A Disturbance Rejection Control Method for Space Manipulator Based on Zero-Sum Differential Game

Abstract

The present invention proposes a disturbance rejection control method for a space manipulator based on zero-sum differential game, comprising the following steps: First, establish a dynamic system of the space manipulator with system disturbance uncertainty; then, represent the established dynamic system of the space manipulator as an affine nonlinear system; immediately afterwards, transform the control problem of the space manipulator into a zero-sum differential game problem and design a continuously differentiable objective function; then, use a single-layer neural network to online learn and solve the Nash equilibrium solution of the objective function; finally, obtain the disturbance rejection controller based on zero-sum game of the present invention, so that the space manipulator can complete space tasks under the condition of having system disturbance uncertainty. The present invention enables the space manipulator control system to have both robustness and optimality without continuous excitation conditions.

Description

Technical Field

[0001] The present invention belongs to the field of anti-disturbance control of robotic arms, and particularly relates to an anti-disturbance control method for a space robotic arm based on zero-sum differential game. Background Technique

[0002] With the continuous development of space technology, space robotic arms have become an irreplaceable part of space missions. As a new type of bionic robotic arm, space robotic arms have high flexibility and can complete typical on-orbit service and maintenance tasks such as in-flight vehicle cabin inspection, capture of failed spacecraft, grasping of space waste and debris, and space station maintenance and repair. In order to enable space robotic arms to complete operations such as capture, clearance, attack, and defense of space non-cooperative targets, it is necessary to control the space mechanical system with high performance and low energy consumption. Therefore, the control problem of space robotic arms has also become a research hotspot.

[0003] In the related research of space robotic arms, there are mainly two ideas to solve the problems related to such tasks. One is to use traditional classical control methods such as sliding mode control and adaptive control to design high-performance robust controllers. Although these methods ensure good control performance, they do not consider the problem of control energy consumption. The other is to combine the idea of optimal control and the control method of reinforcement learning to design an online learning intelligent controller, but this usually requires the designed learning controller to meet strict persistent excitation conditions. Therefore, the existing online learning-based space robotic arm control methods are difficult to ensure high performance, strong robustness, and optimal performance without using persistent excitation conditions. Summary of the Invention

[0004] To achieve the anti-disturbance control of a space robotic arm, the purpose of the present invention is to provide an anti-disturbance control method for a space robotic arm based on zero-sum differential game. Aiming at the problem of system disturbance uncertainty in the robotic arm system, using the idea of reinforcement learning, an anti-disturbance controller based on zero-sum differential game is proposed, so that the robotic arm control system has both robustness and optimality without persistent excitation conditions.

[0005] To achieve the above invention purpose, the technical solution adopted by the present invention is as follows:

[0006] An anti-disturbance control method for a space robotic arm based on zero-sum differential game, comprising the following steps:

[0007] S1: Use sensor elements to obtain the angle information of the joints of the space robotic arm, and then establish the dynamic system of the space robotic arm with the obtained joint angle information. Subsequently, the established dynamic system of the space robotic arm is expressed as an affine nonlinear system;

[0008] S2: Based on the above affine nonlinear system, transform the anti-disturbance control problem of the space manipulator into a zero-sum differential game problem. Divide the anti-disturbance controller and the system disturbance uncertainty of the space manipulator into two parties of the zero-sum game. Design the objective function according to the principle that both parties hope to maximize their own interests and minimize the interests of the other party.

[0009] S3: Use a single-layer neural network to online learn and solve the Nash equilibrium solution of the objective function. The single-layer neural network uses the joint angle values of the space manipulator and the environment's real-time feedback as inputs to online adjust the weight vector of the single-layer neural network, and obtain an anti-disturbance controller based on the zero-sum differential game.

[0010] S4: The drivers of the motors of each joint of the manipulator generate corresponding control torques according to the anti-disturbance controller based on the zero-sum differential game in S3, and then realize the joint rotation and manipulator movement of the space manipulator under the condition of system disturbance uncertainty, so as to complete the space mission.

[0011] Further, in S1, the dynamic system of the space manipulator is expressed as:

[0012]

[0013] where, θ = [θ1 θ2] T ∈R 2×1 represents the angle values of the joints of the space manipulator. θ1 represents the angle value of the first joint of the manipulator, and θ2 is the angle value of the second joint, which are measured using sensor elements. are the angular velocity and angular acceleration of the joints of the manipulator respectively; u is the anti-disturbance controller based on the zero-sum differential game; J0(θ) ∈ R 2×2 is the nominal inertia matrix of the system; is the Coriolis force term, G0(θ) ∈ R 2×1 is the gravity term. The system disturbance uncertainty of the space manipulator system is and satisfies △J(θ) represents the inertia uncertainty, the Coriolis force term uncertainty, △G(θ) represents the gravity term uncertainty, all of which represent the modeling errors between the nominal system and the real system of the space manipulator, represents the unknown torque disturbance;

[0014] Define x1 = θ = [θ1 θ2] T including the angle values of the joints of the manipulator, including the angular velocity values of the joints of the manipulator system, so as to obtain the space manipulator system equation as:

[0015]

[0016] denotes the inverse matrix of the inertia matrix \(J_0(x_1)\in\mathbb{R}\) 2×2 , \(H_0(x_1,x_2)\) and \(G_0(x_1)\) are the Coriolis force term and the gravity term of the space manipulator system;

[0017] The system disturbance uncertainty of the space manipulator system is expressed as and there is Furthermore, the affine nonlinear system of the space manipulator is obtained as:

[0018]

[0019] where \(x = [x_1\ x_2]\) T \(\in\mathbb{R}\) 4×1 , denotes the control allocation matrix of the affine nonlinear system of the space manipulator, is a square matrix of all zeros, is the system term of the affine nonlinear system of the space manipulator.

[0020] Furthermore, in the above \(S_2\), the objective function is:

[0021]

[0022] where the matrices \(Q\), \(R\), and \(B\) are all positive definite matrices, \(\rho\) represents a positive real number, and the superscript \(T\) of the symbol \(\cdot\) T denotes the transpose of a vector or a matrix;

[0023] The optimized objective function is expressed as:

[0024]

[0025] where, denotes the minimum \(u\), that is, it represents the optimal anti-disturbance controller \(u\) * ; denotes the maximum that is, it represents the maximum system disturbance uncertainty \(\tau\) is the integration variable;

[0026] There exists a unique Nash equilibrium solution that makes the objective function optimal and satisfies:

[0027]

[0028] Therefore, the following equation is obtained:

[0029]

[0030] where, denotes the partial derivative of the objective function with respect to \(x\);

[0031] Furthermore, the equation is obtained as follows:

[0032]

[0033] where J * (0) = 0; represents the partial derivative of the optimized objective function with respect to x;

[0034] Therefore, the optimal disturbance rejection controller is obtained as:

[0035]

[0036] The maximum system disturbance uncertainty is expressed as:

[0037]

[0038] where R -1 , B -1 represent the inverse matrices of the positive definite matrices R and B; represents the minimum allowable ρ value that can be selected, satisfying

[0039] Furthermore, the S3 includes:

[0040] Use a single-layer neural network to represent the optimized objective function J * (x) as:

[0041]

[0042] where w c ∈R v is the ideal neural network weight vector, v represents the number of neurons in the single-layer neural network; σ c (x) ∈R v is the basis function, ε c (x) represents the approximation error of the single-layer neural network; respectively represent the partial derivative of the basis function with respect to x and the partial derivative of the approximation error with respect to x; Therefore, the optimal disturbance rejection control represented by the single-layer neural network is:

[0043]

[0044] Use a single-layer neural network to represent the maximum system disturbance uncertainty:

[0045]

[0046] The estimated value of the objective function J(x) is where represents the ideal single-layer neural network weight vector wc The estimated value; the partial derivative of the estimated value of the objective function with respect to x is

[0047] Furthermore, the anti-disturbance controller based on zero-sum differential game is obtained as:

[0048]

[0049] where μ and β both represent positive constants; K represents a positive definite matrix;

[0050] The system disturbance uncertainty estimated by the single-layer neural network is obtained as:

[0051]

[0052] where the weight vector of the single-layer neural network The weight update law is expressed as:

[0053]

[0054] where λ1, λ2, λ3, and ε all represent positive constants; represents the switching term of the weight update law, k represents the sampling times; the Bellman error is expressed as Taking the partial derivative of the Bellman error e can obtain Γ min (·) represents the minimum eigenvalue of a matrix; · represents the two-norm of a vector; tanh(·) represents the hyperbolic tangent function; ∑(·) represents the summation function.

[0055] Furthermore, the designed switching term in the weight update law Relaxes the assumption of the persistent excitation condition for the single-layer neural network and ensures the stability of the manipulator control system in the initial stage;

[0056] The switching term When, the shearing term in the weight update law:

[0057]

[0058] The shearing term ensures the initial stability of the manipulator control system. At the initial stage of the controlled system, that is, when the finite excitation is not satisfied, it ensures the stability of the manipulator control system; during the stable process of the system, the switching term The switching condition matrix of Continuously collects and records the historical data in the control process and judges whether the minimum eigenvalue of the switching condition matrix is greater than 0;

[0059] The switching term When, that is, when the minimum eigenvalue of the switching condition matrix is greater than 0, the shear term in the weight update law is:

[0060]

[0061] At this time, the weight update law changes to and updates the weight vector of the single-layer neural network, so that the estimated weight vector of the single-layer neural network approximates the ideal weight vector w c , and at the same time, it also makes the space manipulator system maintain uniformly ultimately bounded stability.

[0062] The space manipulator working in orbit will be affected by the system disturbance uncertainty. The anti-disturbance controller proposed in the present invention based on the idea of zero-sum differential game can suppress the influence of the system disturbance uncertainty on the manipulator control system; at the same time, the single-layer neural network uses the state information of the system to approximate the objective function to obtain the anti-disturbance controller, and the driver of the joint motor of the manipulator generates the corresponding control torque according to the anti-disturbance controller to drive the motor, so that the space manipulator completes the space in-orbit mission.

[0063] The anti-disturbance control method for a space manipulator based on zero-sum differential game proposed by the present invention adopts the above technical solutions. Compared with the prior art, the advantages are as follows:

[0064] (1) The present invention patent uses a single-layer neural network to approximate the objective function, and the proposed weight update law of the single-layer neural network can not only remove the persistent excitation condition, but also has the effect of initially stabilizing the manipulator control system;

[0065] (2) The present invention designs an anti-disturbance controller with the idea of zero-sum game, and uses an online reinforcement learning method to propose a new stable anti-disturbance controller, which improves the robustness of the space manipulator system and also ensures the optimality. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 is a flow block diagram of an anti-disturbance control method for a space manipulator based on zero-sum differential game of the present invention.

[0067] Figure 2 is a principle block diagram of an anti-disturbance control method for a space manipulator based on zero-sum differential game of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0068] The specific implementation of the present invention will be described in detail below. It is necessary to point out here that the following embodiments are only for further illustration of the present invention and cannot be construed as limiting the protection scope of the present invention. Some non-essential improvements and adjustments made by those skilled in the art to the present invention based on the above content of the present invention still fall within the protection scope of the present invention.

[0069] The present invention first establishes a dynamic system of the space manipulator and formulates it as an affine nonlinear system; designs an objective function based on the idea of zero-sum differential game; further, designs a single-layer neural network to approximate the objective function to obtain an anti-disturbance controller based on zero-sum differential game, so that the space manipulator system has both optimality and strong robustness; finally, the motor driver of the space manipulator generates corresponding control torques according to the designed anti-disturbance controller based on zero-sum differential game to drive the manipulator to move.

[0070] As Figure 1 shown, an anti-disturbance control method for a space manipulator based on zero-sum differential game of the present invention includes the following steps:

[0071] S1: Use a sensor element to obtain the angle information of the joints of the space manipulator, and then establish the dynamic system of the space manipulator with the obtained angle information of the joints. Subsequently, the established dynamic system of the space manipulator is formulated as an affine nonlinear system;

[0072] S2: Based on the affine nonlinear system, transform the anti-disturbance control problem of the space manipulator into a zero-sum differential game problem. Divide the anti-disturbance controller and the system disturbance uncertainty into two parties of the zero-sum game, and design an objective function according to the principle that both parties hope that their own interests are optimal and the interests of the other party are the worst;

[0073] S3: Use a single-layer neural network to online learn and solve the Nash equilibrium solution of the objective function; the single-layer neural network uses the joint angle values of the real-time feedback of the space manipulator and the environment as inputs to online adjust the weight vector of the single-layer neural network to obtain an anti-disturbance controller based on zero-sum differential game;

[0074] S4: The drivers of the motors of each joint of the manipulator generate corresponding control torques according to the anti-disturbance controller based on zero-sum differential game in S3, and then realize the joint rotation and manipulator movement of the space manipulator under the condition of system disturbance uncertainty, so as to complete the space mission.

[0075] Specifically, in S1, considering the dynamic characteristics of the manipulator, the dynamic system of the space manipulator in this embodiment is established as:

[0076]

[0077] where, θ = [θ1 θ2]T ∈R 2×1 is the angular value of the robotic arm joint. θ1 represents the angular value of the first joint of the robotic arm, and θ2 is the angular value of the second joint, which are measured using sensor elements. are the angular velocity and angular acceleration of the robotic arm joint respectively; is the nominal inertia matrix of the system, is the gravity term; u is the anti-disturbance controller based on zero-sum differential game. In this embodiment, the system disturbance uncertainty of the space robotic arm system is:

[0078]

[0079] is the Coriolis force term;

[0080] △J(θ) = -0.1J0(θ) and △G(θ) = -0.2G0(θ) represent the modeling errors between the nominal system and the real system of the space robotic arm.

[0081] Furthermore, the system equation of the space robotic arm is:

[0082]

[0083] where x1 = θ = [θ1 θ2] T contains the angular values of the joints of the robotic arm system, contains the angular velocity values of the joints of the robotic arm system. represents the inverse matrix of the inertia matrix J0(x1) ∈ R 2×2 . H0(x1, x2) and G0(x1) are the Coriolis force term and gravity term of the nominal system of the space robotic arm.

[0084] The system disturbance uncertainty is expressed as Furthermore, the affine nonlinear system of the space robotic arm is obtained as:

[0085]

[0086] where x = [x1 x2] T ∈R 4×1 , represents the control allocation matrix of the affine nonlinear system of the space robotic arm, is a square matrix of all zeros, represents the system term of the affine nonlinear system of the space robotic arm.

[0087] In S2, considering the obtained affine nonlinear system of the space manipulator, the disturbance rejection controller and the system disturbance uncertainty are divided into two parties of a zero-sum game; both the party of the disturbance rejection controller and the party of the system disturbance uncertainty hope that their own interests are optimal and the interests of the other party are the worst.

[0088] The party of the system disturbance uncertainty hopes to destroy the stability of the system, and by destroying the stability of the system, its own interests are optimal, that is, the control performance of the system is reduced. While the party of the disturbance rejection controller hopes to ensure the stability of the system, so that its own interests are optimal, that is, to improve the performance of the system. Therefore, a zero-sum game is formed between the system disturbance uncertainty and the disturbance rejection controller.

[0089] The disturbance rejection control problem of the space manipulator is transformed into a problem of zero-sum differential game. According to the characteristics of the zero-sum game, the objective function designed in this embodiment is:

[0090]

[0091] Among them, the matrices are all positive definite matrices, ρ = 0.5 represents a positive real number, and the superscript T of the symbol · T represents the transpose of a vector or a matrix.

[0092] At this time, the optimized objective function is expressed as:

[0093]

[0094] Among them, represents the minimum u, that is, the optimal disturbance rejection controller is expressed as u * ; represents the maximum That is, it represents that the maximum system disturbance uncertainty is τ is an integration variable; there is a unique Nash equilibrium solution that optimizes the objective function and satisfies:

[0095]

[0096] Furthermore, the following equation is obtained:

[0097]

[0098] Among them, represents the partial derivative of the objective function with respect to x.

[0099] Furthermore, the equation can be obtained:

[0100]

[0101] Among them, J *(0) = 0. Denotes the partial derivative of the optimized objective function with respect to x;

[0102] Therefore, the optimal anti-disturbance controller can be obtained and expressed as:

[0103]

[0104] The maximum system disturbance uncertainty is expressed as:

[0105]

[0106] In S3, a single-layer neural network is used for online learning to solve the objective function, thereby obtaining the anti-disturbance controller described above; a single-layer neural network is used to represent the optimized objective function J * (x) is:

[0107]

[0108] where, w c ∈R v Denotes an ideal neural network weight vector of dimension v, and v represents the number of neurons in the single-layer neural network; σ c (x) ∈ R v is the basis function, and ε c (x) represents the approximation error of the single-layer neural network; Denote the partial derivative of the basis function with respect to x and the partial derivative of the approximation error with respect to x, respectively.

[0109] Therefore, the optimal anti-disturbance control represented by a single-layer neural network is expressed as:

[0110]

[0111] Use a single-layer neural network to represent the maximum system disturbance uncertainty:

[0112]

[0113] The estimated value of the objective function J(x) is where Denotes the estimated value of the optimal single-layer neural network weight vector w c The partial derivative of the estimated value of the objective function with respect to x is

[0114] Furthermore, the anti-disturbance controller obtained in this embodiment based on zero-sum differential game is:

[0115]

[0116] where,

[0117] The system disturbance uncertainty estimated by the single-layer neural network is expressed as:

[0118]

[0119] In this embodiment, the weight vector of the neural network The weight update law is expressed as:

[0120]

[0121] where the Bellman error is expressed as Taking the partial derivative of the Bellman error \(e\) with respect to we can obtain The switching term \(k\) represents the sampling times. \(\Gamma\) min (·) represents the minimum eigenvalue of a matrix. \(\lambda_1 = 0.02\), \(\lambda_2 = 0.2\), \(\lambda_3 = 0.1\), \(\varepsilon = 0.001\) represent positive constants. · represents the two-norm of a vector. \(\tanh(·)\) represents the hyperbolic tangent function. \(\sum(·)\) represents the summation function. Using the above steps to obtain the anti-disturbance controller based on zero-sum differential game can achieve high-performance and low-energy consumption control of the space manipulator.

[0122] Using the manipulator anti-disturbance control method based on zero-sum differential game given above, it can be ensured that the entire manipulator control system is uniformly ultimately bounded stable when the space manipulator is subject to system disturbance uncertainty. Figure 2 This is the principle block diagram of a space manipulator anti-disturbance control method based on zero-sum differential game according to the present invention. The above-mentioned space manipulator anti-disturbance control method based on zero-sum differential game needs to be composed of an objective function, a single-layer neural network, a weight update law, an anti-disturbance controller based on differential game, and a space manipulator dynamics system. The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A disturbance rejection control method for a space manipulator based on zero-sum differential game, characterized in that It includes the following steps: S1: Obtain the angular information of the joints of the space manipulator using a sensor element, and then establish the dynamics system of the space manipulator with the obtained angular information of the joints. Subsequently, express the established dynamics system of the space manipulator as an affine nonlinear system; S2: Based on the affine nonlinear system, transform the anti-disturbance control problem of the space manipulator into a zero-sum differential game problem. Divide the anti-disturbance controller and the system disturbance uncertainty into two parties of the zero-sum game, and design an objective function according to the principle that both parties hope for the optimal interests of their own side and the worst interests of the other side; S3: Use a single-layer neural network to online learn and solve the Nash equilibrium solution of the objective function; The single-layer neural network uses the joint angle values of the real-time feedback between the space manipulator and the environment as inputs to online adjust the weight vector of the single-layer neural network, and obtain an anti-disturbance controller based on the zero-sum differential game; S4: The drivers of the motors of each joint of the manipulator generate corresponding control torques according to the anti-disturbance controller based on the zero-sum differential game in S3, and then realize the joint rotation and manipulator movement of the space manipulator under the condition of system disturbance uncertainty, so as to complete the space mission.

2. A disturbance rejection control method for a space manipulator based on zero-sum differential game according to claim 1, characterized in that, In S1, the dynamics system of the space manipulator is expressed as: where, θ = [θ1 θ2] T ∈R 2×1 represents the angular values of the joints of the space manipulator. θ1 represents the angular value of the first joint of the manipulator, and θ2 is the angular value of the second joint, which are measured using sensor elements. are the angular velocity and angular acceleration of the manipulator joints respectively; u is an anti-disturbance controller based on zero-sum differential game; J0(θ) ∈ R 2×2 is the nominal inertia matrix of the system, is the Coriolis force term, G0(θ) ∈ R 2×1 is the gravity term; the system disturbance uncertainty of the space manipulator system is and satisfies △J(θ) represents the inertia uncertainty, the Coriolis force term uncertainty, △G(θ) represents the gravity term uncertainty, all of which represent the modeling errors between the nominal system and the real system of the space manipulator, represents the unknown disturbance torque acting on the space manipulator system; Define \(x_1 = \theta = [\theta_1\ \theta_2]\) T including the angular values of the robotic arm joints, including the angular velocity values of the robotic arm system joints, so as to obtain the robotic arm system equation as follows: denotes the inverse matrix of the inertia matrix \(J_0(x_1)\in\mathbb{R}\) 2×2 , \(H_0(x_1,x_2)\) and \(G_0(x_1)\) are the Coriolis force terms and gravitational terms of the space manipulator system; The system disturbance uncertainty of the space manipulator system is expressed as and there is Furthermore, the affine nonlinear system of the space manipulator is obtained as follows: where \(x = [x_1\ x_2]\) T ∈ ℝ 4×1 , denotes the control allocation matrix of the affine nonlinear system of the space manipulator, is a square matrix of all zeros, is the system term of the affine nonlinear system of the space manipulator.

3. A disturbance rejection control method for a space manipulator based on zero-sum differential game according to claim 2, characterized in that, In S2, the objective function is: Among them, matrices Q, R, and B are all positive definite matrices, ρ represents a positive real number, and the superscript T of the symbol · T denotes the transpose of a vector or a matrix; The optimized objective function is expressed as: Among them represents the minimum u, that is, it represents the optimal anti-disturbance controller u * ; represents the maximum that is, it represents the maximum system disturbance uncertainty τ is an integral variable; There exists a unique Nash equilibrium solution Optimize the objective function and satisfy: Therefore, the following equation is obtained: Among them, represents the partial derivative of the objective function with respect to x; Further, the equation is obtained: Among them, J * (0) = 0; represents the partial derivative of the optimized objective function with respect to x; Therefore, the optimal anti-disturbance controller is obtained and expressed as: The maximum system disturbance uncertainty is expressed as: Among them, R -1 , B -1 represent the inverse matrices of the positive definite matrices R and B; represents the minimum value of ρ that can be admissibly selected and satisfies 4. A disturbance rejection control method for a space manipulator based on zero-sum differential game according to claim 3, characterized in that: S3 includes: Use a single-layer neural network to represent the optimization objective function J * (x) is as follows: where, w c ∈R v is the ideal neural network weight vector, v represents the number of neurons in a single-layer neural network; σ c (x) ∈ R v is the basis function, and ε c (x) represents the approximation error of the single-layer neural network; respectively represent the partial derivative of the basis function with respect to x and the partial derivative of the approximation error with respect to x; therefore, the optimal anti-disturbance control represented by a single-layer neural network is expressed as: Use a single-layer neural network to represent the maximum system disturbance uncertainty: The estimated value of the objective function J(x) is where represents the estimated value of the weight vector w of the ideal single-layer neural network c ; the partial derivative of the estimated value of the objective function with respect to x is Furthermore, the anti-disturbance controller based on the zero-sum differential game is obtained as: Wherein, both μ and β represent positive constants; K represents a positive definite matrix; Obtain the system disturbance uncertainty estimated by the single-layer neural network as: Among them, the weight vector of the single-layer neural network has its weight update law expressed as: Among them, λ1, λ2, λ3, and ε all represent positive constants; represents the switching term of the weight update law, and k represents the number of sampling times; the Bellman error is expressed as Taking the partial derivative of the Bellman error e gives Γ min (·) represents the minimum eigenvalue of a matrix; ||·|| represents the two-norm of a vector; tanh(·) represents the hyperbolic tangent function; ∑(·) represents the summation function.

5. A disturbance rejection control method for a space manipulator based on zero-sum differential game according to claim 4, characterized in that The switching term designed in the weight update law Relax the assumption of the persistent excitation condition for the single-layer neural network and ensure the stability of the manipulator control system in the initial stage; The switching item When, the switching term in the weight update law: The shear term ensures the initial stability of the manipulator control system. At the initial stage of the controlled system, that is, when the finite excitation is not satisfied, it ensures the stability of the manipulator control system; During the stabilization of the system, the switching term in the weight update law of the switching condition matrix continuously collects and records the historical data in the control process, and determines whether the minimum eigenvalue of the switching condition matrix is greater than 0; The switching term mentioned above When, that is, when the switching condition matrix has the minimum eigenvalue greater than 0, the shearing term in the weight update law is as follows: At this time, the weight update law changes to and update the weight vector of the single-layer neural network to make the estimated weight vector of the single-layer neural network approximate the ideal weight vector w c , and at the same time, it also makes the space manipulator system maintain consistent ultimate bounded stability.

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