Data-driven error tracking iterative learning control method for robot accurate trajectory tracking

Abstract

The invention discloses a data driving error tracking iterative learning control method for robot accurate trajectory tracking. The method comprises the steps of constructing a kinetic model of a repeated operation robot system, determining a discrete dynamic model of a robot, designing a data driving error tracking iterative learning controller, performing experimental verification and the like. According to the method, under the condition that parameters of a robot system are uncertain, the error tracking iterative learning controller is designed only by using input and output data generated by the robot system, and the robot tracking error is converged to an expected error trajectory within a limited number of iterations; wherein the designed error tracking strategy is used for reducing the requirement of the robot control system for the same initial state, so that the robot system can still achieve a better control effect in any initial state. Only input and output measurement data generated by the robot are used in the design and analysis process of the robot controller, the difficulty in the aspect of model decoupling is reduced, and practical engineering popularization is better facilitated.

Description

Technical Field

[0001] The present invention belongs to the technical field of robot control, and in particular relates to a data-driven error tracking iterative learning control method for precise robot trajectory tracking. Background Art

[0002] Robots have attracted wide attention from all walks of life due to their simple structure, high efficiency and easy control. They can assist or replace humans in completing long-term, repetitive and dangerous work. In practical applications, robots often need to repeatedly track the required signal trajectory with high precision within a limited time to complete specific tasks. The accuracy of parameter information is crucial to the tracking performance of robots. However, there may be many uncertain factors in practical applications. It is often challenging to obtain an accurate mathematical model of the robot system or a model with uncertain parameters, which greatly limits the effectiveness of model-based control methods. Therefore, it is of great significance to study the control problem of robot systems in the presence of parameter uncertainty.

[0003] As an important branch of data-driven control methods, iterative learning control does not require too much prior knowledge, is simple and easy to implement, and can effectively improve the transient response and tracking performance of uncertain repetitive dynamic systems. In order to solve the problem of parameter uncertainty, data-driven iterative learning control makes full use of the system input and output information after the previous iteration, iteratively learns the uncertain parameters of the system, continuously improves the tracking performance of the system, and can achieve perfect tracking effects. In the case of nonlinear uncertainty in system parameters or structures, iterative learning control can also effectively ensure the stability of the system and achieve high tracking accuracy within a limited time interval.

[0004] The advantages of iterative learning control largely rely on the repeatability of system operation, which places high demands on the system's repeated positioning accuracy. However, it is difficult to guarantee that the starting position of the system at the beginning of each iteration is located at the starting point of the desired trajectory in practice. The error tracking method converts the traditional output tracking strategy into an error tracking strategy by presetting the desired error trajectory. The control target is converted into the actual tracking error tracking the designed desired error trajectory within a limited time interval, which effectively relaxes the initial consistency condition without excessive requirements on the desired error trajectory. Through this design, arbitrary initial offsets are allowed.

[0005] In the case where the model information of the robot is completely unknown or partially unknown, there are few relevant research results that can well deal with the impact of unknown model information on the robot's control performance, achieve good control performance of the robot, and relax the initial state consistency conditions. Therefore, it is of great theoretical and practical significance to study the use of input and output data and error tracking methods generated by the system in the presence of arbitrary initial states, combined with iterative learning, to achieve high-precision tracking control of the robot. Summary of the invention

[0006] In order to overcome the problems of incompletely known model information and consistent initial state conditions in existing robots, the present invention provides a data-driven error tracking iterative learning control method for precise robot trajectory tracking. When system parameter information is unknown, a purely data-driven learning controller is designed, and an error tracking iterative learning control method is designed by constructing a reference error trajectory. The restriction on the same initial state conditions is relaxed to achieve high-precision tracking of the reference error trajectory by the robot.

[0007] In order to solve the above technical problems, the proposed technical solutions are as follows:

[0008] A data-driven error tracking iterative learning control method for precise robot trajectory tracking, the method comprising the following steps:

[0009] Step 1, establish the dynamic equation of the robot for repeated operation:

[0010]

[0011] Where t represents time, k represents the number of repeated runs; θ k (t), and They represent the actual joint angular displacement, joint angular velocity and joint acceleration of the single-joint robot at the kth iteration, J represents the bounded moment of inertia, C represents the centrifugal force and Coriolis force, m and l represent the mass and length of the robot, g represents the gravitational acceleration, τ k (t) represents the input torque of the robot at the kth iteration;

[0012] Step 2: Establish a discrete dynamic model of the repetitively running robot. The process is as follows:

[0013] 2.1 Select the sampling period T s , for the formula (1) and Discretize it, and get Further, there are:

[0014]

[0015] 2.2 Substituting equation (2) into equation (1), after mathematical transformation, the discrete form of the robot model is obtained:

[0016]

[0017] Where m∈T N is a discrete time point; T N ={0,1,…,N}, N is a finite positive integer, representing the running cycle of each iteration; x 1,k (m) = θ k (m), x 2,k (m) = θ k (m+1),u k (m) = τ k (m), y k (m) = θ k (m);

[0018] 2.3 Define the state variable x k (m) = [x 1,k (m),x 2,k (m)] Τ , then formula (3) is further transformed into the following nonlinear discrete dynamic model:

[0019]

[0020] Among them, u k (m)∈R,y k (m)∈R and x k (m)∈R 2 They represent the control input, position output and state vector of the robot system at time m in the kth iteration respectively; the nonlinear functions f(·) and h(·) represent the relationship between the robot's position output and state and the relationship between the state and control input respectively; considering the case where the information of functions f(·) and h(·) in equation (4) is unknown, only the input and output data information of the robot is used to design the controller to realize data-driven control;

[0021] Step 3: Convert the robot's nonlinear dynamic model into a linear dynamic model. The process is as follows:

[0022] 3.1 According to the robot dynamic system (4), the input and output relationship is expressed by the following algebraic function expression:

[0023]

[0024] Among them, g i , i={1,2,…,N} is an unknown algebraic function that satisfies the global Lipschitz condition, that is:

[0025]

[0026] Among them, α1, α2∈R 2 ; μ a ,μ b are two positive bounded Lipschitz constants;

[0027] 3.2 For Define the following variables:

[0028]

[0029] Furthermore, from formula (5) we know

[0030]

[0031] According to equations (7) and (8), and using the mean value theorem of differential multivariate functions, we get:

[0032]

[0033] 3.3 Definition Combined with formula (7), formula (9) can be restated as

[0034]

[0035] 3.4 y k (i),u k (i)(i∈{1,2,…,N}) can be restated in supervector form:

[0036] y k =[y k (1),y k (2),…,y k (N)] Τ ∈R N ,

[0037] u k =[u k (0),u k (1),…,u k (N-1)] Τ ∈R N ,

[0038] In addition, let ξ k =[ξ k (1);ξ k (2);…;ξ k (N)]∈R N×2 , and Expressed as a matrix Φ k The form of the elements in the i-th row is

[0039]

[0040] Therefore, formula (10) can be further rewritten as

[0041] δy k =Φ k δu k +ξ k δx k (0) (11)

[0042] 3.5 Since the precise information of functions f(·) and h(·) is unknown, the information of function g(·) is also unknown, so Φ k ,ξ k (i) and ξ k The information of is also unknown; let P k =[ξ k ,Φ k ]∈R N×(N+2) Represents the matrix with unknown information, ω k =[δx k (0); δu k ]∈R N+2 According to formula (11), we can get

[0043] δy k =P k ω k (12)

[0044] Furthermore, formula (12) can be written in the following component form:

[0045]

[0046] Step 4: Design of parameter adaptive update strategy. The process is as follows:

[0047] 4.1 For parameter vector For unknown situations, consider estimating them using Representation vector The estimated value at the kth iteration; get the output δy k (i) Corresponding estimated value The form is as follows

[0048]

[0049] 4.2 To reduce the error The value of , construct the following optimization objective function

[0050]

[0051] The objective function (15) is in the variable The gradient at

[0052]

[0053] Therefore, based on the gradient technique, the parameter vector The adaptive update algorithm is constructed as follows:

[0054]

[0055] Among them, the search step length λ is a weight factor, λ>0;

[0056] 4.3 For any small σ>0, if the parameter matrix The diagonal elements of or |δu k (i)|≤σ, then let

[0057]

[0058] Step 5: Design of error tracking iterative learning control scheme.

[0059] Further, the process of step 5 is as follows:

[0060] 5.1 For Define signal y d (i) is the expected position of the robot during motion, then e k (i) = y d (i)-y k (i) The error between the robot's expected position signal and the actual position generated when the robot moves; define the expected error trajectory for

[0061]

[0062] Where Ω is the pre-set time point connecting the transition trajectory and the expected zero error trajectory, ζ(i) is a smooth monotonic decay function on [0, Ω], satisfying ζ(0) = 1 and ζ(Ω) = 0;

[0063] 5.2 Design of error tracking iterative learning control scheme The goal of the control scheme is to generate the control input sequence u k+1 (i)(i∈{1,2,…,N}) attempts to drive the robot system (13) so that as the number of iterations increases, the system outputs a tracking error e k (i) Ability to achieve a given desired error trajectory tracking, that is, when k→∞, there is

[0064]

[0065] Among them, ε k (i) is the difference between the expected error signal and the actual error signal, which is defined as

[0066]

[0067] 5.3 For According to the error information ε k (i) The definition of

[0068]

[0069] in, In addition, according to the input vector The definition of And order express The vector consisting of the first i-1 elements of

[0070]

[0071] Then, substituting equation (21) into (20), we get

[0072]

[0073] 5.4 Iterative learning controller is designed by considering the following objective function

[0074] minJ(u k+1 (i))=|ε k+1 (i+1)| 2 +wδu k (i)| 2 (twenty three)

[0075] Therefore, substituting formula (22) into the objective function (23), we get

[0076]

[0077] 5.5 Objective function (24) with respect to u k+1 (i) Take the derivative and set the partial derivative to zero to obtain the following iterative learning controller:

[0078]

[0079] 5.6 Due to ξ k The information of (i+1) is unknown and cannot be used. In this case, equation (25) will lose its effect. Therefore, the estimated parameter information When applied to the design of iterative learning controller,

[0080] u k+1 (i) = uk (i)+γ k (i+1)(ε k (i+1)-χ k (i+1)) (26)

[0081] in, w>0 is an adjustable parameter;

[0082] 5.7 The data-driven error tracking iterative learning control method of the robot is described as follows:

[0083]

[0084] u k+1 (i) = u k (i)+γ k (i+1)(ε k (i+1)-χ k (i+1)) (28).

[0085] In the present invention, the accuracy of parameter information is crucial to the tracking performance of the robot. There may be many uncertain factors in the actual application process, which makes it impossible for the traditional model-based control method to complete the high-speed and accurate tracking task; however, the present invention designs a data-driven error tracking iterative learning control method for accurate robot trajectory tracking. The design and analysis process of the controller is independent of the robot system model information. Even in the case of incomplete understanding of the system parameter information, the robot can achieve high-precision tracking of the trackable reference error trajectory. In addition, the error tracking strategy designed by the present invention allows the initial state to be set arbitrarily, relaxing the requirement of the same initial state.

[0086] The technical concept of the present invention is: for a robot system with unknown model parameter information, the present invention uses the input and output data generated by the robot system to design an iterative learning controller. The controller can effectively reduce the dependence of the robot control method on accurate model information, while ensuring that the robot system has a fast convergence ability. In addition, the designed error tracking strategy is used to reduce the requirements of the robot controller for the same initial state, so that the robot system can still achieve a good control effect in any initial state.

[0087] Effective effects of the invention: The present application discloses a nonlinear system with repetitive motion characteristics such as a robot, which takes the robot as a controlled object, realizes the rapid tracking of the robot to the desired position in any initial state, and reduces the dependence of the robot control method on precise model information. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] Figure 1 This is the control flow chart of the robot;

[0089] Figure 2 Output tracking for different numbers of iterations;

[0090] Figure 3 The error tracking situation under different iteration numbers;

[0091] Figure 4 The change of the maximum tracking error under different numbers of iterations. DETAILED DESCRIPTION

[0092] The present invention will be further described below in conjunction with the accompanying drawings.

[0093] Reference Figure 1 , showing the model block diagram of the robot disclosed in the present invention. The initial state x of the kth batch k (0) and input u k Acting on the robot can get the actual output y of the kth batch of the system k The input and output data generated by the robot system are stored in the memory to construct the parameter adaptive update algorithm. Furthermore, the actual output y of the robot is k and the expected trajectory y d The difference between the actual tracking error e k , the actual error e k and the expected error signal The difference ε k With parameter estimation information Passed to the iterative learning controller, the controller output u for the next batch can be generated k+1 The loop runs like this until the system reaches the set maximum number of iterations, then the iteration stops. The effectiveness of the proposed data-driven error tracking iterative learning control method is experimentally evaluated on the Universal Robots 5 (UR5) manipulator system.

[0094] Reference Figure 2 to Figure 4 , a data-driven error tracking iterative learning control method for precise robot trajectory tracking, comprising the following steps:

[0095] Step 1, establish the dynamic equation of the robot for repeated operation:

[0096]

[0097] Where t represents time, k represents the number of repeated runs; θ k (t), and They represent the actual joint angular displacement, joint angular velocity and joint acceleration of the single-joint robot at the kth iteration, J represents the bounded moment of inertia, C represents the centrifugal force and Coriolis force, m and l represent the mass and length of the robot, g represents the gravitational acceleration, τ k (t) represents the input torque of the robot at the kth iteration;

[0098] Step 2: Establish a discrete dynamic model of the repetitively running robot. The process is as follows:

[0099] 2.1 Select the sampling period T s , for the formula (1) and Discretize it, and get Further, there are:

[0100]

[0101] 2.2 Substituting equation (2) into equation (1), after mathematical transformation, the discrete form of the robot model is obtained:

[0102]

[0103] Where m∈T N is a discrete time point; T N ={0,1,…,N}, N is a finite positive integer, representing the running cycle of each iteration; x 1,k (m) = θ k (m), x 2,k (m) = θ k (m+1),u k (m) = τ k (m), y k (m) = θ k (m);

[0104] 2.3 Define the state variable x k (m) = [x 1,k (m),x 2,k (m)] Τ , then formula (3) is further transformed into the following nonlinear discrete dynamic model:

[0105]

[0106] Among them, u k (m)∈R,y k (m)∈R and x k (m)∈R 2They respectively represent the control input, position output and state vector of the robot system at time m in the kth iteration; the nonlinear functions f(·) and h(·) respectively represent the relationship between the robot's position output and state and the relationship between the state and control input; there may be many uncertain factors in the actual application process, which makes it impossible for the traditional model-based control method to complete the high-speed and accurate tracking task; therefore, the present invention considers the situation where the information of the functions f(·) and h(·) in formula (4) is unknown, and only uses the input and output data information of the robot to design the controller to realize data-driven control;

[0107] Step 3: Convert the robot's nonlinear dynamic model into a linear dynamic model. The process is as follows:

[0108] 3.1 According to the robot dynamic system (4), the input and output relationship is expressed by the following algebraic function expression:

[0109]

[0110] Among them, g i , i={1,2,…,N} is an unknown algebraic function that satisfies the global Lipschitz condition, that is:

[0111]

[0112] Among them, α1, α2∈R 2 ; μ a ,μ b are two positive bounded Lipschitz constants;

[0113] 3.2 For Define the following variables:

[0114]

[0115] Furthermore, from formula (5) we know

[0116]

[0117] According to equations (7) and (8), and using the mean value theorem of differential multivariate functions, we get:

[0118]

[0119] 3.3 Definition Combined with formula (7), formula (9) can be restated as

[0120]

[0121] 3.4 y k(i),u k (i)(i∈{1,2,…,N}) can be restated in supervector form:

[0122] y k =[y k (1),y k (2),…,y k (N)] Τ ∈R N ,

[0123] u k =[u k (0),u k (1),...,u k (N-1)] Τ ∈R N ,

[0124] In addition, let ξ k =[ξ k (1);ξ k (2);…;ξ k (N)]∈R N×2 , and Expressed as a matrix Φ k The form of the elements in the i-th row is

[0125]

[0126] Therefore, formula (10) can be further rewritten as

[0127] δy k =Φ k δu k +ξ k δx k (0) (11)

[0128] 3.5 Since the precise information of functions f(·) and h(·) is unknown, the information of function g(·) is also unknown, so Φ k ,ξ k (i) and ξ k The information of is also unknown; in particular, let P k =[ξ k ,Φ k ]∈R N×(N+2) Represents the matrix with unknown information, ω k =[δx k (0); δu k ]∈R N+2 According to formula (11), we can get

[0129] δy k =P k ωk (12)

[0130] Furthermore, formula (12) can be written in the following component form:

[0131]

[0132] Step 4: Design of parameter adaptive update strategy. The process is as follows:

[0133] 4.1 For parameter vector For unknown situations, consider estimating them using Representation vector The estimated value at the kth iteration; further, the output δy is obtained k (i) Corresponding estimated value The form is as follows

[0134]

[0135] 4.2 To minimize the error The value of , construct the following optimization objective function

[0136]

[0137] The objective function (15) is in the variable The gradient at

[0138]

[0139] Therefore, based on the gradient technique, the parameter vector The adaptive update algorithm is constructed as follows:

[0140]

[0141] Among them, the search step length λ is a weight factor, λ>0;

[0142] 4.3 In particular, to ensure that the parameter matrix The diagonal elements of And prevent the input signal difference δu k (i) If the parameter matrix is ​​too small, it will cause meaningless changes in the parameter adaptive term, reducing the system response speed. Construct a reset algorithm: For any small σ>0, if the parameter matrix The diagonal elements of or |δu k (i)|≤σ, then let

[0143]

[0144] Step 5, the design of error tracking iterative learning control scheme, the process is as follows:

[0145] 5.1 For Define signal y d (i) is the expected position of the robot during motion, then e k (i) = y d (i)-y k (i) The error between the robot's expected position signal and the actual position generated when the robot moves; define the expected error trajectory for

[0146]

[0147] Where Ω is the pre-set time point connecting the transition trajectory and the expected zero error trajectory, ζ(i) is a smooth monotonic decay function on [0, Ω], satisfying ζ(0) = 1 and ζ(Ω) = 0;

[0148] 5.2 The error tracking iterative learning control scheme designed in this invention aims to generate the control input sequence u k+1 (i)(i∈{1,2,…,N}) attempts to drive the robot system (13) so that as the number of iterations increases, the system outputs a tracking error e k (i) Ability to achieve a given desired error trajectory tracking, that is, when k→∞, there is

[0149]

[0150] Among them, ε k (i) is the difference between the expected error signal and the actual error signal, which is defined as

[0151]

[0152] 5.3 For According to the error information ε k (i) The definition of

[0153]

[0154] in, In addition, according to the input vector From the definition of , we know that And order express The vector consisting of the first i-1 elements of

[0155]

[0156] Then, substituting equation (21) into (20), we get

[0157]

[0158] 5.4 Iterative learning controller is designed by considering the following objective function

[0159] minJ(u k+1 (i))=|ε k+1 (i+1)| 2 +|wδu k (i)| 2 (twenty three)

[0160] Therefore, substituting formula (22) into the objective function (23), we get

[0161]

[0162] 5.5 Objective function (24) with respect to u k+1 (i) Take the derivative and set the partial derivative to zero to obtain the following iterative learning controller:

[0163]

[0164] 5.6 Due to ξ k The information of (i+1) is unknown and cannot be used. In this case, equation (25) will lose its effect. Therefore, the estimated parameter information When applied to the design of iterative learning controller,

[0165] u k+1 (i) = u k (i)+γ k (i+1)(ε k (i+1)-χ k (i+1)) (26)

[0166] in, w>0 is an adjustable parameter;

[0167] 5.7 The data-driven error tracking iterative learning control method of the robot is described as follows:

[0168]

[0169] u k+1 (i) = u k (i)+γ k (i+1)(ε k (i+1)-χ k (i+1)) (28)

[0170] Step 6: Experimentally verify the effectiveness of the robot's data-driven error tracking iterative learning control method. The process is as follows:

[0171] 6.1 To verify the effectiveness of the present invention, an experiment was conducted on the control effect of the data-driven error tracking iterative learning control method of the robot shown in equations (27) and (28). The UR5 manipulator is a type of robot. The experimental platform is briefly described as follows: Figure 2 As shown in the figure, it includes a computer, a manipulator, a router and a controller. In the experiment on the UR5 manipulator, an input torque is applied to the end effector of the manipulator to drive the manipulator to perform a single-axis motion along the z-axis of the tool coordinate axis. Then, the position information of the manipulator is transmitted to the computer and further processed by the controller. In the experiment, the UR5 manipulator was subjected to 5 single-coordinate lifting experiments. The UR5 manipulator only provided the input and output data generated during the movement, and the model parameter information was not involved in the controller design. The control goal is that as the number of iterations increases, the position signal of the end effector can track the required position signal.

[0172] 6.2 The total number of discrete time points is set to N = 600, and the initial state is x k (0)=0.01rand(2,1), initial input u1(m)=0, the expected output trajectory is:

[0173] y d (m) = 50sin(πm / 400) (29)

[0174] Select the parameter w in the controller as 0.01, the parameter λ in the parameter adaptive update algorithm as 0.01, and the initial value of the parameter estimation Select as:

[0175]

[0176] Expected Error Trajectory The switching time point Ω is selected as Ω=5, and ζ(i) is designed as follows

[0177]

[0178] Figure 2 to Figure 4 This is a rendering of the data-driven error tracking iterative learning control method for the robot of the present invention. Figure 2 It can be seen that without relying on the precise model information of the robot, only using the input and output data generated by the robot system, the actual position of the robot in any initial state can gradually track the expected position as the number of iterations increases; Figure 3 It can be seen that the actual tracking error of the robot achieves progressive tracking of the expected error trajectory as the number of iterations increases. Figure 4It is shown that the maximum tracking error of the robot decreases with the increase of iteration number and gradually approaches zero at the 5th iteration. In summary, the data-driven error tracking iterative learning control method of the robot of the present invention can realize the rapid tracking of the robot to the desired position, reduce the dependence of the robot control method on accurate model information, and relax the requirement of the same initial state.

[0179] The above is an experimental result of the present invention. Obviously, the present invention is not limited to the above examples, and various modifications can be made to the present invention without departing from the basic spirit of the present invention and without exceeding the scope of the essential content of the present invention.

Claims

1. A data-driven error tracking iterative learning control method for precise robot trajectory tracking, characterized in that: The method comprises the following steps: Step 1, establish the dynamic equation of the robot for repeated operation: Where t represents time, k represents the number of repeated runs; θ k (t), and They represent the actual joint angular displacement, joint angular velocity and joint acceleration of the single-joint robot at the kth iteration, J represents the bounded moment of inertia, C represents the centrifugal force and Coriolis force, m and l represent the mass and length of the robot, g represents the gravitational acceleration, τ k (t) represents the input torque of the robot at the kth iteration; Step 2: Establish a discrete dynamic model of the repetitively running robot. The process is as follows: 2.1 Select the sampling period T s , for the formula (1) and Discretize it, and get Further, there are: 2.2 Substituting equation (2) into equation (1), after mathematical transformation, the discrete form of the robot model is obtained: in, m∈T N is a discrete time point; T N ={0,1,…,N}, N is a finite positive integer, representing the running cycle of each iteration; x 1,k (m) = θ k (m), x 2,k (m) = θ k (m+1),u k (m) = τ k (m), y k (m) = θ k (m); 2.3 Define the state variable x k (m) = [x 1,k (m),x 2,k (m)] Τ , then formula (3) is further transformed into the following nonlinear discrete dynamic model: Among them, u k (m)∈R,y k (m)∈R and x k (m)∈R 2 They represent the control input, position output and state vector of the robot system at time m in the kth iteration respectively; the nonlinear functions f(·) and h(·) represent the relationship between the robot's position output and state and the relationship between the state and control input respectively; considering the case where the information of functions f(·) and h(·) in equation (4) is unknown, only the input and output data information of the robot is used to design the controller to realize data-driven control; Step 3: Convert the robot's nonlinear dynamic model into a linear dynamic model. The process is as follows: 3.1 According to the robot dynamic system (4), the input and output relationship is expressed by the following algebraic function expression: in, g i , i={1,2,…,N} is an unknown algebraic function that satisfies the global Lipschitz condition, that is: Among them, α1, α2∈R 2 ; μ a ,μ b are two positive bounded Lipschitz constants; 3.2 For Define the following variables: Furthermore, from formula (5) we know According to equations (7) and (8), and using the mean value theorem of differential multivariate functions, we get: 3.3 Definition Combined with formula (7), formula (9) can be restated as 3.4 y k (i),u k (i)(i∈{1,2,…,N}) can be restated in supervector form: and k =[and k (1),and k (2),…,and k (N)] Τ ∈R N , u k =[u k (0),u k (1),…,u k (N-1)] Τ ∈R N , In addition, let ξ k =[ξ k (1);ξ k (2);…;ξ k (N)]∈R N×2 , and Expressed as a matrix Φ k The form of the elements in the i-th row is Therefore, formula (10) can be further rewritten as dy k =Φ k you k +ξ k δx k (0) (11) 3.5 Since the precise information of functions f(·) and h(·) is unknown, the information of function g(·) is also unknown, so Φ k ,ξ k (i) and ξ k The information of is also unknown; let P k =[ξ k ,Φ k ]∈R N×(N+2) Represents the matrix with unknown information, ω k =[δx k (0); δu k ]∈R N+2 According to formula (11), we can get dy k =P k oh k (12) Furthermore, formula (12) can be written in the following component form: Step 4: Design of parameter adaptive update strategy. The process is as follows: 4.1 For parameter vector For unknown situations, consider estimating them using Representation vector The estimated value at the kth iteration; get the output δy k (i) Corresponding estimated value The form is as follows 4.2 To reduce the error The value of , construct the following optimization objective function The objective function (15) is in the variable The gradient at Therefore, based on the gradient technique, the parameter vector The adaptive update algorithm is constructed as follows: Among them, the search step length λ is a weight factor, λ>0; 4.3 For any small σ>0, if the parameter matrix The diagonal elements of or |δu k (i)|≤σ, then let Step 5: Design of error tracking iterative learning control scheme.

2. The data-driven error tracking iterative learning control method for precise robot trajectory tracking according to claim 1, characterized in that: The process of step 5 is as follows: 5.1 For Define signal y d (i) is the expected position of the robot during motion, then e k (i) = y d (i)-y k (i) The error between the robot's expected position signal and the actual position generated when the robot moves; define the expected error trajectory for Where Ω is the pre-set time point connecting the transition trajectory and the expected zero error trajectory, ζ(i) is a smooth monotonic decay function on [0, Ω], satisfying ζ(0) = 1 and ζ(Ω) = 0; 5.2 Design of error tracking iterative learning control scheme The goal of the control scheme is to generate the control input sequence u k+1 (i)(i∈{1,2,…,N}) attempts to drive the robot system (13) so that as the number of iterations increases, the system outputs a tracking error e k (i) Ability to achieve a given desired error trajectory tracking, that is, when k→∞, there is Among them, ε k (i) is the difference between the expected error signal and the actual error signal, which is defined as 5.3 For According to the error information ε k (i) The definition of in, In addition, according to the input vector The definition of And order express The vector consisting of the first i-1 elements of Then, substituting equation (21) into (20), we get 5.4 Iterative learning controller is designed by considering the following objective function min J(u k+1 (i))=|ε k+1 (i+1)| 2 +w|δu k (and)| 2 (23) Therefore, substituting formula (22) into the objective function (23), we get 5.5 Objective function (24) with respect to u k+1 (i) Take the derivative and set the partial derivative to zero to obtain the following iterative learning controller: 5.6 Due to The information of is unknown and cannot be used, so Equation (25) will lose its effect; so the estimated parameter information When applied to the design of iterative learning controller, u k+1 (i)=u k (i)+γ k (i+1)(ε k (i+1)-χ k (i+1)) (26) in, w>0 is an adjustable parameter; 5.7 The data-driven error tracking iterative learning control method of the robot is described as follows: u k+1 (i)=u k (i)+γ k (i+1)(ε k (i+1)-χ k (i+1)) (28)。

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