Mechanical arm motion optimization method for undemanded inverse gradient calculation

Abstract

The invention discloses a mechanical arm motion optimization method for undemanded inverse gradient solving, and relates to the field of mechanical arm motion optimization, comprising the following steps: constructing a mechanical arm motion optimization problem needing to be solved; a mathematical model corresponding to the mechanical arm motion optimization problem needing to be solved is established, and the quadratic programming problem of the mathematical model is converted into a dynamic linear matrix equation; constructing a neurodynamics improved model based on the linear matrix equation; and obtaining parameters for motion optimization of the mechanical arm by solving the neurodynamics improved model so as to complete motion optimization of the mechanical arm. The problems that an existing gradient neurodynamics algorithm cannot effectively solve the mechanical arm motion optimization problem in real time, and an existing zero neurodynamics algorithm is complex in calculation and possibly has no analytical solution when solving the mechanical arm motion optimization problem are solved.

Description

Technical Field

[0001] The present invention relates to the field of robotic arm motion optimization, and particularly to a robotic arm motion optimization method without the need to inverse and calculate gradients. Background Art

[0002] In the fields of automation and intelligent manufacturing, the motion control and optimization of robotic arms are the keys to achieving precise operations. Although traditional control methods such as PID control and model predictive control have been widely applied, they have limitations in terms of response speed, robustness, and adaptability to complex environments. In recent years, neural dynamics technology has provided new solutions for robotic arm control. However, the existing traditional gradient neural dynamics algorithms cannot be solved effectively in real time, and their anti-noise ability is extremely low; the existing nullifying neural dynamics algorithms require an inverse process with a huge amount of calculation, which involves the inverse dynamics solution of a high-dimensional nonlinear system, and the calculation is complex and may have no analytical solution. Therefore, it is of great significance to study a robotic arm motion optimization method that can achieve zero-error control and does not require inverse dynamics solution to improve control efficiency and system robustness. Summary of the Invention

[0003] Aiming at the above deficiencies in the prior art, the robotic arm motion optimization method without the need to inverse and calculate gradients provided by the present invention solves the problems that the existing gradient neural dynamics algorithms cannot effectively solve the robotic arm motion optimization problem in real time, and the existing nullifying neural dynamics algorithms are complex in calculation and may have no analytical solution when solving the robotic arm motion optimization problem.

[0004] In order to achieve the above invention objective, the technical solution adopted by the present invention is as follows:

[0005] Provide a robotic arm motion optimization method without the need to inverse and calculate gradients, which includes the following steps:

[0006] Construct a robotic arm motion optimization problem to be solved;

[0007] Establish a mathematical model corresponding to the robotic arm motion optimization problem to be solved, and transform the quadratic programming problem of the mathematical model into a dynamic linear matrix equation;

[0008] Construct a neural dynamics improved model based on the linear matrix equation;

[0009] Obtain the parameters for robotic arm motion optimization by solving the neural dynamics improved model, and then complete the robotic arm motion optimization.

[0010] Further, the robotic arm motion optimization problem to be solved is a robotic arm control problem based on the minimum angular velocity scheme of the robotic arm, and its expression is:

[0011] ;

[0012] where min represents taking the minimum value; s.t. represents the constraint condition; represents the velocity of the robotic arm joint angle movement; the superscript T represents the transpose of the matrix; is the first derivative of represents the velocity of the end movement of the robotic arm, , represents the Jacobian matrix, t represents time, represents the position of the end of the robotic arm in the three-dimensional Cartesian space; and respectively represent the upper and lower limits of the angular velocity of the robotic arm; represents the angle between the joints of the robotic arm.

[0013] Furthermore, the expression of the mathematical model corresponding to the robotic arm motion optimization problem to be solved is:

[0014] ;

[0015] where is the identity matrix; and are Lagrange multipliers; , is the Hadamard product, is a positive number, .

[0016] Furthermore, the expression of the linear matrix equation is:

[0017] where = , which is the matrix to be solved in the mathematical model; = , = .

[0018] Furthermore, the expression of the improved neural dynamics model is:

[0019] ;

[0020] where represents the derivative of is the convergence coefficient; is the parameter for adjusting the change of the coefficient, ; represents the two-norm.

[0021] Furthermore, the specific method for solving the improved neural dynamics model is:

[0022] Set the convergence coefficient 1. Parameters for the change of the adjustment coefficient , substituting the initial angle of the robotic arm, the angular velocity limit of the robotic arm, and the solution time, to obtain the parameters for the motion optimization of the robotic arm.

[0023] Furthermore, the convergence coefficient , parameters for the change of the adjustment coefficient .

[0024] Furthermore, the initial angle of the robotic arm , where π is 180°.

[0025] Furthermore, the lower and upper limits of the angular velocity of the robotic arm are -0.4 rad / s and +0.4 rad / s respectively.

[0026] Furthermore, the solution time is 10 seconds.

[0027] The beneficial effects of the present invention are as follows:

[0028] 1. The present invention avoids the inverse dynamics solution of the high-dimensional non-linear system required by the traditional method for solving the motion optimization problem of the robotic arm, thereby reducing the computational amount and improving the optimization real-time performance. This method simplifies the calculation process by directly learning the dynamic characteristics of the robotic arm, making the motion control of the robotic arm faster and more efficient, and is particularly suitable for application scenarios that require rapid response.

[0029] 2. The present invention improves the computational ability under noise interference through the neural dynamics improved model, enhancing the robustness of the system. This means that even under changing environmental conditions or in the presence of measurement noise, the robotic arm can maintain precise motion control, thereby improving the reliability and stability of the operation.

[0030] 3. This method can achieve zero-error control of the robotic arm motion planning, which is achieved by transforming the quadratic programming problem into a dynamic linear matrix equation and using the neural dynamics improved model for solution. This method not only avoids the processes of inversion and gradient calculation, but also further simplifies the calculation process through the design of the adaptive coefficient, improving the control accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 is a schematic flow diagram of this method;

[0032] Figure 2 is the trajectory of the end effector of the robotic arm calculated by the designed neural dynamics improved model and the expected trajectory diagram on the two-dimensional plane;

[0033] Figure 3 is the entire robotic arm motion trajectory diagram calculated by the designed neural dynamics improved model in the three-dimensional space;

[0034] Figure 4 is the error of the manipulator motion trajectory calculated by the designed improved neurodynamics model in three dimensions;

[0035] Figure 5 is the change diagram of the angular velocity of each joint of the manipulator;

[0036] Figure 6 is the solution obtained by calculating the time-varying quadratic programming problem using the designed improved neurodynamics model under four random initial values ;

[0037] Figure 7 is the solution obtained by calculating the time-varying quadratic programming problem using the designed improved neurodynamics model under four random initial values ;

[0038] Figure 8 is the error norm diagram obtained by calculating the time-varying quadratic programming problem using the designed improved neurodynamics model under four random initial values;

[0039] Figure 9 is the error norm diagram obtained by calculating the time-varying quadratic programming problem under noise perturbation using the designed improved neurodynamics model under four random initial values. Specific implementation manners

[0040] The specific implementation manners of the present invention will be described below to facilitate those skilled in the art to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation manners. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions and creations using the concept of the present invention are within the scope of protection.

[0041] Example 1:

[0042] As Figure 1 shown, the manipulator motion optimization method without inverse demand for gradient includes the following steps:

[0043] S1. Construct the manipulator motion optimization problem to be solved;

[0044] S2. Establish a mathematical model corresponding to the manipulator motion optimization problem to be solved, and transform the quadratic programming problem of the mathematical model into a dynamic linear matrix equation;

[0045] S3. Construct an improved neurodynamics model based on the linear matrix equation;

[0046] S4. Obtain the parameters for optimizing the manipulator motion by solving the improved neural dynamics model, and then complete the optimization of the manipulator motion.

[0047] In this embodiment, the manipulator control problem can be expressed as:

[0048] ;

[0049] where represents the position of the end effector of the manipulator in the three-dimensional Cartesian space; represents the non-linear mapping relationship between the position of the end effector of the manipulator and the joint angles of the manipulator; represents the angles between the joints of the manipulator.

[0050] Due to the complexity of the non-linear mapping relationship, in this embodiment, the above equation is differentiated, and is represented by the Jacobian matrix as:

[0051] ;

[0052] where represents the velocity of the end effector of the manipulator; represents the velocity of the joint angle motion of the manipulator.

[0053] Based on this, when the manipulator motion optimization problem to be solved is a manipulator control problem based on the minimum angular velocity scheme of the manipulator, its expression is:

[0054] ;

[0055] where min represents taking the minimum value; s.t. represents the constraint condition; the superscript T represents the transpose of the matrix; is the first derivative of, t represents time; and represent the upper and lower limits of the angular velocity of the manipulator respectively.

[0056] To determine the optimal solution in the quadratic programming, where the problem satisfies the Karush-Kuhn-Tucker conditions, then two Lagrange multipliers and are introduced, and the expression of the mathematical model corresponding to the manipulator motion optimization problem to be solved is:

[0057] ;

[0058] where is the identity matrix; , is the Hadamard product, is a positive number, .

[0059] The above system of equations can be simplified to a dynamic linear matrix equation:

[0060] ;

[0061] where = , is the matrix to be solved in the mathematical model; = , = .

[0062] In this embodiment, the process of constructing the improved neural dynamics model in step S3 is as follows:

[0063] S3-1. For the convenience of calculation, an error equation corresponding to the dynamic linear matrix equation is given:

[0064] ;

[0065] where represents the error equation;

[0066] S3-2. Based on the requirements of the gradient algorithm, a loss function of non-negative scalar is constructed for the above equation:

[0067] ;

[0068] where represents the loss function of non-negative scalar; represents the two-norm;

[0069] S3-3. Minimizing along the negative gradient direction of the error function, the gradient neural dynamics model can be obtained as:

[0070] ;

[0071] where is the coefficient for controlling the model convergence, i.e., the convergence coefficient;

[0072] S3-4. Substituting the linear matrix equation into the gradient neural dynamics model and expanding, we can get:

[0073] ;

[0074] S3-5. By designing an adaptive coefficient regarding the error and the correlation matrix, the original gradient neural dynamics model can not only avoid the process of inversion but also enhance the real-time performance. The expression of the obtained improved neural dynamics model is:

[0075] ;

[0076] where denotes the derivative of; is a parameter for adjusting the coefficient change, , and the overall is an adaptive coefficient.

[0077] The specific method for solving the improved neural dynamics model is as follows:

[0078] Set the convergence coefficient and the parameter for adjusting the coefficient change , substitute the initial angle of the robotic arm, the angular velocity limit of the robotic arm, and the solution time, and obtain the parameters for optimizing the motion of the robotic arm.

[0079] Example 2:

[0080] This example is a further expansion based on Example 1. In this Example 2, the convergence coefficient and the parameter for adjusting the coefficient change . The initial angle of the robotic arm , where π is 180°. The lower and upper limits of the angular velocity of the robotic arm are -0.4 rad / s and +0.4 rad / s respectively. The solution time is 10 seconds, which is used to observe the solution situation and can be set to any number. The purpose is to understand the completion degree of the task.

[0081] Figures 2 to 5 All are the results of this Example 2. Figure 2 represents the actual motion trajectory (blue circular dotted line) and the expected trajectory (red solid line) of the end of the robotic arm. In the two-dimensional plane, the calculated trajectory basically coincides with the expected trajectory. Figure 3 represents the posture of the entire robotic arm in three-dimensional space, and it can be seen that the robotic arm is very smooth and stable when performing tasks. Figure 4 represents the error of the robotic arm in each dimension of the three-dimensional space when performing tasks. It can be clearly seen that the error can be controlled within 10 -4 meters. Figure 5 represents the change speed (angular velocity) of the angle between the joints of the robotic arm. It can clearly show that the angular velocity change of each joint is within the angular velocity limit we designed, and the angular velocity is well constrained.

[0082] Example 3:

[0083] This example is a further expansion based on Example 1. In this Example 3, the quadratic programming problem of the mathematical model is a time-varying quadratic programming problem, and its expression is:

[0084] ;

[0085] Among them and are the unknown functions we need to solve. The parameters are designed as , , and the time is set to 1.5 seconds.

[0086] Figure 6 and Figure 7 show that the solution (blue dashed line) obtained by the designed improved neurodynamics model in solving the above time-varying quadratic programming problem, and the solution range is the red solid line. It can be clearly seen that the calculated solutions all converge within 0.5 seconds, and the solution ranges all meet the preset conditions. Figure 8 What is shown is the error norm graph obtained by the designed improved neurodynamics model in solving the above time-varying quadratic programming problem. It can be seen that even with four different initial values, the error norm can finally converge to 10 -6 . Figure 9 What is shown is the error norm graph obtained by the designed improved neurodynamics model in solving the above time-varying quadratic programming problem with noise perturbation. It can be seen that even in the case of noise perturbation, the designed improved neurodynamics model can still maintain high precision and high convergence speed when solving.

[0087] In summary, the present invention avoids the inverse dynamics solution of the high-dimensional non-linear system required by the traditional method for solving the manipulator motion optimization problem, thereby reducing the computational amount and improving the optimization real-time performance. This method simplifies the calculation process by directly learning the dynamic characteristics of the manipulator, making the motion control of the manipulator faster and more efficient, and is particularly suitable for application scenarios that require fast response.

Claims

1. A manipulator motion optimization method without the need to invert and obtain gradients, characterized in that, Including the following steps: Construct the robotic arm motion optimization problem to be solved; Establish a mathematical model corresponding to the robotic arm motion optimization problem to be solved, and transform the quadratic programming problem of the mathematical model into a dynamic linear matrix equation; Construct an improved neurodynamics model based on the linear matrix equation; Complete the acquisition of the parameters for the robotic arm motion optimization by solving the improved neurodynamics model, and thus complete the robotic arm motion optimization.

2. A robotic arm motion optimization method without demand inverse gradient according to claim 1, characterized in that The robotic arm motion optimization problem to be solved is a robotic arm control problem based on the robotic arm minimizing the angular velocity scheme, and its expression is: ； where min represents taking the minimum value; s.t. represents the constraint condition; represents the velocity of the robotic arm joint angle movement; the superscript T represents the transpose of the matrix; is the first derivative of, represents the velocity of the robotic arm end movement, , represents the Jacobian matrix, t represents time, represents the position of the robotic arm end in the three-dimensional Cartesian space; and respectively represent the upper and lower limits of the robotic arm angular velocity; represents the angle between the robotic arm joints.

3. The robotic arm motion optimization method without inverse gradient calculation according to claim 2, characterized in that, The expression of the mathematical model corresponding to the robotic arm motion optimization problem to be solved is: ； wherein is an identity matrix; and are Lagrange multipliers; , is a Hadamard product, is a positive number, .

4. A manipulator motion optimization method without demand inverse gradient according to claim 3, characterized in that The expression of the linear matrix equation is: ； where = , is the matrix to be solved in the mathematical model; = , = .

5. The robotic arm motion optimization method without demand inverse gradient according to claim 4, characterized in that, The expression of the improved neurodynamics model is: ； wherein represents the derivative of; is the convergence coefficient; is a parameter for adjusting the change of the coefficient, ; represents the two - norm.

6. The robotic arm motion optimization method without inverse gradient requirement according to claim 5, characterized in that The specific method for solving the improved neurodynamics model is: Set the convergence coefficient and the parameter for adjusting the coefficient change , substitute the initial angle of the robotic arm, the angular velocity limit of the robotic arm, and the solution time, and obtain the parameters for optimizing the motion of the robotic arm.

7. A manipulator motion optimization method without inverse demand for gradient according to claim 6, characterized in that Convergence coefficient , a parameter for adjusting the change of the coefficient .

8. A manipulator motion optimization method without demand for inverse gradient according to claim 6, characterized in that, Initial included angle of the robotic arm , where π is 180°.

9. A manipulator motion optimization method without requiring inverse gradient calculation according to claim 6, characterized in that The lower and upper limits of the angular velocity of the robotic arm are -0.4 rad / s and +0.4 rad / s respectively.

10. A manipulator motion optimization method without demand inverse gradient according to claim 6, characterized in that The solution time is 10 seconds.

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