Mechanical arm singular point avoiding method based on improved approximation damping reciprocal

Abstract

The invention relates to a mechanical arm singular point avoiding method based on improved approximation damping reciprocal. Comprising the steps that firstly, a base coordinate system and a tool coordinate system of a six-degree-of-freedom mechanical arm are established, kinematics parameters of the six-degree-of-freedom robot are determined, and a forward kinematics model is established; 2, matrix transformation is conducted on the mechanical arm Jacobian matrix, and a singular Jacobian matrix is obtained; the singularity degree is judged according to an operability degree formula; 3, deducing a singular factor, and judging the state of the mechanical arm according to the singular factor; 4, deducing a singular factor of the Jacobian matrix, and solving an improved approximation damping reciprocal by using an improved approximation damping reciprocal method to replace a reciprocal of the singular factor; the precision of the moving trajectory is guaranteed as much as possible while the singular point avoiding capacity of the mechanical arm is improved.

Description

Technical Field

[0001] The invention belongs to the technical field of robot arm control, and in particular is a robot arm singular point avoidance method based on improved approximation of damping inverse. Background Art

[0002] In recent years, with the support of many policy backgrounds, the manufacturing model of traditional manufacturing has changed. What follows is the era of intelligent manufacturing with robotic arms as the main body of work. In the links of grinding, welding, spraying, assembly, etc., due to the mature technology and wide popularity of robotic arms, humans are freed from the harsh environment with high safety risks and high labor intensity, while greatly reducing labor costs and improving production efficiency. In the field of industrial robots, avoiding the singularity of the robotic arm is of great significance, because the singularity will cause the robot to lose one or more degrees of freedom, unable to perform precise operations, and may even cause system instability or damage equipment. By effectively avoiding the singularity, it can ensure that the robotic arm operates smoothly and safely throughout the entire workspace, improving production efficiency and product quality. In addition, the avoidance of singularity helps to extend the service life of the robotic arm, reduce maintenance and downtime, and ensure the continuity and reliability of the automated production line.

[0003] The degree of singularity of the robot arm can be judged by the maneuverability of the robot arm. The Jacobian matrix is ​​used as a mapping between the end velocity of the robot arm and the joint velocity. When the Jacobian matrix is ​​singular, the end effector of the robot arm loses at least one degree of freedom, the maneuverability of the robot arm is reduced, and the joint velocity will suddenly change or become infinite. When the robot arm is near the singular area, a small displacement of the end also requires a large joint angle velocity, which may exceed the driving capacity of the joint, causing the end trajectory to deviate from the expected trajectory and cause damage to the robot itself or the surrounding environment.

[0004] From the above analysis, it can be seen that it is very important for industrial production for the robot to avoid singular points while ensuring the trajectory accuracy during trajectory operation. Therefore, it is of great significance to improve the trajectory accuracy of the robot when avoiding singular points. Summary of the invention

[0005] To solve the above problems, the present invention provides a method for avoiding singularity points of a robot arm based on an improved approximation to the inverse of damping, which improves the ability of the robot arm to avoid singularity points while ensuring the accuracy of the running trajectory as much as possible.

[0006] The technical solution of the present invention is described as follows in conjunction with the accompanying drawings:

[0007] A method for avoiding singular points of a manipulator based on improved approximation of the damping inverse comprises the following steps:

[0008] Step 1: Using the screw theory, establish the base coordinate system and tool coordinate system of the six-degree-of-freedom robot to determine the kinematic parameters of the six-degree-of-freedom robot, and establish a forward kinematics model;

[0009] Step 2: Calculate the object Jacobian matrix of the robot based on the six-degree-of-freedom robot joint motion screw coordinates and the corresponding robot joint coordinate system, and perform matrix transformation on the robot Jacobian matrix to obtain a singular Jacobian matrix; and determine the singularity degree according to the operability formula;

[0010] Step 3: Extract singular factors and determine the state of the robotic arm based on the singular factors;

[0011] Step 4: Derivation of the Jacobian matrix J b The singular factor k i , and use the improved approximate damping inverse method to solve the improved approximate damping inverse to replace the singular factor k i The reciprocal of

[0012] Furthermore, the specific method of step one is as follows:

[0013] 11) Establish the initial configuration of the six-degree-of-freedom robot end tool coordinate system as shown below:

[0014]

[0015] Wherein, a1 is the connecting rod length between joint axis 1 and joint axis 2 and the connecting rod length between joint axis 4 and joint axis 5; a2 is the length between joint axis 6 and the origin of the end tool coordinate system; d1 is the connecting rod length between joint axis 2 and joint axis 3; d2 is the length between joint axis 3 and joint axis 4; d3 is the connecting rod length between joint axis 5 and joint axis 6;

[0016] 12) Establish the screw coordinates B of each screw axis relative to the end tool coordinate system i (ω i ν i ), as follows:

[0017] B1[-1 0 0 0 -(a1+a2) 0]

[0018] B2[0 0 1 0 d1+d2+d3 0]

[0019] B3[0 0 1 0 d2+d3 0]

[0020] B4[0 0 1 0 a1 0]

[0021] B5[-1 0 0 0 -a2 0]

[0022] B6[0 0 1 0 0 0]

[0023] 13) Establish the exponential product expression of the six-degree-of-freedom robot's positive kinematics as follows:

[0024]

[0025] Where θ1…θ6 are the angle values ​​of each joint.

[0026] Furthermore, the specific method of step 2 is as follows:

[0027] 21) The degree to which a given pose approaches singularity is measured by the operable degree ellipsoid, taking

[0028] θ1=π / 4, θ2=π / 6, θ3=π / 2, θ4=π / 4, θ5=0, θ6=π / 3

[0029] It means that the six-degree-of-freedom robot arm is in a singular wrist state;

[0030] 22) According to the manipulator, the operability expression is Calculate the value of the maneuverability A when the six-degree-of-freedom manipulator is in a singular wrist state;

[0031] 23) By substituting the maximum and minimum eigenvalues ​​of A into the formula That is, the singularity of the robot arm in the current posture is obtained, and the numerical value of the singularity is set as the threshold for judging the singularity state of the wrist of the six-degree-of-freedom robot arm. When the μ(A) value is closer to this value, the singularity of the six-degree-of-freedom wrist is greater.

[0032] Furthermore, the specific method of step three is as follows:

[0033] 31) The expression of the object Jacobian matrix of the six-degree-of-freedom robot is:

[0034]

[0035] In the formula, J bi (θ) is the spinor coordinate on the i-th joint axis;

[0036] 32) Decompose the six-dimensional Jacobian matrix into four three-dimensional Jacobian sub-matrices as follows:

[0037]

[0038] Where V1 is the joint velocity of the first three axes of the robot arm, which is a 3×1 matrix; V2 is the joint velocity of the last three axes of the robot arm, which is a 3×1 matrix; J 11 ,J 21 ,J 22The decomposed Jacobian matrix is ​​a 3×3 matrix; 3×3 is a 3×3 zero matrix; θ f and θ g They are all 3×1 matrices, representing a subset of joint velocities;

[0039] The calculation formula of the joint speed of the robot arm is:

[0040]

[0041] In the formula, when the robot arm is in the front end singular posture, including boundary singularity and internal singularity, J 11 The Jacobian submatrix is ​​singular, and the joint velocities of the first three axes do not exist; when the robot is in the wrist singularity, J 22 The Jacobian submatrix is ​​singular, and the joint velocities of the rear three axes do not exist;

[0042] 33) Calculate the Jacobian matrix in the third joint coordinate system to simplify the operation as follows:

[0043]

[0044] In the formula, s i is the sine of the joint angle of joint i (θ i );c i is the cosine of the joint angle of joint i (θ i );s ij is the sine of the sum of the joint angles of joint i and joint j (θ i +θ j );c ij is the cosine of the sum of the joint angles of joint i and joint j (θ i +θ j );

[0045] Calculate the inverse of the Jacobian matrix in the third joint coordinate system as follows:

[0046]

[0047] Where k1 = c 23 d1-s 23 d3+c2a1, k2=d1a1s3+d3a1c3 is the front-end singular factor; a1, d1, d3 are the lengths of the robot arm; a1 corresponds to the length of the connecting rod between joint 1 and joint 2 of the robot arm; d1 corresponds to the length of the connecting rod between joint 2 and joint 3; d3 corresponds to the length of the connecting rod between joint 5 and joint 6;

[0048] Calculate the inverse of the Jacobian matrix in the fifth joint coordinate system as follows:

[0049]

[0050] In the formula, is the inverse of the sub-Jacobian matrix in the fifth joint coordinate system, and k3=s5 is the singular factor of the wrist.

[0051] Furthermore, the specific method of step 4 is as follows:

[0052] 41) A method to improve the approximation of the damping inverse is proposed as follows:

[0053]

[0054] In the formula, λ1 is 0.3; C is 0.0001; ε1 is 1 / 110;

[0055] 42) Use f(k i ) replaces the inverse of k1, k2, and k3. The Jacobian matrix with the new damping inverse ensures that when the robot end velocity in Cartesian space is converted to the velocity in joint space, the obtained joint velocity will not exceed the joint velocity limit due to singularity problems, that is, the joint velocity will not be infinite, thereby avoiding singular points when the robot end trajectory tracking accuracy is partially lost.

[0056] The beneficial effects of the present invention are:

[0057] 1. The present invention adopts the screw theory to establish the screw coordinates of six degrees of freedom, and judges the singularity of the manipulator according to the manipulable ellipsoid, thereby improving the efficiency of judging the singularity of the six-degree-of-freedom manipulator;

[0058] 2. The present invention utilizes an improved approximate damping inverse method to solve the inverse kinematics of a six-degree-of-freedom robotic arm, thereby effectively improving the ability of the six-degree-of-freedom robotic arm to avoid singular points, and effectively improving the accuracy of the trajectory near the singular points while avoiding the singular points, thereby avoiding sudden changes in joint speed due to the robotic arm entering the singular points, ensuring the stability and accuracy of the six-degree-of-freedom robotic arm's movement, and improving the performance of the industrial robot. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for use in the embodiments are briefly introduced below. It should be understood that the following drawings only show certain embodiments of the present invention and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other related drawings can be obtained based on these drawings without creative work.

[0060] Figure 1 This is a schematic diagram of the forward kinematics model of a six-degree-of-freedom robotic arm;

[0061] Figure 2To improve the approximate damping algorithm and the traditional damping algorithm for 1 / k i Comparison chart of curve approximation effect;

[0062] Figure 3 This is a diagram of the bizarre mechanical configuration of the wrist of a six-freedom robotic arm. DETAILED DESCRIPTION

[0063] The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It is to be understood that the specific embodiments described herein are only used to explain the present invention, rather than to limit the present invention. It should also be noted that, for ease of description, only parts related to the present invention, rather than all structures, are shown in the accompanying drawings.

[0064] Embodiment 1

[0065] This embodiment provides a method for avoiding singular points of a manipulator based on an improved approximation of the damping inverse. The singular state of the wrist of a six-degree-of-freedom manipulator is as follows: Figure 3 As shown, the specific method is as follows:

[0066] Step 1: Read Figure 1 , using the screw theory, the base coordinate system and tool coordinate system of the six-degree-of-freedom robot are established to determine the kinematic parameters of the six-degree-of-freedom robot, and a forward kinematic model is established, as follows:

[0067] 11) Taking the Zhongke Shengu six-axis robot arm as an example, the initial configuration of the six-degree-of-freedom robot end tool coordinate system is established as follows:

[0068]

[0069] 12) Establish the screw coordinates B of each screw axis relative to the end tool coordinate system i (ω i ν i ), the following uses the Zhongke Shengu six-axis robot arm as an example to establish the screw coordinates as shown below:

[0070] B1[-1000-212.20]

[0071] B2[00106690]

[0072] B3[00103690]

[0073] B4[0010110.10]

[0074] B5[-1000-102.1 0]

[0075] B6[001000]

[0076] 13) According to the screw theory, the exponential product expression of the six-degree-of-freedom robot positive kinematics is established as follows:

[0077]

[0078] Where θ1…θ6 are the angle values ​​of each joint.

[0079] Step 2: Calculate the object Jacobian matrix of the robot based on the six-degree-of-freedom robot joint motion screw coordinates and the corresponding robot joint coordinate system, and transform the robot Jacobian matrix to obtain the singular Jacobian matrix; and determine the singularity according to the operability formula, as follows:

[0080] 21) The degree to which a given pose approaches singularity is measured by the operable degree ellipsoid, taking

[0081] θ1=π / 4, θ2=π / 6, θ3=π / 2, θ4=π / 4, θ5=0, θ6=π / 3

[0082] It means that the six-degree-of-freedom robot arm is in a singular wrist state;

[0083] 22) According to the manipulator, the operability expression is Calculate the value of the maneuverability A when the six-degree-of-freedom manipulator is in a singular wrist state;

[0084] 23) By substituting the maximum and minimum eigenvalues ​​of A into the formula That is to say, the singularity degree of the six-axis robot arm of Zhongke Shengu in the current posture is 26602.8857;

[0085] λ i (i=1…m) is the eigenvalue of A, μ(A) is the volume formula of the operable sphere A; the larger μ(A) is, the greater the singularity is. When it approaches 1, the singularity is smaller.

[0086] The μ(A) value at this time is the value indicating the singularity when the robot is in a wrist singularity state, and 26602.8857>>1, so the robot is in a wrist singularity state at this time. Set this value as the threshold. The closer the singularity of the robot is to this value, the greater the singularity of the wrist. This feature can be used to compare the improved algorithm with the traditional algorithm. If the difference between the singularity value of the improved algorithm and the threshold is smaller than the difference between the singularity value of the traditional algorithm and the threshold, it can be determined that the performance of the improved algorithm is better than the traditional algorithm. If it is greater, it means that the improved algorithm is not as good as the traditional effect, so as to judge the performance of the singularity avoidance algorithm.

[0087] Step 3: Extract singular factors and judge the state of the robot arm according to the singular factors. The state of the robot arm includes the front end singular states k1 and k2 and the wrist singular state k3; k1 is the internal singular state and k2 is the boundary singular state, as follows:

[0088] 31) The expression of the object Jacobian matrix of the six-degree-of-freedom robot is:

[0089]

[0090] Where J bi The physical meaning of (θ) is the spinor coordinate on the i-th joint axis;

[0091] 32) Decompose the six-dimensional Jacobian matrix into four three-dimensional Jacobian sub-matrices. The purpose of this is to reduce the computational complexity. The formula after decomposition is as follows:

[0092]

[0093] Where V1 is the joint velocity of the first three axes of the robot arm, which is a 3×1 matrix; V2 is the joint velocity of the last three axes of the robot arm, which is a 3×1 matrix; J 11 ,J 21 ,J 22 The decomposed Jacobian matrix is ​​a 3×3 matrix; 3×3 is a 3×3 zero matrix; θ f and θ g They are all 3×1 matrices, representing a subset of joint velocities. The formula for calculating the joint velocities of the robotic arm is:

[0094]

[0095] In the formula, when the robot arm is in a front end singular posture (including boundary singularity and internal singularity), J 11 The Jacobian submatrix is ​​singular, and the joint velocities of the first three axes do not exist. Similarly, when the robot arm is in a singular wrist state (such as Figure 3 State shown), J 22 The Jacobian submatrix is ​​singular and the joint velocities of the rear three axes do not exist.

[0096] 33) Since the three axes at the end only affect the posture of the robot arm, the Jacobian matrix in the third joint coordinate system is calculated to simplify the operation:

[0097]

[0098] In the formula, s i is the sine of the joint angle of joint i (θ i );c i is the cosine of the joint angle of joint i (θ i );sij is the sine of the sum of the joint angles of joint i and joint j (θ i +θ j );c ij is the cosine of the sum of the joint angles of joint i and joint j (θ i +θ j );

[0099] The inverse matrix of the Jacobian matrix in the third joint coordinate system The formula is as follows:

[0100]

[0101] Where k1 = c 23 d1-s 23 d3+c2a1, k2=d1a1s3+d3a1c3 is the front-end singular factor; a1, d1, d3 are the lengths of the robot arm, a1 corresponds to the length of the connecting rod between joint 1 and joint 2; d1 corresponds to the length of the connecting rod between joint 2 and joint 3; d3 corresponds to the length of the connecting rod between joint 5 and joint 6. You can refer to Figure 1 .

[0102] Similarly, the inverse matrix of the Jacobian submatrix in the fifth joint coordinate system is 5 J 22 -1 .

[0103]

[0104] In the formula, is the inverse of the sub-Jacobian matrix in the fifth joint coordinate system, and k3=s5 is the singular factor of the wrist.

[0105] Step 4: Derivation of Jacobian matrix J through step 3 b The singular factor k i Then, the improved approximate damping inverse method is used to solve the improved approximate damping inverse to replace the singular factor k i The reciprocal of is as follows:

[0106] 41) A method to improve the approximation of the damping inverse is proposed as follows:

[0107]

[0108] In the formula, λ1 is taken as 0.3; C is taken as 0.0001; ε1 is taken as 1 / 110, and the three values ​​are empirical values ​​obtained through experimental verification; then the improved approximation damping f(k i ) replaces the original singular factor k i Substitute the reciprocal of the original object Jacobian matrix Jb middle.

[0109] 42) Use f(k i ) replaces the inverse of k1, k2, and k3. The Jacobian matrix with the new damping inverse ensures that when the robot end velocity in Cartesian space is converted to the velocity in joint space, the obtained joint velocity will not exceed the joint velocity limit due to singularity problems, that is, the joint velocity will not be infinite, thereby avoiding singular points when the robot end trajectory tracking accuracy is partially lost.

[0110] Embodiment 2

[0111] In this embodiment, the damping inverse curve and 1 / k are judged. i The approximation degree of the curve determines the quality of the improved approximation of the damped inverse. That is, if the damped inverse curve passes through zero, it means that the singular point can be avoided. And the closer it is to 1 / k i The smaller the curve, the higher the trajectory accuracy. Then, use Matlab to compare the reciprocal curve with 1 / k i The curve is simulated in the same coordinate system, such as Figure 2 As shown, the improved approximation of the damping inverse 1 / f(k i ) passes through zero and is closer to 1 / k than other damping algorithms i Curve. Where 1 / k i The curve represents the inverse curve of the singular factor. When k i When it approaches 0, the curve approaches positive infinity, which is reflected in the movement of the robotic arm, indicating that when the singular factor is very small, the joint speed is infinite. Figure 2 It shows that the proposed improved approximate damping inverse method can avoid singular points while ensuring trajectory accuracy as much as possible. Figure 2 The dotted line in the figure represents the general damping inverse method, which can ensure the avoidance of singular points, that is, ensure that the joint speed does not tend to infinity, but will lose a certain degree of accuracy. The improved approximate damping inverse method in the present invention can ensure that the joint speed does not tend to infinity while ensuring the trajectory accuracy as much as possible, so as to avoid the singular point.

[0112] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for avoiding singular points of a manipulator based on improved approximation of the damping inverse, characterized in that: The following steps are involved: Step 1: Using the screw theory, establish the base coordinate system and tool coordinate system of the six-degree-of-freedom robot to determine the kinematic parameters of the six-degree-of-freedom robot, and establish a forward kinematics model; Step 2: Calculate the object Jacobian matrix of the robot based on the six-degree-of-freedom robot joint motion screw coordinates and the corresponding robot joint coordinate system, and perform matrix transformation on the robot Jacobian matrix to obtain a singular Jacobian matrix; And the degree of singularity is determined according to the operability formula; Step 3: derive the singular factor and determine the state of the robot arm according to the singular factor; Step 4: Derivation of the Jacobian matrix J b The singular factor k i , and use the improved approximate damping inverse method to solve the improved approximate damping inverse to replace the singular factor k i The inverse of , thereby avoiding the singularity point of the robot arm.

2. The method for avoiding singular points of a manipulator based on improved approximation of the damping inverse according to claim 1, characterized in that: The specific method of step one is as follows: 11) Establish the initial configuration of the six-degree-of-freedom robot end tool coordinate system as shown below: Wherein, a1 is the connecting rod length between joint axis 1 and joint axis 2 and the connecting rod length between joint axis 4 and joint axis 5; a2 is the length between joint axis 6 and the origin of the end tool coordinate system; d1 is the connecting rod length between joint axis 2 and joint axis 3; d2 is the length between joint axis 3 and joint axis 4; d3 is the connecting rod length between joint axis 5 and joint axis 6; 12) Establish the screw coordinates B of each screw axis relative to the end tool coordinate system i (ω i ν i ), as follows: B1[-1000-(a1+a2)0] B2[0010d1+d2+d3 0] B3[0010d2+d3 0] B4[0010a1 0] B5[-1000-a2 0] B6[001000] 13) Establish the exponential product expression of the six-degree-of-freedom robot's positive kinematics as follows: Where θ1…θ6 are the angle values ​​of each joint.

3. The method for avoiding singular points of a manipulator based on improved approximation of the damping inverse according to claim 1, characterized in that: The specific method of step 2 is as follows: 21) The degree to which a given pose approaches singularity is measured by the operable degree ellipsoid, taking θ1=π / 4, θ2=π / 6, θ3=π / 2, θ4=π / 4, θ5=0, θ6=π / 3 It means that the six-degree-of-freedom robot arm is in a singular wrist state; 22) According to the manipulator, the operability expression is Calculate the value of the maneuverability A when the six-degree-of-freedom manipulator is in a singular wrist state; 23) By substituting the maximum and minimum eigenvalues ​​of A into the formula That is, the singularity of the robot arm in the current posture is obtained, and the numerical value of the singularity is set as the threshold for judging the singularity state of the wrist of the six-degree-of-freedom robot arm. When the μ(A) value is closer to this value, the singularity of the six-degree-of-freedom wrist is greater.

4. The method for avoiding singular points of a manipulator based on improved approximation of the damping inverse according to claim 1, characterized in that: The specific method of step three is as follows: 31) The expression of the object Jacobian matrix of the six-degree-of-freedom robot is: 32) Decompose the six-dimensional Jacobian matrix into four three-dimensional Jacobian sub-matrices as follows: Where V1 is the joint velocity of the first three axes of the robot arm, which is a 3×1 matrix; V2 is the joint velocity of the last three axes of the robot arm, which is a 3×1 matrix; J 11 ,J 21 ,J 22 The decomposed Jacobian matrix is ​​a 3×3 matrix; 3×3 is a 3×3 zero matrix; θ f and θ g They are all 3×1 matrices, representing a subset of joint velocities; The calculation formula of the joint speed of the robot arm is: In the formula, when the robot arm is in a front end singular posture (including boundary singularity and internal singularity), J 11 The Jacobian submatrix is ​​singular, and the joint velocities of the first three axes do not exist. Similarly, when the robot arm is in a singular wrist state (as shown in Figure 3), J 22 The Jacobian submatrix is ​​singular and the joint velocities of the rear three axes do not exist. 33) Calculate the Jacobian matrix in the third joint coordinate system to simplify the operation as follows: In the formula, s i is the sine of the joint angle of joint i (θ i );c i is the cosine of the joint angle of joint i (θ i );s ij is the sine of the sum of the joint angles of joint i and joint j (θ i +θ j );c ij is the cosine of the sum of the joint angles of joint i and joint j (θ i +θ j ); Calculate the inverse of the Jacobian matrix in the third joint coordinate system as follows: Where k1 = c 23 d1-s 23 d3+c2a1, k2=d1a1s3+d3a1c3 is the front-end singular factor; a1, d1, d3 are the lengths of the robot arm, a1 corresponds to the length of the connecting rod between joint 1 and joint 2 of the robot arm; d1 corresponds to the length of the connecting rod between joint 2 and joint 3; d3 corresponds to the length of the connecting rod between joint 5 and joint 6; Calculate the inverse of the Jacobian matrix in the fifth joint coordinate system as follows: In the formula, is the inverse of the sub-Jacobian matrix in the fifth joint coordinate system, and k3=s5 is the singular factor of the wrist.

5. The method for avoiding singular points of a manipulator based on improved approximation of the damping inverse according to claim 1, characterized in that: The specific method of step 4 is as follows: 41) A method to improve the approximation of the damping inverse is proposed as follows: In the formula, λ1 is 0.3; C is 0.0001; ε1 is 1 / 110; 42) Use f(k i ) replaces the inverse of k1, k2, and k3. The Jacobian matrix with the new damping inverse ensures that when the robot end velocity in Cartesian space is converted to the velocity in joint space, the obtained joint velocity will not exceed the joint velocity limit due to singularity problems, that is, the joint velocity will not be infinite, thereby avoiding singular points when the robot end trajectory tracking accuracy is partially lost.

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