Finite iterative error tracking learning control method for robot precise trajectory tracking

Abstract

The application discloses a limited iterative error tracking learning control method for precise trajectory tracking of a robot, and comprises the following steps: establishing a dynamic equation of a repeatedly operated robot system, determining a discrete dynamic model of the robot system, designing a limited iterative error tracking learning controller, and carrying out simulation verification. According to the application, under the condition that an initial position of the robot system is arbitrary and a desired trajectory changes, an error tracking learning controller is designed, and only error information obtained through first iteration is used to calculate the number of iterations required for realizing arbitrary precision tracking of the robot system output error to the given desired error trajectory. The application effectively overcomes the limitation that realization of zero error convergence needs to rely on infinite iterations in the traditional iterative learning control method. Meanwhile, the application ensures that the robot system can still achieve the predetermined tracking precision with fast convergence ability under the disturbance conditions that the initial position is arbitrary and the desired trajectory changes, and improves the practicability and popularization value of the iterative learning control in actual engineering.

Description

Technical Field

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

[0002] Industrial robots, due to their simple structure, high efficiency, and convenient control, have become indispensable tools in modern manufacturing. They can not only perform repetitive tasks for extended periods in high-risk or harsh environments, but also significantly improve production efficiency and product quality in fields such as automated assembly, welding, and precision machining. To meet the stringent requirements of industrial production for high precision and efficiency, industrial robots need to accurately and repeatedly track predetermined trajectory signals within a limited time. Therefore, researching how robots can achieve stable control and precise trajectory tracking within a finite time has significant theoretical and practical value.

[0003] Finite-time stability is particularly important in industrial applications, as it ensures that control objectives are achieved rapidly within a finite time, thereby significantly reducing production costs and improving operational efficiency. However, relying solely on finite-time stability is insufficient to meet the perfect tracking requirements of repetitive control tasks. Therefore, iterative learning control methods have been proposed and extensively studied. These methods, based on the inherent repeatability of the system, gradually approximate the desired trajectory through multiple iterations, demonstrating significant advantages in improving the transient response and tracking performance of uncertain dynamic systems. Traditional iterative learning control methods typically require strict repeatability between the initial state and the desired state of the system. However, in real-world industrial scenarios, this assumption is often difficult to fully satisfy.

[0004] To overcome the aforementioned limitations, an error tracking strategy proposes a novel approach, transforming the traditional output tracking problem into an error trajectory tracking problem. The goal is no longer to ensure the actual output perfectly matches the expected output, but rather to gradually reduce the tracking error through a pre-defined error trajectory. This strategy effectively relaxes the stringent requirement of initial state consistency, enabling the robot system to achieve rapid error convergence even with arbitrary initial states, thereby further expanding the applicability of iterative learning control in practical applications.

[0005] On the other hand, most existing research on iterative learning control focuses on iterative asymptotic convergence, which assumes that the system can achieve ideal error convergence through an infinite number of iterations. While this iterative asymptotic convergence can theoretically achieve high-precision trajectory tracking, it faces significant limitations in practical engineering. For industrial robots used in high-precision tasks such as assembly and welding, the requirement is to achieve high-quality completion within a finite number of operations in a finite time. Infinite iterations are not only impractical but may also increase control complexity and reduce the operability of engineering implementation. Furthermore, from an engineering efficiency perspective, excessive iterations lead to wasted time, which does not meet the core requirements of efficient production.

[0006] Therefore, in industrial robot systems, how to achieve rapid error convergence within a finite number of iterations by designing a controller under conditions of arbitrary initial states and changing desired trajectories has become an important research topic in the field of robot control. This not only meets the dual requirements of speed and accuracy in practical engineering but also provides practical theoretical guidance and a feasible path for solving control challenges in complex industrial tasks. Summary of the Invention

[0007] To address the problem that traditional iterative learning control methods for robots rely on an infinite number of iterations to achieve zero-error convergence, this invention proposes a finite iterative error tracking learning control method for precise robot trajectory tracking. Under conditions of arbitrary initial robot position and varying desired trajectory, this method, through the design of a finite iterative error tracking learning controller, can calculate the number of iterations required to achieve arbitrary precision tracking of the robot system's output tracking error on a given desired error trajectory using only the error information obtained in the first iteration. Compared to traditional methods, this method effectively eliminates the dependence on infinite iterations, ensuring that the robot system can achieve rapid convergence of the target error within a finite number of iterations. This provides theoretical support and technical assurance for precise and efficient trajectory tracking tasks for industrial robots.

[0008] The proposed technical solution to address the above-mentioned technical problems is as follows:

[0009] A finite iterative error tracking learning control method for precise trajectory tracking of robots, the method comprising the following steps:

[0010] Step 1: Establish the continuous-time dynamic equations of the repetitive robot:

[0011]

[0012] Where s represents time, k represents the number of repetitions; θ k (s), and Let G(θ) represent the joint angular displacement, joint angular velocity, and joint acceleration of the robot in the k-th iteration, respectively; D represent the moment of inertia; C represent the centrifugal force and Coriolis force; and G(θ) represent the joint angular displacement, joint angular velocity, and joint acceleration of the robot in the k-th iteration. k (s))=mglcosθ k (s) represents the bounded gravitational term, m and l represent the robot's mass and length respectively, g represents the gravitational acceleration, and τ k (s) represents the input torque of the robot in the kth iteration;

[0013] Step 2: Establish a discrete dynamic model of the repetitive robot, as follows:

[0014] 2.1 The system equations of the robot (1) are transformed into mathematical equations.

[0015]

[0016] 2.2 Using the Euler discretization method, the discretization of equation (2) is performed. and Discretization is performed, and there is

[0017]

[0018] Where, t∈T N Indicates step size; T N = {0, 1, ..., N}, where N is a finite positive integer representing the running period of each iteration; h > 0 indicates the sampling period;

[0019] 2.3 Substituting equation (3) into equation (2), we obtain the discrete form of the robot model:

[0020]

[0021] Where, θ k (t)=θ k (th) is the sampled output signal at step size t; τ k (t)=τ k (th) is the actual input torque maintained by the zero-order hold;

[0022] 2.4 Define parameters variable Nonlinear variable ψ(θ) k (t))=[θ k (t),θ k (t-1),cosθ k (t)] Τ Then equation (4) can be restated as the following nonlinear discrete dynamic model:

[0023] θ k (t+1)=ζ Τ ψ(θk (t))+bτ k (t) (5)

[0024] Where, θ k (0) represents an arbitrary but bounded initial position, θ k (-1) = 0;

[0025] 2.5 Further, equation (5) is restated in vector form:

[0026] θ k =ΞΦ k +Bτ k (6)

[0027] Where, θ k =[θ k (1),θ k (2),…,θ k (N+1)] Τ , τ k =[τ k (0),τ k (1),…,τ k (N)] Τ A diagonal matrix B = diag(b,b,…,b) ∈ R (N+1)×(N+1) Composed of b; Ξ = diag(ζ) Τ ,ζ Τ ,…,ζ Τ )∈R (N+1)×3(N+1) It is made by ζ Τ The constructed block matrix, Φ k =[ψ(θ) k (0)),ψ(θ k (1)),…,ψ(θ k (N))] Τ ∈R 3(N+1) ;

[0028] Step 3: Design of a finite iterative error tracking iterative learning controller.

[0029] Furthermore, the process of step 3 is as follows:

[0030] 3.1 Consider the case where the desired trajectory of the robot's motion may change. For any t∈T N Define signal θ d,k If (t) is the desired trajectory of the robot during its movement, then e k (t)=θ d,k (t)-θ k (t) represents the error between the robot's desired position signal and the actual position generated during robot movement; the desired error trajectory is defined. for

[0031]

[0032] Where Ω represents the pre-defined time points of the connection transition trajectory and the desired zero-error trajectory, and μ(t) is a smooth monotonically decaying function on [0,Ω], satisfying μ(0)=1 and μ(Ω)=0; define vector e k =[e k (0),e k (1),…,e k (N)] Τ , It is the difference between the expected error signal and the actual error signal;

[0033] 3.2 The goal of designing a finite iterative error tracking learning control scheme is to generate the control input sequence τ. k+1 Drive the robot to move so that after K iterations, the robot system outputs a tracking error e. k For a given expected error trajectory It can achieve tracking with a specified precision, that is, when k→K, it has

[0034]

[0035] Where max|·| represents the function as k→K, t∈T N The maximum absolute value of , K is a finite positive integer representing the number of error iterations; φ represents the desired error convergence accuracy.

[0036] 3.3 According to equation (6), we know

[0037] θ k+1 =ΞΦ k+1 +Bτ k+1 (8)

[0038] Therefore, based on equations (6), (8), and the error information ε k The definition has

[0039]

[0040] in, Δθ d,k+1 =θ d,k+1 -θ d,k , ΔΦ k+1 =Φ k+1 -Φ k ,Δτ k+1 =τ k+1 -τ k ;

[0041] 3.4 Further, the finite iterative error tracking learning controller is designed as follows:

[0042]

[0043] Among them, B -1 ρ is the inverse matrix of B, and 0 < ρ < 1 are parameters to be set.

[0044] Furthermore, the method also includes the following steps:

[0045] Step 4, error ε k After K iterations, the convergence to within the desired accuracy φ is as follows:

[0046] 4.1 Substituting the controller (10) into the error equation (9), we get

[0047] ε k+1 =(1-ρ)ε k (11)

[0048] According to equation (11), we have

[0049]

[0050] Since 0 < ρ < 1, when the number of iterations k approaches infinity, we have Therefore, the error ε k Converging to zero;

[0051] 4.2 According to (12), when Sometimes, Therefore, the number of iterations required for the error to converge to the desired accuracy φ is:

[0052]

[0053] in, Represents the smallest integer greater than or equal to x;

[0054] 4.3 Therefore, when the first iteration ends, the error ε1 information is obtained, that is, the error ε is calculated. k The specific value of the number of iterations K required to converge to the desired accuracy φ.

[0055] 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, which is impractical in real-world applications. Since infinite iterations are not feasible in real-world scenarios, error tracking performance analysis often lacks predictability, making it difficult to determine the specific number of iterations required to complete high-speed, accurate tracking tasks. To address the problem of error convergence relying on infinite iterations, this invention proposes a finite-iterative error tracking learning control method for precise robot trajectory tracking. This method, through reasonable controller design, utilizes only the error information from the first iteration to calculate the number of iterations required for the robot system to output tracking error with arbitrary accuracy for a given desired error trajectory, given an arbitrary initial position and changing desired trajectory.

[0056] The technical concept of this invention is as follows: For robot systems with arbitrary initial positions and varying desired trajectories, this invention designs a finite iterative error tracking learning controller. This controller utilizes the error information obtained in the first iteration to directly calculate the number of iterations required for the robot system to achieve arbitrary given accuracy tracking of the desired trajectory with the output tracking error. This method effectively overcomes the dependence of traditional control methods on infinite iterations, while ensuring that the robot system still possesses the ability to quickly converge errors even under conditions of arbitrary initial positions and varying desired trajectories.

[0057] The advantages of this invention are as follows: For robot systems with repetitive motion characteristics, this invention treats the robot as the controlled object and proposes a method to achieve error convergence without relying on an infinite number of iterations. Compared to traditional methods, this technique not only solves the requirement of an infinite number of iterations for zero-error convergence, but also ensures that the robot can efficiently and quickly track the desired trajectory under perturbation conditions with arbitrary initial positions and changing desired trajectories. This method provides a new technical path for the precise control of robot systems, while improving their usability and adaptability in practical engineering. Attached Figure Description

[0058] Figure 1 This is the control flowchart for the robot;

[0059] Figure 2 The output tracking results are shown for different iteration numbers;

[0060] Figure 3 Error tracking results at different iteration numbers;

[0061] Figure 4 This shows the variation of the maximum absolute value of the error under different iteration numbers. Detailed Implementation

[0062] The invention will now be further described with reference to the accompanying drawings.

[0063] Reference Figure 1 The robot model block diagram disclosed in this invention describes the working principle of the system. In the kth batch, the initial position θ k (0) and initial input τ k When applied to the robot system, it produces the actual output θ. k The input and output data of the robot system are stored in the system memory for subsequent processing. Furthermore, the actual output θ of the robot is calculated. k With the expected trajectory θ d,k The difference between the two values ​​can be used to obtain the actual tracking error e. k The actual error e k With expected error signal The difference ε k Along with the input information τ k The robot system parameters, along with the data, are transmitted to the iterative learning controller. The iterative learning controller then uses this information to generate the control input τ for the next batch. k+1 The above process runs in a loop until the system reaches the preset maximum number of iterations, at which point the iteration stops, thus achieving the controller design goal.

[0064] Reference Figure 2 — Figure 4 A finite iterative error tracking learning control method for precise trajectory tracking of robots, the method comprising the following steps:

[0065] Step 1: Establish the continuous-time dynamic equations of the repetitive robot:

[0066]

[0067] Where s represents time, k represents the number of repetitions; θ k (s), and Let G(θ) represent the joint angular displacement, joint angular velocity, and joint acceleration of the robot in the k-th iteration, respectively; D represent the moment of inertia; C represent the centrifugal force and Coriolis force; and G(θ) represent the joint angular displacement, joint angular velocity, and joint acceleration of the robot in the k-th iteration. k (s))=mglcosθ k (s) represents the bounded gravitational term, m and l represent the robot's mass and length respectively, g represents the gravitational acceleration, and τ k (s) represents the input torque of the robot in the kth iteration;

[0068] Step 2: Establish a discrete dynamic model of the repetitive robot, as follows:

[0069] 2.1 The system equations of the robot (1) are transformed into mathematical equations.

[0070]

[0071] 2.2 Using the Euler discretization method, the discretization of equation (2) is performed. and Discretization is performed, and there is

[0072]

[0073] Where, t∈T N Indicates step size; T N = {0, 1, ..., N}, where N is a finite positive integer representing the running period of each iteration; h > 0 indicates the sampling period;

[0074] 2.3 Substituting equation (3) into equation (2), we obtain the discrete form of the robot model:

[0075]

[0076] Where, θ k (t)=θ k (th) is the sampled output signal at step size t; τ k (t)=τ k (th) is the actual input torque maintained by the zero-order hold;

[0077] 2.4 Define parameters variable Nonlinear variable ψ(θ) k (t))=[θ k (t),θ k (t-1),cosθ k (t)] Τ Then equation (4) can be restated as the following nonlinear discrete dynamic model:

[0078] θ k (t+1)=ζ Τ ψ(θ k (t))+bτ k (t) (5)

[0079] Where, θ k (0) represents an arbitrary but bounded initial position, θ k (-1) = 0;

[0080] 2.5 Further, equation (5) can be restated in vector form:

[0081] θ k =ΞΦ k +Bτ k (6)

[0082] Where, θ k =[θk (1),θ k (2),…,θ k (N+1)] Τ , τ k =[τ k (0),τ k (1),…,τ k (N)] Τ A diagonal matrix B = diag(b,b,…,b) ∈ R (N+1)×(N+1) Composed of b; Ξ = diag(ζ) Τ ,ζ Τ ,…,ζ Τ )∈R (N+1)×3(N+1) It is made by ζ Τ The constructed block matrix, Φ k =[ψ(θ) k (0)),ψ(θ k (1)),…,ψ(θ k (N))] Τ ∈R 3(N+1) ;

[0083] Step 3, the design of the finite iterative error tracking iterative learning controller, is as follows:

[0084] 3.1 Consider the case where the desired trajectory of the robot's motion may change. For any t∈T N Define signal θ d,k If (t) is the desired trajectory of the robot during its movement, then e k (t)=θ d,k (t)-θ k (t) represents the error between the robot's desired position signal and the actual position generated during robot movement; the desired error trajectory is defined. for

[0085]

[0086] Where Ω represents the pre-defined time points of the connection transition trajectory and the desired zero-error trajectory, and μ(t) is a smooth monotonically decaying function on [0,Ω], satisfying μ(0)=1 and μ(Ω)=0; define vector e k =[e k (0),e k (1),…,e k (N)] Τ , It is the difference between the expected error signal and the actual error signal;

[0087] 3.2 The goal of designing a finite iterative error tracking learning control scheme is to generate the control input sequence τ. k+1Drive the robot to move so that after K iterations, the robot system outputs a tracking error e. k For a given expected error trajectory It can achieve tracking with a specified precision, that is, when k→K, it has

[0088]

[0089] Where max|·| represents the function as k→K, t∈T N The maximum absolute value of , K is a finite positive integer representing the number of error iterations; φ represents the desired error convergence accuracy.

[0090] 3.3 According to equation (6), we know

[0091] θ k+1 =ΞΦ k+1 +Bτ k+1 (8)

[0092] Therefore, based on equations (6), (8), and the error information ε k The definition has

[0093]

[0094] in, Δθ d,k+1 =θ d,k+1 -θ d,k , ΔΦ k+1 =Φ k+1 -Φ k ,Δτ k+1 =τ k+1 -τ k ;

[0095] 3.4 Further, the finite iterative error tracking learning controller is designed as follows:

[0096]

[0097] Among them, B -1 ρ is the inverse matrix of B, and 0 < ρ < 1 are parameters to be set.

[0098] Step 4, error ε k After K iterations, the convergence to within the desired accuracy φ is as follows:

[0099] 4.1 Substituting the controller (10) into the error equation (9), we get

[0100] ε k+1 =(1-ρ)ε k (11)

[0101] According to equation (11), we have

[0102]

[0103] Since 0 < ρ < 1, when the number of iterations k approaches infinity, we have Therefore, the error ε k Converging to zero;

[0104] 4.2 According to (12), when Sometimes, Therefore, the number of iterations required for the error to converge to the desired accuracy φ is:

[0105]

[0106] in, Represents the smallest integer greater than or equal to x;

[0107] 4.3 Therefore, when the first iteration ends, the error ε1 information is obtained, that is, the error ε is calculated. k The specific value of the number of iterations K required to converge to the desired accuracy φ.

[0108] The method in this embodiment further includes the following steps:

[0109] Step 5: Simulation verification of the effectiveness of the finite iterative error tracking learning control method for precise robot trajectory tracking. The process is as follows:

[0110] 5.1 To verify the effectiveness of the present invention, the control effect of the finite iterative error tracking learning control method for precise trajectory tracking of robots shown in equations (10) and (13) is verified by simulation.

[0111] 5.2 The model parameters for the continuous-time dynamic equation (1) of the robot are selected as m = 1, g = 9.8, l = 0.25. C = 2; sampling period h = 0.02s; total number of discrete time points set to N = 100; initial angular position θ k (0) Choose any value between [-0.1, 0.1], and the initial input τ k (t)=1(t∈T N ), select controller parameter ρ = 0.6; the desired angular position is expressed as Noise term w k The error trajectory iterates between [-0.1, 0.1] with a desired convergence accuracy φ = 0.001. The switching time point Ω is selected as Ω = 20, and μ(t) is designed as follows:

[0112]

[0113] 5.3 Figures 2-4 This is a schematic diagram illustrating the effect of the finite iterative error tracking learning control method for precise trajectory tracking of robots proposed in this invention. Figure 2 It can be seen that, under the condition that the robot's initial position is arbitrary and the desired trajectory changes, the proposed method can achieve effective tracking of the robot's actual position to the desired trajectory. Figure 3 This demonstrates how the actual tracking error gradually approaches the expected error trajectory with increasing iterations, achieving an error convergence accuracy of 10^6 in the 15th iteration. -5 Within. Figure 4 This further demonstrates the maximum value of the absolute value of the robot's error. The trend of gradually decreasing and approaching zero with increasing iteration number. This is analyzed... Figure 4 As can be seen, the error value can be calculated after the first iteration. Combining equation (13) with parameters ρ = 0.6 and φ = 0.001, the error ε can be determined. k Upper bound on the number of iterations required to converge to the desired accuracy range φ Calculation results and Figure 4 The actual convergence process shown is consistent with the results, verifying the effectiveness and accuracy of the method.

[0114] In summary, the finite iterative error tracking learning control method of this embodiment enables the robot to quickly and accurately track the target position even when the initial position is arbitrary and the desired trajectory changes. By designing the desired error trajectory, the constraint that the robot needs to maintain the same initial position in each iteration is effectively relaxed. More importantly, after the first iteration, the upper bound of the number of iterations required to achieve convergence of the desired error can be calculated based on the error information. This provides a clear theoretical basis and operational guidance for the application of iterative learning control methods in practical engineering, significantly improving its practicality and applicability.

[0115] The above describes the experimental results given by this invention. Obviously, this invention is not limited to the above examples. Various modifications can be made to it without departing from the basic spirit of this invention or exceeding the scope of its substantive content.

Claims

1. A finite iterative error tracking learning control method for precise trajectory tracking of robots, characterized in that, The method includes the following steps: Step 1: Establish the continuous-time dynamic equations of the repetitive robot: (1)； in, Indicates time, Indicates the number of times the operation is repeated; , and They represent the robot's number 1 and 2. Joint angular displacement, joint angular velocity, and joint acceleration at each iteration Indicates the moment of inertia. Representing centrifugal force and Coriolis force, This represents the bounded gravitational term. and These represent the robot's mass and length, respectively. Represents gravitational acceleration. Indicates the robot's first Input torque at the next iteration; Step 2: Establish a discrete dynamic model of the repetitive robot, as follows: 2.1 The system equations of the robot (1) are transformed into mathematical equations. (2)； 2.2 Using the Euler discretization method, the discretization of equation (2) is performed. and Discretization is performed, and there is (3)； in, Indicates the step size; , It is a finite positive integer, representing the running period of each iteration; Indicates the sampling period; 2.3 Substituting equation (3) into equation (2), we obtain the discrete form of the robot model: (4)； in, In step size The sampled output signal at the location; It is the actual input torque maintained by the zero-order hold; 2.4 Define parameters ,variable nonlinear variables Then equation (4) can be restated as the following nonlinear discrete dynamic model: (5)； in, This represents an arbitrary but bounded initial position. ; 2.5 Furthermore, equation (5) is restated in vector form: (6)； in, , diagonal matrix Depend on constitute; It is by The constructed block matrix, ; Step 3, the design of the finite iterative error tracking iterative learning controller, is as follows: 3.1 Considering the possibility that the desired trajectory of the robot's motion may change, for any... Define signal If it is the desired trajectory of the robot's movement, then... This represents the error between the robot's desired position signal and its actual position during movement; the desired error trajectory is defined. for ； in, For the pre-defined connection transition trajectory and the expected zero-error trajectory, for A smooth, monotonically decaying function that satisfies and Define vector , , It is the difference between the expected error signal and the actual error signal; 3.2 The goal of designing a finite iterative error tracking learning control scheme is to generate a control input sequence. Drive the robot to move, so that in After the next iteration update, the robot system outputs the tracking error. For a given expected error trajectory Capable of achieving tracking with a specified precision, i.e., when Sometimes, (7)； in, express When, the function is The maximum absolute value of , ; It is a finite positive integer representing the number of error iterations; This represents the expected error convergence accuracy. 3.3 According to equation (6), we know (8)； Therefore, based on equations (6), (8), and error information The definition has (9)； in, , , , ; 3.4 Further, the finite iterative error tracking learning controller is designed as follows: (10)； in, yes The inverse matrix, These are the parameters to be set.

2. The finite iterative error tracking learning control method for precise trajectory tracking of robots as described in claim 1, characterized in that, The method further includes the following steps: Step 4, Error go through After the second iteration, it converges to the desired accuracy. Within this scope, the process is as follows: 4.1 Substituting the controller (10) into the error equation (9), we get (11)； According to equation (11), we have (12) ； because ,when As it approaches infinity, we have And thus error Converging to zero; 4.2 According to (12), when Sometimes, This allows the error to converge to the desired accuracy. The required number of iterations is (13)； in, Indicates greater than or equal to The smallest integer; 4.3 Therefore, when the first iteration ends, the error is obtained. Information, i.e., the calculated error Converging to the desired accuracy The number of iterations required within The specific value.

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