Unequal-length iterative learning control optimization method for robot impedance control system

Abstract

The invention discloses an unequal-length iterative learning control optimization method for a robot impedance control system, and relates to the field of robot control, and the method comprises the steps: building a high-dimensional linear state space model capable of describing nonlinear dynamic characteristics based on a Koopman operator theory; based on the model, an approximate linear repetitive process model of the robot impedance control system is further constructed, and an input and output matrix expression form of a time sequence is obtained through a lifting technology. For the unequal-length problem, a prediction compensation method is used, missing information caused by the unequal-length problem is predicted, and the track length is adjusted to the expected length. Under a norm optimization framework, the performance of future batches is considered, a predictive iterative learning control algorithm is designed, and finally the predictive iterative learning control algorithm is applied to an actual robot impedance control system. Control input obtained through the method can better adapt to an original nonlinear system, and high-precision tracking control over an expected trajectory is basically achieved.

Description

Technical Field

[0001] This invention relates to the field of robot control, and in particular to an unequal-length iterative learning control optimization method for robot impedance control systems. Background Technology

[0002] With the development of intelligent manufacturing, robots are widely used in contact tasks such as grinding, polishing, assembly, and machining. In these scenarios, the robot's end effector needs to be in direct contact with the environment, and its operation typically involves unknown environmental stiffness, continuously changing contact forces, complex dynamic characteristics, and significant nonlinear features. Under these conditions, traditional control methods struggle to effectively address the uncertainties and nonlinear effects generated during contact. Therefore, impedance control is widely used to achieve compliant robot interaction, enabling the robot to proactively adapt to environmental changes during task execution by adjusting equivalent stiffness, damping, and mass.

[0003] For robotic systems with repetitive motion characteristics, iterative learning control (ILC), as a control strategy suitable for repetitive tasks, has become an important method for improving robot accuracy and stability by gradually improving control performance by utilizing historical execution information. In highly repetitive tasks, ILC can significantly reduce error accumulation, improve execution accuracy, and enable robots to maintain high stability and reliability during long-term operation. However, traditional ILC methods generally assume that the execution time and trajectory length are the same for each task, an assumption that is often difficult to meet in practical applications. When faced with varying trajectory lengths, traditional ILC suffers from learning information failure and poor convergence, limiting its application in complex contact tasks. Therefore, designing iterative learning controllers to achieve trajectory tracking for robot impedance control systems with unequal length problems is a research area of ​​practical significance.

[0004] Therefore, in this application, by utilizing historical task information for learning, a controller is designed for the robot impedance control system to achieve effective learning and optimization of repetitive tasks with unequal lengths. Summary of the Invention

[0005] The purpose of this invention is to solve the trajectory tracking problem of robot impedance control systems under unequal length conditions. A nonlinear iterative learning control optimization method for robot impedance control systems is proposed. This method designs a Koopman operator to linearize the nonlinear system, uses a predictive compensation method to solve the unequal length problem, considers the performance index of future batches, and designs a predictive iterative learning control law.

[0006] The technical solution of the present invention is as follows: A non-uniform length iterative learning control optimization method for robot impedance control systems includes the following steps: The first step is to establish a dynamic model of the robot's impedance control system, including: To describe the dynamic characteristics of a robot during its interaction with the external environment, an impedance control system is modeled. The core idea of ​​impedance control is to construct a desired dynamic model of mechanical impedance so that the force and displacement generated by the robot's end effector satisfy a predetermined dynamic relationship when subjected to external forces. This dynamic model is represented by the following second-order differential equation: (1) in, For the quality of the robot, For the motion displacement of the robot's end effector, For control signals, The impedance coefficient, It is a smooth nonlinear function representing displacement. The force; We select the displacement and velocity of the robot's end effector as state variables, denoted as . Select the system output as the robot's end effector speed. Select the input variable as the control signal Based on this, the second-order differential equation (1) is discretized using the Euler approximation method to obtain the discrete state equation form of the system: (2) in, The sampling time interval, index Indicates batch; Indicates time, Indicates the first The runtime length of each batch is defined as the expected runtime length. The minimum running length is ,therefore, ; , and These are the system's first batch The state, output, and input vectors at each time step; assuming that the initial state of each batch is the same, i.e., the initial conditions are satisfied. .

[0007] The second step, based on Koopman operator theory, is to establish a high-dimensional linear state-space model describing the nonlinear dynamic model, which is used to obtain estimates of the robot's impedance control system output. This step specifically includes: For the discrete state equation (2) of the nonlinear dynamic model, the augmented state is defined as follows: Then the dynamic equation in the augmented state is: (3) Define an infinite-dimensional function space. Observation function yes The function in the dynamic model (1) and the Koopman operator Defined as: (4) Koopman operator It is an infinite-dimensional operator, solved using the EDMD (Extended Dynamic Mode Decomposition) method. A finite-dimensional approximation; select a suitable set of observation functions to combine into a finite-dimensional lifting function. Since the future input of the system does not need to be predicted, the lifting function has the following form: (5) in, As basis functions, The dimension of the system in the higher-dimensional space after the system state is boosted; After each iteration batch, the state and input data generated during the batch process are collected to form a dataset, and a Koopman operator approximation solution is performed. The dataset is defined as follows: (6) (7) (8) Under the action of the lifting function (5), the dataset and After the upgrade and : (9) (10) Based on the EDMD method, solve the following two minimization problems: (11) (12) in, Expressing the Frobenius norm, we obtain the discrete state equation (2) in the th... Approximate high-dimensional linear model after each batch: (13) In the formula, It is the system state in a higher-dimensional space. It is an estimated value output by the system. , C It is a mapping between the system output and state in the original space. C xk This represents the mapping from the system state in the higher-dimensional space to the original state; , , Let be a constant matrix, assuming .

[0008] The third step involves constructing a predictive compensation correction model based on the estimated output of the robot's impedance control system. This correction model is used to adjust the output of the robot's impedance control system to the desired length. This step specifically includes: Discrete state equation (2) in the first The approximate high-dimensional linear model after the completion of each batch is shown in equation (13). If we use model (13) (i.e., the estimated value of the output of the robot impedance control system) Directly predicting the system output for missing time periods in variable batch length problems, because Compared with the actual output of the system There is an error between them. If the error between the two is large, use directly. Using this as the system output for a missing time period may result in poor control performance; therefore, it is necessary to adjust the output accordingly. Make corrections; define and The prediction error between them is: (14) Therefore, in the first In batches, at the estimated value Based on the prediction error, predict the time period System output on: (15) The system output and tracking error are redefined as follows: (16) (17) in, , To utilize equation (15) and input estimation u k ( t )exist t The system output value at that moment; For the predicted system tracking error, The reference trajectory for the robot end effector is given; Equations (16) and (17) adjust the system output and tracking error to the desired length for subsequent predictive iterative learning (ILC) control law design.

[0009] The fourth step, within the norm optimization framework, considers the performance metrics of future batches, designs a predictive iterative learning control law, and solves for its optimal control gain. This step specifically includes: Since the runtime of each batch is finite, using boosting techniques, the expected output, input, actual output, tracking error, and predicted output of the boosted system are defined as follows: (18) (19) (20) (twenty one) (twenty two) The estimated value output by the system is represented in boosted form, as shown below: (twenty three) in, (twenty four) (25) The ILC control law is designed in the following form: (26) In the formula, To control the gain, a performance index function is designed and solved. Considering the tracking error and input variations in future batches, the performance index function is designed as follows: (27) in, , As a discount factor, , , and It is a positive definite symmetric matrix; In the performance index function (27) and All of this information pertains to future batches, and can be obtained using equation (23) and control law (26): (28) (29) in, This is the improvement form of the prediction error; further, combining equations (28) and (29), the performance index function (27) is rewritten in the following form: (30) in, It is a positive definite symmetric matrix; Solving the performance index function (30), the optimal control gain is obtained: (31) in, The solution is the following discrete algebraic Riccati equation: (32) Therefore, the first The control law for the next iteration of learning is: (33) The solution to the discrete algebraic Riccati equation (32) is obtained by the following iterative method: (34) After obtaining a stable solution through iterative solution, substitute it into equation (31) to obtain the optimal control gain.

[0010] The fifth step is to implement the tracking control of the predictive iterative learning control method for unequal length problems, including: the iterative learning control gain based on the predictive iterative learning control law obtained in the previous steps. The control signals for each time step in the next batch are calculated to perform tracking control on the impedance control system of a real nonlinear robot with unequal length problems.

[0011] A further technical solution is that the method also includes analyzing the convergence of the predictive iterative learning control law, specifically including: Since the original nonlinear system and the approximate high-dimensional linear model obtained by the EDMD method have modeling errors, and there are prediction errors when predicting missing information, the prediction error and modeling error are uniformly classified as modeling error in the convergence analysis. The discrete state equation (2) of the nonlinear dynamic model is expressed as a system with modeling error related to equation (23): (35) in, Indicates modeling error. , , , Represents the Euclidean norm. b G , b d , This is the upper bound of the relevant parameters; Based on the control law (26) and the system (35), we can obtain: (36) Taking the norm of both sides of equation (36) yields: (37) Pick The upper bound is ,Pick The upper bound is , , We can obtain: (38) Similarly, we can conclude that , If the conditions are met. , Equation (37) can be further derived as follows: (39) Pick The upper bound is , , Bounded, Take the upper bound as ; Equation (39) gives the boundedness of the control input, and based on this, the convergence characteristics of the tracking error are further given; The tracking error is rewritten as: (40) in, , , ; Taking the norm of both sides of equation (40) yields: (41) in, (42) Similarly, we can conclude that , ;right Taking the norm, we have: (43) Pick The upper bound is If the conditions are met , Equation (41) can be further derived to obtain: (44) Thus obtain Bounded, that is ,Pick ; when hour, Therefore: (45) Therefore, the tracking error after prediction compensation It can achieve bounded convergence; definition ,because Therefore, tracking error Bounded convergence can be achieved.

[0012] The beneficial technical effects of this invention are: For systems like robots exhibiting repetitive motion, this paper treats the robot's impedance-controlled nonlinear system as the controlled object and models it using the Koopman operator, obtaining an approximate high-dimensional linear model. Addressing the issue of inconsistent task execution cycles in practical applications, the high-dimensional linear model is used to predict missing information, adjusting the system output and tracking error to the desired length. By filling in these missing data, the iterative learning control law can fully utilize historical trajectories and predicted information in each task execution, enabling reasonable correction and optimization of unexecuted parts. Based on this, within the norm optimization framework, and considering the performance indicators of future batches, a predictive iterative learning control law is designed. The resulting control input can better adapt to the original nonlinear system, essentially achieving high-precision tracking control of the desired trajectory. Attached Figure Description

[0013] Figure 1 This is a block diagram illustrating the principle of the unequal-length iterative learning control optimization method for the robot impedance control system provided in this application. Figure 2 This is the actual output curve of the robot impedance control system provided in this application; Figure 3 This is the actual input curve of the robot impedance control system provided in this application; Figure 4 This is the root mean square error convergence plot of the tracking error of the robot impedance control system provided in this application; Figure 5 This is a comparison chart showing the root mean square error of the control algorithm proposed in this application and two existing iterative optimization algorithms as a function of batch size. Detailed Implementation

[0014] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0015] One embodiment of this application provides a non-uniform length iterative learning control optimization method for a robot impedance control system, the principle block diagram of which is shown below. Figure 1 As shown, the specific implementation steps of this method refer to the contents of steps one through five in the invention description. Among them, using the nonlinear robot impedance control system model shown in equation (2), the parameters in the robot impedance control system are respectively set as: , , The system's operating cycle is Sampling time interval =0.02s, expected runtime Actual running length of each batch In the interval The values ​​change randomly between the given values, and the number of iterations is [number missing]. The initial state of the system is Initial control quantity The control objective is:

[0016] When using the EDMD method to solve the approximate high-dimensional linear model of a nonlinear robot impedance control system, the basis function set is chosen as the state itself, i.e. , And the Gaussian function. The Gaussian function has the following form:

[0017] in, Indicates the first The center of the Gaussian function, For width parameter, . For interval Random values ​​are selected for the width parameter. .

[0018] Design discount factor The symmetric positive definite matrix is , An impedance control system for robots with unequal length problems (2) is applied, and the output response diagram is shown below. Figure 2 As shown in the figure, under the control method proposed in this application, the system output can achieve high-precision tracking of the desired trajectory, verifying the effectiveness of the method in solving the system's nonlinear characteristics and unequal length problems. The input curve is shown in the figure. Figure 3 As shown in the figure, the control input generated by the system during the simulation remains within a bounded range, without over-excitation or oscillation. This further demonstrates that the algorithm proposed in this application meets the convergence and stability requirements, and can achieve reasonable allocation of input energy while ensuring control performance. The root mean square error curve of the system tracking error is shown in the figure. Figure 4 As shown in the figure, the results indicate that the tracking error gradually decreases with the increase of the number of iterations and reaches a stable state after a few iterations, indicating that the system output achieves bounded convergence and the learning process has a relatively fast convergence speed. Furthermore, the iterative learning control algorithm proposed in this application is compared with two other methods, and the root mean square error varies with the batch size as shown in the figure. Figure 5As shown, the algorithm proposed in this application has significantly smaller errors and faster convergence speed in the same batch.

[0019] This application presents an unequal-length iterative learning control optimization method for robot impedance control systems with repetitive motion characteristics. The method samples the input and output data of the robot impedance control system and uses the Koopman operator to linearize the nonlinear robot impedance control system, obtaining an approximate high-dimensional linear model. Using the obtained approximate linear model, model prediction compensates for missing state and output information under unequal length conditions, adjusting the system output and tracking error to the desired length. Based on this, a control performance index incorporating future batch information is designed, and the control law gain is solved. Simulation results show that the proposed method can achieve high-precision tracking of the desired output in robot impedance control systems.

[0020] The above descriptions are merely preferred embodiments of this application, and the present invention is not limited to the above embodiments. It is understood that other improvements and variations directly derived or conceived by those skilled in the art without departing from the spirit and concept of the present invention should be considered to be included within the protection scope of the present invention.

Claims

1. A method for optimizing unequal-length iterative learning control of a robot impedance control system, the method comprising: The method comprises: a dynamic model of the robot impedance control system is established, the dynamic model is used for characterizing that when a robot end is subjected to external force, a generated force and a displacement satisfy an expected dynamic relationship; a high-dimensional linear state space model describing the nonlinear dynamic model is established based on Koopman operator theory, and an estimated value of an output of the robot impedance control system is obtained; a prediction compensation correction model is constructed based on the estimated value of the output of the robot impedance control system, and the correction model is used for adjusting the output of the robot impedance control system to a desired length; in a norm optimization framework, considering performance indicators of future batches, a prediction iterative learning control law is designed and optimal control gains are solved; according to the prediction iterative learning control law, control signals at each time of the next batch are calculated, and tracking control is performed on an actual nonlinear robot impedance control system with unequal length problems.

2. The unequal-length iterative learning control optimization method of claim 1, wherein, The dynamic model of the robot impedance control system comprises: The dynamic model is represented by the following second-order differential equation: (1) wherein, is the mass of the robot, is the motion displacement of the robot end, is the control signal, is the impedance coefficient, is a smooth nonlinear function representing the force of displacement ; The motion displacement and motion speed of the robot end are selected as state variables, denoted as The motion speed of the robot end is selected as the system output The control signal is selected as the input variable ; on this basis, the second-order differential equation (1) is discretized by using the Euler approximation method to obtain the discrete state equation form of the system: (2) where, is the sampling time interval, the subscript denotes the batch number; denotes time, denotes the length of the run of the th batch, the desired length of the run is defined as , and the minimum length of the run is , thus, ; , and are the state, output and input vectors of the system at the th batch time, respectively. It is assumed that the initial state of each batch is the same, i.e., the initial condition satisfies .

3. The unequal-length iterative learning control optimization method of claim 1, wherein, The high-dimensional linear state space model describing the nonlinear dynamic model is established based on Koopman operator theory, and an estimated value of an output of the robot impedance control system is obtained. For the discrete state equations of the aforementioned nonlinear dynamic model, the augmented state is defined as follows: , x k ( t ), u k ( t ) indicates the system's first batch Given the state and input vector at time t, the dynamic equation in the augmented state is: (3) Defining an infinite-dimensional function space , the observation function is a function in the Koopman operator of a dynamic model in the form of a second-order differential equation is defined as: (4) The Koopman operator is an infinite-dimensional operator, and the finite-dimensional approximation is solved by EDMD method ; a set of suitable observation functions is selected to form a finite-dimensional lifting function, and since the future input of the system does not need to be predicted, the lifting function is in the form of (5) wherein, is the basis function, is the dimension of the system state after lifting in the high-dimensional space; After each iteration batch, state and input data generated during the batch process are collected to form a data set, and Koopman operator approximation is solved once, and the data set is defined as follows: (6) (7) (8) wherein denotes the length of the run time of the th batch; under the action of the boosting function (5), the data set and is boosted to obtain and : (9) (10) According to the EDMD method, the following two minimization problems are solved: (11) (12) wherein, denotes the Frobenius norm, yields an approximate high-dimensional linear model of the discrete state equation of the nonlinear dynamic model at the end of the th batch. (13) wherein is the system state in a high-dimensional space, is an estimate of the system output, , C is a mapping of the system output and state in the original space, C xk denotes a mapping from the system state in the high-dimensional space to the original state; , , is a constant matrix, assuming .

4. The unequal-length iterative learning control optimization method of claim 1, wherein, The prediction compensation correction model is constructed based on the estimated value of the output of the robot impedance control system, and the system output and tracking error are redefined as follows: If the estimated value output by the robot impedance control system is used Directly predicting the system output for missing time periods in variable batch length problems, because With the actual output of the system There is an error between them. If the error between the two is large, use directly. Using this as the system output for a missing time period will result in poor control performance; therefore, it is necessary to... Make corrections; define and The prediction error between them is: (14) wherein denotes the length of the run time of the batch, is the desired length of the run time; thus, at the batch, the system output over the time period is predicted on the basis of the estimate combined with the prediction error. (15) In a norm optimization framework, considering performance indicators of future batches, a prediction iterative learning control law is designed and optimal control gains are solved. (16) (17) where , is the system output value at time instant t using the input estimate of formula (15) and t is the predicted system tracking error, is the robot end-effector reference trajectory; formulas (16) and (17) adjust the system output and tracking error to a desired length for subsequent prediction iterative learning control law design.​ 5. The unequal-length iterative learning control optimization method of claim 1, wherein, Since the running time length of each batch is limited, the lifting technique is used, and the expected output, input, actual output, tracking error and predicted output of the estimated value of the system output of the system after lifting are defined as follows: Wherein, (18) (19) (20) (21) (22) wherein is the desired run length; the estimate of the system output is expressed in lifted form as follows: (23) The form of the prediction iterative learning control law is designed as follows: (24) (25) The optimal control gains are obtained by solving the performance indicator function (30): (26) wherein To control the gain, a performance index function is designed to solve; considering the tracking error and input variation of future batches, the performance index function is designed as: (27) wherein , is a discount factor, , , and is a positive definite symmetric matrix; The performance index function (27) in and are information of future batches, which are obtained using equation (23) and the control law (26): (28) (29) wherein is the lifting form of the prediction error; and in combination with equations (28) and (29), the performance index function (27) is rewritten as follows: (30) wherein is a positive definite symmetric matrix; The solution of the discrete algebraic Riccati equation (32) is solved by the following iterative method: (31) wherein is the solution of the following discrete algebraic Riccati equation: (32) Thus, the first The second iterative learning control law is: (33) After the stable solution is obtained by iteration, the optimal control gains are obtained by substituting equation (31). (34) The method further comprises analyzing the convergence of the prediction iterative learning control law, comprising:

6. The unequal-length iterative learning control optimization method of claim 5, wherein, Since there are modeling errors between the original nonlinear system and the approximate high-dimensional linear model obtained by the EDMD method, and prediction errors exist when prediction information is missing, in the convergence analysis, the prediction errors and the modeling errors are unified as modeling errors, and the discrete state equation of the nonlinear dynamic model is represented as a system with modeling errors related to equation (23): According to the control law (26) and the system (35), we have: (35) wherein, denotes the modeling error, , , , denotes the Euclidean norm, b G , b d , is an upper bound for the correlation parameter; Taking the norm of both sides of equation (36) gives: (36) Equation (39) gives the bounded control input, and on this basis, the convergence characteristics of the tracking error are further given; (37) Take the upper bound of , take the upper bound of , , , we get: (38) By analogy, we get , ; if the condition , , formula (37) is further derived: (39) Take the upper bound of , , bounded, , take the upper bound of ; The tracking error is rewritten as: Taking the norm of both sides of equation (40) gives: (40) wherein , , ; Wherein, (41) ​ (42) By the same token, we have , ; and the norm of (43) Take the upper bound of ; if the condition is met, , formula (41) is further derived: (44) Thus obtaining bounded, i.e. , take ; When time, therefore, there are: (45) Thus, the tracking error after the prediction compensation is To achieve bounded convergence; define Since Thus, the tracking error Achieve bounded convergence.

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