A robotic arm control method with input saturation compensation based on bias neural networks

Abstract

A robotic arm control method based on a bias neural network with input saturation compensation comprises the following steps: 1. Establishing a robotic arm system considering control input saturation; 2. Establishing a fixed-time stable control input saturation compensation system; 3. Establishing the joint tracking error of the robotic arm system; 4. Establishing the fixed-time end-effector sliding surface of the robotic arm system; 5. Establishing the input of the bias neural network, the basis functions of the bias neural network, and the adaptive rate of the bias neural network; 6. Establishing a robotic arm controller based on the bias neural network with input saturation compensation; 7. Inputting the established robotic arm controller based on the bias neural network with input saturation compensation into the robotic arm system considering control input saturation, thereby achieving fast and accurate trajectory tracking of each joint of the robotic arm system. This invention overcomes the influence of the limitation of the joint motor torque output on system stability and tracking performance, ensuring that the robotic arm system reaches the preset trajectory within a fixed time, and providing a higher level of robustness and tracking performance.

Description

Technical Field

[0001] This invention belongs to the field of robotic arm control technology, and particularly relates to a robotic arm control method with input saturation compensation based on a bias neural network. Background Technology

[0002] In today's rapidly developing technological era, robotic arms have become indispensable technical equipment across various industries, bringing revolutionary improvements in productivity through their high degree of freedom and efficiency. From automated manufacturing on production lines to precise operations in operating rooms, and even assisting astronauts in space, robotic arm technology is widely used in fields such as industrial automation, medicine, and aerospace exploration. Therefore, there is an urgent need for a high-precision, robust trajectory tracking control method to meet the ever-growing application demands.

[0003] However, as a highly complex nonlinear multi-input multi-output system, robotic arms are often affected by model parameter uncertainties and external disturbances, leading to a decline in system control performance. Disturbance observers based on the system model and control law can be used to estimate model uncertainties and external disturbances in the robotic arm system, but they are highly dependent on the model during the design process and their expressions are quite complex. Furthermore, due to the limitations of the torque output of the robotic arm joint motors, control inputs can easily exceed the motor's operating range. While treating control inputs exceeding the motor's operating range as part of the external disturbance is relatively convenient in operation, it also means that larger estimates of uncertainties and disturbance upper bounds are needed, which may lead to a decrease in system stability. Summary of the Invention

[0004] To address the problems existing in the prior art, the present invention aims to provide a robotic arm control method with input saturation compensation based on a bias neural network. This method overcomes the impact of the limitation on the torque output of the robotic arm joint motor on system stability and tracking performance, ensuring that the robotic arm system reaches the preset trajectory within a fixed time, and providing a higher level of robustness and tracking performance.

[0005] To achieve the above objectives, the present invention employs the following technical means:

[0006] A robotic arm control method with input saturation compensation based on a bias neural network includes the following steps:

[0007] Step 1: Establish a robotic arm system considering control input saturation, represented as:

[0008]

[0009] Where, q, and Let be the joint angle vector, joint angular velocity vector, and joint angular acceleration vector of the robotic arm system, respectively, and M(q) be the inertia matrix of the robotic arm system. Let be the Coriolis force and centrifugal force matrices of the robotic arm system, g(q) be the gravity vector of the robotic arm system, τ be the robotic arm controller based on a bias neural network with input saturation compensation, and Δu be the difference between the control input before and after saturation. Let τ be the sum of the model uncertainty and external disturbances of the robotic arm system. d J is the load vector of the robotic arm system. T Let be the transpose of the Jacobian matrix of the robotic arm system;

[0010] Step 2: Based on the established robotic arm system considering control input saturation, establish a fixed-time stable control input saturation compensation system, expressed as:

[0011]

[0012] in, For a fixed-time stable control input saturation compensation system, the gain matrix has all diagonal elements as positive numbers, λ1=[λ 11 ,λ 12 ],λ2=[λ 21 ,λ 22 Let M(q) be the state variable of a fixed-time stable control input saturated compensation system, representing the tracking error caused by Δu. -1 Let M(q) be the inverse matrix of the inertia matrix of the robotic arm system. The time derivative of the state variable λ1 of a fixed-time stable control input saturated compensation system. Let λ² be the time derivative of the state variable λ² of a fixed-time stable control input saturated compensation system. Let m, n, p, and q be the fractional power parameters of the fixed-time stable control input saturated compensation system, satisfying m > n and q > p. The sig function satisfies... κ>0, x is any n-dimensional vector, x1,…,x n Let x be the components of vector x, and sign be the sign function;

[0013] Step 3: Based on the established fixed-time stable control input saturation compensation system, establish the joint tracking error of the robotic arm system, expressed as:

[0014]

[0015] Where e1 and e2 are the joint angle tracking error and joint angular velocity tracking error of the robotic arm system, respectively, q d and These are the predetermined trajectories for the joint angles and joint angular velocities of the robotic arm system, respectively.

[0016] Step 4: Based on the joint tracking error of the established robotic arm system, establish the fixed-time terminal sliding surface of the robotic arm system, represented as:

[0017] s = e² + α·sig ε (e1)+β·sig μ (e1)

[0018] Where s is the fixed-time terminal sliding surface of the robotic arm system, α and β are the gain coefficient matrices of the fixed-time terminal sliding surface, both of which are diagonal positive definite matrices, and ε,μ are the fractional power parameters of the fixed-time terminal sliding surface, satisfying 0<μ<1, ε>1, sig ε (e1) and sig μ (e1) represents the sliding component of the fixed-time terminal sliding surface;

[0019] Step 5: Based on the joint tracking error of the established robotic arm system and the fixed-time terminal sliding surface of the established robotic arm system, establish the bias neural network input, bias neural network basis functions, and the adaptive rate of the bias neural network.

[0020] The input to the biased neural network is represented as:

[0021]

[0022] Where z is the input to the biased neural network, [e1, e2] T This is the transpose of the joint angle tracking error and the joint angular velocity tracking error of the robotic arm system.

[0023] The basis functions of a biased neural network are expressed as follows:

[0024]

[0025] Where the number of bias neural network basis function layers is m+1, σ is the width of the bias neural network basis function, and μ j b is the center of the j-th biased neural network basis function network. g Let φ be a global bias scalar, n be the number of joints in the robotic arm system, and φ be a variable. i (z) is the i-th bias neural network basis function of the robotic arm system, [S1(z),…,S m+1 [(z)] represents the components of the i-th biased neural network basis function of the robotic arm system;

[0026] The adaptive rate of a biased neural network is expressed as:

[0027]

[0028] in, Let j be the weight of the bias neural network at the i-th joint of the robotic arm system. Let be the adaptive rate of the j-th weight of the bias neural network at the i-th joint of the robotic arm system. Let be the norm of the bias neural network weights at the i-th joint of the robotic arm system, and let Γ be the symmetric positive definite matrix representing the learning rate of the bias neural network. i Let μ be the component of the fixed-time terminal sliding surface of the robotic arm system at the i-th joint. i The adjustment function for the biased neural network is expressed as:

[0029]

[0030] in, Let be the upper bound of the norm of the bias neural network weights at the i-th joint of the robotic arm system. and d i For the robustness parameters of the biased neural network, 0 < d i <1, This allows the biased neural network to more quickly approximate the ideal biased neural network weights, while when the biased neural network weights are greater than... hour, d i It can slow down the rate of change in the biased neural network and improve its robustness.

[0031] Step 6: Based on the established fixed-time terminal sliding surface of the robotic arm system, the input of the bias neural network, the basis functions of the bias neural network, and the adaptive rate of the bias neural network, establish a robotic arm controller with input saturation compensation based on the bias neural network, represented as:

[0032]

[0033] in, To bias the neural network weights, Adaptive rate of bias neural network renew, Let φ(z) be the transpose of the bias neural network weights, φ(z) be the basis functions of the bias neural network for the robotic arm system, and K2 and K3 be the gain matrices of the robotic arm controller with input saturation compensation based on the bias neural network. For the fractional power parameters of a robotic arm controller with input saturation compensation based on a bias neural network, satisfying... And sign(s) are components of a bias neural network-based robotic arm controller with input saturation compensation;

[0034] Step 7: Input the established bias neural network-based robotic arm controller with input saturation compensation into the robotic arm system considering control input saturation. This enables fast and accurate trajectory tracking of each joint in the robotic arm system. The upper bound of the convergence time of the joint tracking error of the robotic arm system is expressed as:

[0035]

[0036] Among them, T max Let ξ1, ξ2, v be the upper bound of the convergence time of the joint tracking error of the robotic arm system, and let ξ1, ξ2, v be constant parameters of the upper bound of the convergence time of the joint tracking error of the robotic arm system.

[0037] Compared with the prior art, the present invention has the following advantages:

[0038] 1. This invention, by designing a fixed-time stable control input saturation compensation system, satisfies the requirement for faster trajectory tracking of robotic arm joints under the condition of limited motor torque output;

[0039] 2. By establishing a fixed-time terminal sliding surface for the robotic arm system, this invention enables the joint tracking error of the robotic arm system to converge at a faster rate throughout the entire process.

[0040] 3. This invention improves the tracking speed and robustness of the robotic arm system in the face of unknown system uncertainties and disturbance boundaries by designing a bias neural network to estimate the system. Attached Figure Description

[0041] Figure 1 A flowchart illustrating a robotic arm control method with input saturation compensation based on a bias neural network, provided in an embodiment of the present invention;

[0042] Figure 2 A schematic diagram of the state variable variation curves of a fixed-time stable control input saturation compensation system provided in an embodiment of the present invention;

[0043] Figure 3 This is a schematic diagram of the angle tracking error variation curve of the two-joint robotic arm system provided in an embodiment of the present invention;

[0044] Figure 4 This is a schematic diagram of the angular velocity tracking error variation curve of a two-joint robotic arm system provided in an embodiment of the present invention;

[0045] Figure 5 This is a schematic diagram of the variation curve of a robotic arm controller with input saturation compensation based on a bias neural network, provided for an embodiment of the present invention. Detailed Implementation

[0046] To enable those skilled in the art of robotic arm control to better understand the method of this invention, the invention will be described in more detail below through embodiments and accompanying drawings. It should be emphasized that the embodiments of this invention are based on the technical solutions of this invention and are only used to illustrate the invention, not to limit its scope. The implementation of this invention is not limited thereto.

[0047] A robotic arm control method with input saturation compensation based on a bias neural network is invented. Figure 1 This is a flowchart illustrating a robotic arm control method with input saturation compensation based on a bias neural network, provided in an embodiment of the present invention. The embodiment uses a two-joint robotic arm for trajectory tracking control. It should be noted that although this embodiment uses a two-joint robotic arm as an example, the trajectory tracking control method proposed in this invention is applicable to robotic arm trajectory control technology with any number of joints, and these applications are all within the scope of protection of this invention.

[0048] A robotic arm control method with input saturation compensation based on a bias neural network includes the following steps:

[0049] Step 1: Establish a robotic arm system considering control input saturation, represented as:

[0050]

[0051] Where, q, and Let be the joint angle vector, joint angular velocity vector, and joint angular acceleration vector of the robotic arm system, respectively, and M(q) be the inertia matrix of the robotic arm system. Let be the Coriolis force and centrifugal force matrices of the robotic arm system, g(q) be the gravity vector of the robotic arm system, τ be the robotic arm controller based on a bias neural network with input saturation compensation, τ is limited between [-100, 100], Δu be the difference before and after control input saturation, and be the sum of the model uncertainty of the robotic arm system and external disturbances. The load vector τ of the robotic arm system d =[10; 20], J T Let be the transpose of the Jacobian matrix of the robotic arm system. The individual matrices are represented as follows:

[0052]

[0053]

[0054]

[0055]

[0056] In this system, the joint angle vector and joint angular velocity vector of the two-joint robotic arm are q = [q1, q2]. The acceleration due to gravity is g = 9.8 m / s². 2 p1, p2, p3, p4, and p5 are all functions of the link lengths l1 and l2, link masses m1 and m2, and link moments of inertia J1 and J2 of the two-joint robotic arm system. Let p1 = 2, p2 = 0.448, p3 = 0.56, p4 = 2.56, and p5 = 0.56. The length of the first link in the two-joint robotic arm system is l1 = 1m, the length of the second link is l2 = 0.8m, the mass of the first link is m1 = 2kg, the mass of the second link is m2 = 1.5kg, and the moment of inertia of the first link is J1 = 0.25kg·m. 2 The moment of inertia of the second connecting rod is J2 = 0.112 kg·m. 2 ;

[0057] Step 2: Based on the established robotic arm system considering control input saturation, establish a fixed-time stable control input saturation compensation system, expressed as:

[0058]

[0059] in, Let C1 = diag([5,5]), C2 = diag([25,25]), C3 = diag([100,100]), C1 = diag([1,1]), and λ1 and λ2 be the state variables of the fixed-time stable control input saturation compensation system, representing the tracking error generated by Δu, M(q). -1 Let M(q) be the inverse matrix of the inertia matrix of the robotic arm system. The time derivative of the state variable λ1 of a fixed-time stable control input saturated compensation system. Let λ² be the time derivative of the state variable λ² of a fixed-time stable control input saturated compensation system. Let m, n, p, and q be the fractional power parameters of the fixed-time stable control input saturated compensation system, where m = 5, n = 3, q ​​= 9, and p = 7. The sig function satisfies... κ>0, x is any 2-dimensional vector, x1, x2 are the components of vector x, and sign is the sign function;

[0060] Figure 2 The figure shows the change curve of the state variables of the fixed-time stable control input saturation compensation system provided in the embodiment of the present invention. It can be seen from the figure that λ1 and λ2 can both converge to 0 stably after t = 0.23s, which is fast and stable.

[0061] Step 3: Based on the established fixed-time stable control input saturation compensation system, establish the joint tracking error of the robotic arm system, expressed as:

[0062]

[0063] Where e1 and e2 are the joint angle tracking error and joint angular velocity tracking error of the robotic arm system, respectively, q d and These are the predetermined trajectories for the joint angles and joint angular velocities of the robotic arm system, respectively. d =[sin(t); sin(t)],

[0064] Step 4: Based on the joint tracking error of the established robotic arm system, establish the fixed-time terminal sliding surface of the robotic arm system, represented as:

[0065] s = e² + α·sig ε (e1)+β·sig μ (e1)

[0066] Where s is the fixed-time terminal sliding surface of the robotic arm system, α and β are the gain coefficient matrices of the fixed-time terminal sliding surface, α = diag([200,200]), β = diag([20,20]), and ε,μ are the fractional power parameters of the fixed-time terminal sliding surface. sig ε (e1) and sig μ (e1) represents the sliding component of the fixed-time terminal sliding surface;

[0067] Step 5: Based on the joint tracking error of the established robotic arm system and the fixed-time end-effector sliding surface of the established robotic arm system, establish the bias neural network input, bias neural network basis functions, and bias neural network adaptive rate. The bias neural network input is expressed as:

[0068] z = [e1, e2] T

[0069] Where z is the input to the biased neural network, [e1, e2] T The bias neural network basis function is expressed as the transpose of the joint angle tracking error and joint angular velocity tracking error of the robotic arm system:

[0070]

[0071] The biased neural network has m+1=16 basis function layers, a width σ=2, and a network center μ for the j-th biased neural network basis function. j= [0,1]×[0,1]×[0,1]×[0,1], global bias scalar b g =4, φ i (z) is the i-th bias neural network basis function of the robotic arm system, [S1(z),…,S m+1 [z] represents the components of the i-th bias neural network basis function of the robotic arm system, and the adaptive rate of the bias neural network is expressed as:

[0072]

[0073] in, Let j be the weight of the bias neural network at the i-th joint of the robotic arm system. Let be the adaptive rate of the j-th weight of the bias neural network at the i-th joint of the robotic arm system. Let be the norm of the bias neural network weights at the i-th joint of the robotic arm system, and let Γ be the symmetric positive definite matrix representing the learning rate of the bias neural network, Γ = diag([200, 200]). i Let δ be the component of the fixed-time terminal sliding surface of the robotic arm system at the i-th joint. i The adjustment function for the biased neural network is expressed as:

[0074]

[0075] in, Let be the upper bound of the norm of the bias neural network weights at the i-th joint of the robotic arm system. and d i These are the robustness parameters for the biased neural network. d i =0.01, This allows the biased neural network to more quickly approximate the ideal biased neural network weights, while when the biased neural network weights are too large, the numerical values ​​become very small. d i It can slow down the rate of change in the biased neural network and improve its robustness;

[0076] Step 6: Based on the established fixed-time terminal sliding surface of the robotic arm system, the input of the bias neural network, the basis functions of the bias neural network, and the adaptive rate of the bias neural network, establish a robotic arm controller with input saturation compensation based on the bias neural network, represented as:

[0077]

[0078] in, To bias the neural network weights, Adaptive rate of bias neural network renew, Let φ(z) be the transpose of the bias neural network weights, φ(z) be the basis functions of the bias neural network for the robotic arm system, and K2 and K3 be the gain matrices of the robotic arm controller with input saturation compensation based on the bias neural network. For the fractional power parameters of a robotic arm controller with input saturation compensation based on a bias neural network, satisfying... And sign(s) are components of a bias neural network-based robotic arm controller with input saturation compensation;

[0079] Step 7: Input the established bias neural network-based robotic arm controller with input saturation compensation into the robotic arm system considering control input saturation. This enables fast and accurate trajectory tracking of each joint in the robotic arm system. The upper bound of the convergence time of the joint tracking error of the robotic arm system is expressed as:

[0080]

[0081] Among them, T max Let ξ1, ξ2, v be the upper bound of the convergence time of the joint tracking error of the robotic arm system, and let ξ1, ξ2, v be constant parameters of the upper bound of the convergence time of the joint tracking error of the robotic arm system.

[0082] Figure 3 This is a schematic diagram of the angle tracking error variation curve of the two-joint robotic arm system provided in an embodiment of the present invention. Figure 4 The figure shows a schematic diagram of the angular velocity tracking error variation curve of the two-joint robotic arm system provided in the embodiment of the present invention. As can be seen from the figure, both the angle tracking error and the angular velocity tracking error of the robotic arm system converge to zero within 0.25s. This time is less than the upper limit of the convergence time of the joint tracking error of the robotic arm system calculated, and the speed is fast and stable.

[0083] Figure 5 The figure shows a schematic diagram of the variation curve of the robotic arm controller with input saturation compensation based on the bias neural network provided in the embodiment of the present invention. As can be seen from the figure, the output of the robotic arm controller with input saturation compensation based on the bias neural network is stable and within the control limit, without any chattering or sudden step phenomenon.

[0084] Throughout the simulation, the robotic arm system remained stable without any sudden step changes or crashes. The control method of this invention enables the robotic arm system to execute the required trajectory tracking task efficiently and stably, demonstrating effectiveness and speed in achieving angle and angular velocity tracking within a fixed time. Even when faced with external disturbances, the robotic arm trajectory tracking did not exhibit significant deviation, demonstrating excellent stability.

[0085] The application of this invention is not limited to the previously disclosed technical means, but includes technical solutions constituted by various combinations of these technical features. Although the preferred embodiments of this invention have been described in detail above, it is worth emphasizing that those skilled in the art can still make various improvements and modifications without departing from the principles of this invention. These improvements and modifications are also considered to be part of the protection scope of this invention. Therefore, the protection scope of this patent should cover all equivalent embodiments that conform to the combination of technical features, and not be limited to the specifically disclosed embodiments.

Claims

1. A robotic arm control method with input saturation compensation based on a bias neural network, characterized in that: Includes the following steps: Step 1: Establish a robotic arm system considering control input saturation, represented as: in, , and These are the joint angle vector, joint angular velocity vector, and joint angular acceleration vector of the robotic arm system, respectively. Here is the inertia matrix of the robotic arm system. The matrix of Coriolis force and centrifugal force for the robotic arm system. The gravity vector of the robotic arm system. This is a robotic arm controller with input saturation compensation based on a bias neural network. Control the difference between the input before and after saturation. This represents the sum of the model uncertainties and external disturbances of the robotic arm system. Let this be the load vector of the robotic arm system. Let be the transpose of the Jacobian matrix of the robotic arm system; Step 2: Based on the established robotic arm system considering control input saturation, establish a fixed-time stable control input saturation compensation system, expressed as: in, This is the gain matrix of a fixed-time stable control input saturation compensation system, where all diagonal elements are positive. For a fixed-time stable control input saturated compensation system, the state variables are expressed by... The resulting tracking error, The number of joints in a robotic arm system. The inertial matrix of the robotic arm system The inverse matrix, State variables of a fixed-time stable control input saturation compensation system The derivative with respect to time, State variables of a fixed-time stable control input saturation compensation system The derivative with respect to time, a non-negative integer. For a fixed-time stable control input saturation compensation system, the fractional power parameters must satisfy... , The function satisfies , , , For vectors The various components, It is a symbolic function; Step 3: Based on the established fixed-time stable control input saturation compensation system, establish the joint tracking error of the robotic arm system, expressed as: in, and These are the joint angle tracking error and joint angular velocity tracking error of the robotic arm system, respectively. and These are the predetermined trajectories for the joint angles and joint angular velocities of the robotic arm system, respectively. Step 4: Based on the joint tracking error of the established robotic arm system, establish the fixed-time terminal sliding surface of the robotic arm system, represented as: in, For the fixed-time terminal sliding surface of the robotic arm system, and The gain coefficient matrix of the fixed-time terminal sliding surface. and All are diagonal positive definite matrices. For a fixed-time terminal sliding surface, the fractional power parameter satisfies... , , and The sliding component of the fixed-time terminal sliding surface; Step 5: Based on the joint tracking error of the established robotic arm system and the fixed-time terminal sliding surface of the established robotic arm system, establish the bias neural network input, bias neural network basis functions, and the adaptive rate of the bias neural network. The input to the biased neural network is represented as: in, For the input of the biased neural network, This is the transpose of the joint angle tracking error and the joint angular velocity tracking error of the robotic arm system. The basis functions of a biased neural network are expressed as follows: The number of basis function layers in the biased neural network is . , To determine the width of the basis functions of the bias neural network, For the first A biased neural network basis function network center It is a global bias scalar. The number of joints in a robotic arm system. For the first robotic arm system A biased neural network basis function For the first robotic arm system Each component of a biased neural network basis function; The adaptive rate of a biased neural network is expressed as: in, For the first robotic arm system The bias neural network at the nth joint Each weight, For the first robotic arm system The bias neural network at the nth joint The adaptive rate of each weight For the first robotic arm system The norm of the bias neural network weights at each joint, a symmetric positive definite matrix. The learning rate is used to bias the neural network. For the fixed-time terminal sliding surface of the robotic arm system at the first The components at each joint. The adjustment function for the biased neural network is expressed as: in, For the first robotic arm system The upper bound of the norm of the biased neural network weights at each joint. and These are the robustness parameters for the biased neural network. , , This allows the biased neural network to more quickly approximate the ideal biased neural network weights, while when the biased neural network weights are greater than... hour, It can slow down the rate of change in the biased neural network and improve its robustness. Step 6: Based on the established fixed-time terminal sliding surface of the robotic arm system, the input of the bias neural network, the basis functions of the bias neural network, and the adaptive rate of the bias neural network, establish a robotic arm controller with input saturation compensation based on the bias neural network, represented as: in, To bias the neural network weights, renew, For the transpose of the weights of the biased neural network, For the bias neural network basis functions of the robotic arm system, and This is the gain matrix of a robotic arm controller with input saturation compensation based on a bias neural network. For the fractional power parameters of a robotic arm controller with input saturation compensation based on a bias neural network, satisfying... , and These are components of a bias neural network-based robotic arm controller with input saturation compensation. Step 7: Input the established bias neural network-based robotic arm controller with input saturation compensation into the robotic arm system considering control input saturation. This enables fast and accurate trajectory tracking of each joint in the robotic arm system. The upper bound of the convergence time of the joint tracking error of the robotic arm system is expressed as: in, This is the upper bound of the convergence time for the joint tracking error of the robotic arm system. The number of joints in a robotic arm system. The fractional power parameter is the parameter of the bias neural network-based robotic arm controller with input saturation compensation defined in step six. This is a constant parameter representing the upper bound of the convergence time of the joint tracking error in the robotic arm system.

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