Robot adaptive neural optimal tracking control method and device considering dynamic constraints

Abstract

An adaptive neural optimal tracking control method and device for robots considering dynamic constraints are disclosed. The method includes: establishing a dynamic model of the robot in joint space, considering the robot system's input torque, inertia matrix, centripetal force, Coriolis force matrix, and gravitational torque; mapping the robot system's dynamics to Cartesian space, defining position and velocity vectors, and establishing a state-space model of the robot system; and designing an adaptive optimal neural tracking controller based on the optimal backstepping method and the "actor-critic" neural network method to handle dynamic constraints and achieve optimal tracking performance under dynamic constraints. This invention enables the robot to track the desired trajectory more accurately under dynamic constraints while ensuring motion safety.

Description

Technical Field

[0001] This invention relates to the field of robot control technology, and more specifically to a robot adaptive neural optimal tracking control method and apparatus that takes into account dynamic constraints. Background Art

[0002] With the development of robotics technology, robots are increasingly widely used in industry, medicine, and service sectors, especially in performing high-precision tasks in complex and dynamic environments. However, existing robot control methods still face many challenges in achieving stable and safe trajectory tracking while addressing dynamic constraints and optimal control performance. Robot systems typically exhibit highly nonlinear and multivariable coupling characteristics. Traditional control methods, such as proportional-integral-derivative (PID), fuzzy control, and sliding mode control, while providing some stability, struggle to achieve precise control over complex constraints in dynamically changing environments. Furthermore, these methods are ill-suited to responding to rapid changes in robot dynamics, failing to meet the system's real-time optimal performance requirements.

[0003] In recent years, neural network-based adaptive control methods for robots have gained widespread attention in academia and industry due to their strong learning and nonlinear approximation capabilities. Neural networks can model complex dynamics of systems with limited prior knowledge and adapt to changes in system state. In particular, the "actor-critic" neural network architecture, with its ability to approximate unknown internal dynamics of a system, has become an effective means of solving optimal control problems for complex nonlinear systems. However, existing neural network control schemes, when faced with dynamic state constraints, typically rely on preset excitation conditions, making it difficult to ensure optimal trajectory tracking performance under dynamic constraints. Furthermore, most neural network-based adaptive control methods may fail to maintain the robustness of the control system when constraints change significantly, thus failing to meet safety control requirements.

[0004] In constraint control, barrier Lyapunov functions are widely used to ensure that the system does not violate constraints in terms of output and state. By incorporating barrier functions into Lyapunov control design, it is possible to prevent the state from exceeding preset limits. However, traditional barrier Lyapunov functions are mainly applied to static constraint scenarios, and are difficult to apply directly to robot systems with time-varying and dynamic asymmetric constraints. Although symmetric, asymmetric, time-invariant, and time-varying barrier Lyapunov functions have evolved from barrier Lyapunov functions, existing models still struggle to address the stability and control accuracy issues of robot systems under highly dynamic asymmetric constraints.

[0005] Existing optimal control schemes have made some progress in meeting specific performance indicators, but they typically require solving complex Hamilton-Jacobi-Bellman equations, which is computationally burdensome for complex nonlinear systems. While reinforcement learning-based adaptive optimization methods offer a way to avoid precisely solving the Hamilton-Jacobi-Bellman equations, they place high demands on the system's control excitation conditions and lack effective designs to handle dynamic constraints, making them prone to constraint violations during control. Therefore, existing control schemes cannot meet the requirements of system safety and optimal performance under dynamic state constraints.

[0006] In summary, existing technologies have significant shortcomings in robot dynamic constraint control and optimal performance balancing, making it difficult to cope with the complex dynamic changes of highly nonlinear systems, and lacking efficient dynamic constraint processing methods in practical applications. Therefore, there is an urgent need for an adaptive control method for robots that can achieve optimal control while satisfying dynamic constraints, in order to meet the requirements of robot safety, stability, and optimal trajectory tracking in complex environments. Summary of the Invention

[0007] The purpose of this invention is to overcome the above-mentioned shortcomings of the prior art and to provide a robot adaptive neural optimal tracking control method and apparatus that takes into account dynamic constraints.

[0008] To address the existing problems of robot motion safety and optimal tracking performance, this invention designs a robot adaptive neural optimal control method with dynamic constraints, enabling the system to track the desired trajectory. All closed-loop signals are semi-globally consistent and eventually bounded, and never violate the dynamic constraints.

[0009] To achieve the above objectives, the present invention provides the following technical solution:

[0010] An adaptive neural optimal tracking control method for robots considering dynamic constraints includes the following steps:

[0011] S1. Establish a dynamic model of the robot in the joint space, considering the input torque, inertia matrix, centripetal force and Coriolis force matrix of the robot system, as well as the gravitational torque;

[0012] S2. Map the dynamics of the robot system to Cartesian space, define position and velocity vectors, and establish the state space model of the robot system;

[0013] S3. Under dynamic constraints, an adaptive optimal neural tracking controller is designed based on the optimal backstepping method and the "actor-critic" god network method to handle dynamic constraints and achieve optimal tracking performance.

[0014] As a preferred embodiment of the present invention, step S1 includes the following steps:

[0015] In the joint space The dynamic equations of the linkage robot are:

[0016] (1)

[0017] In the formula, These represent the joint position vector, velocity vector, and acceleration vector, respectively. This represents the input torque of the robot system. It is a positive definite symmetric inertial matrix. Represents the centripetal force and Coriolis force matrix. Represents the gravity matrix;

[0018] As a preferred embodiment of the present invention, step S2 includes the following steps:

[0019] Joint angle It can be mapped to the end-effector position of the robot's forward kinematics. Differentiating the forward kinematics with respect to time yields , The Jacobian matrix is ​​obtained from the inverse kinematics of the robot. , , for The false reversal; , The dimension of the end effector coordinate vector; These are the robot's end-effector position, end-effector velocity, and acceleration vector, respectively.

[0020] The robot dynamics model in Cartesian space is as follows:

[0021] (2)

[0022] In the formula, , , , ;

[0023] definition , Let represent the robot's position vector and velocity vector in Cartesian space; the general nonlinear equations describing the robot's dynamics model are:

[0024] (3)

[0025] In the formula, , , Represents the upper and lower bounds of dynamic constraints;

[0026] As a preferred embodiment of the present invention, S31 in S3 includes the following steps:

[0027] Decomposing and defining the tracking error of a complex system: The core idea of ​​the optimal backstepping method is to decompose the complex nonlinear system into subsystems that do not exceed the system order. For each subsystem... Define control tracking error

[0028] (4)

[0029] In the formula, , For the desired trajectory, , It is a virtual controller to be designed, and The relationship with other variables is ;

[0030] As a preferred embodiment of the present invention, S32 in S3 includes the following steps:

[0031] Based on the error defined in step S31 and The asymmetric time-varying integral barrier Lyapunov function is designed as follows:

[0032] (5)

[0033] In the formula, It is positive definite and continuously differentiable within the constraints. Represents the integral variable.

[0034] As a preferred embodiment of the present invention, S33 in S3 includes the following steps:

[0035] Subsystem Define the optimal performance index function :

[0036] (6)

[0037] To reflect the performance of the system in achieving the desired trajectory;

[0038] right Subsystem design Hamilton-Jacobi-Bellman equations To achieve optimal control:

[0039] (7)

[0040] In the formula, , , These are all design parameters. , , For process variables; this function can ensure that the designed controller meets the corresponding optimal performance requirements;

[0041] As a preferred embodiment of the present invention, S34 in S3 includes the following steps:

[0042] Solution To obtain the ideal virtual controller :

[0043] (8)

[0044] In the formula, It can be approximated as

[0045] (9)

[0046] In the formula, For the ideal weight vector, It is a kernel function vector. This is an approximation error;

[0047] As a preferred embodiment of the present invention, S35 in S3 includes the following steps:

[0048] Based on the optimal virtual controller obtained in step S34 of step S3 An "actor-critic" neural network is introduced to approximate ideal adaptive weights. ;Utilizing the function fitting capabilities of neural networks to solve unknown, uncertain, and nonlinear functions in robot systems. Approximation is performed to find the optimal solution to the equation. :

[0049] Define Hamiltonian approximation error :

[0050] (10)

[0051] Used to measure the accuracy of approximate solutions.

[0052] To ensure the convergence of neural network weights, a critic neural network is designed. Actor Neural Network The adaptive weight update law is expressed as:

[0053] (11)

[0054] (12)

[0055] This leads to the approximate optimal virtual control law. for:

[0056] (13)

[0057] As a preferred embodiment of the present invention, S36 in S3 includes the following steps:

[0058] Subsystem based on step S1 Repeat steps S32 to S35 in S3, in In the subsystem, based on Define the optimal performance index Hamilton-Jacobi-Bellman equations :

[0059] (14)

[0060] (15)

[0061] Solution To obtain the ideal optimal controller :

[0062] (16)

[0063] In the formula, For design parameters, It can be approximated by a neural network as

[0064] (17)

[0065] In the formula, Represents the ideal weight vector, It is a kernel function vector. Represents approximation error;

[0066] Introducing an "actor-critic" neural network to approximate the solution This yields an approximate optimal solution to the equation. ;

[0067] Define Hamiltonian approximation error :

[0068] (18)

[0069] Obtaining an approximate optimal controller :

[0070] (19)

[0071] Minimize approximation error The derivation yields and The adaptive weight update law is expressed as:

[0072] (20)

[0073] (twenty one)

[0074] A second aspect of the present invention relates to a robot adaptive neural optimal control device under dynamic constraints, comprising a memory and one or more processors, wherein the memory stores executable code, and the one or more processors execute the executable code to implement the robot adaptive neural optimal control method under dynamic constraints of the present invention.

[0075] A third aspect of the invention relates to a computer-readable storage medium having a program stored thereon that, when executed by a processor, implements the adaptive neural optimal control method for robots under dynamic constraints of the present invention.

[0076] The innovation of this invention is:

[0077] This invention introduces an asymmetric time-varying integral barrier Lyapunov function, which enables more accurate trajectory tracking in dynamic environments while ensuring system safety.

[0078] This invention employs an optimal backstepping method to design performance index functions for different subsystems, achieving hierarchical dynamic optimization. This strategy balances dynamic constraints and optimal performance, ensuring both control accuracy and safety.

[0079] This invention utilizes the nonlinear approximation and self-learning capabilities of neural networks to approximate the dynamic function of an unknown system through an "actor-critic" structure, and updates the optimal control law in real time.

[0080] The device of this invention, when combined with a computer-readable storage medium, provides a complete hardware and software solution, enabling the method to be deployed in practical systems. This invention has the following beneficial effects:

[0081] 1. To address the challenge of achieving optimal trajectory tracking and ensuring motion safety in robots under dynamic constraints, this invention employs an asymmetric time-varying integral barrier Lyapunov function to handle dynamic constraints, and combines this with an optimal backstepping method and an "actor-critic" neural network structure to design an adaptive neural optimal controller. Under dynamic constraints, the robot can more accurately track the desired trajectory while ensuring motion safety, providing a new technology and method for precise control and safe operation of robots in the field of safety constraints.

[0082] 2. To address the highly nonlinear characteristics of robot control systems and the difficulty of precise control using traditional methods, this invention utilizes the self-learning and nonlinear approximation capabilities of neural networks. It approximates unknown and uncertain functions in the robot system through an "actor-critic" neural network and updates the optimal control law online. This invention not only improves the control capability for complex nonlinear systems but also ensures a certain level of real-time performance and accuracy. Attached Figure Description

[0083] Figure 1 This is a schematic diagram of the structure of an adaptive neural optimal control method for robots with dynamic constraints according to the present invention.

[0084] Figure 2 This is a position state trajectory diagram under dynamic constraints for a robot adaptive neural optimal control method with dynamic constraints according to the present invention.

[0085] Figure 3 This is a velocity state trajectory diagram under dynamic constraints for a robot adaptive neural optimal control method with dynamic constraints according to the present invention.

[0086] Figure 4 This is an adaptive updated weight trajectory diagram of an adaptive neural optimal control method for a robot with dynamic constraints according to the present invention.

[0087] Figure 5 This is a control input trajectory diagram for an adaptive neural optimal control method for robots with dynamic constraints according to the present invention.

[0088] Figure 6 This is a trajectory diagram of the Hamiltonian function approximation value for an adaptive neural optimal control method for robots with dynamic constraints according to the present invention.

[0089] Figure 7 This is a schematic diagram of the device of the present invention. Detailed Implementation

[0090] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0091] Example 1

[0092] Please see Figure 1 The diagram shown is a framework diagram of the invention's control method. This invention provides a technical solution: a robot adaptive neural optimal tracking control method considering dynamic constraints, comprising the following steps:

[0093] S1. Establish a dynamic model of the robot in the joint space, considering the input torque, inertia matrix, centripetal force and Coriolis force matrix of the robot system, as well as the gravitational torque;

[0094] S2. Map the dynamics of the robot system to Cartesian space, define position and velocity vectors, and establish the state space model of the robot system;

[0095] S3. Under dynamic constraints, an adaptive optimal neural tracking controller is designed based on the optimal backstepping method and the "actor-critic" god network method to handle dynamic constraints and achieve optimal tracking performance.

[0096] S1 includes the following steps:

[0097] In joint space, the robot's dynamic model is as follows:

[0098] (1)

[0099] In the formula, These are the joint positions, velocities, and acceleration vectors of the interactive robot. This is the control input for the robot's joints.

[0100] The dynamic parameters of the two-bar linkage robotic arm in the joint space are as follows:

[0101] (2)

[0102] (3)

[0103] (4)

[0104] In the formula, the corresponding dynamic parameters of the robotic arm model are: the mass of link 1 The mass of connecting rod 2 is 2kg. The weight is 0.85 kg, and the length of connecting rod 1 is... The length of connecting rod 2 is 0.35m. The inertia of link 1 is 0.31m. for Inertia of link 2 for ;

[0105] S2 includes the following steps:

[0106] Joint angle It can be mapped to the end-effector position of the robot's forward kinematics. Differentiating the forward kinematics with respect to time yields , The Jacobian matrix is ​​obtained from the inverse kinematics of the robot. , , for The false reversal; , The dimension of the end effector coordinate vector; These are the robot's end-effector position, end-effector velocity, and acceleration vector, respectively.

[0107] The robot dynamics model in Cartesian space is as follows:

[0108] (5)

[0109] In the formula, , , , ;

[0110] In Cartesian space, the standard nonlinear kinematic model of the robot is established as follows:

[0111] (6)

[0112] In the formula, and These represent the position and velocity vectors of the robot's end effector, respectively. This represents the dimension of the end effector coordinate vector. , , ;

[0113] S3 includes the following steps:

[0114] S31 includes the following steps:

[0115] Decomposing and defining the tracking error of a complex system: The core idea of ​​the optimal backstepping method is to decompose the complex nonlinear system into subsystems that do not exceed the system order. For each subsystem... Define control tracking error

[0116] (7)

[0117] In the formula, , For the desired trajectory, , It is a virtual controller to be designed, and and The relationship is ;

[0118] S32 includes the following steps:

[0119] Based on the error defined in step S31 and The asymmetric time-varying integral barrier Lyapunov function is designed as follows:

[0120] (8)

[0121] In the formula, It is positive definite and continuously differentiable within the constraints. This represents the integral variable. The upper and lower bounds of the dynamic constraints are... and ,Right now and .

[0122] S33 includes the following steps:

[0123] Define subsystem The optimal performance index function is as follows:

[0124] (9)

[0125] In the formula, Represents the cost function. It is the ideal and optimal virtual controller. It is a design parameter.

[0126] right Subsystem design Hamilton-Jacobi-Bellman equations To achieve optimal control:

[0127] (10)

[0128] In the formula, , , These are all design parameters. , , For process variables; this function can ensure that the designed controller meets the corresponding optimal performance requirements;

[0129] S34 includes the following steps:

[0130] Solution To obtain the ideal virtual controller :

[0131] (11)

[0132] In the formula, It can be approximated as:

[0133] (12)

[0134] In the formula, For the ideal weight vector, It is a kernel function vector. This is an approximation error;

[0135] S35 includes the following steps:

[0136] Based on the optimal virtual controller obtained in step S34 of step S3 An "actor-critic" neural network is introduced to approximate ideal adaptive weights. ;Utilizing the function fitting capabilities of neural networks to solve unknown, uncertain, and nonlinear functions in robot systems. Approximation is performed to find the optimal solution to the equation. :

[0137] Define Hamiltonian approximation error :

[0138] (13)

[0139] To ensure the convergence of neural network weights, a critic neural network is designed. Actor Neural Network The adaptive weight update law is expressed as:

[0140] (14)

[0141] (15)

[0142] This leads to the approximate optimal virtual control law. for:

[0143] (16)

[0144] S36 includes the following steps:

[0145] Subsystem based on step S1 Repeat steps S32 to S35 in S3, in In the subsystem, based on Define the optimal performance index Hamilton-Jacobi-Bellman equations :

[0146] (17)

[0147] (18)

[0148] Solution To obtain the ideal optimal controller :

[0149] (19)

[0150] In the formula, For design parameters, It can be approximated by a neural network as:

[0151] (20)

[0152] In the formula, Represents the ideal weight vector, It is a kernel function vector. Represents approximation error;

[0153] Introducing an "actor-critic" neural network to approximate the solution This yields an approximate optimal solution to the equation. ;

[0154] Define Hamiltonian approximation error :

[0155] (twenty one)

[0156] Obtaining an approximate optimal controller :

[0157] (twenty two)

[0158] Minimize approximation error The derivation yields and The adaptive weight update law is expressed as:

[0159] (twenty three)

[0160] (twenty four)

[0161] To verify the effectiveness of the proposed method, a simulation was conducted on a robotic arm system. The system initialization parameters were set as follows:

[0162] Expected trajectory is Given the asymmetric time-varying state constraints: , The initial conditions are: , , , The control parameters are given as follows: , , , , , The initial values ​​of the "actor-critic" neural network were preset to... And the width of the Gaussian kernel function is Simulation time .

[0163] Simulation results are as follows Figure 2-6 As shown: Figure 2 and Figure 3 The robot's state trajectory under dynamic state constraints was plotted, and it can be seen that it achieved good optimal tracking performance for the desired state and never violated the dynamic constraint boundary. Figure 4 The adaptive weight update under the “actor-critic” neural network architecture is described. Figure 5 This represents the state error of the robot system. Figure 6 Approximate values ​​for the Hamilton-Jacobi-Bellman equations were plotted. For example... Figure 2-6 As shown, all signals in the closed-loop system are bounded and convergent.

[0164] In summary, this invention can achieve the goal of adaptive neural optimal trajectory tracking control while ensuring robot motion safety.

[0165] Example 2

[0166] This embodiment relates to a computer-readable storage medium storing a program that, when executed by a processor, implements the adaptive neural optimal control method for robots under dynamic constraints according to the present invention. The computer-readable storage medium can provide control code and store data generated during the experiment. Based on the experimental data stored in the computer-readable storage medium, plotting can be performed. Figure 2-6 . Figure 2 and Figure 3 The robot's state trajectory under dynamic state constraints was plotted. Figure 4 The adaptive weight update under the “actor-critic” neural network architecture is described. Figure 5 This represents the state error of the robot system. Figure 6 Approximate values ​​for the Hamilton-Jacobi-Bellman equations were plotted.

[0167] Example 3

[0168] like Figure 7 This embodiment relates to a robot adaptive neural optimal control device under dynamic constraints, including a memory and one or more processors. The memory stores executable code, and when the one or more processors execute the executable code, they are used to implement the robot adaptive neural optimal control method under dynamic constraints of Embodiment 1.

[0169] The above describes the excellent optimization effect shown by one embodiment of the present invention. Obviously, the present invention is not limited to the above embodiment. Various modifications can be made to it without departing from the basic spirit of the present invention and without exceeding the scope of the substantive content of the present invention.

Claims

1. A robot adaptive neural optimal tracking control method considering dynamic constraints, characterized in that, The following steps are involved: S1. Establish a dynamic model of the robot in the joint space, considering the input torque, inertia matrix, centripetal force and Coriolis force matrix of the robot system, as well as the gravitational torque; S2. Map the dynamics of the robot system to Cartesian space, define position and velocity vectors, and establish the state space model of the robot system; S3. Under dynamic constraints, an adaptive neural network tracking controller is designed based on the optimal backstepping method and the "actor-critic" neural network method to handle dynamic constraints and achieve optimal tracking performance; specifically including: The S31 system reduces the order of the original complex nonlinear system by decomposing it into optimal subsystems that do not exceed the system order. S32 Based on the subsystem established in step S31, design the corresponding asymmetric time-varying integral barrier Lyapunov function for the subsystem to handle the corresponding asymmetric time-varying dynamic constraints. S33 Based on the subsystem established in step S31, define the corresponding optimal performance index function for the subsystem, and obtain the Hamilton-Jacobi-Bellman equation for each optimal subsystem to meet the optimality requirement. Based on the functions established in steps S32 and S33, S34 derives the theoretically optimal virtual controller corresponding to the subsystem. Based on the optimal virtual controller obtained in step S34, S35 designs an "actor-critic" neural network architecture to approximate the unknown uncertain function of the optimal virtual controller and update the optimal control law online, thereby obtaining an approximate optimal virtual controller. S36 reverts from the lowest-order subsystem to the original system, repeating steps S32 to S35 for all subsystems, integrating all near-optimal virtual controllers to obtain the final adaptive neural network tracking controller; specifically including: Based on the subsystem z2 defined in step S31, repeating steps S32 to S35 will yield the theoretically optimal controller. for: In the formula, Φ 1,i k is a process variable. 2,i For design parameters; It can be approximated by NNs as Represents the ideal adaptive weights, It is the kernel function vector, ε 2,i (z 2,i ) represents the approximation error; Design a corresponding "actor-critic" neural network architecture with approximately ideal adaptive weights. The adaptive weight update laws for the critic neural network and the actor neural network are obtained as follows: Substituting the above results, we can finally obtain the approximate adaptive optimal neural control law u. i for:

2. The robot adaptive neural optimal tracking control method considering dynamic constraints according to claim 1, characterized in that, In step S1, In joint space, the robot's dynamic model is as follows: In the formula, These are the joint positions, velocities, and acceleration vectors of the interactive robot; It is a symmetric inertial matrix. The vector of Coriolis force and centripetal force. It is the gravity vector; This is the control input for the robot's joints.

3. The robot adaptive neural optimal tracking control method considering dynamic constraints according to claim 2, characterized in that, Step S2 specifically includes: The joint position q can be mapped to the end-effector position x of the robot's forward kinematics. Differentiating the forward kinematics with respect to time yields... The Jacobian matrix is ​​obtained from the inverse kinematics of the robot. The pseudoinverse of J(q); x = [x1, x2, ..., x n ] T n is the dimension of the end effector coordinate vector; These are the robot's end-effector position, end-effector velocity, and acceleration vector, respectively. The robot dynamics model in Cartesian space is as follows: in Define x1 = [x 11 ,x 12 ,...,x 1n ] T , x2=[x 21 ,x 22 ,...,x 2n ] T Let represent the robot's position vector and velocity vector in Cartesian space; the general nonlinear equations describing the robot's dynamics model are: In the formula, g2=M(x) -1 .

4. The robot adaptive neural optimal tracking control method considering dynamic constraints according to claim 3, characterized in that, Step S31 specifically includes: Define tracking error and Define virtual control law z1 and z2 are defined as follows: In the formula, x r For the desired trajectory; The differential relationship between subsystems z1 and z2, i.e., the derivative of z1 with respect to time, is:

5. The robot adaptive neural optimal tracking control method considering dynamic constraints according to claim 4, characterized in that, Step S32 specifically includes: Based on the subsystem z1 defined in step S31, the asymmetric time-varying integral barrier Lyapunov function is designed as follows: In the formula, It is positive definite and continuously differentiable within the constraints, where δ represents the integral variable and k a1,i ,k b1,i This represents the upper and lower bounds of dynamic constraints; this function can effectively handle dynamic constraints and ensure the safety of the robot during its movement.

6. The robot adaptive neural optimal tracking control method considering dynamic constraints according to claim 5, characterized in that, Step S33 specifically includes: Based on the subsystem z1 defined in step S31, the optimal performance index function is designed as H. 1,i for: In the formula, η 1,i , k 1,i All are design parameters, Φ 1,i I 1,i Ψ 1,i This is a process variable; this function can ensure that the designed controller meets the corresponding optimal performance requirements.

7. The robot adaptive neural optimal tracking control method considering dynamic constraints according to claim 6, characterized in that, Step S34 specifically includes: Based on the optimal index function H defined in step S33 1,i By solving The optimal virtual controller can be determined. for: In the formula, Approximated as For ideal adaptive weights, It is the kernel function vector, ε 1,i (z 1,i ) represents the approximate error.

8. The robot adaptive neural optimal tracking control method considering dynamic constraints according to claim 7, characterized in that, Step S35 specifically includes: Based on the optimal virtual controller obtained in step S34 Design an "actor-critic" neural network architecture to approximate ideal adaptive weights. By minimizing the approximation error The adaptive weight update laws for the critic neural network and the actor neural network are derived. This leads to the approximate optimal virtual control law. for: Achieve adaptive weights and The convergence, to ensure right An effective approximation.

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