Mechanical arm iterative learning control method based on radial basis function network

Abstract

The invention discloses a mechanical arm iterative learning control method based on a radial basis function network. The method solves the limitation that a traditional mechanical arm control method deals with nonlinear characteristics and external disturbance problems in a complex dynamic environment. The method provided by the invention comprises the following steps: firstly, constructing a mechanical arm system dynamic model, and dynamically correcting and optimizing a reference track by adopting a dynamic correction strategy; then designing a radial basis function neural network to construct a nonlinear compensation item, and designing a dynamic weight parameter and a dynamic learning gain for optimizing the performance of the controller; and finally, designing an iterative learning controller, and verifying the stability and error convergence of a control algorithm. Through the radial basis function network, the dynamic adjustment strategy and the iterative learning strategy, the control precision, the adaptive capacity and the error convergence speed of the mechanical arm system are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of robot arm control and nonlinear control, and in particular to a robot arm iterative learning control method based on a radial basis function network. Background Art

[0002] Efficient control of robotic arms is one of the core research directions in the field of automation, and the key lies in achieving high-precision trajectory tracking. In this context, iterative learning control is widely used in repetitive task scenarios because it can dynamically adjust and gradually optimize control performance using historical data. However, traditional iterative learning control methods have limited performance when dealing with nonlinear systems, external disturbances, and measurement noise. In addition, existing methods often lack effective modeling and control methods for unknown nonlinear characteristics in complex dynamic systems, which limits their robustness and accuracy in practical applications.

[0003] Among them, the trajectory alignment problem is one of the key challenges for the robot to achieve high-precision control. Existing technologies usually rely on random modeling methods with strict assumptions (such as Markov chains) to describe trajectory changes and alignment conditions. However, these methods not only rely on the assumption that the probability distribution is known, but are also difficult to meet in practical applications. On the other hand, the rule design with fixed or decreasing learning gain in traditional iterative learning control methods also has limitations. Although simple to implement, this rule lacks the ability to adapt to dynamic changes in the system when faced with complex nonlinearities and external disturbances, which may lead to low learning efficiency and even cause system divergence, further affecting the robustness and stability of iterative learning control.

[0004] Therefore, it is necessary to study new control strategies to address the trajectory alignment problems in robotic arm systems and the limitations of traditional methods, so as to improve the trajectory tracking accuracy and robustness of nonlinear systems under external disturbances, thereby meeting application requirements in complex scenarios. Summary of the invention

[0005] In view of the problems existing in the above-mentioned prior art, the present invention proposes an iterative learning control method for a manipulator based on a radial basis function network. Compared with the prior art, this method fully considers the influence of the nonlinear dynamic behavior of the system, external disturbances and unmodeled dynamics, and aims to overcome the limitation of the traditional control method in the complex dynamic environment, and improve the trajectory tracking accuracy and the adaptability of the system.

[0006] In order to achieve the above technical objectives, the present invention provides the following technical solutions:

[0007] The iterative learning control method of a robotic arm based on a radial basis function network specifically includes the following steps:

[0008] S1. Construct the dynamic model of the robot system, clarify the dynamic characteristics of the robot system and set applicable assumptions;

[0009] S2. Combined with the initial trajectory data of the robot arm, a dynamic correction strategy is adopted to iteratively optimize the reference trajectory of the robot arm multiple times and define the trajectory error; at the same time, ensure that the starting and target states of the reference trajectory meet the task requirements and optimize the smoothness and accessibility of the trajectory path;

[0010] S3. Design a radial basis function (RBF) network and construct a nonlinear compensation term that adapts to the dynamic characteristics by selecting the center position and width parameters of the basis function to compensate for the influence of unmodeled dynamics and external disturbances on the robotic arm system.

[0011] S4. Design estimated dynamic weight parameters and dynamic learning gains to optimize the response speed and output smoothness of the controller during the iteration process, reduce the oscillation problem of the control input, and ensure the gradual convergence of the weight error;

[0012] S5. Combining feedback control, nonlinear compensation and iterative learning methods, design an iterative learning controller to make full use of historical data, adjust the control law in real time, and gradually reduce the trajectory error to enhance the adaptability of the control system;

[0013] S6. Based on Lyapunov stability theory and composite energy function CEF, the stability of the control system in a complex dynamic environment is analyzed, and the global asymptotic stability of the controller and the gradual convergence of the robot trajectory error are verified.

[0014] By means of the above technical solution, the method proposed in the present invention can be used for iterative learning control of robotic arm systems with different test lengths and alignment conditions, and has at least the following beneficial effects:

[0015] 1. The present invention achieves high-precision and fast error convergence performance of robot arm trajectory tracking by combining radial basis function network and dynamic adjustment mechanism.

[0016] 2. While ensuring trajectory tracking accuracy, the present invention can effectively compensate for unmodeled dynamics and external disturbances, providing strong adaptability and robustness support for robotic arm control in complex dynamic environments. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] The drawings described herein are used to provide a further understanding of the present application and constitute a part of the present application. The illustrative embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0018] Figure 1 It is a graph showing the trajectory changes of the robot arm in the 1st, 3rd, 5th and 20th iterations of the method proposed in the present invention; Figure 1 (a) is the curve diagram of the trajectory change in the first dimension. Figure 1 (b) is the curve diagram of the trajectory change in the second dimension;

[0019] Figure 2 It is a curve diagram showing the change of the mean value of the trajectory error modulus in the method proposed in the present invention in the first 20 iterations;

[0020] Figure 3 It is a dynamic response curve diagram of the nonlinear compensation term estimated in the method proposed in the present invention in the 1st, 3rd, 5th and 20th iterations;

[0021] Figure 4 It is a dynamic change curve diagram of the control input in the method proposed by the present invention in the 1st, 3rd, 5th and 20th iterations; Figure 4 (a) is the curve diagram of the change of control input in the first dimension. Figure 4 (b) is the curve diagram of the change of the control input in the second dimension;

[0022] Figure 5 A comparison curve diagram of the method proposed in the present invention and a variable gain iterative learning control method in terms of the mean value of the trajectory error modulus;

[0023] Figure 6 A comparison curve diagram of the trajectory error modulus mean between the method proposed by the present invention and a nonlinear control method including an error compensation factor;

[0024] Figure 7 A comparison curve diagram of the trajectory error modulus mean between the method proposed in the present invention and an improved adaptive iterative learning control method;

[0025] Figure 8 A comparison diagram of the control input between the method proposed in the present invention and an improved adaptive iterative learning control method; Figure 8 (a) is a comparison diagram of the control input in the first dimension. Figure 8 (b) is a comparison diagram of the control input in the second dimension;

[0026] Fig. 9 A comparison diagram of the method proposed by the present invention and an improved adaptive iterative learning control method in terms of the mean value of the control input modulus value;

[0027] Fig.10 The flowchart of the method proposed in the present invention is shown in FIG. DETAILED DESCRIPTION

[0028] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the following Figure 1-10It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0029] Although the steps in the present invention are arranged with numbers, they are not used to limit the order of the steps. Unless the order of the steps is clearly stated or the execution of a certain step requires other steps as a basis, the relative order of the steps can be adjusted. It is understood that the term "and / or" used in this article involves and covers any and all possible combinations of one or more of the associated listed items.

[0030] like Fig.10 As shown, the iterative learning control method for a manipulator based on a radial basis function network proposed in the present invention solves the problem of insufficient accuracy of the traditional manipulator control method in dealing with nonlinear dynamic compensation and disturbance influence. The method specifically includes the following steps:

[0031] S1. Construct the dynamic model of the robot system, clarify the dynamic characteristics of the robot system and set applicable assumptions;

[0032] As a preferred implementation, the mechanical arm system dynamics model constructed in step S1 is:

[0033]

[0034] in, Respectively represent the actual displacement, velocity and acceleration of the robot arm; is the number of iterative learning, t∈[0,T k ] is the time index; T k is the actual execution time of the kth iteration, T min is a constant and satisfies 0<T min ≤T k ≤T; is the inertia matrix, is the centripetal-Coriolis force matrix, is the gravity vector; τ k (t) is the torque input, i.e. the control input, d k (t) is the comprehensive disturbance vector including unmodeled dynamics and external disturbances;

[0035] In addition, two points should be explained for the above formula:

[0036] ① In the present invention, the actual displacement, velocity and acceleration of the robot arm correspond to the actual trajectory of the robot arm, the first-order derivative and the second-order derivative of the actual trajectory, respectively, and can be collectively referred to as trajectory variables. These three variables constitute the state vector of the robot arm system, which is used for subsequent trajectory error calculation and controller design; in addition, since the robot arm usually includes multiple joints, and the displacement, velocity and acceleration of each joint are independent, in this application, these three variables are actually in the form of multi-dimensional vectors, and each dimension in the vector corresponds to the displacement, velocity and acceleration of a joint, respectively. The present invention does not highlight each joint in order to ensure the mathematical simplicity of the dynamic equation;

[0037] ②, “τ k (t) is the torque input, that is, the "control input" is due to the fact that in the dynamic equation (emphasizing physical modeling) τ k (t) represents the torque acting on the joint of the robot arm, which directly affects the angular acceleration of the joint; therefore, if it is used in the dynamic equation, it is more accurate to use "torque input"; if it is used in the control method and algorithm description, it is more general to use "control input", indicating that the control method can be applied to different actuators.

[0038] Next, to simplify the expression, the time and state dependence symbols of the system are omitted without causing confusion; for example: k (t) as an example, some formulas in this embodiment are simplified to q k , and the meanings expressed by these two expressions are consistent;

[0039] Based on the simplified expression of the robot arm system dynamics model, in order to further clarify the system dynamics characteristics and assumptions involved in the present invention, the dynamics model must meet the following characteristics:

[0040] Characteristics 1. Inertia matrix M(q k ) is symmetric and positive definite, and its value remains bounded during the operation of the robot system;

[0041] Feature 2: Matrix is an antisymmetric matrix; that is, it satisfies:

[0042]

[0043] Where x is an arbitrary real matrix, x T is the transposed matrix of x; is the first-order derivative of the inertia matrix;

[0044] Feature 3: Centripetal-Coriolis Force Matrix and the gravity vector G(q k ) satisfies the following boundedness conditions:

[0045]

[0046] G(q k )≤κ g ;

[0047] Among them, κ c and κ g is a positive parameter;

[0048] On the basis of clarifying the dynamic characteristics, in order to ensure the rationality of the control strategy design and the rigor of the theoretical analysis, the following assumptions are introduced:

[0049] Assumption 1: The target trajectory and its first and second order time derivatives, i.e., q d , and and the comprehensive disturbance vector d k exist are bounded under the conditions of ;

[0050] Assumption 2: In each iterative learning control, the initial state of the robot system satisfies the alignment condition, that is, q k (0) = q k-1 (T k )and At this point, the dynamic model of the robotic arm system is completed.

[0051] It should be noted that in the iterative learning control of this embodiment, the robot arm trajectory can be divided into the following three types:

[0052] Target trajectory q d : The pre-set ideal trajectory, that is, the final trajectory that the robot hopes to achieve; q d It is generally fixed and not updated with iterations;

[0053] Reference trajectory q m,k : The optimized trajectory used for control in the iterative process is gradually adjusted at each iteration to approach the target trajectory q d ;q m,k In the present invention, the correction function is used to make the adjustment more consistent with the actual operation of the system;

[0054] Actual trajectory q k : The trajectory actually executed by the robot in the kth iteration; q k Affected by control input, unmodeled dynamics, external disturbances, etc., usually related to the target trajectory q d There is a deviation.

[0055] S2. Combined with the initial trajectory data of the robot arm, a dynamic correction strategy is adopted to iteratively optimize the reference trajectory of the robot arm multiple times and define the trajectory error; at the same time, ensure that the starting and target states of the reference trajectory meet the task requirements and optimize the smoothness and accessibility of the trajectory path;

[0056] As a preferred implementation, step S2 specifically includes:

[0057] S21. In this embodiment, under non-equal length test conditions, in order to meet the needs of iterative learning control and ensure the continuity and adaptability of dynamic adjustment of the trajectory, a trajectory optimization method based on a correction function is proposed; a correction function is constructed to dynamically optimize the reference trajectory of the manipulator, and the formula is expressed as:

[0058]

[0059] Among them, q m,k (t) represents the reference trajectory generated by the kth iteration optimization; and It is a second-order differentiable correction function; it is used to dynamically adjust the target trajectory to ensure smooth transition of the trajectory in different time periods;

[0060] S22. In order to achieve dynamic adjustment of the trajectory and ensure the design performance of the controller, it is necessary to set the constraints of the correction function; specifically, the following are included:

[0061] Setting the correction function The constraints are:

[0062] When t∈[T a ,T], and

[0063] is bounded on [0,T];

[0064] When t∈[T a ,T], and

[0065] When t∈[T a ,T], and

[0066] Setting the correction function The constraints are:

[0067] When t∈[T a ,T], and

[0068] is bounded on [0,T];

[0069] When t∈[T a ,T], and

[0070] When t∈[T a ,T], and

[0071] in, The correction functions are The first and second time derivatives of ; The correction functions are The first and second time derivatives of T a is the design parameter, satisfying 0<T a ≤T k ; The reference trajectory q generated in each iteration m,k The starting state of (t) is consistent with the end state of the previous iteration, that is, q m,k (0) = q m,k-1 (T k )and

[0072] S23. To quantitatively describe the deviation between the actual trajectory of the system and the reference trajectory and provide a theoretical basis for controller design and performance analysis, this embodiment defines trajectory error, which includes displacement error and speed error The formula is:

[0073]

[0074] Among them, q m,k (t) represents the reference trajectory generated by the kth iteration optimization, q k (t) represents the actual trajectory during the kth iteration; and They represent the speed of the robot on the reference trajectory and the actual trajectory in the kth iteration, that is, q m,k (t), q k The first derivative of (t).

[0075] S3. Design a radial basis function (RBF) network and construct a nonlinear compensation term that adapts to the dynamic characteristics by selecting the center position and width parameters of the basis function to compensate for the influence of unmodeled dynamics and external disturbances on the robotic arm system.

[0076] As a preferred implementation, step S3 specifically includes:

[0077] S31. In the dynamics model of the robot system, the dynamic characteristics of the trajectory error may be affected by the comprehensive disturbance vector d k , gravity vector G(q k) and control input and other factors; In order to constrain the dynamic behavior of the error and ensure the control performance of the system, the method proposed in the present invention introduces a nonlinear compensation term to describe the dynamic characteristics of the trajectory error and impose constraints; First, according to the dynamic properties, the dynamic model of the manipulator system is reconstructed as:

[0078]

[0079] Combined with the optimized reference trajectory and error definition, the above formula can be further expressed as

[0080]

[0081] Then, based on the nonlinear compensation term, the following constraints are established for the trajectory error:

[0082]

[0083] Among them, α k (t) is the real nonlinear compensation term; is the velocity error, sign(.) is the sign function; q k They represent the acceleration of the manipulator on the reference trajectory and the actual trajectory of the manipulator during the kth iteration, respectively; ∈ is the error smoothing parameter; the left side of the constraint inequality It reflects the dynamics of trajectory error and the system inertia matrix and gravity vector G(q k ), and the comprehensive perturbation vector d k The relationship between the real nonlinear compensation term α k (t) and the sign function sign(·) impose constraints on the error, which can ensure that the influence of the speed error is not too large and gradually converges with the iteration;

[0084] More specifically, according to known inequalities, for any ∈>0 and χ∈R, there is always:

[0085]

[0086] The constraints on the velocity error based on the nonlinear compensation term can be further divided into:

[0087]

[0088] S32. In order to accurately express and compensate for the nonlinear dynamic behavior and disturbance of the robotic arm system, this example uses the RBF network to compensate for the real nonlinear compensation term α k (t) is approximated and the formula is expressed as:

[0089] α k (t) = θ T (t)μ(Ψ(t),t)+ε;

[0090] in, is the input feature vector, q k They represent the acceleration of the robot on the reference trajectory and the actual trajectory of the robot during the kth iteration respectively; is the ideal weight parameter of the RBF network, but it is usually unknown in actual calculations. p is the number of nodes of the basis function, ε is the approximation error, and increasing the basis function node p can reduce the approximation error ε to a sufficiently small value; is the basis function vector of the RBF network; each basis function is defined as:

[0091]

[0092] Among them, Λ i , γ i are the center and width of the basis function respectively; the RBF network can dynamically adapt to the characteristics of the error and estimate the nonlinear characteristics of the system and its changes by learning the basis function vector and weight parameter parameters.

[0093] S4. Design estimated dynamic weight parameters and dynamic learning gains to optimize the response speed and output smoothness of the controller during the iteration process, reduce the oscillation problem of the control input, and ensure the gradual convergence of the weight error;

[0094] As a preferred implementation, step S4 specifically includes:

[0095] S41, designing estimated dynamic weight parameters to achieve dynamic adjustment and gradual optimization of weight parameters in radial basis function RBF network in the iterative process; the update rule of the estimated dynamic weight parameters is:

[0096]

[0097] in, The dynamic weight parameters estimated in the kth and k-1th iterations respectively; set the initial estimated dynamic weight parameters Ensure that the algorithm is optimized from scratch; through iterative learning, Gradually approach θ(t) to achieve optimal control; μ(Ψ(t), t) is used as the basis function vector of the RBF network, which significantly improves the fitting ability and flexibility of the system by embedding the error characteristics; dynamic learning gain η k (t)>0 is responsible for adjusting the speed of parameter update, thereby effectively controlling the speed and stability of the convergence process; tanh(.) is a hyperbolic sine function, is the speed error in the kth iteration, ∈ is the error smoothing parameter, where Used to limit the impact range of trajectory error to avoid excessive errors affecting adjustment stability;

[0098] S42. The method proposed in the present invention is to match the dynamic weight parameters and design a dynamic learning gain at the same time; that is, to design a dynamic adjustment and gradual optimization mechanism, and to control the iterative process to take into account both efficiency and stability by adjusting the dynamic learning gain at different moments in the iterative process;

[0099] The adjustment rule of dynamic learning gain is:

[0100]

[0101] Among them, η k (t) is the dynamic learning gain during the kth iteration; η0>0 is the initial gain, Δ η is the gain, which can be any large value and is used to accelerate the initial parameter learning; k (t) is the gain parameter in the kth iteration process, which ensures the stability of the adjustment process;

[0102] The specific definition of the gain parameter is:

[0103]

[0104] Among them, the initial value ζ1(t) = 1, indicating that the system starts from the most basic adjustment starting point; in the dynamic adjustment process, the initial learning gain η k (t) and a lower incremental parameter ζ k (t), so that the system can quickly respond to errors and achieve initial adjustments; in the middle and late stages, with the increasing parameter ζ k (t) gradually increases, the learning gain η k (t) Automatically reduce, and the adjustment range gradually decreases, so as to ensure that the adjustment process of the system tends to be stable and ensure the robustness and accuracy of the overall control;

[0105] The method proposed in the present invention designs the estimated dynamic weight parameters and dynamic learning gain η k (t), realizing adaptive adjustment of the dynamic characteristics of trajectory error; thus, the controller can dynamically optimize parameters according to the real-time error, thereby improving the trajectory tracking accuracy and adaptability to disturbances of the robot system.

[0106] S5. Combining feedback control, nonlinear compensation and iterative learning methods, design an iterative learning controller to make full use of historical data, adjust the control law in real time, and gradually reduce the trajectory error to enhance the adaptability of the control system;

[0107] As a preferred implementation, according to the contents described in steps S1-S4, the design of the robot arm system controller in this embodiment includes the following steps:

[0108] First, by analyzing the dynamic characteristics of the robotic arm system, a suitable feedback gain matrix K is selected. P and K D , to adjust the dynamic response of displacement error and velocity error;

[0109] Next, the nonlinear compensation term is designed, and the basis function vector is constructed using the RBF network. By selecting the appropriate basis function center and width, the network is ensured to have sufficient approximation ability. At the same time, the estimated dynamic weight parameters are initially optimized to zero so as to achieve dynamic adjustment in subsequent iterations.

[0110] Then, the current reference trajectory is dynamically corrected based on the trajectory data in each iteration, and the starting point is ensured to be consistent with the end point of the previous round;

[0111] Finally, the control input is calculated using the trajectory error of the current iteration, and the real-time update of the control law is achieved by combining the feedback term and the nonlinear compensation term.

[0112] More specifically, the specific structure of the control law is expressed as:

[0113]

[0114] Among them, τ k (t) is the control input in the kth iteration, K P and K D are the positive feedback gain matrices of displacement error and velocity error respectively, and represent the displacement error and velocity error of the kth iteration respectively; is the estimated dynamic nonlinear compensation term, the hyperbolic tangent function It is used to limit the amplitude of the compensation term, ∈ is the error smoothing parameter, which is used to limit the influence range of the trajectory error;

[0115] The estimated dynamic nonlinear compensation term is calculated by the RBF network, namely:

[0116]

[0117] in, is the estimated dynamic weight parameter of the RBF network; μ(Ψ(t), t) is the basis function vector of the RBF network, and Ψ(t) is the input feature vector of the RBF network;

[0118] Based on the control law, the controller combines the synergy of feedback control, nonlinear compensation and iterative learning to dynamically adjust the trajectory error and ensure that the error gradually converges; the dynamic equation of the trajectory error is expressed as:

[0119]

[0120] In this embodiment, the inertia matrix M in the dynamics model of the robotic arm system -1 (q k ), centripetal-Coriolis force matrix Gravity term G(q k ) and the perturbation vector d k The controller adjusts the error dynamic response through feedback control and uses the estimated dynamic nonlinear compensation term generated by the RBF network. Realize real-time compensation for unmodeled dynamics and external disturbances, ultimately ensuring the gradual reduction of displacement error and velocity error, and improving the trajectory tracking accuracy and robustness of the system;

[0121] The controller combines feedback control and nonlinear compensation to accurately adjust trajectory errors and improve tracking performance. It also dynamically responds to external disturbances and unmodeled dynamics to enhance system adaptability. It makes full use of historical data through iterative learning methods to accelerate error convergence and demonstrate dynamic response capabilities that can flexibly adapt to different task requirements, thereby ensuring the efficiency and stability of the control system.

[0122] At this point, the dynamic model and control law of the robotic arm system in the method proposed in the present invention have been constructed; and in the design of the iterative learning control method of the robotic arm, verifying its stability and error convergence is the key to ensuring the feasibility and theoretical integrity of the system; the stability analysis aims to prove that the control law guarantees the global asymptotic stability of the system in a complex dynamic environment; the error convergence analysis focuses on whether the trajectory error (displacement error and velocity error) can converge to zero when the number of iterations tends to infinity; for this reason, the present invention proposes the use of Lyapunov stability theory combined with the composite energy function CEF to strictly derive and prove the stability and convergence of the control algorithm.

[0123] S6. Based on Lyapunov stability theory and composite energy function CEF, analyze the stability of the control system in a complex dynamic environment, and verify the global asymptotic stability of the controller and the gradual convergence of the robot trajectory error;

[0124] As a preferred implementation, step S6 specifically includes:

[0125] S61. Construct a CEF including displacement error, velocity error and network weight error as a Lyapunov function to analyze the overall energy change of the robotic arm system. To simplify the expression, the time and state dependent symbols of the robotic arm system are omitted. The constructed CEF is expressed by the formula:

[0126] V k (t) = V 1,k (t)+V2,k (t)+V 3,k (t);

[0127] Among them, V 1,k (t) represents the inertia matrix M(q k ) and speed error The related error energy term is used to characterize the effect of error dynamics on the system energy; V 2,k (t) represents the displacement error term, which is used to describe the displacement error in combination with the feedback gain in the control law. V 3,k (t) represents the weight error term, which reflects the estimated weight in the RBF network The deviation from the ideal weight θ(t);

[0128] The error energy term, displacement error term and weight error term are expressed as follows:

[0129]

[0130] Among them, the weight error s is an auxiliary scalar and satisfies Used to regulate the dynamic behavior of the composite energy function; its core design intention is to ensure the rapid decay of the energy function, weaken the cumulative impact of historical errors, and meet the convergence conditions of the Lyapunov function, thereby achieving the global stability of the system and effective convergence of errors;

[0131] It should be noted that in the method proposed in the present invention, θ(t) is used as an ideal weight parameter, but it is usually unknown; the controller performs online estimation through iterative learning and adjusts Make it gradually approach θ(t); the error term reflects the gap between the estimated weight and the ideal weight. Mathematically, θ(t) and The two exist at the same time, but the estimation calculation of controller iterative learning can only use θ(t) is only used as a theoretical reference; through iterative optimization, the error between the two Gradually decrease, and finally Converge to θ(t), ensuring the performance of the control system.

[0132] S62. Combined with the dynamics model and control law of the robot system, according to the Lyapunov stability theory, the kinetic energy error term, displacement error term and weight error term are derived respectively, the monotonic boundedness of the composite energy function is verified, and the stability of the robot system and the convergence of the robot trajectory error are analyzed;

[0133] The specific derivation process is as follows:

[0134] S621. Analyze the energy change of the system at the kth iteration, and its difference is defined as:

[0135] ΔV k (t) = ΔV 1,k (t)+ΔV 2,k (t)+ΔV 3,k (t);

[0136] Where, ΔV 1,k (t) is the same as the inertia matrix M(q k ) and speed error The associated error energy term, ΔV 2,k (t) is the energy change of the displacement error term, ΔV 3,k (t) is the energy change of the weight error term;

[0137] Then, this embodiment performs a detailed analysis of the energy change of each error term;

[0138] S622, analyze the energy change of the kinetic energy error term; according to the dynamics model of the manipulator system and the properties of the Lyapunov function, the error energy term ΔV 1,k The mathematical expression of (T) is as follows:

[0139]

[0140] According to Assumption 2, in the iterative learning process, it is assumed that the initial state of each iteration is consistent with the terminal state of the previous iteration, that is, and Then V 1,k (0) and the energy V at the end of the previous iteration 1,k-1 (T k-1 ) can establish a relationship, namely V 1,k (0) = V 1,k-1 (T k-1 );

[0141] Then replace the derivative in the integral term of the above equation with the corresponding dynamic expression, and according to the velocity error constraint inequality Scaling it, we finally get:

[0142]

[0143] Among them, α k represents the true nonlinear compensation term, which is usually unknown, and is the nonlinear compensation term estimated by the controller through the RBF network. is the nonlinear compensation term estimated by the RBF network and the true nonlinear compensation term α k The deviation between them defines the nonlinear compensation term error This error term is used in Lyapunov convergence analysis to ensure that as the optimization is iterated, Gradually reduce Converges to α k , to ensure control accuracy; and α k The relationship also refers to and θ(t); in addition, δ k =0.2785n∈η k α k .

[0144] S623, analyze the energy change of the displacement error term; the energy change of the displacement error term ΔV 2,k The mathematical expression of (t) is as follows:

[0145] ΔV 2,k (T) = V 2,k (T k )-V 2,k-1 (T k-1 );

[0146] According to Assumption 2, V 2,k (0) = V 2,k-1 (T k-1 ), and combined with the definition of the energy function derivative, the energy change ΔV of the process reference kinetic energy error term is derived 1,k (T); In addition, due to η k The effect time is limited, e -st The rapid decay of the displacement error As η decreases, its influence on the overall energy change gradually weakens; therefore, η can be reasonably ignored. k The secondary contribution of , focuses on the core role of displacement error and feedback gain; the derivation process of the energy change of the displacement error term is finally obtained as:

[0147]

[0148] S624, analyze the energy change of the weight error term; the derivation of the energy change of the weight error term is based on the dynamic characteristics of the weight error. By gradually decomposing the integral term and combining the constraints of the Lyapunov energy function, the nonlinear mapping ability of the RBF network is used to constrain the influence range of the weight adjustment, and finally ΔV can be derived. 3,k (T) has an upper bound. The specific derivation process is:

[0149]

[0150] S625: According to steps S621-S624, the composite energy function change ΔV can be obtained. k (t) satisfies the following inequality:

[0151]

[0152] From the inequality, we know that ΔV k (t) is a non-increasing sequence during the iteration process; and when it satisfies When ΔV k (t)≤0; therefore, as long as V1(t) is bounded, the composite energy function V k (t) It remains bounded throughout the iteration process.

[0153] Therefore, in this embodiment, we further prove that V1(t) is bounded; the specific process includes:

[0154] 1) Define the derivative of V1(t) as:

[0155]

[0156] Substituting the control law and feedback gain, we can get the upper bound form after sorting:

[0157]

[0158] 2) Analyze V1(t) using the integral inequality; at the same time, considering that V1(0) = 0, we can further conclude that:

[0159]

[0160] Therefore, a parameter M is introduced to represent the maximum energy change of the system, so that:

[0161]

[0162] It is proved that V1(t) The interior is bounded, that is:

[0163]

[0164] According to the inequality satisfied by the change of the composite energy function in step S625, we obtain:

[0165]

[0166] Since V k The non-negativity of (t) leads to:

[0167]

[0168] This inequality shows that as the number of iterations k increases, the accumulated trajectory error energy will gradually approach zero, that is, Therefore, we can further conclude that Based on the continuity of error dynamics, the L2 norm of the displacement error will also tend to zero with the number of iterations, that is, So far, the stability and convergence of the control method proposed in the present invention have been verified.

[0169] In order to enable those skilled in the art to better understand the implementation of the present invention, this embodiment will use Matlab software to perform simulation experiments to verify the reliability of the method proposed in the present invention.

[0170] The specific information of the simulation software is as follows:

[0171] Software name: MATLAB;

[0172] Version information: 9.8.0.1380330 (R2020a) Update 2;

[0173] License number: 919961;

[0174] Operating system: Microsoft Windows 11 Home Chinese Edition Version 11.0 (Build 22621.2861);

[0175] Java version: Java 1.8.0_202-b08with Oracle Corporation Java HotSpot(TM)64-Bit Server VM mixed mode;

[0176] Special toolboxes: Statistics and Machine Learning Toolbox-11.7(R2020a), Aircraft Control Toolbox-1.0;

[0177] In this embodiment, in order to verify the effectiveness of the iterative learning control method for a robotic arm based on a radial basis function network proposed in the present invention and its adaptability in a complex dynamic environment, multiple groups of simulation experiments were carried out, and a comparative analysis was performed with three existing methods.

[0178] In order to verify the effectiveness of the proposed method and its adaptability in complex dynamic environments, multiple simulation experiments based on a two-degree-of-freedom robotic arm system were conducted. The experiment adopted a classical dynamic model, and the specific forms of its inertia matrix M, centripetal-Coriolis force matrix C and gravity vector G are as follows:

[0179]

[0180] The comprehensive disturbance vector d = [d1, d2] TSet d1 = d2 = rand(k)sin(t) to simulate unknown disturbances in complex dynamic environments, where rand(k) is a random function with a value range of [0,1]. Target trajectory q d By q d,1 and q d,2 Composition, that is Set the execution time T = 1s, and the first iteration is when the robot arm's initial state is q1(0) = [0,0] T . Introducing non-equal length test conditions, assuming that T k Uniformly distributed in the interval [0.7,1] with probability

[0181] The control law adopts the iterative learning control algorithm proposed in the present invention, and the parameters include the initial learning gain η0=10, the gain Δ η =1000, feedback gain matrix K of displacement error P =diag{500,500}, positive feedback gain matrix K of velocity error D =diag{150,500}, error smoothing parameter∈=0.1;

[0182] The correction functions are: where t∈[0,T a ), T a =0.1s.

[0183] The RBF neural network contains 6 basis functions, the central parameter Λ i They are Λ1=[1,1,1,1] T ,Λ2=[1,1,-1,-1] T ,Λ3=[1,-1,1,-1] T ,Λ4=[-1,-1,-1,-1] T ,Λ5=[-1,-1,1,1] T ,Λ6=[-1,1,-1,1] T ; The width parameter is γ i =6;

[0184] In addition, using φ e Indicates displacement error modulus The average value of all sampling points reflects the trajectory tracking accuracy of the robotic arm system; τ represents the control input modulus ||τ k (t)||The average value at all sampling points is used to evaluate the smoothness of the control input; the sampling period is set to T=0.001s.

[0185] By analyzing the trajectory q of the 1st, 3rd, 5th and 20th iterations k(t)(q k,1 ,q k,2 denote the trajectory of the first dimension and the second dimension respectively) and the control input τ k (t) Record and draw the corresponding trend graph. The results are as follows Figure 1 and Figure 4 In addition, Figure 2 Shows the first 20 iterations of φ e The change curve of Figure 3 The estimated nonlinear compensation term is shown The changing trend of Figure 1 and Figure 2 It can be seen that the trajectory error converges quickly, indicating that the system has high tracking performance; Figure 3 and Figure 4 It shows that the estimated nonlinear compensation term and control input gradually stabilize, and there is no obvious jitter phenomenon in the control law. The above simulation results verify the effectiveness of the control method proposed in the present invention. By introducing the dynamic gain adjustment mechanism and nonlinear compensation term, the tracking accuracy and control stability of the system are effectively improved, while adapting to complex dynamic environments.

[0186] In order to verify the superiority of the control method proposed in the present invention, a comparative simulation analysis is carried out between the method and three existing methods, specifically including the following contents:

[0187] Existing method 1: a variable gain iterative learning control method

[0188] This method uses the initial condition q k (0) = [0, 1] T In the case of , the iterative learning control method based on variable gain is adopted. Its control law expression is:

[0189]

[0190] Among them, S t is the sampling time index, ζ k (S t ) is the dynamic gain parameter. The simulation results are shown in Figure 5 As shown, compared with the mean value of the trajectory error modulus of the first 20 iterations, φ e , the method of the present invention shows a faster convergence speed and a lower error level.

[0191] Existing method 2: A nonlinear control method including error compensation factor

[0192] For non-equal length test conditions, a nonlinear control method including error compensation factor is adopted, and the design of the control law is based on the following form:

[0193]

[0194] in, is part of the system dynamics, including the inertia matrix M(q k ), the centripetal-Coriolis force matrix and the gravity vector G(q k (t)). Compensation factor The update rules are as follows:

[0195]

[0196] The simulation results are as follows Figure 6 As shown, compared with the mean value of the trajectory error modulus of the first 20 iterations, φ e ,The method of the present invention is significantly superior to the second method and exhibits higher trajectory tracking accuracy and dynamic adaptability.

[0197] Existing method 3: an improved adaptive iterative learning control method

[0198] Assuming that the operation can always be fully executed, an improved adaptive iterative learning control method is adopted, and its control law is:

[0199]

[0200] in, The update rule is:

[0201]

[0202] Figure 7 The mean value of trajectory error modulus φ of different methods in the first 20 iterations is shown. e The results show that the proposed method is superior to the existing method 3 in terms of the convergence of the trajectory error modulus and the final error, showing a faster convergence speed and a lower error level. Figure 8 Shows the control input τ for different methods at iteration 20 k (t). The results show that by introducing the hyperbolic tangent function, the method of the present invention can effectively suppress the input jitter phenomenon, thereby significantly improving the smoothness of the control input. Fig. 9 Shows the mean control input modulus φ for different methods in the first 20 iterations τ The results show that the proposed method successfully limits the variation of the control input amplitude through the dynamic adjustment mechanism of the dynamic learning gain, thus avoiding the over-learning problem.

[0203] In summary, the control method proposed in the present invention has superior performance in trajectory tracking accuracy, control input smoothness and control input amplitude optimization, and also has effectiveness and adaptability in complex dynamic environments.

[0204] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

[0205] In addition, it should be understood that although the present specification is described according to implementation modes, not every implementation mode contains only one independent technical solution. This description of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation modes that can be understood by those skilled in the art.

Claims

1. A robotic arm iterative learning control method based on a radial basis function network, characterized in that: The specific steps include: S1. Construct the dynamic model of the robot system, clarify the dynamic characteristics of the robot system and set applicable assumptions; S2. Combined with the initial trajectory data of the robot arm, a dynamic correction strategy is adopted to iteratively optimize the reference trajectory of the robot arm multiple times and define the trajectory error; at the same time, ensure that the starting and target states of the reference trajectory meet the task requirements and optimize the smoothness and accessibility of the trajectory path; S3. Design a radial basis function (RBF) network and construct a nonlinear compensation term that adapts to the dynamic characteristics by selecting the center position and width parameters of the basis function to compensate for the influence of unmodeled dynamics and external disturbances on the robotic arm system. S4. Design estimated dynamic weight parameters and dynamic learning gains to optimize the response speed and output smoothness of the controller during the iteration process, reduce the oscillation problem of the control input, and ensure the gradual convergence of the weight error; S5. Combining feedback control, nonlinear compensation and iterative learning methods, design an iterative learning controller to make full use of historical data, adjust the control law in real time, and gradually reduce the trajectory error to enhance the adaptability of the control system; S6. Based on Lyapunov stability theory and composite energy function CEF, the stability of the control system in a complex dynamic environment is analyzed, and the global asymptotic stability of the controller and the gradual convergence of the robot trajectory error are verified.

2. The iterative learning control method for a robotic arm based on a radial basis function network according to claim 1, characterized in that: The dynamic model of the robotic arm system constructed in step S1 is: in, Respectively represent the actual displacement, velocity and acceleration of the robot arm; is the number of iterative learning, t∈[0,T k ] is the time index; T k is the actual execution time of the kth iteration, and satisfies 0<T min ≤T k ≤T,T min is the shortest execution time of a single iteration, and T is the maximum execution time of a single iteration; is the inertia matrix, is the centripetal-Coriolis force matrix, is the gravity vector; τ k (t) is the torque input, i.e. the control input, d k (t) is the comprehensive disturbance vector including unmodeled dynamics and external disturbances.

3. The iterative learning control method for a robotic arm based on a radial basis function network according to claim 2, characterized in that: Step S1 defines the dynamic characteristics of the robotic arm system and sets applicable assumptions, including: To simplify the expression, the time and state dependent symbols are omitted; then the dynamic model of the robotic arm system constructed in step S1 needs to satisfy the following dynamic characteristics: Characteristics 1. Inertia matrix M(q k ) is symmetric and positive definite, and its value remains bounded during the operation of the robot system; Feature 2: Matrix is an antisymmetric matrix; is the first-order derivative of the inertia matrix; Feature 3: Centripetal-Coriolis Force Matrix and the gravity vector G(q k ) satisfies the following boundedness conditions: G(q k )≤κ g ; Among them, κ c and κ g is a positive parameter; On the basis of clarifying the dynamic characteristics, the following assumptions are introduced: Assumption 1: The target trajectory and its first and second order time derivatives, i.e., q d , and and the comprehensive disturbance vector d k exist are bounded under the conditions of ; Assumption 2: In each iterative learning control, the initial state of the robot system satisfies the alignment condition, that is, q k (0) = q k-1 (T k )and 4. The iterative learning control method for a robotic arm based on a radial basis function network according to claim 3, characterized in that: Step S2 specifically includes the following steps: S21. Construct a correction function to dynamically optimize the reference trajectory of the robot arm. The formula is: Among them, q m,k (t) represents the reference trajectory generated by the kth iteration optimization; and is a second-order differentiable correction function; S22, setting the constraint conditions of the correction function; specifically including: Setting the correction function The constraints are: When t∈[T a ,T], and is bounded on [0,T]; When t∈[T a ,T], and When t∈[T a ,T], and Setting the correction function The constraints are: When t∈[T a ,T], and is bounded on [0,T]; When t∈[T a ,T], and When t∈[T a ,T], and in, The correction functions are The first and second time derivatives of ; The correction functions are The first and second time derivatives of T a is the design parameter, satisfying 0<T a ≤T k ; The reference trajectory q generated in each iteration m,k The starting state of (t) is consistent with the end state of the previous iteration, that is, q m,k (0) = q m,k-1 (T k )and S23. Define trajectory error; trajectory error includes displacement error and speed error The formula is: Among them, q m,k (t) represents the reference trajectory generated by the kth iteration optimization, q k (t) represents the actual trajectory during the kth iteration; and They represent the speed of the robot on the reference trajectory and the actual trajectory in the kth iteration, that is, q m,k (t), q k The first derivative of (t).

5. The iterative learning control method for a robotic arm based on a radial basis function network according to claim 2, characterized in that: Step S3 specifically includes: S31. Nonlinear compensation terms are introduced to describe the dynamic characteristics of the trajectory error and impose constraints. The time and state dependent symbols of the system are omitted to simplify the expression. Based on the nonlinear compensation terms, the following constraints are established for the trajectory error: Among them, α k (t) is the real nonlinear compensation term; is the velocity error, sign(.) is the sign function; q k They represent the acceleration of the manipulator on the reference trajectory and the actual trajectory of the manipulator during the kth iteration respectively; ∈ is the error smoothing parameter; S32, through the RBF network to the real nonlinear compensation term α k (t) is approximated and the formula is expressed as: a k (t)=θ T (t)μ(Ψ(t),t)+ε; in, is the input feature vector, q k They represent the acceleration of the robot on the reference trajectory and the actual trajectory of the robot during the kth iteration respectively; is the ideal weight parameter of the RBF network, θ T (t) is the transpose of θ(t); p is the number of basis function nodes, ε is the approximation error, is the basis function vector of the RBF network; each basis function is defined as: Among them, Λ i , γ i are the center and width of the basis function respectively.

6. The iterative learning control method for a robotic arm based on a radial basis function network according to claim 5, characterized in that: Step S4 specifically includes: S41, designing estimated dynamic weight parameters to achieve dynamic adjustment and gradual optimization of weight parameters in the radial basis function RBF network during the iteration process; the update rule of the estimated dynamic weight parameters is: in, are the dynamic weight parameters estimated in the kth and k-1th iterations respectively; the initial value of the estimated dynamic weight parameter is set to Through iterative learning, Gradually approach θ(t) to achieve optimal control; μ(Ψ(t), t) is the basis function vector of the RBF network; dynamic learning gain η k (t)>0 is responsible for adjusting the speed of parameter update; tanh(.) is the hyperbolic sine function, is the velocity error in the kth iteration process, ∈ is the error smoothing parameter, which is used to limit the influence range of trajectory error; S42. Design dynamic adjustment and gradual optimization mechanisms; by adjusting the dynamic learning gain at different moments in the iteration process, control the iteration process to balance efficiency and stability; The adjustment rule of dynamic learning gain is: Among them, η k (t) is the dynamic learning gain during the kth iteration; η0>0 is the initial gain, Δ η is the gain; k (t) is the gain parameter in the kth iteration process; The specific definition of the gain parameter is: Among them, the initial value ζ1(t)=1, indicating that the system starts from the most basic adjustment starting point.

7. The iterative learning control method for a robotic arm based on a radial basis function network according to claim 2, characterized in that: In step S5: The specific structure of the control law is expressed as: Among them, τ k (t) is the control input in the kth iteration, K P and K D are the positive feedback gain matrices of displacement error and velocity error respectively, and represent the displacement error and velocity error of the kth iteration respectively; is the estimated dynamic nonlinear compensation term, the hyperbolic tangent function It is used to limit the amplitude of the compensation term, ∈ is the error smoothing parameter, which is used to limit the influence range of the trajectory error; The estimated dynamic nonlinear compensation term is calculated by the RBF network, namely: in, is the transpose of the estimated dynamic weight parameters; μ(Ψ(t), t) is the basis function vector of the RBF network, and Ψ(t) is the input feature vector of the RBF network; The trajectory error of the robot arm is dynamically adjusted through the control law. The dynamic equation of the trajectory error is expressed as:

8. The iterative learning control method for a robotic arm based on a radial basis function network according to claim 7, characterized in that: Step S6 specifically includes: S61. Construct a CEF including displacement error, velocity error and network weight error as a Lyapunov function to analyze the overall energy change of the robotic arm system. To simplify the expression, the time and state dependent symbols of the robotic arm system are omitted. The constructed composite energy function is expressed as follows: V k (t)=V 1,k (t)+V 2,k (t)+V 3,k (t); Among them, V 1,k (t) represents the inertia matrix M(q k ) and speed error The related error energy term is used to characterize the effect of error dynamics on the system energy; V 2,k (t) represents the displacement error term, which is used to describe the displacement error in combination with the feedback gain in the control law. V 3,k (t) represents the weight error term, which reflects the dynamic weight parameters estimated in the RBF network The deviation from the ideal weight parameter θ(t); The error energy term, displacement error term and weight error term are expressed as follows: in, s is an auxiliary scalar and satisfies Δ η , η k are the gain and dynamic learning gain in the k-th iteration process respectively; S62. Combined with the dynamic model and control law of the robot system, according to the Lyapunov stability theory, the kinetic energy error term, displacement error term and weight error term are derived respectively, the monotonic boundedness of the composite energy function is verified, and the stability of the robot system and the convergence of the robot trajectory error are analyzed.

Images