Under-actuated AUV optimal trajectory tracking control method based on data driving

Abstract

The invention discloses an under-actuated AUV optimal trajectory tracking control method based on data driving. The method comprises the following steps: constructing a mathematical model of an under-actuated AUV; acquiring a tracking error dynamic equation and an under-actuated speed dynamic equation of the under-actuated AUV according to the mathematical model; obtaining a virtual control speed based on the tracking error dynamic equation in combination with a backstepping method; obtaining a speed tracking error according to an under-actuated speed dynamic equation and the virtual control speed; and based on an optimal control theory of reinforcement learning and an approximation technology of a neural network, designing an optimal trajectory tracking control law based on data driving in combination with the speed tracking error, and realizing trajectory tracking control of the under-actuated AUV based on the optimal tracking control law. The optimal trajectory tracking control law based on data driving is designed, the control performance is optimized online through real-time data driving, and high-precision and high-stability optimal trajectory tracking control of the under-actuated AUV in a complex marine environment is achieved.

Description

Technical Field

[0001] The present invention relates to the field of autonomous underwater vehicle trajectory tracking control, and in particular to a data-driven optimal trajectory tracking control method for an underactuated AUV. Background Art

[0002] An AUV is a self-powered, autonomous, and self-propelled, cable-free underwater robot. It boasts a wide range of motion, high maneuverability, and a high degree of intelligence. It has become an essential piece of equipment for underwater missions in a variety of fields. For example, in the military, AUVs are used for military hydrographic surveys, intelligence gathering and reconnaissance, anti-submarine warfare, and time-sensitive target strikes. Based on their propulsion configuration, AUVs are generally classified into underactuated, overactuated, and fully actuated. However, due to manufacturing cost, energy consumption, and load considerations, most AUVs adopt an underactuated configuration. Therefore, this paper focuses on underactuated AUVs.

[0003] Motion control of underactuated autonomous underwater vehicles (AUVs) is an important prerequisite for completing various complex missions. Trajectory tracking is more difficult to achieve in underactuated AUVs due to its strong time constraints compared to waypoint tracking and path planning control. Trajectory tracking can be summarized as: under any given initial conditions, using a trajectory tracking controller to drive the underactuated AUV to reach and track a pre-planned trajectory. According to the definition of trajectory tracking, when performing trajectory tracking tasks, underactuated AUVs must not only track the path in space but also meet the time requirements of the task. Therefore, stable, efficient, and fast trajectory tracking control for underactuated AUVs has very high research value and application significance. Trajectory tracking control of underactuated AUVs faces enormous challenges due to the influence of the external environment, complex ocean environment disturbances, and strong nonlinear dynamic characteristics. Traditional model-based control methods rely on precise system parameters, and in practical applications, performance often degrades due to modeling errors. Summary of the Invention

[0004] The present invention provides a data-driven optimal trajectory tracking control method for an under-actuated AUV to overcome the above problems.

[0005] In order to achieve the above object, the technical solution of the present invention is:

[0006] A data-driven optimal trajectory tracking control method for an underactuated AUV comprises the following steps:

[0007] S1. Construct a mathematical model of the underactuated AUV;

[0008] S2. Obtaining a tracking error dynamic equation and an underactuated velocity dynamic equation of the underactuated AUV according to the mathematical model; obtaining a virtual control velocity based on the tracking error dynamic equation and in combination with a backstepping method;

[0009] S3. Obtaining a speed tracking error according to an under-actuated speed dynamic equation and the virtual control speed;

[0010] S4. Based on the optimal control theory of reinforcement learning and the approximation technology of neural networks, combined with the speed tracking error, a data-driven optimal trajectory tracking control law is designed, and the trajectory tracking control of the under-actuated AUV is realized based on the optimal tracking control law.

[0011] Furthermore, the mathematical model of the underactuated AUV is expressed as:

[0012]

[0013] Where x, y, z represent the position of the underwater vehicle in the inertial coordinate system; θ, ψ represent the attitude angle of the underwater vehicle; ν = [u, v, ω, q, r] Τ , where u, v, ω are linear velocities, q, r are angular velocities; M = diag (m 11 ,m 22 ,m 33 ,m 55 ,m 66 ) is the inertia matrix; C(v) is the Coriolis matrix and the centripetal matrix; D(v) = diag(d 11 ,d 22 ,d 33 ,d 55 ,d 66 ) is the fluid damping matrix; τ d =[τ du ,τ dv ,τ dω ,τ dq ,τ dr ] Τ is the unknown external disturbance; τ=[τ u ,0,0,τ q ,τ r ] Τ For control input.

[0014] Furthermore, the specific steps of obtaining the tracking error dynamic equation and the under-actuated velocity dynamic equation of the under-actuated AUV according to the mathematical model include:

[0015] S21. Select corresponding coordinates based on the under-actuated characteristics of the under-actuated AUV, and redefine the output of the under-actuated AUV by selecting the coordinates:

[0016]

[0017] Where, l b is the virtual control point P L and AUV center of gravity O b the distance between them;

[0018] S22. According to the output of the under-actuated AUV and in combination with the mathematical model of the under-actuated AUV, a dynamic equation of the output of the under-actuated AUV is obtained, which is expressed as follows:

[0019]

[0020] Where, v a =[u,q,r] Τ Indicates driving speed; v u =[v,ω] Τ represents the under-actuated speed; J1(η) and J2(η,v u ) are respectively expressed as:

[0021]

[0022] S23. Define the expected trajectory, which is expressed as:

[0023] η d (t) = [x d (t),y d (t),z d (t)] T ; (4)

[0024] The tracking error between the dynamic equation output by the underactuated AUV and the desired trajectory is redefined as:

[0025] e1=η r -η d (t); (5)

[0026] According to the dynamic equation output by the under-actuated AUV, the desired trajectory and the redefined tracking error, the dynamic equation of the tracking error is obtained, which is expressed as follows:

[0027]

[0028] Where, v α For virtual control speed;

[0029] S24. Based on the backstepping method and the dynamic equation of the tracking error, the expression of the virtual control speed is designed as follows:

[0030]

[0031] Where, k>0 is the parameter to be designed;

[0032] S25, according to the mathematical model The under-actuated velocity dynamic equation is obtained, and its expression is:

[0033]

[0034] Where M n1 =diag(m 11 ,m 55 ,m 66 );D n1 =diag(d 11, d 55, d 66 ); is the total model uncertainty and external disturbance, where τ d1 =[τ du ,τ dq ,τ dr ] Τ and M d1 , C d1 (v u ), D d1 are the corresponding model uncertainty parameters, τ a =[τ u ,τ q ,τ r ] Τ Indicates a valid control input.

[0035] Furthermore, according to the under-actuated speed dynamic equation and the virtual control speed, the specific steps of obtaining the speed tracking error are as follows:

[0036] S31. Based on the virtual control speed, the speed tracking error is designed to be:

[0037] e2=v a -v α ; (9)

[0038] S32. Derivative the virtual control speed to obtain the virtual control speed dynamic equation, which is expressed as follows:

[0039]

[0040] The speed tracking error is derived according to the virtual control speed dynamic equation to obtain the speed tracking error, which is expressed as:

[0041]

[0042] Where,

[0043]

[0044] Furthermore, based on the optimal control theory of reinforcement learning and the approximation technology of neural networks, combined with the speed tracking error, a data-driven optimal trajectory tracking control law is designed. The trajectory tracking control of the underactuated AUV is realized based on the optimal tracking control law. The specific steps include:

[0045] S41. Based on the optimal control theory of reinforcement learning, the optimal tracking control law is obtained. The process is as follows:

[0046] S411. Rewrite the tracking error into the form of collected data. Its expression is:

[0047]

[0048] Where, τ a,i (e2) is the artificially defined iterative tracking control law to be updated; τ a (e2) = τ a，0 (e2)+n(t);τ a，0 (e2) and n(t) are the initial tracking control law and noise, respectively;

[0049] S412. Define the optimal cost function based on the tracking error, which is expressed as:

[0050]

[0051] Where P and Q are positive definite matrices of appropriate dimensions; r is the discount parameter, 0<r<1;

[0052] S413: Based on the tracking error in the form of the collected data and the optimal cost function, define the Hamiltonian function as:

[0053]

[0054] Where, Represents the partial derivative of the optimal cost function with respect to e2;

[0055] S414. Calculate the partial derivative of the Hamiltonian function to obtain the expression of the optimal tracking control law:

[0056]

[0057] S42. Since the optimal cost function and the optimal tracking control law are unknown functions, the general approximate characteristics of the neural network are used for approximation to obtain the optimal trajectory tracking control law;

[0058] S421, define the expression of the neural network as

[0059]

[0060] Where, is the weight of the neural network; and is the basis function of the neural network; N v and N a is the number of basis functions of the neural network; ε v,i and ε a,i+1 is the approximate error at the i-th iteration;

[0061] S422. Based on the approximate characteristics of the neural network, the total error is obtained as:

[0062]

[0063] Where k represents the number of sampling times;

[0064] S423. Define the total error function according to the total error as:

[0065]

[0066] Where C represents the maximum number of sampling times;

[0067] S424, iteratively updating the weights of the neural network according to the function value of the total error function;

[0068] The iterative update expression of the neural network weight is:

[0069]

[0070] Where, P i and Q i It is the collected data;

[0071] When || W i+1 -W i When ||<ε, stop the iteration and get the ideal weight ω a,i+1 , bring the ideal weights into the neural network to obtain the optimal trajectory tracking control law

[0072] Beneficial effects:

[0073] This paper proposes a data-driven optimal trajectory tracking control method for underactuated AUVs. This method leverages the AUV's actual operational data to achieve adaptive robust control without the need for a precise mathematical model. Furthermore, it combines the optimal control theoretical framework of reinforcement learning with adaptive control using neural networks to learn control strategies directly from the AUV's motion data, achieving model-free adaptive control. The data-driven and online learning nature of this method enables it to effectively adapt to complex and changing underwater environments, particularly those subject to unknown ocean current disturbances and actuator physical constraints. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following is a brief introduction to the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.

[0075] Figure 1 This is a flow chart of the data-driven optimal trajectory tracking control method for under-actuated AUVs of the present invention. DETAILED DESCRIPTION

[0076] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0077] This embodiment provides a data-driven optimal trajectory tracking control method for underactuated AUVs. Figure 1 As shown, the following steps are included:

[0078] S1. Construct a mathematical model of the underactuated AUV;

[0079] S2. Obtaining a tracking error dynamic equation and an underactuated velocity dynamic equation of the underactuated AUV according to the mathematical model; obtaining a virtual control velocity based on the tracking error dynamic equation and in combination with a backstepping method;

[0080] S3. Obtaining a speed tracking error according to an under-actuated speed dynamic equation and the virtual control speed;

[0081] S4. Based on the optimal control theory of reinforcement learning and the approximation technology of neural networks, combined with the speed tracking error, a data-driven optimal trajectory tracking control law is designed, and the trajectory tracking control of the under-actuated AUV is realized based on the optimal tracking control law.

[0082] Specifically, first, a mathematical model of the under-actuated AUV is constructed as the controlled object for subsequent control design; according to the under-actuated characteristics, the tracking error dynamic equation and the velocity dynamic equation are established, and the virtual control speed is generated based on the tracking error dynamic equation combined with the backstepping method. The speed instruction can decouple posture tracking and dynamic control, and guide the AUV to move along the desired trajectory; secondly, the speed tracking error is calculated according to the velocity dynamic equation and the virtual control speed, and the under-actuated speed tracking problem is transformed into an error adjustment problem; thirdly, based on the optimal control theory of reinforcement learning and the neural network approximation technology, a data-driven optimal trajectory tracking control law is designed; wherein, a cost function with the speed tracking error as the state is constructed through reinforcement learning, and the Hamiltonian equation is solved online through adaptive dynamic programming to obtain the optimal control strategy; the neural network is used as a function approximator to compensate for model uncertainty online and fit the optimal control law at the same time, eliminating the dependence on precise dynamic parameters; finally, by integrating the stability framework of the backstepping method with the data-driven optimal controller, high-precision and energy-optimal tracking of the AUV on the desired trajectory is achieved.

[0083] Specifically, the mathematical model of the underactuated AUV is expressed as:

[0084]

[0085] Where x, y, z represent the position of the underwater vehicle in the inertial coordinate system; θ, ψ represent the attitude angle of the underwater vehicle; ν = [u, v, ω, q, r] Τ , where u, v, ω are linear velocities, q, r are angular velocities; M = diag (m 11 ,m 22 ,m 33 ,m 55 ,m 66 ) is the inertia matrix; C(v) is the Coriolis matrix and the centripetal matrix; D(v) = diag(d 11 ,d 22 ,d 33 ,d 55 ,d 66 ) is the fluid damping matrix; τ d =[τ du ,τ dv ,τ dω ,τ dq ,τ dr ] Τ is the unknown external disturbance; τ=[τ u ,0,0,τ q ,τ r ] Τ For control input.

[0086] Specifically, the specific steps of obtaining the tracking error dynamic equation and the under-actuated velocity dynamic equation of the under-actuated AUV according to the mathematical model include:

[0087] S21. Select corresponding coordinates based on the under-actuated characteristics of the under-actuated AUV, and redefine the output of the under-actuated AUV by selecting the coordinates:

[0088]

[0089] Where, l b is the virtual control point P L and AUV center of gravity O b the distance between them;

[0090] S22. According to the output of the under-actuated AUV and in combination with the mathematical model of the under-actuated AUV, a dynamic equation of the output of the under-actuated AUV is obtained, which is expressed as follows:

[0091]

[0092] Where, v a =[u,q,r] Τ Indicates driving speed; v u =[v,ω] Τ represents the under-actuated speed; J1(η) and J2(η,v u ) are respectively expressed as:

[0093]

[0094] S23. Define the expected trajectory, which is expressed as:

[0095] η d (t) = [x d (t),y d (t),z d (t)] T ; (4)

[0096] The tracking error between the dynamic equation output by the underactuated AUV and the desired trajectory is redefined as:

[0097] e1=η r -η d (t); (5)

[0098] According to the dynamic equation output by the under-actuated AUV, the desired trajectory and the redefined tracking error, the dynamic equation of the tracking error is obtained, which is expressed as follows:

[0099]

[0100] Where, vα For virtual control speed;

[0101] S24. Based on the backstepping method and the dynamic equation of the tracking error, the expression of the virtual control speed is designed as follows:

[0102]

[0103] Where, k>0 is the parameter to be designed;

[0104] S25, according to the mathematical model The under-actuated velocity dynamic equation is obtained, and its expression is:

[0105]

[0106] Where M n1 =diag(m 11 ,m 55 ,m 66 );D n1 =diag(d 11, d 55, d 66 ); is the total model uncertainty and external disturbance, where τ d1 =[τ du ,τ dq ,τ dr ] Τ and M d1 , C d1 (v u ), D d1 are the corresponding model uncertainty parameters, τ a =[τ u ,τ q ,τ r ] Τ Indicates a valid control input.

[0107] Specifically, according to the under-actuated speed dynamic equation and the virtual control speed, the specific steps of obtaining the speed tracking error are as follows:

[0108] S31. Based on the virtual control speed, the speed tracking error is designed to be:

[0109] e2=v a -v α ; (9)

[0110] S32. Derivative the virtual control speed to obtain the virtual control speed dynamic equation, which is expressed as follows:

[0111]

[0112] The speed tracking error is derived according to the virtual control speed dynamic equation to obtain the speed tracking error, which is expressed as:

[0113]

[0114] Where,

[0115]

[0116] Specifically, based on the optimal control theory of reinforcement learning and the approximation technology of neural networks, combined with the speed tracking error, a data-driven optimal trajectory tracking control law is designed. The trajectory tracking control of the underactuated AUV is realized based on the optimal tracking control law. The specific steps include:

[0117] S41. Based on the optimal control theory of reinforcement learning, the optimal tracking control law is obtained. The process is as follows:

[0118] S411. Rewrite the tracking error into the form of collected data. Its expression is:

[0119]

[0120] Where, τ a,i (e2) is the artificially defined iterative tracking control law to be updated; τ a (e2) = τ a，0 (e2)+n(t);τ a，0 (e2) and n(t) are the initial tracking control law and noise, respectively;

[0121] S412. Define the optimal cost function based on the tracking error, which is expressed as:

[0122]

[0123] Where P and Q are positive definite matrices of appropriate dimensions; r is the discount parameter, 0<r<1;

[0124] S413: Based on the tracking error in the form of the collected data and the optimal cost function, define the Hamiltonian function as:

[0125]

[0126] Where, Represents the partial derivative of the optimal cost function with respect to e2;

[0127] S414. Calculate the partial derivative of the Hamiltonian function to obtain the expression of the optimal tracking control law:

[0128]

[0129] S42. Since the optimal cost function and the optimal tracking control law are unknown functions, the general approximate characteristics of the neural network are used for approximation to obtain the optimal trajectory tracking control law;

[0130] S421, define the expression of the neural network as

[0131]

[0132] Where, is the weight of the neural network; and is the basis function of the neural network; N v and N a is the number of basis functions of the neural network; ε v,i and ε a,i+1 is the approximate error at the i-th iteration;

[0133] S422. Based on the approximate characteristics of the neural network, the total error is obtained as:

[0134]

[0135] Where k represents the number of sampling times;

[0136] S423. Define the total error function according to the total error as:

[0137]

[0138] Where C represents the maximum number of sampling times;

[0139] S424, iteratively updating the weights of the neural network according to the function value of the total error function;

[0140] The iterative update expression of the neural network weight is:

[0141]

[0142] Where, P i and Q i It is the collected data;

[0143] When || W i+1 -W i When ||<ε, stop the iteration and get the ideal weight ω a,i+1 , bring the ideal weights into the neural network to obtain the optimal trajectory tracking control law

[0144] This example utilizes actual AUV operational data to achieve adaptive robust control without the need for precise mathematical models. It also combines the optimal control theoretical framework of reinforcement learning with adaptive control using neural networks to learn control strategies directly from AUV motion data, achieving model-free adaptive control. The data-driven and online learning nature of this method enables it to effectively adapt to complex and changing underwater environments, particularly unknown ocean current disturbances and actuator physical constraints.

[0145] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A data-driven optimal trajectory tracking control method for underactuated AUV, characterized by: The following steps are involved: S1. Construct a mathematical model of the underactuated AUV; S2. Obtaining a tracking error dynamic equation and an underactuated speed dynamic equation of the underactuated AUV according to the mathematical model; obtaining a virtual control speed based on the tracking error dynamic equation and in combination with a backstepping method; S3. Obtaining a speed tracking error according to an under-actuated speed dynamic equation and the virtual control speed; S4. Based on the optimal control theory of reinforcement learning and the approximation technology of neural networks, combined with the speed tracking error, a data-driven optimal trajectory tracking control law is designed, and the trajectory tracking control of the under-actuated AUV is realized based on the optimal tracking control law.

2. The data-driven optimal trajectory tracking control method for an underactuated AUV according to claim 1, characterized in that: The mathematical model of the underactuated AUV is expressed as: Where x, y, z represent the position of the underwater vehicle in the inertial coordinate system; θ, ψ represent the attitude angle of the underwater vehicle; ν = [u, v, ω, q, r] Τ , where u, v, ω are linear velocities, q, r are angular velocities; M = diag (m 11 ,m 22 ,m 33 ,m 55 ,m 66 ) is the inertia matrix; C(v) is the Coriolis matrix and the centripetal matrix; D(v) = diag(d 11 ,d 22 ,d 33 ,d 55 ,d 66 ) is the fluid damping matrix; τ d =[τ du ,τ dv ,τ dω ,τ dq ,τ dr ] Τ is the unknown external disturbance; τ=[τ u ,0,0,τ q ,τ r ] Τ For control input.

3. The data-driven optimal trajectory tracking control method for an underactuated AUV according to claim 1, characterized in that: The specific steps of obtaining the tracking error dynamic equation and the under-actuated velocity dynamic equation of the under-actuated AUV according to the mathematical model include: S21. Select corresponding coordinates based on the under-actuated characteristics of the under-actuated AUV, and redefine the output of the under-actuated AUV by selecting the coordinates: Where, l b is the virtual control point P L and AUV center of gravity O b the distance between them; S22. According to the output of the under-actuated AUV and in combination with the mathematical model of the under-actuated AUV, a dynamic equation of the output of the under-actuated AUV is obtained, which is expressed as follows: Where, v a =[u,q,r] Τ Indicates driving speed; v u =[v,ω] Τ represents the under-actuated speed; J1(η) and J2(η,v u ) are respectively expressed as: S23. Define the expected trajectory, which is expressed as: η d (t)=[x d (t),y d (t),z d (t)] T ; (4) The tracking error between the dynamic equation output by the underactuated AUV and the desired trajectory is redefined as: e1=η r -or d (t); (5) According to the dynamic equation output by the underactuated AUV, the desired trajectory and the redefined tracking error, the dynamic equation of the tracking error is obtained, which is expressed as follows: Where, v α For virtual control speed; S24. Based on the backstepping method and the dynamic equation of the tracking error, the expression of the virtual control speed is designed as follows: Where, k>0 is the parameter to be designed; S25, according to the mathematical model The under-actuated velocity dynamic equation is obtained, and its expression is: Where M n1 =diag(m 11 ,m 55 ,m 66 );D n1 =diag(d 11, d 55, d 66 ); is the total model uncertainty and external disturbance, where τ d1 =[τ du ,τ dq ,τ dr ] Τ and M d1 , C d1 (v u ), D d1 are the corresponding model uncertainty parameters, τ a =[τ u ,τ q ,τ r ] Τ Indicates a valid control input.

4. The data-driven optimal trajectory tracking control method for an underactuated AUV according to claim 1, characterized in that: According to the under-actuated velocity dynamic equation and the virtual control velocity, the specific steps for obtaining the velocity tracking error are as follows: S31. Based on the virtual control speed, the speed tracking error is designed to be: e2=v a -v α ; (9) S32. Derivative the virtual control speed to obtain the virtual control speed dynamic equation, which is expressed as follows: The speed tracking error is derived according to the virtual control speed dynamic equation to obtain the speed tracking error, which is expressed as: Where, 5. The data-driven optimal trajectory tracking control method for an underactuated AUV according to claim 1, characterized in that: Based on the optimal control theory of reinforcement learning and the approximation technology of neural networks, combined with the speed tracking error, a data-driven optimal trajectory tracking control law is designed. The trajectory tracking control of the under-actuated AUV is realized based on the optimal tracking control law. The specific steps include: S41. Based on the optimal control theory of reinforcement learning, the optimal tracking control law is obtained. The process is as follows: S411. Rewrite the tracking error into the form of collected data. Its expression is: Where, τ a,i (e2) is the artificially defined iterative tracking control law to be updated; τ a (e2) = τ a，0 (e2)+n(t);τ a，0 (e2) and n(t) are the initial tracking control law and noise, respectively; S412. Define the optimal cost function based on the tracking error, which is expressed as: Where P and Q are positive definite matrices of appropriate dimensions; r is the discount parameter, 0<r<1; S413: Based on the tracking error in the form of the collected data and the optimal cost function, define the Hamiltonian function as: Where, Represents the partial derivative of the optimal cost function with respect to e2; S414. Calculate the partial derivative of the Hamiltonian function to obtain the expression of the optimal tracking control law: S42. Since the optimal cost function and the optimal tracking control law are unknown functions, the general approximate characteristics of the neural network are used for approximation to obtain the optimal trajectory tracking control law; S421, define the expression of the neural network as Where, is the weight of the neural network; and is the basis function of the neural network; N v and N a is the number of basis functions of the neural network; ε v,i and ε a,i+1 is the approximate error at the i-th iteration; S422. Based on the approximate characteristics of the neural network, the total error is obtained as: Where k represents the number of sampling times; S423. Define the total error function according to the total error as: Where C represents the maximum number of sampling times; S424, iteratively updating the weights of the neural network according to the function value of the total error function; The iterative update expression of the neural network weight is: Where, P i and Q i It is the collected data; When || W i+1 -W i When ||<ε, stop the iteration and get the ideal weight ω a,i+1 , bring the ideal weights into the neural network to obtain the optimal trajectory tracking control law

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