1.47 <![CDATA[d m ]]> 3.95 0.27 <![CDATA[L tool ]]> 29.22 1.63

[0101] The measurement results of repeatable positioning accuracy are shown in Table 2 below:

[0102] Table 2 Repeat positioning accuracy table

[0103] Average value / um Maximum value / um Repeat positioning accuracy 2.989 4.610

[0104] The absolute positioning accuracy measurement results are shown in Table 3 below:

[0105] Table 3 Absolute positioning accuracy table

[0106] Axis of motion average value Maximum Standard Deviation Range of motion Linear motor 1 1, linear motor 3 3 translation / um 2.78 5.00 1.37 -10000～10000 Linear motor 55 translation / um 2.56 5.00 1.67 -10000～10000 Linear motor one 1, linear motor three 3 rotation / ° 0.070 0.322 0.083 -38.459～38.459

[0107] It can be seen from the calibration results that the present invention can obtain high-precision calibration parameters; and it can be seen from the motion accuracy measurement results that when high-precision calibration parameters are obtained, the robot can exhibit higher motion accuracy.

[0108] The above embodiments describe the present invention in detail, but the contents are only preferred embodiments of the present invention and cannot be considered to limit the scope of the present invention. All equivalent changes and improvements made within the scope of the present invention should still fall within the scope of the present invention.

Claims

1. A parameter calibration method for a parallel structure micro-manipulation robot based on microscopic vision, characterized in that: The method comprises the following steps: S1: Modeling and kinematic analysis of the parallel structure robot to be calibrated, and constructing a kinematic model; S2: designing a calibration method based on the kinematic model described in S1, setting a micromanipulator and a camera, and using the micromanipulator as a standard reference object to calibrate the camera to correct the pitch, roll and yaw angles of the camera; S3: Fix the camera calibrated in S2, place a robot with a micrometer on the end effector in front of the camera, adjust the distance between the robot and the camera until the scale on the micrometer is clearly imaged in the field of view, and save the image to determine the corresponding relationship between the real length and the image pixels; S4: Move the end effector to obtain a picture of the end effector under the camera, process the image to obtain data, calculate the kinematic parameters of the robot, and then measure the repeatability, translation and rotation accuracy of the robot.

2. The parameter calibration method of a microscopic vision-based parallel structure micro-manipulation robot according to claim 1, characterized in that: In S1, the design of the parallel structure robot is a planar five-bar structure, which includes two parallel linear motors, two rotary joints and an end effector. When the displacements of the two linear motors are both zero, the end effector is located at an initial position; when the displacements of the two linear motors are equal, the end effector translates; when the displacements of the two linear motors are unequal, the end effector rotates.

3. The parameter calibration method of a microscopic vision-based parallel structure micro-manipulation robot according to claim 2, characterized in that: In S1, the displacement of the linear motor located on the upper side is L1, the displacement of the linear motor located on the lower side is L2, the distance between the two linear motor motion axes is d, and the distance between the end effector and the rotation axis of the rotary joint parallel to the rod direction is d m , the distance between the end effector and the rotation axis of the rotary joint perpendicular to the rod direction is L tool , the relationship between the translation distance L, the rotation angle θ and L1, L2 is: L=L1=L2 A s (x s ,y s ) is the initial position of the end point of the end effector, A f (x f ,y f ) is the final position of the end point of the end effector. From the analysis, we can know that the position coordinates of the end point movement (x f ,y f ) and the relationship between L1 and L2 is:

4. The parameter calibration method of a microscopic vision-based parallel structure micro-manipulation robot according to claim 1, characterized in that: In S2, in order to solve the problem that the camera's viewing plane is not parallel to the robot's motion plane, resulting in defocus and large measurement errors, a standard xyz three-degree-of-freedom electric micromanipulator is used as a standard reference object, and its y-axis and z-axis motion directions are set as orthogonal horizontal and vertical directions to calibrate the camera and correct its pitch, roll and yaw angles. After the correction is completed, it is considered that the camera's viewing plane is parallel to the yz plane of the robot's base coordinate system.

5. The parameter calibration method of a microscopic vision-based parallel structure micro-manipulation robot according to claim 1, characterized in that: In S4, the image processing and accuracy measurement specifically include the following steps: S401: Correct the orthogonality of the robot: Due to the error in the assembly of the robot, the motion axes of the robot are likely to be non-orthogonal, so the orthogonality between the axes needs to be corrected under the microscope lens; S402: Parameter calibration: Move the robot's motion axis to make the displacement difference ΔL of two parallel linear motors equal to different values, save the image of the end effector, and obtain the coordinates of the motion point through image processing for calculating kinematic parameters; S403: Measuring repeated positioning accuracy: moving the robot's motion axis to make the end effector repeatedly reach a certain position, repeating n times, saving the image, and using it to calculate the repeated positioning error; S404: measuring translation accuracy: making the linear motors performing translation motion translate the same distance each time, for a total of n times, and saving images for translation accuracy measurement; S405: Measure the rotation accuracy: Use the kinematic parameters calculated in S402 to calculate the theoretical rotation angle at each ΔL, and subtract it from the actual rotation angle to obtain the rotation accuracy.

6. The parameter calibration method of a microscopic vision-based parallel structure micro-manipulation robot according to claim 5, characterized in that: In S401, in order to solve the problem of large measurement errors caused by errors during robot assembly resulting in non-orthogonality between the robot's motion axes, the orthogonality between the axes is corrected under the microscope lens, the robot's motion axes are moved, and the orthogonality between the robot's motion axes is corrected based on the vertical and horizontal directions in the camera's field of view.

7. The parameter calibration method of a microscopic vision-based parallel structure micro-manipulation robot according to claim 5, characterized in that: In S402, the image is subjected to straight line detection based on Hough transform to obtain the two end point coordinates M (x m ,y m )、N(x n ,y n ), image processing can obtain the coordinates A of the end point of the micrometer before and after movement s (x s ,y s ), A f (x f ,y f ), then the kinematic parameters d, d m and L tool The expression is: in,

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