A fixed sub-iterative learning control method for robot precise trajectory tracking

Abstract

A kind of fixed sub-iterative learning control method for robot precise trajectory tracking belongs to industrial robot control technical field, the dynamic equation of repeated running robot system is established, its discrete time dynamic model is determined, fixed sub-iterative learning controller is designed, and the control effect is verified by simulation;For the situation that the initial position of robot system is uncertain, by constructing the sliding mode approach law based on hyperbolic tangent function, a controller capable of converging sliding variable to within the predetermined steady-state error band within fixed iteration number is designed, and the upper bound of convergence iteration number can be directly set and adjusted by controller parameters.The application can still guarantee the rapid convergence of tracking error under the condition that the initial state is uncertain, and reach the specified precision within fixed iteration number, so as to effectively improve the applicability and popularization value of iterative learning control in practical engineering application.

Description

Technical Field

[0001] This invention belongs to the field of industrial robot control technology, specifically relating to a fixed-iteration iterative learning control method for precise trajectory tracking of robots. Background Technology

[0002] Industrial robots, with their simple structure, rapid response capabilities, and excellent control precision, have become core equipment in modern manufacturing, widely used in high-precision operations such as welding, precision assembly, and painting. These applications typically require robots to repeatedly and accurately track a predetermined trajectory within a finite time interval, thus placing higher demands on the convergence speed and tracking accuracy of the control method. Iterative learning control, by iteratively correcting the control input, can gradually converge the trajectory tracking error within a finite time interval, significantly improving the robot's control performance in repetitive tasks. Therefore, achieving stable control and high-precision trajectory tracking of a robot system within a finite time interval not only has significant theoretical importance but also outstanding value for practical engineering applications.

[0003] Traditional iterative learning control methods typically require an infinite number of iterations to achieve ideal zero-error tracking performance. In industrial production environments, excessive robot iterations not only increase time costs and control complexity but also reduce overall system efficiency, failing to meet the demands of high-efficiency production. To overcome these limitations, finite-iteration iterative learning control methods aim to converge the tracking error to a pre-defined error range within a finite number of iterations. Current implementation methods mainly include dynamically adjusting the learning gain based on linear matrix inequalities and constructing a controller using obstacle composite energy functions. This method enables rapid error convergence after a finite number of repetitions, thus expanding the applicability of iterative learning control in engineering practice.

[0004] In practical robot control scenarios, initial tracking errors are susceptible to model uncertainties and external disturbances, and larger initial errors significantly increase the number of iterations required for convergence. Although existing finite-time iterative learning control methods can achieve error convergence within a finite number of iterations, the number of iterations required for finite-time iterative learning control methods at the same tracking accuracy depends on the initial error; that is, the number of iterations required for convergence increases with the increase of the initial error. In time-domain control theory, fixed-time control methods, by constructing a sliding mode reaching law to design a controller, achieve convergence of the sliding mode variable to the allowable range within a fixed time, and even if the initial error value increases, its convergence time can still remain within the expected range. However, how to achieve convergence of the sliding mode variable to the allowable range within a fixed number of iterations within the iterative learning control framework is currently a relatively rare area of ​​research.

[0005] Therefore, in industrial robot systems, designing a control strategy that can converge the sliding mode variable to a preset steady-state error band within a fixed number of iterations, with the upper bound of the convergence iterations independent of the initial error, under conditions of initial state uncertainty, has become a key focus in the field of robot control. This not only simultaneously meets the dual requirements of fast convergence and high-precision tracking in practical engineering, but also improves the practicality and robustness of the control strategy, providing a reliable control method for industrial robots in high-precision, high-efficiency application scenarios. Summary of the Invention

[0006] To overcome the problem that the convergence performance of existing finite-iteration iterative learning control methods for industrial robot systems is significantly affected by initial errors, this paper proposes a fixed-iteration iterative learning control method for precise robot trajectory tracking. This method, under conditions where the robot's initial state is uncertain, constructs a sliding mode reaching law based on the hyperbolic tangent function and designs a fixed-iteration iterative learning controller. This controller can converge the sliding mode variable to a specified steady-state error band within a fixed number of iterations, and the upper bound of the convergence iteration number can be directly calculated and adjusted through controller parameters. Compared with traditional methods, this invention overcomes the limitation of convergence performance in finite-iteration iterative learning control that depends on initial errors, ensuring that the robot system achieves rapid and stable convergence of the sliding mode variable within a fixed number of iterations. This provides effective theoretical support and technical assurance for high-precision and high-efficiency trajectory tracking tasks for industrial robots.

[0007] The proposed technical solution to address the above-mentioned technical problems is as follows:

[0008] A fixed-number iterative learning control method for precise robot trajectory tracking is proposed. Under the condition that the robot has an uncertain initial state, a sliding mode reaching law based on the hyperbolic tangent function is constructed, and a fixed-number iterative learning controller is designed. This allows the sliding mode variable to converge to a specified steady-state error band within a pre-calculated upper bound of the number of iterations that is independent of the initial error, thereby achieving high-precision and fast-convergence tracking control of the robot trajectory.

[0009] Furthermore, the fixed-iteration learning control method includes the following steps:

[0010] Step 1: Establish the dynamic model of the repetitive robot and construct its discrete dynamic model based on the discretization method;

[0011] Step 2: Based on the discrete dynamic model, define the trajectory tracking error and sliding mode variables, and construct the error dynamic equation;

[0012] Step 3: Design a sliding mode reaching law based on the hyperbolic tangent function, and derive the update law of the fixed-number-iteration learning controller based on this reaching law;

[0013] Step 4: By performing variable transformation on the sliding mode variable, establish the recursive relationship of the transformed variable, and determine the upper bound of the steady-state error band of the sliding mode variable accordingly;

[0014] Step 5: Based on the recursive relationship, analyze and calculate the upper bound of the number of convergence iterations required for the sliding mode variable to enter the steady-state error band. This upper bound is independent of the initial error of the system.

[0015] Furthermore, step 1 includes the following sub-steps:

[0016] 1.1 Establishing the continuous-time dynamic equations of a repetitive robot based on the Euler-Lagrange equations:

[0017] (1);

[0018] in, Indicates the number of iterations; , and They represent the robot's number 1 and 2. Joint angular displacement, joint angular velocity, and joint acceleration at each iteration; Represents the bounded moment of inertia; Indicates the coefficients of centrifugal force and Coriolis force; This represents the bounded gravitational term. and These are the robot's mass and length, respectively. It is the acceleration due to gravity; Indicates the input torque;

[0019] 1.2 The Euler discretization method is adopted, and the sampling period is selected. Discretizing the differential terms in equation (1) yields the discrete dynamic model of the robot:

[0020] (2);

[0021] in, For the first The joint angular displacement of the step; For the first Step input torque; For discrete time steps, , It is a finite positive integer, representing the total number of steps in a single iteration;

[0022] 1.3 To facilitate controller design, the discrete dynamic model (2) is expressed in a compact form containing parameter vectors and nonlinear function vectors:

[0023] (3);

[0024] in, , , , It is by The constructed block matrix; vector ,parameter Nonlinear function vector , Represents the transpose of a vector. Let be any but bounded initial position.

[0025] Furthermore, step 2 includes the following sub-steps:

[0026] 2.1 For Define the expected trajectory If the desired position is the robot's position during movement, then the tracking error is... Introducing sliding mode variables ,in , ,and satisfy Furthermore, based on equation (3) and error information... By defining the error dynamic equation, we obtain:

[0027] (4);

[0028] in, , , ;

[0029] 2.2 If there exists a positive integer This makes it possible for all Sliding mode variables The absolute values ​​are all less than or equal to a preset steady-state error band. And the number of iterations required to converge to the steady-state error band. If it is independent of the initial error, it is called a sliding mode variable. It converges within a fixed number of iterations; the goal of designing a fixed-iteration learning control scheme is to generate a control input sequence. The motion of the driving robot's dynamic equation (1) causes the sliding mode variable to change. It converges to within a fixed number of iterations. .

[0030] Furthermore, step 3 includes the following sub-steps:

[0031] 3.1 According to the definition of sliding mode variables And the error dynamic equation (4), the sliding mode reaching law is designed as

[0032] (5);

[0033] in, , , , For symbolic functions, and Let these represent the hyperbolic tangent function and the inverse hyperbolic tangent function, respectively.

[0034] 3.2 The fixed-number iteration learning controller is designed based on the reaching law (5).

[0035] (6);

[0036] in, .

[0037] Furthermore, step 4 includes the following sub-steps:

[0038] 4.1 Defining Variables and Represent the transformed sliding mode variables and parameters respectively, and denote the sliding mode variables. The upper bound of the steady-state error band is denoted as Then there is Therefore, by changing the original sliding mode variable... Transformed into a new range of values sliding mode variables When calculated Enter Upper bound on the number of convergence iterations At that time, the corresponding sliding mode variable can be obtained. Enter Upper bound on the number of convergence iterations ;

[0039] 4.2 According to and ,Know and The approach law (6) is transformed into Furthermore, by taking the absolute value of both sides and recursively applying the formula, we obtain the recursive inequality for the transformed sliding mode variable.

[0040] (7);

[0041] Combining the definition of the steady-state error band and equation (7), we get

[0042] (8);

[0043] Solving this inequality yields the upper bound of the steady-state error band. Furthermore, based on the upper bound of the steady-state error band... and Relationship Calculate the upper bound of the steady-state error band corresponding to the original sliding mode variable. The range is .

[0044] Furthermore, in step 5, firstly, the sliding mode variable after conversion is calculated. Enter Upper bound on the number of convergence iterations Due to the initial value Using the recurrence relation, inequality (7) is transformed into

[0045] (9);

[0046] Furthermore, in order to achieve Inequality (9) is transformed into Solving this inequality yields an upper bound on the number of convergent iterations. ;in, Indicates greater than or equal to The smallest integer; therefore, the sliding mode variable Enter Upper bound on the number of convergence iterations The upper bound of the number of iterations With initial error Irrelevant; thus, sliding mode variables are obtained. It can converge to the steady-state error band within a fixed number of iterations. The conclusion.

[0047] In this invention, error tracking accuracy is a key indicator for evaluating robot trajectory tracking performance. However, traditional iterative learning control methods require an infinite number of iterations to achieve zero-error convergence, while existing finite-number iterative learning control methods are significantly dependent on the initial error, with the number of iterations required for tracking error convergence increasing with the initial error. This invention proposes a fixed-number iterative learning control method for precise robot trajectory tracking, which can converge the sliding mode variable to a specified steady-state error band within a fixed number of iterations, and the upper bound of the convergence iteration count can be directly calculated and adjusted through controller parameters.

[0048] The technical concept of this invention is as follows: For robot systems with uncertain initial positions, a fixed-iteration learning controller is designed by constructing a reaching law based on the hyperbolic tangent function. This controller ensures that, even with uncertain initial positions, the sliding mode variable constructed from the tracking error can quickly and stably converge to a preset steady-state error band within a fixed number of iterations. Furthermore, the upper bound of the convergence iteration count can be directly adjusted through the controller parameters.

[0049] The main advantages of this invention are as follows: For robot systems with repetitive motion characteristics, a fixed-iteration learning control method is proposed, where the upper bound of the convergence iteration count is unaffected by the initial error magnitude. Compared with traditional methods, this technique not only enables the sliding mode variable to enter the steady-state error band within a predetermined number of iterations, but also allows for direct adjustment of convergence performance through controller parameters, facilitating engineering applications. This method provides a new technical approach for high-precision trajectory tracking of robots, while simultaneously enhancing the system's usability and adaptability in complex engineering environments. Attached Figure Description

[0050] Figure 1 This is a block diagram of the robot control system.

[0051] Figure 2 The output tracking results are shown for different iteration numbers;

[0052] Figure 3 The error variation under different iteration numbers;

[0053] Figure 4 For the sliding mode variable at the initial angular position The changes;

[0054] Figure 5 For the sliding mode variable at the initial angular position The changes;

[0055] Figure 6 For the sliding mode variable at the initial angular position The changes. Detailed Implementation

[0056] The invention will now be further described with reference to the accompanying drawings.

[0057] Reference Figure 1 The block diagram of the robot control system provided by this invention illustrates its basic working principle. Initial position during the next iteration With initial input Acting on the robot system, the drive system generates the actual output trajectory. All input and output data generated in the system are stored in the system memory in real time for subsequent control law updates and learning processes. Furthermore, the output is calculated... With the expected trajectory The deviation between them is used to obtain the tracking error of the current iteration. .error Together with the current control input The robot's parameter information is passed to the fixed-number-times-iteration learning controller, which generates the control input for the next batch based on the reaching law and the update law. The above process is executed cyclically until the number of iterations reaches the preset maximum value.

[0058] Reference Figures 2-6 A fixed-iteration learning control method for precise trajectory tracking of robots is proposed. Under the condition of uncertain initial state of the robot, a fixed-iteration learning controller is designed by constructing a reaching law based on the hyperbolic tangent function. This method can converge the sliding mode variable to a specified steady-state error band within a fixed number of iterations, and the upper bound of the convergence iteration number can be directly calculated and adjusted by the controller parameters.

[0059] The fixed-iteration learning control method for precise trajectory tracking of robots includes the following steps:

[0060] Step 1: Establish a dynamic model of the repetitive robot and construct its discrete dynamic model based on the discretization method; Step 1 in this embodiment includes the following sub-steps:

[0061] 1.1 Establishing the continuous-time dynamic equations of a repetitive robot based on the Euler-Lagrange equations:

[0062] (1);

[0063] in, Indicates the number of iterations; , and They represent the robot's number 1 and 2. Joint angular displacement, joint angular velocity, and joint acceleration at each iteration; Represents the bounded moment of inertia; Indicates the coefficients of centrifugal force and Coriolis force; This represents the bounded gravitational term. and These are the robot's mass and length, respectively. It is the acceleration due to gravity; Indicates the input torque;

[0064] 1.2 The Euler discretization method is adopted, and the sampling period is selected. Discretizing the differential terms in equation (1) yields the discrete dynamic model of the robot:

[0065] (2);

[0066] in, For the first The joint angular displacement of the step; For the first Step input torque; For discrete time steps, , It is a finite positive integer, representing the total number of steps in a single iteration;

[0067] 1.3 To facilitate controller design, the discrete dynamic model (2) is expressed in a compact form containing parameter vectors and nonlinear function vectors:

[0068] (3);

[0069] in, , , , It is by The constructed block matrix; vector ,parameter Nonlinear function vector , Represents the transpose of a vector. Let be any but bounded initial position.

[0070] Step 2: Based on the discrete dynamic model, define the trajectory tracking error and sliding mode variables, and construct the error dynamic equation;

[0071] Step 2 of this embodiment includes the following sub-steps:

[0072] 2.1 For Define the expected trajectory If the desired position is the robot's position during movement, then the tracking error is... Introducing sliding mode variables ,in , ,and satisfy Furthermore, based on equation (3) and error information... By defining the error dynamic equation, we obtain:

[0073] (4);

[0074] in, , , ;

[0075] 2.2 If there exists a positive integer This makes it possible for all Sliding mode variables The absolute values ​​are all less than or equal to a preset steady-state error band. And the number of iterations required to converge to the steady-state error band. If it is independent of the initial error, it is called a sliding mode variable. It converges within a fixed number of iterations; the goal of designing a fixed-iteration learning control scheme is to generate a control input sequence. The motion of the driving robot's dynamic equation (1) causes the sliding mode variable to change. It converges to within a fixed number of iterations. .

[0076] Step 3: Design a sliding mode reaching law based on the hyperbolic tangent function, and derive the update law of the fixed-number-iteration learning controller based on this reaching law; Step 3 in this embodiment includes the following sub-steps:

[0077] 3.1 According to the definition of sliding mode variables And the error dynamic equation (4), the sliding mode reaching law is designed as follows:

[0078] (5);

[0079] in, , , , For symbolic functions, and Let these represent the hyperbolic tangent function and the inverse hyperbolic tangent function, respectively.

[0080] 3.2 The fixed-number iteration learning controller is designed based on the reaching law (5) as follows:

[0081] (6);

[0082] in, .

[0083] Step 4: By performing variable transformation on the sliding mode variables, a recursive relationship is established for the transformed variables, thereby determining the upper bound of the steady-state error band of the sliding mode variables; Step 4 in this embodiment includes the following sub-steps:

[0084] 4.1 Defining Variables and Represent the transformed sliding mode variables and parameters respectively, and denote the sliding mode variables. The upper bound of the steady-state error band is denoted as Then there is Therefore, by changing the original sliding mode variable... Transformed into a new range of values sliding mode variables When calculated Enter Upper bound on the number of convergence iterations At that time, the corresponding sliding mode variable can be obtained. Enter Upper bound on the number of convergence iterations ;

[0085] 4.2 According to and ,Know and The approach law (6) is transformed into Furthermore, by taking the absolute value of both sides and recursively applying the formula, we obtain the recursive inequality for the transformed sliding mode variable:

[0086] (7);

[0087] Combining the definition of the steady-state error band and equation (7), we get:

[0088] (8);

[0089] Solving this inequality yields the upper bound of the steady-state error band. Furthermore, based on the upper bound of the steady-state error band... and Relationship Calculate the upper bound of the steady-state error band corresponding to the original sliding mode variable. The range is .

[0090] Step 5: Based on the recursive relationship, analyze and calculate the upper bound of the number of convergence iterations required for the sliding mode variable to enter the steady-state error band. This upper bound is independent of the initial error of the system.

[0091] In step 5 of this embodiment, firstly, the sliding mode variable after transformation is calculated. Enter Upper bound on the number of convergence iterations Due to the initial value Using the recurrence relation, inequality (7) is transformed into:

[0092] (9);

[0093] Furthermore, in order to achieve Inequality (9) is transformed into Solving this inequality yields an upper bound on the number of convergent iterations. ;in, Indicates greater than or equal to The smallest integer; therefore, the sliding mode variable Enter Upper bound on the number of convergence iterations The upper bound of the number of iterations With initial error Irrelevant; thus, sliding mode variables are obtained. It can converge to the steady-state error band within a fixed number of iterations. The conclusion.

[0094] In this embodiment, the effectiveness of the fixed-iteration learning control method for precise trajectory tracking of a robot is verified through simulation. The process is as follows:

[0095] 6.1 To verify the effectiveness of the present invention, the control effect of the fixed-iteration learning control method for precise trajectory tracking of the robot described above was simulated and verified. The model parameters of the robot system (1) were selected as follows: , , , , Sampling period Total number of discrete time points To demonstrate output tracking and error changes, an initial input is selected. External interference Initial angular position , The desired angle and position are selected as follows:

[0096] ;

[0097] For the reaching law (95) and the fixed-number-iteration learning controller (6), select parameters , , , Therefore, through calculation, , , , Sliding mode variables Enter Upper bound on the number of convergence iterations ;

[0098] 6.2 Figures 2-6 The simulation results demonstrate the effectiveness of the fixed-iteration iterative learning control method for precise trajectory tracking of robots proposed in this invention. From... Figure 2 It can be seen that even when the robot's initial position fluctuates, this method can still effectively track the desired trajectory from the actual position. Figure 3 This demonstrates how the actual tracking error gradually converges to the zero neighborhood as the number of iterations increases. Figures 4-6 The sliding mode variables are given under different initial position conditions. It monotonically decreases with increasing iteration number and eventually enters the steady-state error band. The trend of change. Simulation results show that the sliding mode variable Entering the error band The actual number of iterations required were 18, 20, and 20, respectively, which is less than the theoretically calculated value of 21, verifying the effectiveness of the proposed method in terms of convergence performance in a fixed number of iterations.

[0099] In summary, the fixed-iteration learning control method proposed in this invention can achieve fast and high-precision trajectory tracking even when the robot's initial position is uncertain. This method constructs a reaching law based on the hyperbolic tangent function and designs a fixed-iteration learning controller to ensure that the sliding mode variable converges to a specified steady-state error band within a fixed number of iterations. In particular, the upper bound of the convergence iteration number can be directly calculated and adjusted through controller parameters, providing a clear theoretical basis and operable method for the application of iterative learning control in practical engineering.

[0100] The above description is only for illustrating the technical solution of the present invention and is limited to the above examples. In practical applications, the present invention may have various modifications and improvements. Any modifications, alterations or substitutions made without departing from the basic principles and essence of the present invention shall be considered to fall within the protection scope of the present invention.

Claims

1. A fixed sub-iterative learning control method for robot precision trajectory tracking, characterized in that, In the case that the robot has initial state uncertainty, a fixed-order iterative learning controller is designed by constructing a hyperbolic tangent function-based sliding mode reaching law, so that the sliding mode variable converges to a specified steady-state error band within a pre-calculable iteration number upper bound independent of the initial error, thereby realizing high-precision and fast-convergence tracking control of the robot trajectory; the fixed-order iterative learning control method comprises the following steps: Step 1, establishing a dynamic equation of the robot running repeatedly, and constructing a discrete dynamic model based on a discretization method; Step 2, defining a trajectory tracking error and a sliding mode variable based on the discrete dynamic model, and constructing an error dynamic equation; Step 3, designing a hyperbolic tangent function-based sliding mode reaching law, and deriving a fixed-order iterative learning controller based on the reaching law for updating the control input in each iteration; including the following sub-steps: 3.1 Definition of sliding variable and error dynamics, the sliding mode reaching law is designed as (5)； where , , , is a sign function, and denote hyperbolic tangent and inverse hyperbolic tangent functions, respectively, , , and for , is a discrete time step, , is a finite positive integer, representing the total number of steps for a single iteration, defining the desired trajectory is the desired position of the robot motion, then the tracking error is , is the joint angle displacement at the th step; 3.2 The fixed-order iterative learning controller is designed based on the reaching law (5) as (6)； wherein , is the input torque of the first step, the parameter , denotes a bounded moment of inertia, the sampling period , is a block matrix composed of ; the vector , denotes the centrifugal and Coriolis force coefficients; , , the nonlinear function vector , denotes the transpose of a vector, is an arbitrary but bounded initial position, ; Step 4, establishing a recursive relationship of the transformed variable by performing a variable transformation on the sliding mode variable, and determining an upper bound of the steady-state error band of the sliding mode variable; Step 5, based on the recursive relationship, analyzing and calculating the upper bound of the convergence iteration number required for the sliding mode variable to enter the steady-state error band, which is independent of the initial error of the system.

2. The fixed sub-iterative learning control method for robot precision trajectory tracking according to claim 1, characterized in that, In step 1, a continuous-time dynamic model of the robot is established based on the Euler-Lagrange equation; the continuous-time dynamic model is converted into a discrete dynamic model by using a discretization method; the discrete dynamic model is expressed in a compact form containing a parameter vector and a nonlinear function vector.

3. The fixed-iteration learning control method for precise trajectory tracking of a robot according to claim 1, wherein In step 2, the difference between the desired trajectory and the actual trajectory of the robot motion is defined as a tracking error; based on the discrete dynamic model, an error dynamic equation of the tracking error changing with iterations is derived; a linear combination of the tracking error is defined as a sliding mode variable; if there is a positive integer such that for all iteration times the absolute value of the sliding mode variable is less than or equal to a preset steady state error band, and the iteration number required for the sliding mode variable to converge to the steady state error band is independent of the initial error, then the sliding mode variable is said to be fixed iteration number convergence.

4. The fixed-iteration learning control method for precise trajectory tracking of a robot according to claim 1, wherein In step 4, a variable transformation is performed on the sliding mode variable, which maps the original variable to a bounded interval; based on the sliding mode reaching law, a recursive inequality of the transformed variable is established; by solving the inequality, an analytical expression of the upper bound of the steady-state error band corresponding to the original sliding mode variable is obtained.

5. The fixed-iteration learning control method for precise trajectory tracking of a robot according to claim 1, wherein In step 5, the recursive relationship of the transformed sliding mode variable is used to derive an inequality relationship of the iteration number required for the transformed sliding mode variable to enter the transformed steady-state error band; by solving the inequality, an explicit convergence iteration number upper bound is obtained, which is dependent on the controller parameters and independent of the initial state of the system, thereby ensuring that the robot accurately tracks the desired trajectory within the iteration number.

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