A mechanical arm trajectory tracking control method of optimizing fuzzy sliding mode control

Abstract

A kind of mechanical arm trajectory tracking control method of optimizing fuzzy sliding mode control belongs to industrial robot control technical field, solve the technical problem that traditional mechanical arm trajectory control method system output produces chattering phenomenon, including the following steps:1, obtain the desired trajectory information of mechanical arm;2, establish n degree of freedom mechanical arm dynamics model;3, design sliding surface s and exponential approach law 4, design three coefficients as the gain between input and output of fuzzy controller and sliding mode controller, and use the solution three coefficients using reptile algorithm to optimize membership function;5, design the input of fuzzy controller is sliding surface and the derivative of sliding surface, the switching gain of exponential approach law is output, and the corresponding membership function and fuzzy logic are designed;6, obtain the controller based on reptile algorithm optimization fuzzy control, realize trajectory tracking control.The application effectively realizes mechanical arm trajectory tracking control, has faster response speed, reduces the chattering phenomenon of control output.

Description

Technical Field

[0001] This invention belongs to the field of industrial robot control technology, specifically relating to a robotic arm trajectory tracking control method based on reptile algorithm-optimized fuzzy sliding mode control. Background Technology

[0002] Due to their high flexibility and environmental adaptability, serial robotic arms are widely used in manufacturing, medical, and aerospace fields, playing an increasingly important role. With the expansion of applications, the increasing complexity of mechanical structures, and the rising performance requirements of tasks, higher demands are being placed on the trajectory control of robotic arms.

[0003] Robotic arms are susceptible to various disturbances and model uncertainties during operation, which significantly impact their motion accuracy. Currently, many control methods suffer from unstable control output, system chattering, and slow convergence of system errors. Summary of the Invention

[0004] The main objective of this invention is to overcome the shortcomings of the prior art and solve the technical problem that the sliding mode control method in traditional robotic arm trajectory control methods causes the system output to jitter when approaching the sliding surface. This invention provides a robotic arm trajectory tracking control method based on reptile algorithm-optimized fuzzy sliding mode control.

[0005] This invention is achieved through the following technical solution:

[0006] A method for optimizing fuzzy sliding mode control to track the trajectory of a robotic arm includes the following steps:

[0007] S1. Obtain the desired joint angle q of each joint of the robotic arm. d and expected joint angular velocity This information will be used later.

[0008] S2. Establish the initial dynamic model of the n-degree-of-freedom robotic arm. Take the uncertain parts and other uncertain factors in the initial dynamic model as disturbance terms f, and perform formal transformation on the initial dynamic model to obtain the final dynamic model of the robotic arm.

[0009] S3. Input the desired joint angles and desired joint angular velocities of each joint of the robotic arm obtained in step S1 into the robotic arm dynamics model determined in step S2, calculate the angular position tracking error and angular velocity tracking error, and then design the sliding surface s and select the exponential control law. Perform trajectory tracking control on the robotic arm;

[0010] S4. First, design three coefficients k1, k2, and k3 as the input and output gains between the fuzzy controller and the sliding mode controller; then, use the reptile algorithm to solve for the values ​​of k1, k2, and k3, and thus optimize the membership function; finally, use the dynamic performance response index ITAE as the fitness function of the reptile algorithm, expressed as follows:

[0011]

[0012] In equation (8), t represents the output response time, and e(t) represents the deviation of the actual output from the expected output;

[0013] S5. Using the sliding surface s and exponential control rate designed in step S3. In addition to the three coefficients k1, k2, and k3 designed in step S4, the inputs of the fuzzy controller are designed as the sliding surface k1·s and the derivative of the sliding surface. The output is the switching gain k3·ε with exponential reaching law, and the corresponding membership function and fuzzy logic are designed.

[0014] S6. Determine a controller based on the reptile algorithm to optimize fuzzy control and realize trajectory tracking control.

[0015] Further, step S2 includes the following sub-steps:

[0016] S2-1. Establish the initial dynamic model of the n-degree-of-freedom robotic arm:

[0017]

[0018] In equation (1), q represents the joint angle position. Indicates joint angular velocity, M(q) represents the joint angular acceleration, and M(q) represents the inertia matrix. Let G(q) represent the centrifugal force and Coriolis force matrices, G(q) represent the gravity term, and τ represent the controller input.

[0019] S2-2. The initial dynamic model parameters include nominal and uncertain parts, as shown below:

[0020]

[0021] In equation (2), M0(q), And G0(q) is the nominal part, △M(q), And △G(q) represents the uncertain part;

[0022] S2-3. Taking the modeling error, parameter changes, and uncertainties as the interference term f, rewrite formula (1) in step S2-1 as follows:

[0023]

[0024] S2-4. Rewrite the initial dynamics model into the final robotic arm dynamics model based on step S2-3:

[0025]

[0026] Further, step S3 includes the following sub-steps:

[0027] S3-1, The sliding surface s is designed as follows:

[0028]

[0029] In equation (5), e represents the joint angle position tracking error, e = q d -q; This indicates the joint angular velocity tracking error. q represents the joint angle position, q d Λ represents the desired joint angle;

[0030] S3-2, Differentiating the sliding surface s in step S3-1 yields... As shown below:

[0031]

[0032] In equation (6), This indicates the joint angular acceleration tracking error. This indicates the joint angular velocity tracking error;

[0033] S3-3, Design Exponential Approach Law: The exponential approach law is numerically equal to the derivative of the sliding surface s. The exponential law of convergence is shown below:

[0034]

[0035] In equation (7), ε represents the switching gain of the reaching law, ε>0; k represents the positive definite diagonal matrix, k>0; and s represents the sliding surface.

[0036] Further, step S5 includes the following sub-steps:

[0037] S5-1, Design the fuzzy set as follows:

[0038] {NB,NM,NS,ZO,PS,PM,PB};(9)

[0039] In equation (9), NB is negative large, NM is negative medium, NS is negative small, ZO is zero, PS is positive small, PM is positive medium, and PB is positive large;

[0040] S5-2, Design the membership functions for input and output:

[0041] Based on the fuzzy set designed in step S5-1, the input and output membership functions based on Gaussian functions are constructed using the Fuzzy toolbox in MATLAB.

[0042] Further, step S6 includes the following sub-steps:

[0043] S6-1. Design the Lyapunov function V as follows:

[0044]

[0045] In equation (10), s is the sliding surface;

[0046] S6-2. By stabilizing the Lyapunov function V, substitute equations (4), (6), and (7) into equation (3) to design the controller as shown below:

[0047]

[0048] The beneficial effects of this invention are as follows:

[0049] This invention utilizes a sliding mode controller based on a reptile algorithm to adjust the controller's parameters in real time according to the system's state. This results in a faster convergence speed throughout the entire convergence process on the sliding surface, improving system robustness. Furthermore, the sliding mode controller iteratively optimizes the membership function of the fuzzy controller using the reptile algorithm, and further optimizes the sliding mode controller's parameters through the fuzzy controller, achieving even faster convergence speed and effectively reducing output chattering, thus providing more accurate tracking performance.

[0050] In summary, this invention, based on reptile algorithms, fuzzy control, and sliding mode control, can effectively achieve trajectory tracking control of a robotic arm. Attached Figure Description

[0051] Figure 1 This is a flowchart of the present invention;

[0052] Figure 2 This is a schematic diagram of the motion principle of a two-degree-of-freedom robotic arm; where m1 is joint 1 and m2 is joint 2.

[0053] Figure 3 Membership function graph for fuzzy control input is sliding surface k1·s;

[0054] Figure 4 The input for fuzzy control is the derivative of the sliding surface. Membership function graph at time;

[0055] Figure 5Membership function graph for fuzzy control output ε;

[0056] Figure 6 This is a trajectory tracking curve for the position of joint 1 angle;

[0057] Figure 7 This is a trajectory tracking curve of the joint's two corner positions;

[0058] Figure 8 The angular velocity trajectory tracking curve for joint 1;

[0059] Figure 9 The graph shows the angular velocity trajectory tracking curve for joint 2.

[0060] Figure 10 The diagram shows the change in control torque of joint 1;

[0061] Figure 11 The diagram shows the change in control torque of joint 2. Detailed Implementation

[0062] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0063] like Figure 2 As shown in the figure, the motion trajectory of the two-degree-of-freedom robotic arm is tracked and controlled in this specific embodiment.

[0064] like Figure 1 The method for optimizing fuzzy sliding mode control for tracking the trajectory of a robotic arm, as shown, includes the following steps:

[0065] S1. Obtain the desired joint angle q of each joint of the robotic arm. d and expected joint angular velocity Information to be used later; in this specific embodiment, the desired joint angle q of joints 1 and 2. 1d q 2d and expected joint angular velocity Specifically:

[0066]

[0067]

[0068] S2. Establish an initial dynamic model for the two-degree-of-freedom robotic arm. Treat the uncertainties and other uncertainties in the initial dynamic model as disturbance terms f, and perform a formal transformation on the initial dynamic model to obtain the final robotic arm dynamic model. This includes the following sub-steps:

[0069] S2-1. Establish the initial dynamic model of the two-degree-of-freedom robotic arm:

[0070]

[0071] In equation (1), q represents the joint angle position. Indicates joint angular velocity, M(q) represents the joint angular acceleration, and M(q) represents the inertia matrix. Let G(q) represent the centrifugal force and Coriolis force matrices, G(q) represent the gravity term, and τ represent the controller input.

[0072] S2-2. The initial dynamic model parameters include nominal and uncertain parts, as shown below:

[0073]

[0074] In equation (2), M0(q), And G0(q) is the nominal part, △M(q), And △G(q) represents the uncertain part;

[0075] S2-3. Taking the modeling error, parameter changes, and uncertainties as the interference term f, rewrite formula (1) in step S2-1 as follows:

[0076]

[0077] In equation (3), the specific form of each matrix is ​​as follows:

[0078] in:

[0079] ;

[0080] The physical parameters of the robotic arm are as follows:

[0081] Mass: m1 = 0.8 kg, m2 = 1.8 kg;

[0082] Length: l1 = 1.2m, l2 = 1m;

[0083] Acceleration due to gravity: g = 9.8 N / kg;

[0084] S2-4. Rewrite the initial dynamics model into the final robotic arm dynamics model based on step S2-3:

[0085]

[0086] S3. Input the desired joint angles and desired joint angular velocities of each joint of the robotic arm obtained in step S1 into the robotic arm dynamics model determined in step S2, calculate the angular position tracking error and angular velocity tracking error, and then design the sliding surface s and select the exponential control law. Perform trajectory tracking control on the robotic arm; including the following sub-steps:

[0087] S3-1, The sliding surface s is designed as follows:

[0088]

[0089] In equation (5), e represents the joint angle position tracking error, e = q d -q; This indicates the joint angular velocity tracking error. q represents the joint angle position, q d Let Λ represent the desired joint angle, and let Λ represent the diagonal matrix. The diagonal matrix is:

[0090] S3-2, Differentiating the sliding surface s in step S3-1 yields... As shown below:

[0091]

[0092] In equation (6), This indicates the joint angular acceleration tracking error. This indicates the joint angular velocity tracking error;

[0093] S3-3, The design exponent convergence law is shown below:

[0094]

[0095] In equation (7), ε represents the switching gain of the reaching law, ε>0; k represents the positive definite diagonal matrix, which is:

[0096] k>0; s represents the sliding surface.

[0097] S4. First, design three coefficients k1, k2, and k3 as the input and output gains between the fuzzy controller and the sliding mode controller; then, use the reptile algorithm to solve for the values ​​of k1, k2, and k3, and thus optimize the membership function; finally, use the dynamic performance response index ITAE as the fitness function of the reptile algorithm, expressed as follows:

[0098]

[0099] In equation (8), t represents the output response time, and e(t) represents the deviation of the actual output from the expected output;

[0100] The reptile algorithm parameters are set as follows:

[0101] Population size: N = 30;

[0102] Number of iterations: T = 300;

[0103] Variable range: 0–1;

[0104] S5. Using the sliding surface s and exponential control rate designed in step S3. In addition to the three coefficients k1, k2, and k3 designed in step S4, the inputs of the fuzzy controller are designed as the sliding surface k1·s and the derivative of the sliding surface. The output is the switching gain k3·ε with exponential reaching law, and the corresponding membership function and fuzzy logic are designed; including the following sub-steps:

[0105] S5-1, Design the fuzzy set as follows:

[0106] {NB,NM,NS,ZO,PS,PM,PB};(9)

[0107] In equation (9), NB is negative large, NM is negative medium, NS is negative small, ZO is zero, PS is positive small, PM is positive medium, and PB is positive large; the fuzzy rules are shown in Table 1 below.

[0108] Table 1 Fuzzy Rules

[0109]

[0110] S5-2, Design the membership functions for input and output:

[0111] Based on the fuzzy set designed in step S5-1, the input and output membership functions based on Gaussian functions are constructed using the Fuzzy toolbox in MATLAB.

[0112] S6. Determine a controller based on the reptile algorithm to optimize fuzzy control and achieve trajectory tracking control. This includes the following sub-steps:

[0113] S6-1. Design the Lyapunov function V as follows:

[0114]

[0115] In equation (10), s is the sliding surface;

[0116] S6-2. By stabilizing the Lyapunov function V, substitute equations (4), (6), and (7) into equation (3) to design the controller as shown below:

[0117]

[0118] This specific implementation is compared with the conventional sliding mode controller (controller 1) and the fuzzy sliding mode controller (controller 2). Figure 6 This is the control diagram for tracking the position trajectory of joint 1. Figure 7 This is the control diagram for tracking the position trajectory of joint 2. Figure 8 This is the angular velocity tracking control diagram for joint 1. Figure 9 This is the control diagram for tracking the angular velocity trajectory of joint 2. Figure 10 The diagram shows the change in control torque of joint 1. Figure 11 This is a diagram showing the change in control torque at joint 2. (Combined with...) Figures 6-11 This verifies that the present invention has a faster convergence speed, more accurate convergence precision, and effectively reduces the jitter generated by the output, thus realizing trajectory tracking control of the robotic arm.

[0119] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for optimizing fuzzy sliding mode control to track the trajectory of a robotic arm, characterized in that, Includes the following steps: S1. Obtain the desired joint angle q of each joint of the robotic arm. d and expected joint angular velocity This information will be used later. S2. Establish the initial dynamic model of the n-degree-of-freedom robotic arm. Take the uncertain parts and other uncertain factors in the initial dynamic model as disturbance terms f, and perform formal transformation on the initial dynamic model to obtain the final dynamic model of the robotic arm. S3. Input the desired joint angles and desired joint angular velocities of each joint of the robotic arm obtained in step S1 into the robotic arm dynamics model determined in step S2, calculate the angular position tracking error and angular velocity tracking error, and then design the sliding surface s and select the exponential control law. Perform trajectory tracking control on the robotic arm; S4. First, design three coefficients k1, k2 and k3 as the input and output gains between the fuzzy controller and the sliding mode controller; Then, the reptile algorithm is used to solve for the values ​​of k1, k2, and k3, thereby optimizing the membership function; finally, the dynamic performance response index ITAE is used as the fitness function of the reptile algorithm, as shown below: In equation (8), t represents the output response time, and e(t) represents the deviation of the actual output from the expected output; S5. Using the sliding surface s and exponential control rate designed in step S3. In addition to the three coefficients k1, k2, and k3 designed in step S4, the inputs of the fuzzy controller are designed as the sliding surface k1·s and the derivative of the sliding surface. The output is the switching gain k3·ε with exponential reaching law, and the corresponding membership function and fuzzy logic are designed. S6. Determine a controller based on the reptile algorithm to optimize fuzzy control and realize trajectory tracking control.

2. The robotic arm trajectory tracking control method with optimized fuzzy sliding mode control according to claim 1, characterized in that: Step S2 includes the following sub-steps: S2-1. Establish the initial dynamic model of the n-degree-of-freedom robotic arm: In equation (1), q represents the joint angle position. Indicates joint angular velocity, M(q) represents the joint angular acceleration, and M(q) represents the inertia matrix. Let G(q) represent the centrifugal force and Coriolis force matrices, G(q) represent the gravity term, and τ represent the controller input. S2-2. The initial dynamic model parameters include nominal and uncertain parts, as shown below: In equation (2), M0(q), And G0(q) is the nominal part, △M(q), And △G(q) represents the uncertain part; S2-3. Taking the modeling error, parameter changes, and uncertainties as the interference term f, rewrite formula (1) in step S2-1 as follows: S2-4. Rewrite the initial dynamics model into the final robotic arm dynamics model based on step S2-3:

3. The robotic arm trajectory tracking control method with optimized fuzzy sliding mode control according to claim 1, characterized in that: Step S3 includes the following sub-steps: S3-1, The sliding surface s is designed as follows: In equation (5), e represents the joint angle position tracking error, e = q d -q; This indicates the joint angular velocity tracking error. q represents the joint angle position, q d Λ represents the desired joint angle; S3-2, Differentiating the sliding surface s in step S3-1 yields... As shown below: In equation (6), This indicates the joint angular acceleration tracking error. This indicates the joint angular velocity tracking error; S3-3, The design exponent convergence law is shown below: In equation (7), ε represents the switching gain of the reaching law, ε>0; k represents the positive definite diagonal matrix, k>0; and s represents the sliding surface.

4. The robotic arm trajectory tracking control method with optimized fuzzy sliding mode control according to claim 1, characterized in that: Step S5 includes the following sub-steps: S5-1, Design the fuzzy set as follows: {NB,NM,NS,ZO,PS,PM,PB};(9) In equation (9), NB is negative large, NM is negative medium, NS is negative small, ZO is zero, PS is positive small, PM is positive medium, and PB is positive large; S5-2, Design the membership functions for input and output: Based on the fuzzy set designed in step S5-1, the input and output membership functions based on Gaussian functions are constructed using the Fuzzy toolbox in MATLAB.

5. A robotic arm trajectory tracking control method based on optimized fuzzy sliding mode control according to any one of claims 1, 2, or 3, characterized in that: Step S6 includes the following sub-steps: S6-1. Design the Lyapunov function V as follows: In equation (10), s is the sliding surface; S6-2. By stabilizing the Lyapunov function V, substitute equations (4), (6), and (7) into equation (3) to design the controller as shown below:

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