Inverse kinematics minimum norm solving method for mechanical arm with redundant degree of freedom

Abstract

The invention belongs to the technical field of redundant robots, and particularly relates to an inverse kinematics minimum norm solving method for a mechanical arm with redundant degrees of freedom, which comprises the following steps of: establishing a D-H coordinate system and D-H parameters according to parameters of the mechanical arm, establishing a forward kinematics model according to the D-H coordinates and the D-H parameters of the mechanical arm, and calculating the inverse kinematics minimum norm of the mechanical arm with the redundant degrees of freedom; randomly selecting a group of joint angle values as initial attitude input, converting the group of joint angles into an end effector pose matrix through forward kinematics, establishing an inverse kinematics solution model of the mechanical arm based on Jacobian pseudo-inverse, and optimizing the inverse kinematics solution model by adopting a damping least square method and a gradient projection method, so as to obtain an optimal position of the mechanical arm. And carrying out iterative calculation on the optimized model to obtain a minimum norm solution. According to the method, the singular configuration of the mechanical arm configuration in the movement process is avoided through the damping least square method, the introduced gradient projection method is matched to avoid the limiting position of the joint corner of the mechanical arm, and the obtained minimum norm solution has high resolving efficiency and accuracy.

Description

Technical Field

[0001] The invention relates to the technical field of redundant robots, and in particular to a method for solving the minimum norm of inverse kinematics of a mechanical arm with redundant degrees of freedom. Background Art

[0002] In extreme operating environments with narrow terrain and harsh environments, such as earthquake rescue, deep-sea exploration, and nuclear power plant exploration, traditional articulated robotic arms are restricted by their structural design and environmental conditions and are difficult to meet task requirements. Therefore, redundant robotic arms with higher degrees of freedom and slimmer bodies can form different robotic arm configurations under reasonable path planning to complete operational requirements in various narrow environments. They are expected to become the main operating force in the above-mentioned environments and have great application prospects.

[0003] Solving the inverse kinematics of a robot arm is the basis and key to controlling the robot arm. It is mainly used in robots with articulated structures. By mapping the position and posture of the robot arm's end effector from the three-dimensional Cartesian space to the joint space inside the robot arm, the corresponding joint angle is obtained, thereby reducing the difficulty of controlling the robot arm. Solving the inverse kinematics of a robot arm has its own problems of uncertainty in the existence, uniqueness, and rationality of the solution, which makes the configuration of the robot arm with redundant degrees of freedom more complex, usually with more or infinite solutions, and the difficulty and time of solving the inverse kinematics will increase exponentially.

[0004] At present, the solutions for the inverse kinematics of manipulators are mainly divided into two categories: analytical solutions and numerical solutions. The analytical solution is mainly for manipulator mechanisms that meet the Pieper criterion, that is, the axes of three adjacent joints of the manipulator intersect at one point or are parallel to each other. Such inverse kinematics equations of manipulators are mainly polynomials of degree no higher than four, with small amount of calculation and relatively accurate. However, in the process of solving redundant degrees of freedom manipulators with infinite inverse solutions, the solution given is often not the minimum norm solution, and the joint angles may jump or reverse when the end motion trajectory of the manipulator moves to certain quadrants.

[0005] The numerical solution is mainly obtained by numerical iteration to obtain the minimum norm solution of inverse kinematics. Among them, the more typical one is to solve the robot inverse kinematics based on the Jacobi pseudo-inverse method. This method mainly uses the error of the robot in the end point workspace to project it to each joint for proportional control, and converges to one of the feasible solutions after several iterations. However, this method has a large amount of calculation. When the robot moves near a singular configuration, the joint speed solved by the pseudo-inverse method will increase sharply, making the iterative calculation results uncontrollable, and the joint angle of the robot arm is easy to exceed the working range. Summary of the invention

[0006] The object of the present invention is to provide a minimum norm solution method for inverse kinematics of a manipulator with redundant degrees of freedom, so as to improve the calculation efficiency and accuracy of the minimum norm solution.

[0007] To achieve the above object, the present invention adopts the following technical solution:

[0008] A method for solving the minimum norm of inverse kinematics of a manipulator with redundant degrees of freedom comprises the following steps:

[0009] Step 1: Establish the DH coordinate system and DH parameters according to the robot's own parameters;

[0010] Step 2: Establish a forward kinematics model based on the DH coordinate system and DH parameters obtained in step 1;

[0011] Step 3: Based on the forward kinematics model, an inverse kinematics model of the robot arm based on Jacobi pseudo-inverse is established;

[0012] Step 4: Use the damped least squares method and the gradient projection method to solve and optimize the inverse kinematics model obtained in step 3, and solve it to obtain the minimum norm of the inverse kinematics of the robot arm.

[0013] Furthermore, the implementation method of step 1 of establishing the DH coordinate system and DH parameters according to the robot arm's own parameters includes:

[0014] Establish DH coordinate system: Establish DH coordinate system: define the base as the 0th link, the joint close to the base is named the first joint, followed by the second joint, the third joint, and so on, until the number is 2-n. Two mutually orthogonal Cartesian coordinate systems are established on each joint, where the origin coincides with the cross axis of the universal joint, z i The axis is parallel to the axis direction of the i-th joint, x i Axis perpendicular to z i axis and points away from it, y i The axes are determined according to the right-hand rule in the Cartesian coordinate system;

[0015] Establish DH parameters: According to the structural characteristics of the robot, at the center of gravity of the {i}th and {i-1}th joints, follow the coordinate axis x i With coordinate axis z i-1 The perpendicular intersection principle abstracts the size parameter into the connecting rod length a required to establish the DH parameter. i , connecting rod torsion angle α i , joint distance d i and the joint angle θ i , establish DH parameters.

[0016] Furthermore, the method of establishing a forward kinematics model according to the DH coordinate system and DH parameters in step 2 includes:

[0017] According to the DH coordinate system and DH parameters of the robot, the formula of the forward kinematics model of the robot that represents the functional relationship between the end position and posture of the robot and the variables of each joint is established by using the homogeneous transformation matrix formula of two adjacent coordinate systems in the kinematic space.

[0018] The homogeneous transformation matrix formula is as follows:

[0019]

[0020] Where cθ i =cos(θ i ), sθ i = sin(θ i ); cα i = cos(α i ), sα i = sin(a i ), a i Indicates the connecting rod length, d i represents the joint distance, θ i represents the joint rotation angle, i T i+1 Represents the homogeneous transformation matrix of the link coordinate system {i+1} relative to the coordinate system {i};

[0021] According to the chain rule, the homogeneous transformation matrix is ​​continuously multiplied on the right to obtain the forward kinematic model of the robot:

[0022]

[0023] In the formula, Represents the homogeneous transformation matrix of the end effector of the robot arm relative to the base coordinate system, R 3*3 represents the end effector position matrix, R 3*3 represents the end effector posture matrix.

[0024] Furthermore, the method of establishing the inverse kinematics model of the robot arm based on Jacobi pseudo-inverse based on the forward kinematics model in step 3 includes:

[0025] 3.1 Randomly select a set of angle data q0=(θ1, θ2, …, θ n ) is used as the initial posture. Based on the initial posture, the forward kinematics model established in step 2 is used to calculate the pose matrix of the end effector of the robot arm. Recorded as X, and calculate the difference between the current pose of the end effector of the robot arm and the expected pose, recorded as dX;

[0026] 3.2. Establish inverse kinematics of the robot based on Jacobi pseudo-inverse

[0027] a. Based on the Jacobi equation of the joint space and the end effector workspace, establish the relationship dX between the difference between the current end pose and the desired pose and the joint angle increment;

[0028] dX=J(0)dθ

[0029] Where dX represents the difference between the current end position and the desired position, J(θ) represents the Jacobian matrix of the robot joint motion, and dθ represents the joint angle increment;

[0030] b. Use the relationship obtained in step a to calculate the joint angle increment, and use the iteration method to establish the inverse kinematics model dθ based on the joint angle increment:

[0031] dθ=J + dX

[0032] Among them, J + represents the pseudo-inverse equation of the Jacobi equation, which is expressed as: + =J T (JJ T ) -1 ; J represents the Jacobian matrix, J T Represents the transposed matrix of the Jacobian matrix.

[0033] Furthermore, in step 4, a method for solving and optimizing the inverse kinematics model obtained in step 3 by combining the damped least squares method and the gradient projection method to obtain the minimum norm of the inverse kinematics of the manipulator includes:

[0034] 4.1. For the pseudo-inverse equation J + Perform singular value decomposition to obtain the minimum singular value:

[0035] J + =U T D + V

[0036] Among them, U T and V are orthogonal matrices obtained after singular value decomposition; D + The pseudo-inverse of the Jacobian matrix J + A diagonal matrix composed of singular values ​​of , and satisfying

[0037] Among them, σ m+1-i Refers to the minimum singular value obtained by decomposition, which is used to indicate the singularity of the robot configuration during the current motion process;

[0038] 4.2. Based on the minimum singular value, construct the Jacobi pseudo-inverse equation J based on the damped least squares method # :

[0039] J #=(J T J+λ 2 I) -1 J T

[0040]

[0041] Where I represents the identity matrix; J represents the Jacobian matrix; J T represents the transposed matrix of the Jacobian matrix; λ represents the damping coefficient and λ ≥ 0. By introducing a variable damping coefficient for adaptive adjustment, the end tracking error of the robot arm can be reduced; ε represents the minimum error range, which is used to determine whether the robot enters a singular state, σ m represents the minimum singular value of the Jacobian matrix, which is given by the pseudo-inverse J + Decomposed by singular value; m Indicates the preset maximum damping value;

[0042] 4.3. In the Jacobi pseudo-inverse equation J # The gradient projection method is introduced in order to avoid the situation where the joint rotation limit is exceeded during the solution process. In the gradient projection method, the index function related to the joint space position θ is set as:

[0043]

[0044] Where H(θ) represents θ i is the current rotation angle of the i-th joint of the robot link, θ i represents the rotation angle of the i-th joint, θ imax ,θ imin is the upper and lower rotation limits of the i-th joint, a i represents the median of the upper and lower rotation angle limits of the joint, It means to find the partial derivative of H(θ);

[0045] 4.4. According to step 4.2, the Jacobi pseudo-inverse equation J is obtained # The inverse kinematics model of the manipulator based on Jacobi pseudo-inverse is optimized using the gradient projection method in step 4.3:

[0046]

[0047] In the formula, Represents the partial derivative matrix of the indicator function H(θ), I n represents the n-order unit matrix, K is the optimization coefficient;

[0048] 4.5. Iteratively solve the inverse kinematics model of the manipulator based on Jacobi pseudo-inverse obtained in step 4.4 to obtain the joint angle matrix θ i That is, the minimum norm of the inverse kinematics of the robot.

[0049] Furthermore, the above method also includes the step of verifying the obtained minimum norm of inverse kinematics of the robotic arm, specifically:

[0050] The minimum norm joint rotation matrix θ is solved i Substitute it into the forward kinematics model obtained in step 2 to verify whether the error between the end effector pose matrix obtained by solving the forward kinematics model and the expected position is within the allowable range. If so, it indicates that the inverse kinematics model is established correctly.

[0051] The present invention significantly reduces the amount of calculation by iterating the solution process based on the classic Jacobi pseudo-inverse method, ensuring that the movement speed of the robot arm joint is minimum in most working areas. The damped least squares method is used to avoid the singular configuration of the robot arm configuration during movement, and the gradient projection method is introduced to avoid the extreme position of the robot arm joint angle, so that the minimum norm solution obtained has high solution efficiency and accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a DH coordinate system of a manipulator with redundant degrees of freedom;

[0053] Figure 2 The invention is a flowchart of the minimum norm solution of the inverse kinematics of a manipulator with redundant degrees of freedom. DETAILED DESCRIPTION

[0054] In order to make the purpose, technical solution and advantages of the present invention clearer, the following will be described in detail with reference to the embodiments and drawings. It should be understood that the drawings in the present invention are only for the purpose of illustration and description, and are not used to limit the scope of protection of the present application. The schematic drawings are not drawn in proportion to the actual objects. The flowcharts used show the operations implemented according to the embodiments. For the flowcharts, those skilled in the art can add one or more other operations under the guidance of the content of the present invention, and can also remove one or more operations from the flowcharts.

[0055] It should also be noted that the term "comprising" will be used in the embodiments of the present application to indicate the existence of the features declared thereafter, but does not exclude the addition of other features.

[0056] like Figure 1-2 As shown, this embodiment provides a method for solving the minimum norm of inverse kinematics of a manipulator with redundant degrees of freedom, comprising the following steps:

[0057] Step 1: Establish DH coordinate system and DH parameters according to standard DH method;

[0058] According to the structural characteristics of the robot, the center of gravity of the {i}th and {i-1}th joints is calculated according to the coordinate axis x i With coordinate axis z i-1The perpendicular intersection principle abstracts the size parameter into the connecting rod length a required to establish the DH parameter table. i , connecting rod torsion angle α i , joint distance d i and the joint angle θ i , establish DH parameters. The DH parameters obtained in this embodiment are shown in Table 1.

[0059] Table 1 shows the DH parameters established according to the redundant degree of freedom manipulator parameters in this embodiment:

[0060] Link i <![CDATA[θ i (°)]]> <![CDATA[α i (°)]]> <![CDATA[a i (m)]]> <![CDATA[d i (m)]]> 1 0 0 L 0 2 0 90 0 0 3 0 0 L 0 4 0 -90 0 0 5 0 0 L 0 6 0 90 0 0 7 0 0 L 0 8 0 -90 0 0 9 0 0 L 0 10 0 90 0 0 11 0 0 L 0 12 0 -90 0 0

[0061] Step 2: Establish a forward kinematics model based on the robot arm DH coordinate system and DH parameters, and the implementation method includes:

[0062] Based on the DH coordinate system and DH parameters of the robot, the formula of the forward kinematics model of the robot that represents the functional relationship between the end position and posture of the robot and the variables of each joint is established by using the homogeneous transformation matrix formula of two adjacent coordinate systems in the kinematic space.

[0063] The homogeneous transformation matrix formula is shown below (1):

[0064]

[0065] Where cθ i =cos(θ i ), sθ i = sin(θ i ); cα i = cos(α i ), sα i = sin(α i ), a i Indicates the connecting rod length, d i represents the joint distance, θ i represents the joint rotation angle, i T i+1 Represents the homogeneous transformation matrix of the link coordinate system {i+1} relative to the coordinate system {i};

[0066] By continuously multiplying on the right according to the chain rule, we can obtain the forward kinematics model of the robot arm as shown in formula (2):

[0067]

[0068] In the formula, Represents the homogeneous transformation matrix of the end effector of the robot arm relative to the base coordinate system, R 3*3 represents the end effector position matrix, R 3*3 represents the end effector posture matrix.

[0069] Step 3: Establish an inverse kinematics model of the robot arm based on Jacobi pseudo-inverse, and the implementation method includes:

[0070] 3.1 Randomly select a set of angle data q0=(θ1, θ2, …, θ n ) is used as the initial posture. Based on the initial posture, the forward kinematics model established in step 2 is used to calculate the pose matrix of the end effector of the robot arm. Recorded as X, and calculate the difference between the current pose of the end effector of the robot arm and the expected pose, recorded as dX;

[0071] 3.2. Establish the inverse kinematics of the robot based on Jacobi pseudo-inverse:

[0072] a. Based on the Jacobi equation of the joint space and the end effector workspace, establish the relationship dX between the difference between the current end pose and the desired pose and the joint angle increment:

[0073] dX=J(θ)dθ

[0074] Among them, dX represents the difference between the current end pose and the desired pose, J(θ) represents the Jacobi equation of the robot joint motion, and dθ represents the joint angle increment, which is the difference between the joint angles outputted by the previous and next iterations.

[0075] b. Use the relationship obtained in step a to calculate the joint angle increment, and use the iteration method to establish the inverse kinematics model dθ based on the joint angle increment:

[0076] dθ=J + dX

[0077] Among them, J + represents the pseudo-inverse equation of the Jacobi equation, which is expressed as: + =J T (JJ T ) -1 ; J represents the Jacobian matrix, J T Represents the transposed matrix of the Jacobian matrix.

[0078] Step 4: Combine the damped least squares method and the gradient projection method to solve and optimize the inverse kinematics model obtained in step 3. The implementation method is as follows:

[0079] 4.1. For the pseudo-inverse equation J + Perform singular value decomposition to obtain the minimum singular value:

[0080] J + =U T D + V

[0081] Among them, UT and V are orthogonal matrices obtained after singular value decomposition; D + The pseudo-inverse of the Jacobian matrix J + A diagonal matrix composed of singular values ​​of , and satisfying

[0082] Among them, σ m+1-i Refers to the minimum singular value obtained by decomposition, which is used to indicate the singularity of the robot configuration during the current motion process;

[0083] 4.2. Based on the minimum singular value, construct the Jacobi pseudo-inverse equation J based on the damped least squares method # :

[0084] J # =(J T J-λ 2 I) -1 J T

[0085]

[0086] Where I represents the identity matrix, J represents the Jacobian matrix; J T represents the transposed matrix of the Jacobian matrix; λ represents the damping coefficient and λ ≥ 0. By introducing a variable damping coefficient for adaptive adjustment, the end tracking error of the robot arm can be reduced; ε represents the minimum error range, which is used to determine whether the robot enters a singular state; σ m represents the minimum singular value of the Jacobian matrix, which is given by the pseudo-inverse J + Decomposed by singular value; m Indicates the preset maximum damping value;

[0087] 4.3. In the Jacobi pseudo-inverse equation J # The gradient projection method is introduced in order to avoid the situation where the joint rotation limit is exceeded during the solution process. In the gradient projection method, the index function related to the joint space position θ is set as:

[0088]

[0089] Where H(θ) represents θ i is the current rotation angle of the i-th joint of the robot link, θ i represents the rotation angle of the i-th joint, θ imax ,θ imin is the upper and lower rotation limits of the i-th joint, a i represents the median of the upper and lower rotation angle limits of the joint, It means to find the partial derivative of H(θ);

[0090] 4.4. According to step 4.2, the Jacobi pseudo-inverse equation J is obtained #The gradient projection method in step 4.3 is used to solve and optimize the inverse kinematics model of the manipulator based on Jacobi pseudo-inverse, and the minimum norm of the inverse kinematics of the manipulator is obtained:

[0091]

[0092] In the formula, Represents the partial derivative matrix of the indicator function H(θ), I n represents the n-order unit matrix, K is the optimization coefficient;

[0093] 4.5. Iteratively solve the inverse kinematics model of the manipulator based on Jacobi pseudo-inverse obtained in step 4.4 to obtain the joint angle matrix θ i That is, the minimum norm of the inverse kinematics of the robot.

[0094] The above method also includes the step of verifying the obtained minimum norm of inverse kinematics of the robot arm, specifically:

[0095] The minimum norm joint rotation matrix θ is solved i Substitute it into the forward kinematics model obtained in step 2 to verify whether the end effector pose matrix obtained by solving the forward kinematics model and the expected position are within the allowable error range. If so, it means that the inverse kinematics model is established correctly.

[0096] The above formula iterates the solution process based on the classic Jacobi pseudo-inverse method, avoids the singular configuration of the robot arm during movement through the damped least squares method, and avoids the extreme position of the robot arm joint angle through the gradient projection method, which has high solution efficiency and accuracy.

[0097] In summary, the present invention significantly reduces the amount of calculation by iterating the solution process based on the classic Jacobi pseudo-inverse method, ensuring that the movement speed of the robot arm joint is minimum in most working areas. The damped least squares method is used to avoid the singular configuration of the robot arm configuration during the movement, and the gradient projection method is introduced to avoid the extreme position of the robot arm joint angle, so that the minimum norm solution obtained has high solution efficiency and accuracy.

Claims

1. A method for solving the minimum norm of inverse kinematics of a manipulator with redundant degrees of freedom, characterized in that: The following steps are involved: Step 1: Establish the DH coordinate system and DH parameters according to the robot's own parameters; Step 2: Establish a forward kinematics model based on the DH coordinate system and DH parameters obtained in step 1; Step 3: Based on the forward kinematics model, an inverse kinematics model of the robot arm based on Jacobi pseudo-inverse is established; Step 4: Use the damped least squares method and the gradient projection method to solve and optimize the inverse kinematics model obtained in step 3, and solve it to obtain the minimum norm of the inverse kinematics of the robot arm.

2. The method for solving the minimum norm of inverse kinematics of a manipulator with redundant degrees of freedom according to claim 1, characterized in that: The implementation method of step 1 to establish the DH coordinate system and DH parameters according to the robot arm's own parameters includes: Establish DH coordinate system: define the base as the 0th link, the joint close to the base is named the first joint, followed by the second joint, the third joint, and so on, until the number is 2-n. Two mutually orthogonal Cartesian coordinate systems are established on each joint, where the origin coincides with the cross axis of the universal joint, z i The axis is parallel to the axis direction of the i-th joint, x i Axis perpendicular to z i axis and points away from it, y i The axes are determined according to the right-hand rule in the Cartesian coordinate system; Establish DH parameters: According to the structural characteristics of the robot, at the center of gravity of the {i}th and {i-1}th joints, follow the coordinate axis x i With coordinate axis z i-1 The perpendicular intersection principle abstracts the size parameter into the connecting rod length a required to establish the DH parameter. i , connecting rod torsion angle α i , joint distance d i and the joint angle θ i , establish DH parameters.

3. The method for solving the minimum norm of inverse kinematics of a manipulator with redundant degrees of freedom according to claim 2, characterized in that: The method of establishing a forward kinematics model according to the DH coordinate system and DH parameters in step 2 includes: According to the DH coordinate system and DH parameters of the robot, the formula of the forward kinematics model of the robot that represents the functional relationship between the end position and posture of the robot and the variables of each joint is established by using the homogeneous transformation matrix formula of two adjacent coordinate systems in the kinematic space. The homogeneous transformation matrix formula is as follows: Where cθ i =cos(θ i ), sθ i = sin(θ i );ca i = cos(α i ), i = sin(α i ), a i Indicates the connecting rod length, d i represents the joint distance, θ i represents the joint rotation angle, i T i+1 Represents the homogeneous transformation matrix of the link coordinate system {i+1} relative to the coordinate system {i}; According to the chain rule, the homogeneous transformation matrix is ​​continuously multiplied on the right to obtain the forward kinematic model of the robot: In the formula, Represents the homogeneous transformation matrix of the end effector of the robot relative to the base coordinate system, R 3*3 represents the end effector position matrix, R 3*3 represents the end effector posture matrix.

4. The method for solving the minimum norm of inverse kinematics of a manipulator with redundant degrees of freedom according to claim 3, characterized in that: The method of establishing the inverse kinematics model of the robot arm based on Jacobi pseudo-inverse based on the forward kinematics model in step 3 includes: 3.1 Randomly select a set of angle data q0=(θ1, θ2, …, θ n ) is used as the initial posture. Based on the initial posture, the forward kinematics model established in step 2 is used to calculate the pose matrix of the end effector of the robot arm. Recorded as X, and calculate the difference between the current pose of the end effector of the robot arm and the expected pose, recorded as dX; 3.

2. Establish inverse kinematics of the robot based on Jacobi pseudo-inverse a. Based on the Jacobi equation of the joint space and the end effector workspace, establish the relationship dX between the difference between the current end pose and the desired pose and the joint angle increment; dX=J(θ)dθ Where dX represents the difference between the current end position and the desired position, J(θ) represents the Jacobian matrix of the robot joint motion, and dθ represents the joint angle increment; b. Use the relationship obtained in step a to calculate the joint angle increment, and use the iteration method to establish the inverse kinematics model dθ based on the joint angle increment: dθ=J + dX Among them, J + represents the pseudo-inverse equation of the Jacobi equation, which is expressed as: + =J T (JJ T ) -1 ; J represents the Jacobian matrix, J T Represents the transposed matrix of the Jacobian matrix.

5. The method for solving the minimum norm of inverse kinematics of a manipulator with redundant degrees of freedom according to claim 4, characterized in that: Step 4: Combining the damped least squares method with the gradient projection method to solve and optimize the inverse kinematics model obtained in step 3 to obtain the minimum norm of the inverse kinematics of the manipulator, including: 4.

1. For the pseudo-inverse equation J + Perform singular value decomposition to obtain the minimum singular value: J + =U T D + V Among them, U T and V are orthogonal matrices obtained after singular value decomposition; D + The pseudo-inverse of the Jacobi matrix J + A diagonal matrix composed of singular values ​​of , and satisfying Among them, σ m+1-i Refers to the minimum singular value obtained by decomposition, which is used to indicate the singularity of the robot configuration during the current motion process; 4.

2. Based on the minimum singular value, construct the Jacobi pseudo-inverse equation J based on the damped least squares method # : J # =(J T J+λ 2 I) -1 J T Where I represents the identity matrix; J represents the Jacobian matrix; J T represents the transposed matrix of the Jacobian matrix; λ represents the damping coefficient and λ ≥ 0. By introducing a variable damping coefficient for adaptive adjustment, the end tracking error of the robot arm can be reduced; ε represents the minimum error range, which is used to determine whether the robot enters a singular state, σ m represents the minimum singular value of the Jacobian matrix, which is given by the pseudo-inverse J + Derived by singular value decomposition; λ m Indicates the preset maximum damping value; 4.

3. In the Jacobi pseudo-inverse equation J # The gradient projection method is introduced in order to avoid the situation where the joint rotation limit is exceeded during the solution process. In the gradient projection method, the index function related to the joint space position θ is set as: Where H(θ) represents θ i is the current rotation angle of the i-th joint of the robot link, θ i represents the rotation angle of the i-th joint, θ imax ,θ imin is the upper and lower rotation limits of the i-th joint, a i represents the median of the upper and lower rotation angle limits of the joint, It means to find the partial derivative of H(θ); 4.

4. According to step 4.2, the Jacobi pseudo-inverse equation J is obtained # The inverse kinematics model of the manipulator is optimized based on Jacobi pseudo-inverse using the gradient projection method in step 4.3: In the formula, Represents the partial derivative matrix of the indicator function H(θ), I n represents the n-order unit matrix, K is the optimization coefficient; 4.

5. Iteratively solve the inverse kinematics model of the manipulator based on Jacobi pseudo-inverse obtained in step 4.4 to obtain the joint angle matrix θ i That is, the minimum norm of the inverse kinematics of the robot.

6. A method for solving the minimum norm of inverse kinematics of a manipulator with redundant degrees of freedom according to any one of claims 1 to 5, characterized in that: The solution step also includes the step of verifying the minimum norm of the inverse kinematics of the robot arm, specifically: The minimum norm joint rotation matrix θ is solved i Substitute it into the forward kinematics model obtained in step 2 to verify whether the error between the end effector pose matrix obtained by solving the forward kinematics model and the expected position is within the allowable range. If so, it means that the inverse kinematics model is established correctly.

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