Multi-mechanical-arm adaptive dynamic planning optimal sliding mode control method

Abstract

The invention discloses a multi-mechanical-arm adaptive dynamic planning optimal sliding mode control method, and relates to the field of trajectory tracking control of multi-mechanical-arm, and the method comprises the steps: building a two-degree-of-freedom dynamic model of a multi-mechanical-arm system; defining a two-degree-of-freedom position error of the multi-mechanical-arm system; a multi-mechanical-arm system self-adjusting performance control method based on a specified time performance function and a self-adjusting function is designed; based on a sliding mode control principle, a multi-mechanical-arm system sliding mode control method with flexible constraint capacity is designed; and based on self-adaptive dynamic programming, an optimal sliding mode controller of the multi-mechanical-arm system is designed. According to the method, the problems of performance constraint, limited control precision and optimal control of trajectory tracking during application of an existing multi-mechanical-arm system control method are solved.

Description

Technical Field

[0001] The present invention relates to the field of trajectory tracking control of multiple robotic arms, and in particular to an optimal sliding mode control method for adaptive dynamic programming of multiple robotic arms. Background Art

[0002] With the continuous advancement of technologies such as intelligence, big data, and cloud computing, the application of robotic arms in the fields of intelligent manufacturing, medical care, logistics, etc. will continue to expand from single functions to multiple functions and from fixed scenarios to complex scenarios. Therefore, multi-robotic arms will develop in a more intelligent and complex direction. In the development process of multi-robotic arms, due to their complexity and the interdependence between robotic arms, control performance is particularly important, which is also one of the research focuses of multi-robotic arm collaborative control in recent years.

[0003] Specified performance control is an effective method to achieve good performance. The principle of specified performance control is to provide constraints for the tracking error by specifying the performance function, ensuring that the convergence of the error meets the predetermined conditions, so as to achieve the desired transient and steady-state performance. In recent years, through the unremitting efforts of researchers, many specified performance control methods have been proposed. Using specified performance control, the tracking error converges to the predetermined area, ensuring the transient and steady-state performance of the system. However, the traditional performance control method has a priori conditions, that is, the initial state must be within the specified range. At the same time, the traditional performance control does not consider the threshold requirements when the system enters the steady-state stage, and may be too conservative in parameter selection, affecting the control efficiency. Therefore, designing a more flexible specified performance control method for the multi-manipulator system is crucial to ensure the performance of multi-manipulator collaborative control.

[0004] Sliding mode control has been increasingly valued due to its advantages such as fast response speed, strong resistance to external interference, and simple physical implementation. Therefore, the research on sliding mode control is very practical, and many methods have been proposed to solve the control problem. The boundary layer method has a good effect in improving robustness. However, this method is difficult to ensure the existence of sliding modes in the boundary layer, which means that this method has to sacrifice a certain tracking accuracy. High-order sliding modes can improve the robustness of the system without affecting the tracking performance. However, the complex time derivatives of high-order sliding modes need to be considered. Therefore, designing a suitable sliding mode controller for multi-manipulator collaborative control is still a challenging research.

[0005] The goal of optimal control is to minimize the performance indicator function while pursuing system stability, so as to achieve a balance between system performance and control resources. Reinforcement learning has attracted widespread attention from researchers due to its success in solving optimal control problems. Given that the system dynamics are known, an action-critic framework of reinforcement learning is constructed on the basis of neural networks or fuzzy logic systems, which can solve the optimal control problem of the system. However, the system often has uncertain dynamics, which can be estimated by identifiers. However, the use of the identifier-action-critic framework in a multi-manipulator system greatly increases the computational intensity of the system. Dynamic programming is considered to be an effective and classic tool for solving the optimization control problem of nonlinear systems. In the 1970s, researchers developed adaptive dynamic programming through neural networks or fuzzy logic systems. In recent decades, the research on adaptive dynamic programming has achieved great success and has received widespread attention from researchers. Therefore, one of the goals of the present invention is to solve the optimal control problem of multi-manipulators through adaptive dynamic programming. Summary of the invention

[0006] In view of the above-mentioned deficiencies in the prior art, the present invention provides a multi-manipulator system adaptive dynamic programming optimal sliding mode control method which solves the performance constraint problems, limited control accuracy problems, and optimal control problems of trajectory tracking when applying the current multi-manipulator system control method.

[0007] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is: a multi-manipulator adaptive dynamic programming optimal sliding mode control method, comprising the following steps:

[0008] S1: Establish a 2-DOF dynamic model of the multi-manipulator system;

[0009] S2: Based on the 2-DOF dynamic model, define the 2-DOF position error of the multi-manipulator system;

[0010] S3: Based on the 2-DOF position error, design a self-adjusting performance control method for a multi-manipulator system based on a specified time performance function and a self-adjusting function;

[0011] S4: Based on the self-adjusting performance control method of the multi-manipulator system and the sliding mode control principle, a sliding mode control method for the multi-manipulator system with flexible constraints is designed;

[0012] S5: Based on the sliding mode control method and adaptive dynamic programming of multi-manipulator system, design the optimal sliding mode controller of the multi-manipulator system and complete the optimal sliding mode control of multi-manipulator adaptive dynamic programming.

[0013] Furthermore, the 2-DOF dynamic model in S1 is:

[0014]

[0015] For simplicity, let The expression is rewritten as:

[0016]

[0017] Let H ti Denotes the matrix H i The diagonal matrix of , we get:

[0018] H i =H ti +H Ti

[0019] therefore, Rewritten as:

[0020]

[0021] Among them, M i (·) is the inertia matrix function of the i-th manipulator, C i (·) is the vector function of the centripetal force and Coriolis force of the i-th manipulator, G i (·) is the gravity vector function of the i-th manipulator, q i is the position vector of the i-th robot arm, is the velocity vector of the i-th robot arm, is the acceleration vector of the i-th robot arm, τ i is the input torque vector of the i-th robot, d is the unknown bounded disturbance vector, Ψ i =H Ti τ i +H i D i +H i d;

[0022]

[0023] in is the inertia matrix of the i-th robot arm, C i is the centripetal force and Coriolis force vector of the i-th robot arm, is the velocity vector of the i-th manipulator in the first degree of freedom, q i,2 is the position vector of the i-th robot arm in the second degree of freedom, is the velocity vector of the i-th manipulator in the second degree of freedom, J i,1 is the moment of inertia of the i-th robot arm in the first degree of freedom, J i,2 is the moment of inertia of the i-th robot arm in the second degree of freedom, m i,1 is the mass of the i-th manipulator in the first degree of freedom, m i,2is the mass of the i-th robot in the second degree of freedom, R i,1 is the center of mass of the i-th manipulator in the first degree of freedom, R i,2 is the center of mass of the i-th manipulator in the second degree of freedom, r i,1 is the length of the i-th robot arm in the first degree of freedom.

[0024] Furthermore, the 2-DOF position error of the multi-manipulator system is defined in S2, including the following sub-steps:

[0025] S21: using a directed graph χ=(K, E, A) to represent the relationship between the robot arms in exchanging state information;

[0026] A=[a ij ]∈R n×n

[0027] when a ij =1 means that robot i can receive information from robot j, otherwise a ij =0;

[0028] K={k 1 ,k 2 ,...,k n}

[0029] Among them, A is the communication between the manipulators, R represents the real number set, R n×n is an n×n dimensional Euclidean space, n is a constant, K is the set of followers, k 1 ,k 2 ,…,k n is the number of followers, E is the position error;

[0030] S22: Based on the relationship between the state information exchanged between the robotic arms, the 2-DOF position error of the multi-robotic arm system is obtained as:

[0031]

[0032] The derivative of it is:

[0033]

[0034] Among them, E i,s is the position error of the ith robot arm in the sth degree of freedom, q i,s is the position vector of the ith robot arm in the sth degree of freedom, q j,s is the position vector of the j-th robot arm in the s-th degree of freedom, b i,s Indicates that the i-th robot arm receives information from the leader in the s-th degree of freedom. When the i-th robot arm can receive information from the leader in the s-th degree of freedom, b i,s=1, otherwise b i,s =0,q L,s is the position of the reference manipulator in the sth degree of freedom, N is the total number of manipulators, For E i,s The first derivative of a ij,s It means that the i-th robot can receive information from the j-th robot in the s-th degree of freedom. ij,s =1, otherwise a ij,s =0, for q i,s The first derivative of Q j,s The first derivative of Q L,s The first derivative of .

[0035] Furthermore, S3 includes the following sub-steps:

[0036] S31: The design specifies the time performance function as:

[0037]

[0038] in, is the specified time performance function, is the initial value of the specified time performance function, is the predetermined domain of the specified time performance function, T a is the specified time, t is the current time;

[0039] S32: Design the self-adjusting function as:

[0040]

[0041] The derivative of it is:

[0042]

[0043] The self-tuning parameters are designed as:

[0044]

[0045] in, is a self-adjusting function, is a constant, is the self-adjusting parameter, T b For the specified time, for The first derivative of m is the positive parameter of the design, E i (T a ) is T b Error in time;

[0046] S33: Based on the specified time performance function and the self-adjustment function, a self-adjustment performance control method for the multi-manipulator system is obtained as follows:

[0047]

[0048] Among them, γ i,s (·) Specify performance functions for self-tuning of multi-manipulator systems.

[0049] Furthermore, the sliding mode control method of the multi-manipulator system with flexible constraint capability designed in S4 is:

[0050]

[0051] in, is the sliding mode variable, a 1 、a 2 、c 1 、c 2 and c 3 is a constant, sig(·) is a symbol, γ i,s is a self-adjusting performance function;

[0052] The derivative of it is:

[0053]

[0054] in F i,s =a 1 (d i +b i )H ti , is a sliding mode controller, for The first derivative of For E i,s The second-order derivative of for b i,s The first derivative of is the acceleration vector of the j-th robot, is the acceleration vector of the reference trajectory.

[0055] Furthermore, S5 includes the following sub-steps:

[0056] S51: Based on the sliding mode control method of the multi-manipulator system, the Bellman equation is derived using Leibniz's law:

[0057]

[0058]

[0059] in, is the gradient, r represents the integral variable, w is the attenuation coefficient, N i,s (·) is the cost function, is the partial derivative;

[0060] S52: Based on the Bellman equation, the Hamiltonian function is obtained as:

[0061]

[0062] Among them, H i,s (·) is the Hamiltonian function;

[0063] S53: Based on the Hamiltonian function and the optimal cost function, the Hamilton-Jacobi-Bellman equation is obtained;

[0064] S54: Based on the Hamilton-Jacobi-Bellman equation, by solving Define the optimal sliding mode controller;

[0065] S55: Use the critical neural network to approximate the defined optimal sliding mode controller, obtain the approximate optimal sliding mode controller, and complete the design of the optimal sliding mode controller for the multi-manipulator system.

[0066] Furthermore, the optimal cost function in S53 is:

[0067]

[0068] Among them, the superscript * indicates the optimal, τ i,s is the control rate, min is the minimum value, Ω(A) is the allowed control set for A, and r is the integral variable;

[0069] The Hamilton-Jacobi-Bellman equation is:

[0070]

[0071] Furthermore, the optimal sliding mode controller is defined in S54 as:

[0072]

[0073] Among them, Γ i,s is a positive number, is an unknown function.

[0074] Furthermore, the criticized neural network in S55 is:

[0075]

[0076] in, For unknown functions The estimated value of is the activation function vector, the superscript T is the transpose of the matrix, To criticize the neural network weight θ i,s An estimated value of

[0077] The approximately optimal sliding mode controller is:

[0078]

[0079] Furthermore, in said S55, reinforcement learning Bellman residual and experience replay technology are used to update the critic neural network in real time;

[0080] The reinforcement learning Bellman residual p i,s for:

[0081]

[0082] t 1 =tT w

[0083]

[0084] Among them, t 1 For a certain moment in the past, T w is a normal number, C i,s The estimated value of (·), and is the difference;

[0085] In order to minimize the Bellman residual of reinforcement learning, the following update rules are obtained using gradient descent method, parallel learning technology and experience replay technology:

[0086]

[0087] in, for The first derivative of 0,s is the learning rate, for In the past l The state of the moment, For p i,s In the past l The state at the moment, l is the number of indexes, and L is the total number of indexes;

[0088] according to get:

[0089]

[0090] in, for The first derivative of Fi,s In the past l The state at the moment, ξ i,s is the approximation error of the cost function, ξ i,s (t 1 ) is ξ i,s In t 1 status, For i,s In the past moment, for In t 1 status.

[0091] The beneficial effects of the present invention are:

[0092] (1) The present invention combines the traditional specified performance function and the self-adjusting function to design a self-adjusting performance function for the multi-manipulator, and enables the convergence boundary of the manipulator system to be automatically adjusted according to the system state, thereby ensuring better control performance of the multi-manipulator.

[0093] (2) The present invention designs a sliding mode control method with constraint capability based on self-adjusting performance and inverse tangent function, which not only realizes flexible performance constraints but also enhances the robustness of the multi-manipulator system.

[0094] (3) The present invention combines the gradient descent method, parallel learning technology and experience replay technology to construct a critical neural network dynamic programming controller, which not only reduces the amount of calculation but also achieves optimal control. BRIEF DESCRIPTION OF THE DRAWINGS

[0095] Figure 1 This is a flow chart of an optimal sliding mode control method for multi-manipulator adaptive dynamic programming.

[0096] Figure 2 Schematic diagram of the interaction of two-DOF multi-manipulators.

[0097] Figure 3 This is the trajectory tracking diagram of multiple robotic arms.

[0098] Figure 4 Error diagram of the multi-manipulator.

[0099] Figure 5 Estimated weight map of 1-critic neural network for multi-robot joints.

[0100] Figure 6 Estimated weight map of 2-critic neural network for multi-robot arm joints.

[0101] Figure 7 This is the error diagram of the multi-manipulator under the method of the present invention at 35-300s.

[0102] Figure 8This is the input diagram of the multi-manipulator in 35-300s under the method of the present invention.

[0103] Fig. 9 This is the error diagram of the multi-manipulator under the existing method in 35-300s.

[0104] Fig.10 This is the input diagram of multiple robotic arms in 35-300s under the existing method. DETAILED DESCRIPTION

[0105] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0106] like Figure 1 As shown, a multi-manipulator adaptive dynamic programming optimal sliding mode control method includes the following steps:

[0107] S1: Establish a 2-DOF dynamic model of the multi-manipulator system;

[0108] S2: Based on the 2-DOF dynamic model, define the 2-DOF position error of the multi-manipulator system;

[0109] S3: Based on the 2-DOF position error, design a self-adjusting performance control method for a multi-manipulator system based on a specified time performance function and a self-adjusting function;

[0110] S4: Based on the self-adjusting performance control method of the multi-manipulator system and the sliding mode control principle, a sliding mode control method for the multi-manipulator system with flexible constraints is designed;

[0111] S5: Based on the sliding mode control method and adaptive dynamic programming of multi-manipulator system, design the optimal sliding mode controller of the multi-manipulator system and complete the optimal sliding mode control of multi-manipulator adaptive dynamic programming.

[0112] The 2-DOF dynamic model in S1 is:

[0113]

[0114] Among them, M i (·) is the inertia matrix function of the i-th manipulator, C i (·) is the vector function of the centripetal force and Coriolis force of the i-th manipulator, G i (·) is the gravity vector function of the i-th manipulator, q i is the position vector of the i-th robot arm, is the velocity vector of the i-th robot arm, is the acceleration vector of the i-th robot arm, τ i is the input torque vector of the i-th robot arm, and d is the unknown bounded disturbance vector;

[0115] Combining the dynamic equations of the multi-manipulator, we can conclude that:

[0116]

[0117] For simplicity, let The expression can be rewritten as Let H ti Denotes the matrix H i The diagonal matrix of i =H ti +H Ti ,therefore, Rewritten as:

[0118]

[0119] Where Ψ i =H Ti τ i +H i D i +H i d;

[0120] Among them, H Ti =H i -H ti ,Ψ i is a nonlinear function;

[0121] The dynamic model of the i-th robotic arm is:

[0122]

[0123] in P i,2 =m i,2 r i,1 R i,2 , is the inertia matrix of the i-th robot arm, C i is the centripetal force and Coriolis force vector of the i-th robot arm, is the velocity vector of the i-th manipulator in the first degree of freedom, q i,2 is the position vector of the i-th robot arm in the second degree of freedom, is the velocity vector of the i-th manipulator in the second degree of freedom, J i,1 is the moment of inertia of the i-th robot arm in the first degree of freedom, J i,2 is the moment of inertia of the i-th robot arm in the second degree of freedom, m i,1 is the mass of the i-th manipulator in the first degree of freedom, m i,2 is the mass of the i-th robot in the second degree of freedom, R i,1 is the center of mass of the i-th manipulator in the first degree of freedom, Ri,2 is the center of mass of the i-th manipulator in the second degree of freedom, r i,1 is the length of the i-th robot arm in the first degree of freedom.

[0124] The 2-DOF position error of the multi-manipulator system is defined in S2, including the following sub-steps:

[0125] S21: using a directed graph χ=(K, E, A) to represent the relationship between the robot arms in exchanging state information;

[0126] A=[a ij ]∈R n×n

[0127] when a ij =1 means that robot i can receive information from robot j, otherwise a ij =0;

[0128] K={k 1 ,k 2 ,...,k n}

[0129] Among them, A is the communication between the manipulators, R represents the real number set, R n×n is an n×n dimensional Euclidean space, n is a constant, K is the set of followers, k 1 ,k 2 ,…,k n is the number of followers, E is the position error;

[0130] S22: Based on the relationship between the state information exchanged between the robotic arms, the 2-DOF position error of the multi-robotic arm system is obtained as:

[0131]

[0132] The derivative of it is:

[0133]

[0134] Among them, E i,s is the position error of the ith robot arm in the sth degree of freedom, q i,s is the position vector of the ith robot arm in the sth degree of freedom, q j,s is the position vector of the j-th robot arm in the s-th degree of freedom, b i,s Indicates that the i-th robot arm receives information from the leader in the s-th degree of freedom. When the i-th robot arm can receive information from the leader in the s-th degree of freedom, b i,s =1, otherwise b i,s =0,q L,s is the position of the reference manipulator in the sth degree of freedom, N is the total number of manipulators, For E i,s The first derivative of a ij,s It means that the i-th robot can receive information from the j-th robot in the s-th degree of freedom. ij,s =1, otherwise a ij,s =0, Q i,s The first derivative of Q j,s The first derivative of Q L,s The first derivative of .

[0135] The S3 includes the following sub-steps:

[0136] S31: The design specifies the time performance function as:

[0137]

[0138] in, is the specified time performance function, is the initial value of the specified time performance function, is the predetermined domain of the specified time performance function, T a is the specified time, t is the current time;

[0139] S32: Design the self-adjusting function as:

[0140]

[0141] The derivative of it is:

[0142]

[0143] The self-tuning parameters are designed as:

[0144]

[0145] in, is a self-adjusting function, is a constant, is the self-adjusting parameter, T b For the specified time, for The first derivative of m is the positive parameter of the design, E i (T a ) is T b Error in time;

[0146] S33: Based on the specified time performance function and the self-adjustment function, a self-adjustment performance control method for the multi-manipulator system is obtained as follows:

[0147]

[0148] Among them, γ i,s (·) Specify performance functions for self-tuning of multi-manipulator systems.

[0149] The sliding mode control method of the multi-manipulator system with flexible constraint capability designed in S4 is:

[0150]

[0151] in is the sliding mode variable, a 1 、a 2 、c 1 、c 2 and c 3 is a constant, sig(·) is a symbol, γ i,s is a self-adjusting performance function;

[0152] The derivative of it is:

[0153]

[0154] in F i,s =a 1 (d i +b i )H ti , is a sliding mode controller, for The first derivative of For E i,s The second-order derivative of for b i,s The first derivative of is the acceleration vector of the j-th robot, is the acceleration vector of the reference trajectory.

[0155] The S5 comprises the following sub-steps:

[0156] S51: Based on the sliding mode control method of the multi-manipulator system, the Bellman equation is derived using Leibniz's law:

[0157]

[0158] in, is the gradient, r represents the integral variable, w is the attenuation coefficient, N i,s (·) is the cost function, is the partial derivative;

[0159] S52: Based on the Bellman equation, the Hamiltonian function is obtained as:

[0160]

[0161] Among them, H i,s (·) is the Hamiltonian function;

[0162] S53: Based on the Hamiltonian function and the optimal cost function, the Hamilton-Jacobi-Bellman equation is obtained;

[0163] S54: Based on the Hamilton-Jacobi-Bellman equation, by solving Define the optimal sliding mode controller;

[0164] S55: Use the critical neural network to approximate the defined optimal sliding mode controller, obtain the approximate optimal sliding mode controller, and complete the design of the optimal sliding mode controller for the multi-manipulator system.

[0165] The optimal cost function in S53 is:

[0166]

[0167] Among them, the superscript * is optimal, τ i,s is the control rate, min is the minimum value, Ω(A) is the allowed control set for A, where A∈R n represents a compact set (in mathematics, a compact set means that any open cover in a set can find a finite subset covering the entire set. Simply put, a compact set is a set that can be covered by a finite number of open sets. For example, a finite interval or a finite curve is a compact set), r is the integral variable;

[0168] The Hamilton-Jacobi-Bellman equation is:

[0169]

[0170] Considered to be the best By solving The optimal controller can be defined as:

[0171]

[0172] Obviously, due to There are complex nonlinearities in the , and it is very difficult to directly calculate the optimal controller. In order to achieve the best performance control goal, is constructed as where Γ i,s Is a positive number.

[0173] Therefore, we can get:

[0174]

[0175] therefore, can be rewritten as:

[0176]

[0177] The optimal sliding mode controller is defined in S54 as:

[0178]

[0179] Among them, Γ i,s is a positive number, is an unknown function.

[0180] Considering and are unknown functions, so a critical neural network is used to approximate them. The unknown function is approximated as and where θ i,s is the ideal neural network parameter vector, is the activation function vector, ξ i,s represents the approximation error and is a constant.

[0181] The neural network is criticized in S55 as follows:

[0182]

[0183] in, For unknown functions The estimated value of is the activation function vector, the superscript T is the transpose of the matrix, To criticize the neural network weight θ i,s An estimated value of

[0184] The approximately optimal sliding mode controller is:

[0185]

[0186] In the S55, reinforcement learning Bellman residual and experience replay technology are used to update the critic neural network in real time;

[0187] By using reinforcement learning algorithms, we can obtain:

[0188]

[0189] Therefore, the reinforcement learning Bellman residual p i,s for:

[0190]

[0191] t 1 =tT w

[0192]

[0193] Among them, t 1 For a certain moment in the past, T w is a normal number, C i,s The estimated value of (·), and is the difference;

[0194] In order to minimize the Bellman residual of reinforcement learning, the following update rules are obtained using gradient descent method, parallel learning technology and experience replay technology:

[0195]

[0196] in, for The first derivative of 0,s is the learning rate, for In the past l The state of the moment, For p i,s In the past l The state at the moment, l is the number of indexes, and L is the total number of indexes;

[0197] according to get:

[0198]

[0199] in, for The first derivative of F i,s In the past l The state at the moment, ξ i,s is the approximation error of the cost function, ξ i,s (t 1 ) is ξ i,s In t 1 status, For i,s In the past moment, for In t 1 status.

[0200] In one embodiment of the present invention, the stability of the optimal sliding mode control method of multi-manipulator adaptive dynamic programming is proved based on steps S4 and S5, including the following steps:

[0201] S61: According to and V s Taking the derivative, we can get:

[0202]

[0203] Assumptions is a candidate for a continuously differentiable Lyapunov function that satisfies:

[0204]

[0205] in, Yes V s Along The gradient of

[0206] Therefore, there exists a positive definite matrix ψ that satisfies the following condition i,s ∈R n×n :

[0207]

[0208] Among them, η min (·) represents the smallest eigenvalue of the matrix.

[0209] You can get:

[0210]

[0211] S62: According to θ i,s and V 1 Taking the derivative, we get:

[0212]

[0213] According to Young's inequality, we can get:

[0214]

[0215] You can get:

[0216]

[0217]

[0218] You can get:

[0219]

[0220] S63: According to and V 2 Taking the derivative, we can get:

[0221]

[0222] S64: Define V = V s +V 1 +V 2 , taking the derivative of V, we can get:

[0223]

[0224] Finally, we can get:

[0225]

[0226] According to the standard Lyapunov extension theorem, the trajectories of multi-manipulator systems can be considered to be eventually uniformly bounded.

[0227] In one embodiment of the present invention, Figure 2 As shown in Figure 1, a two-DOF multi-manipulator system consisting of a leader and five followers is used to verify the effectiveness of the proposed method. Table 1 shows the initial state and parameters of the multi-manipulator system, and Table 2 shows the multi-manipulator model parameters:

[0228] Table 1 Initial state and parameters of multi-manipulator system

[0229]

[0230] Table 2 Multi-manipulator model parameters

[0231]

[0232] from Figure 3 and Figure 4 It can be seen that the controller designed by the present invention has good tracking performance, and the tracking error of the system can converge to the predetermined area within the predetermined time without violating the constraints, highlighting the independence of the predefined control scheme to the initial conditions and controller parameters and the robustness of the system. Figure 5 and Figure 6 The learning process of the multi-arm critic neural network estimating weights is shown, which eventually becomes stable.

[0233] Specifically, the performance constraint method shown in the present invention enhances the flexibility of the multi-manipulator state and has better control performance. First, in the initial period of time, due to the properties of the inverse tangent function, it can be seen that the system error is in an unconstrained state. After a period of time, when When Constraint. a Later, under the action of the self-adjusting function, the constraint boundary of the system error becomes smaller again, which makes the system error smaller again. bAfter that, the constraint boundary of the system error is designed by humans. and self-adjusting The structure makes the system error enter a smaller constraint range, thus obtaining better control performance.

[0234] In order to highlight the advantages of the sliding mode control method proposed in the present invention, this embodiment will be compared with the existing sliding mode control method. The existing sliding mode control method is as follows:

[0235]

[0236] Where a 2 , a 3 and c 4 is a constant.

[0237] Figure 7 and Figure 8 They respectively represent the error diagram and control input diagram within 35-300s under the control method of the present invention, Fig. 9 and Fig.10 They respectively represent the error graph and control input graph within 35-300s under the existing control method. By comparison, it can be seen that the control method proposed in the present invention can keep the multi-manipulator system stable, while the multi-manipulator system under the existing control method has serious jitter and poor control performance. In order to comprehensively evaluate the effectiveness of these two methods, three indicators are used to quantify their performance. They are the integral of the absolute value of the error, the integral of the time multiplied by the absolute value of the error, and the integral of the square value of the control input, which are defined as follows:

[0238]

[0239] In order to facilitate the comparison of the two methods, the average values ​​of the corresponding indicators of the five robotic arms are directly taken. The control performance indicators of the five robotic arms are shown in Table 3. It can be seen from Table 3 that the integral of the absolute value of the error and the integral of the time multiplied by the absolute value of the error of the control method proposed by the present invention are both the smallest, indicating that the tracking error of the control method proposed by the present invention is smaller than that of the existing control method, and the dynamic tracking accuracy is higher than that of the existing method; the integral index of the square value of the control input of the control method proposed by the present invention is the smallest, indicating that the control cost of the controller proposed by the present invention is the smallest. Therefore, compared with the existing control method, the control method proposed by the present invention not only has better tracking performance, but also can save more costs.

[0240] Table 3 Comparison of three control performance indicators of the two methods

[0241]

[0242] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the invention.

Claims

1. A multi-manipulator adaptive dynamic programming optimal sliding mode control method, characterized in that: The following steps are involved: S1: Establish a 2-DOF dynamic model of the multi-manipulator system; S2: Based on the 2-DOF dynamic model, define the 2-DOF position error of the multi-manipulator system; S3: Based on the 2-DOF position error, design a self-adjusting performance control method for a multi-manipulator system based on a specified time performance function and a self-adjusting function; S4: Based on the self-adjusting performance control method of the multi-manipulator system and the sliding mode control principle, a sliding mode control method for the multi-manipulator system with flexible constraints is designed; S5: Based on the sliding mode control method and adaptive dynamic programming of multi-manipulator system, design the optimal sliding mode controller of the multi-manipulator system and complete the optimal sliding mode control of multi-manipulator adaptive dynamic programming.

2. The multi-manipulator adaptive dynamic programming optimal sliding mode control method according to claim 1, characterized in that: The 2-DOF dynamic model in S1 is: For simplicity, let The expression is rewritten as: Let H ti Denotes the matrix H i The diagonal matrix of , we get: H i =H ti +H Ti ; therefore, Rewritten as: Among them, M i (·) is the inertia matrix function of the i-th manipulator, C i (·) is the vector function of the centripetal force and Coriolis force of the i-th manipulator, G i (·) is the gravity vector function of the i-th manipulator, q i is the position vector of the i-th robot arm, is the velocity vector of the i-th robot arm, is the acceleration vector of the i-th robot arm, τ i is the input torque vector of the i-th robot, d is the unknown bounded disturbance vector, Ψ i =H Ti τ i +H i D i +H i d; in M i is the inertia matrix of the i-th robot arm, C i is the centripetal force and Coriolis force vector of the i-th robot arm, is the velocity vector of the i-th manipulator in the first degree of freedom, q i,2 is the position vector of the i-th robot arm in the second degree of freedom, is the velocity vector of the i-th manipulator in the second degree of freedom, J i,1 is the moment of inertia of the i-th robot arm in the first degree of freedom, J i,2 is the moment of inertia of the i-th robot arm in the second degree of freedom, m i,1 is the mass of the i-th manipulator in the first degree of freedom, m i,2 is the mass of the i-th robot in the second degree of freedom, R i,1 is the center of mass of the i-th manipulator in the first degree of freedom, R i,2 is the center of mass of the i-th manipulator in the second degree of freedom, r i,1 is the length of the i-th robot arm in the first degree of freedom.

3. The multi-manipulator adaptive dynamic programming optimal sliding mode control method according to claim 1, characterized in that: The 2-DOF position error of the multi-manipulator system is defined in S2, including the following sub-steps: S21: using a directed graph χ=(K, E, A) to represent the relationship between the robot arms in exchanging state information; A=[a ij ]∈R n×n ; when a ij =1 means that robot i can receive information from robot j, otherwise a ij =0; K={k1,k2,...,k n }; Among them, A is the communication between the manipulators, R represents the real number set, R n×n is an n×n dimensional Euclidean space, n is a constant, K is the set of followers, k1, k2,…, k n is the number of followers, E is the position error; S22: Based on the relationship between the state information exchanged between the robotic arms, the 2-DOF position error of the multi-robotic arm system is obtained as: The derivative of it is: Among them, E i,s is the position error of the ith robot arm in the sth degree of freedom, q i,s is the position vector of the ith robot arm in the sth degree of freedom, q j,s is the position vector of the j-th robot arm in the s-th degree of freedom, b i,s Indicates that the i-th robot arm receives information from the leader in the s-th degree of freedom. When the i-th robot arm can receive information from the leader in the s-th degree of freedom, b i,s =1, otherwise b i,s =0,q L,s is the position of the reference manipulator in the sth degree of freedom, N is the total number of manipulators, For E i,s The first derivative of a ij,s It means that the i-th robot can receive information from the j-th robot in the s-th degree of freedom. ij,s =1, otherwise a ij,s =0, for q i,s The first derivative of for q j,s The first derivative of for q L,s The first derivative of .

4. The multi-manipulator adaptive dynamic programming optimal sliding mode control method according to claim 3 is characterized in that: The S3 includes the following sub-steps: S31: The design specifies the time performance function as: in, is the specified time performance function, is the initial value of the specified time performance function, is the predetermined domain of the specified time performance function, T a is the specified time, t is the current time; S32: Design the self-adjusting function as: The derivative of it is: The self-tuning parameters are designed as: in, is a self-adjusting function, is a constant, is the self-adjusting parameter, T b For the specified time, for The first derivative of m is the positive parameter of the design, E i (T a ) is T b Error in time; S33: Based on the specified time performance function and the self-adjustment function, a self-adjustment performance control method for the multi-manipulator system is obtained as follows: Among them, γ i,s (·) Specify performance functions for self-tuning of multi-manipulator systems.

5. The multi-manipulator adaptive dynamic programming optimal sliding mode control method according to claim 4 is characterized in that: The sliding mode control method of the multi-manipulator system with flexible constraint capability designed in S4 is: in is the sliding mode variable, a1, a2, c1, c2 and c3 are constants, sig(·) is the symbol, γ i,s is a self-adjusting performance function; The derivative of it is: in F i,s =a1(d i +b i )H ti , is a sliding mode controller, for The first derivative of For E i,s The second-order derivative of for b i,s The first derivative of is the acceleration vector of the j-th robot, is the acceleration vector of the reference trajectory.

6. The multi-manipulator adaptive dynamic programming optimal sliding mode control method according to claim 5, characterized in that: The S5 comprises the following sub-steps: S51: Based on the sliding mode control method of the multi-manipulator system, the Bellman equation is derived using Leibniz's law: in, is the gradient, r represents the integral variable, w is the attenuation coefficient, N i,s (·) is the cost function, is the partial derivative; S52: Based on the Bellman equation, the Hamiltonian function is obtained as: Among them, H i,s (·) is the Hamiltonian function; S53: Based on the Hamiltonian function and the optimal cost function, the Hamilton-Jacobi-Bellman equation is obtained; S54: Based on the Hamilton-Jacobi-Bellman equation, by solving Define the optimal sliding mode controller; S55: Use the critical neural network to approximate the defined optimal sliding mode controller, obtain the approximate optimal sliding mode controller, and complete the design of the optimal sliding mode controller for the multi-manipulator system.

7. The multi-manipulator adaptive dynamic programming optimal sliding mode control method according to claim 6 is characterized in that: The optimal cost function in S53 is: Among them, the superscript * indicates the optimal, τ i,s is the control rate, min is the minimum value, Ω(A) is the allowed control set for A, and r is the integral variable; The Hamilton-Jacobi-Bellman equation is:

8. The multi-manipulator adaptive dynamic programming optimal sliding mode control method according to claim 7, characterized in that: The optimal sliding mode controller is defined in S54 as: Among them, Γ i,s is a positive number, is an unknown function.

9. The multi-manipulator adaptive dynamic programming optimal sliding mode control method according to claim 8, characterized in that: The neural network is criticized in S55 as follows: in, For unknown functions The estimated value of is the activation function vector, the superscript T is the transpose of the matrix, To criticize the neural network weight θ i,s An estimated value of The approximately optimal sliding mode controller is:

10. The multi-manipulator adaptive dynamic programming optimal sliding mode control method according to claim 9, characterized in that: In the S55, reinforcement learning Bellman residual and experience replay technology are used to update the critic neural network in real time; The reinforcement learning Bellman residual p i,s for: t1=tT w ; Among them, t1 is a certain moment in the past, T w is a normal number, C i,s The estimated value of (·), and is the difference; In order to minimize the Bellman residual of reinforcement learning, the following update rules are obtained using gradient descent method, parallel learning technology and experience replay technology: in, for The first derivative of 0,s is the learning rate, for In the past l The state of the moment, For p i,s In the past l The state at the moment, l is the number of indexes, and L is the total number of indexes; according to get: in, for The first derivative of F i,s In the past l The state at the moment, ξ i,s is the approximation error of the cost function, ξ i,s (t1) is ξ i,s In the state of t1, For i,s In the past moment, for State at t1.

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