A Tracking Control Method for a Motion-Constrained Robot System with Finite-Time Convergence

Abstract

The present invention belongs to the technical field of robot system control, and specifically relates to a tracking control method for a motion-limited robot system with finite-time convergence. By introducing a non-linear bijective mapping function, the tracking control problem with motion constraints is transformed into an optimal control problem of an unconstrained auxiliary error tracking system. By introducing a dynamic event-triggering mechanism, a tracking control method for a motion-limited robot system based on adaptive dynamic programming is proposed, realizing the solution of the optimal control problem. On this basis, by designing an update rule for the evaluation network weight parameters that enables the control algorithm to converge in finite time and giving the upper limit value of finite-time convergence, the response speed of the robot system tracking control is improved. The present invention takes into account the stability and overall performance of the robot system.

Description

Technical Field

[0001] The present invention belongs to the technical field of robot system control, and particularly relates to a tracking control method for a motion-limited robot system with finite-time convergence. Background Art

[0002] With the development of artificial intelligence technology, robot systems have been widely used in social life and production manufacturing, which puts forward higher requirements for the control performance of robot systems. Due to the strong non-linearity, strong coupling and other characteristics of robot systems, it is challenging to complete complex control tasks according to preset performance requirements. Therefore, the motion control problem of robot systems has been widely studied and concerned.

[0003] In many robot application scenarios, due to the limitations of the robot system's own structure and operation space requirements, its motion space often needs to meet a certain constraint range. Most of the existing research on the control of motion-limited robot systems is carried out in the sense of infinite time. However, with the wide application of robots in various fields, the requirements for motion control performance are getting higher and higher. It is necessary to accurately track a specific trajectory within a limited time, while satisfying the motion constraint conditions and system performance optimization.

[0004] Therefore, considering the limited range of robot joint motion, how to design a robot motion tracking controller that takes into account both finite-time convergence and performance optimization has become an urgent problem to be solved. Summary of the Invention

[0005] The purpose of the present invention is to propose a tracking control method for a motion-limited robot system with finite-time convergence, aiming at the improvement requirements of the above-mentioned existing technologies, so that the robot system realizes the finite-time trajectory tracking control goal under the condition of meeting the motion constraint conditions.

[0006] To achieve the above purpose, the present invention adopts the following technical solutions:

[0007] A tracking control method for a motion-limited robot system with finite-time convergence, comprising the following steps:

[0008] Step 1, based on the dynamic model of the robot system and the system state constraints, establish a tracking error system with constraints;

[0009] Step 2, through introducing a non-linear bijective function, equivalently transform the motion control problem of the tracking error system with constraints to obtain the optimal control problem of an unconstrained auxiliary error tracking system;

[0010] Step 3: Construct a performance cost function based on the unconstrained auxiliary error tracking system, and then solve the cost function using the Hamilton function and the Bellman optimality principle to obtain an optimal control model for the unconstrained auxiliary error tracking system;

[0011] Step 4: Introduce an event-triggering mechanism into the optimal control model obtained in Step 3 to obtain an event-triggered optimal control model;

[0012] Step 6: Use an evaluation network to approximate the Bellman error to design a weight parameter update rule for finite-time convergence, and establish an adaptive event-triggered optimal control model based on this; given a dynamic variable, design a dynamic event-triggering condition based on this dynamic variable;

[0013] Step 6: Apply the dynamic event-triggering condition and the adaptive event-triggered optimal control model obtained in Step 5 to the tracking control of the robot motion constraints to achieve the tracking control of the motion-limited robot system with finite-time convergence.

[0014] Furthermore, Step 1 is based on the dynamic model of the robot system and the system state constraints to establish a constrained error tracking system, and its implementation method includes:

[0015] Step 1.1: The dynamic model of the robot system is:

[0016]

[0017] where, H(q), and g(q) represent the inertia matrix, the Coriolis force matrix, and the gravity vector respectively; q, represent the joint state vector, the velocity vector, and the acceleration vector of the robot system respectively; u represents the control input of the system;

[0018] Define the motion tracking error as z1 = q - q d and where, q d and represent the target joint motion angle and angular velocity respectively; according to the motion tracking errors z1 and z2, the state variables x and the desired state x d , obtain the tracking error state where, Τ represents the transpose;

[0019] Step 1.2: According to the tracking error state z and the dynamic model formula, establish a tracking error dynamic model :

[0020]

[0021] In formula (2),

[0022] Among them, \(I\) represents the identity matrix, \(H\) is the inertia matrix in the system dynamics model, \(C\) represents the Coriolis force matrix in the system dynamics model, \(g\) represents the gravity vector, and \(H\) -1 is the inverse matrix of \(H\).

[0023] Define the range of robot joint motion constraints as:

[0024]

[0025] In Equation (3), \(n\) represents the degrees of freedom of the robot, and x i represent the upper and lower bounds of the robot's motion state;

[0026] Step 1.3, according to the defined range of robot joint motion constraints and the tracking error dynamics model obtain the range of tracking error: where and z i are the upper and lower bounds of the tracking state; thus, a tracking error system with constraints is obtained.

[0027] Furthermore, in Step 2, by introducing a nonlinear bi-directional mapping function, the motion control problem of the error tracking system with constraints is equivalently transformed into the optimal control problem of an unconstrained auxiliary error tracking system. The implementation method includes:

[0028] Step 2.1, introduce the Barrier transformation equation and find its inverse function \(z\) i :

[0029] The introduced Barrier transformation equation is:

[0030]

[0031] The inverse function \(z\) of the Barrier transformation equation i is:

[0032]

[0033] In Equations (4) and (5), \(\chi\) i is the transformed error state, is a bi-directional reversible mapping function, and \(a \gt 1\) is a positive constant.

[0034] Step 2.2, according to the tracking error dynamics model and the inverse function \(z\) i , construct an unconstrained auxiliary error tracking system. The expression of the unconstrained tracking error system is as follows:

[0035]

[0036] In Equation (6),

[0037] Furthermore, the implementation method of the said Step 3 includes:

[0038] Step 3.1: Based on the unconstrained auxiliary error tracking system, set the cost function for evaluating its performance as shown in Equation (7):

[0039]

[0040] In Equation (7), R(χ,u) = χ T Eχ + u T Gu is the utility function, E and G are given positive definite matrices, and τ represents the variable of the integral function;

[0041] Step 3.2: Solve the cost function J(χ,u) using the Hamilton function and the Bellman optimality principle to obtain the optimal control problem model for the auxiliary tracking error system. The implementation method is as follows:

[0042] Step 3.2.1: Define the Hamilton function as:

[0043]

[0044] In Equation (8), represents the partial derivative of the cost function with respect to the state χ;

[0045] Using the Bellman optimality principle, obtain the Hamilton-Jacobi-Bellman (HJB) equation as:

[0046]

[0047] Step 3.2.2: Solve the cost function J(χ,u) using Equation (8) and Equation (9) to obtain the optimal control model for the auxiliary tracking error system. The expression of this model is as follows:

[0048]

[0049] where, G -1 is the inverse of the given positive definite matrix G, is the partial derivative of the optimal cost function, and u * is the expression of the optimal control output.

[0050] Furthermore, the implementation method of the said Step 4 is as follows:

[0051] Step 4.1: Define a monotonically increasing set of event triggering times The system state at the event triggering time is denoted as Set a continuous function σ i (t) to evaluate the state error between the sampling time and the current state;

[0052]

[0053] Step 4.2: Obtain the corresponding event-triggered optimal control model based on the continuous function The expression of is:

[0054]

[0055] Furthermore, use the evaluation network to approximate the Bellman error to design a weight parameter update rule for finite-time convergence, and establish an adaptive event-triggered optimal control model based on this; given a dynamic variable, design a dynamic event-triggering condition based on this dynamic variable. The implementation method includes:

[0056] Step 5.1: Use the evaluation network to approximately obtain the updated error function of the unconstrained auxiliary error tracking system by approximating the optimal cost function online Specifically:

[0057] Use the evaluation network to approximately express the cost function. The cost function is approximated as:

[0058]

[0059] In Equation (13), W c represents the evaluation network weight, ψ(χ) is the activation function, represents the evaluation network approximation error.

[0060] Since it is difficult to directly and accurately obtain the evaluation network weight and approximation error, the following true approximation expression can be used:

[0061]

[0062] In Equation (14), and respectively represent the estimated values of the cost function and the evaluation network, which is obtained from the definition of the weight estimate value.

[0063] Substitute Equation (14) into Equation (8) to obtain the error function of the evaluation network as:

[0064]

[0065] In Equation (15):

[0066]

[0067] As can be seen from Equation (8), H = 0, from which we can obtain:

[0068]

[0069] For perform normalization to obtain Equation (17), and Equation (17) is as follows:

[0070]

[0071] In Equation (17),

[0072] Step 5.2: Design a weight parameter update rule with finite-time convergence based on this error function The expression of

[0073]

[0074] In Equation (18), 0 < γ < 1; Λ is the learning rate matrix, and both P and Λ are positive definite matrices and P > 0.

[0075] Step 5.3: Based on the weight parameter update rule with finite-time convergence in Step 5.2, establish an adaptive event-triggered optimal control model, and this model is as shown in Equation (19):

[0076]

[0077] In Equation (19), is the estimated value of W c in the evaluation network.

[0078] Step 5.4: Design a dynamic event-triggering condition. Specifically:

[0079] Step 5.4.1 Define a dynamic variable Design the filtering dynamic equation for this variable as:

[0080]

[0081]

[0082] In Equation (20), l is a positive filtering constant, and λ min (E) represents the minimum eigenvalue of matrix E;

[0083] Step 5.4.2: Design the dynamic event-triggering condition based on the filtering dynamic equation as:

[0084]

[0085] Among them, θ∈(0,1) is a design parameter.

[0086] Furthermore, when the step 6 realizes the finite-time convergent robot motion constraint tracking control, the convergence event also satisfies the following conditions:

[0087]

[0088] Among them β = 2λ min (Λ), κ∈(0,1).

[0089] After adopting the above technical solutions, the present invention has the following advantages:

[0090] 1. For the robot system with motion state constraints, the present invention transforms the constrained error tracking control problem into an unconstrained optimal control problem of the auxiliary error tracking system by introducing a nonlinear bi-directional mapping function, so that the adaptive dynamic programming method can be effectively applied to the solution of this problem.

[0091] 2. By introducing a dynamic event-triggering mechanism and combining the evaluation network technology, the present invention constructs an adaptive control model based on dynamic event triggering, realizes the online solution of the optimal control model, takes into account the stability and performance optimization of the robot system at the same time, fully considers the actual requirements of limited computing power and energy of the robot system, and has feasibility.

[0092] 3. By reconstructing the weight parameter update rule of the evaluation network, the present invention ensures the finite-time convergence of the parameter learning of the adaptive dynamic programming algorithm, speeds up the convergence speed of the optimal controller, and meets the real-time requirements of the robot system. BRIEF DESCRIPTION OF THE DRAWINGS

[0093] Figure 1 It is a flowchart of a tracking control method for a motion-limited robot system with finite-time convergence in Embodiment 1. DETAILED DESCRIPTION OF THE INVENTION

[0094] The technical solution of the present invention will be described in detail below with reference to the drawings and embodiments.

[0095] As Figure 1 shown, a tracking control method for a motion-limited robot system with finite-time convergence provided in this embodiment includes the following steps:

[0096] Step 1. Based on the dynamic model of the robot system and the system state constraints, establish a constrained error tracking system, and its implementation method includes:

[0097] Step 1.1. The dynamic model of the robot system is:

[0098]

[0099] In Equation (1), H(q), and g(q) represent the inertia matrix, the Coriolis force matrix, and the gravity vector respectively; q, represent the joint state vector, the velocity vector, and the acceleration vector of the robot system respectively; u represents the control input of the system.

[0100] Define the motion tracking error as z1 = q - q d and where, q d and represent the target joint motion angle and angular velocity respectively; According to the motion tracking errors z1 and z2, the state variables x and the desired state x d , obtain the tracking error state where, T represents the transpose.

[0101] Step 1.2, According to the tracking error state z and the dynamic model formula, establish the tracking error dynamic model:

[0102]

[0103]

[0104] In Equation (2), I represents the identity matrix, H is the inertia matrix in the system dynamic model, C represents the Coriolis force matrix in the system dynamic model, g represents the gravity vector, H -1 is the inverse matrix of H.

[0105] Define the range of the robot joint motion constraint as:

[0106]

[0107] In Equation (3), n represents the degree of freedom of the robot, and x i represent the upper and lower bounds of the robot motion state.

[0108] Step 1.3, According to the defined range of the robot joint motion constraint and the tracking error dynamic model, obtain the range of the tracking error: where and z i are the upper and lower bounds of the tracking state, thereby obtaining a tracking error system with constraints.

[0109] Step 2, By introducing a nonlinear bi-directional mapping function, equivalently transform the motion control problem of the error tracking system with constraints to obtain the optimal control problem of an unconstrained auxiliary error tracking system, and its implementation method includes:

[0110] Step 2.1: Introduce the Barrier transformation equation and find its inverse function:

[0111] The introduced equation is:

[0112]

[0113] In Equation (4), χ i is the transformed error state, is a bi-directional reversible mapping function, and a > 1 is a positive constant.

[0114] The form of the inverse function of Equation (4) is:

[0115]

[0116] Step 2.2: Based on the tracking error dynamics model (2) and Equation (5), construct an unconstrained auxiliary error tracking system. The expression of the unconstrained auxiliary error tracking system is shown in Equation (6):

[0117]

[0118] Step 3: Construct a performance cost function based on the unconstrained auxiliary error tracking system, and then use the Hamilton function and Bellman optimality principle to solve the cost function to obtain the optimal control model for the unconstrained auxiliary error tracking system. The implementation method includes:

[0119] Step 3.1: Based on the unconstrained auxiliary error tracking system, set the cost function for evaluating its performance as shown in Equation (7):

[0120]

[0121] In Equation (7), R(χ, u) = χ T Eχ + u T Gu is the utility function, E and G are given positive definite matrices, and τ represents the variable of the integral function.

[0122] Step 3.2: Use the Hamilton function and Bellman optimality principle to solve the cost function J(χ, u) to obtain the optimal control problem model for the unconstrained auxiliary error tracking system. The implementation method is as follows:

[0123] Step 3.2.1: Define the Hamilton function as:

[0124]

[0125] In Equation (8), represents the partial derivative of the cost function with respect to the state χ;

[0126] Using the Bellman optimality principle, the Hamilton-Jacobi-Bellman (HJB) equation is obtained as follows:

[0127]

[0128] Step 3.2.2: Solve the cost function J(χ, u) using Equations (8) and (9) to obtain the optimal control model for the unconstrained auxiliary error tracking system. The expression of this model is as follows:

[0129]

[0130] In Equation (10), G -1 is the inverse of the given positive definite matrix G. is the partial derivative of the optimal cost function.

[0131] Step 4: Introduce an event-triggering mechanism into the optimal control model obtained in Step 3 to obtain an event-triggered optimal control model. The implementation method is as follows:

[0132] Step 4.1: Define a monotonically increasing set of event-triggering times The system state at the event-triggering time is denoted as Set a continuous function σ i (t) that evaluates the state error between the sampling time state and the current state. The form of the continuous function is expressed as:

[0133]

[0134] Step 4.2: Obtain the corresponding event-triggered optimal controller based on the continuous function. Its expression is:

[0135]

[0136] Step 5: Design a weight parameter update rule for finite-time convergence using an evaluation network to approximate the Bellman error, and establish an adaptive event-triggered optimal control model based on this. Given a dynamic variable, design a dynamic event-triggering condition. The implementation method is as follows:

[0137] Step 5.1: Use the evaluation network to approximately obtain the optimal cost function online to obtain the updated error function of the unconstrained auxiliary error tracking system. In this embodiment, a 3-layer backpropagation BP neural network is used as the evaluation network. The operations are as follows:

[0138] In this embodiment, the evaluation network is used to approximately represent the cost function. The cost function is approximated as:

[0139]

[0140] In Equation (13), W c represents the evaluation network weight, ψ(χ) is the activation function, and represents the approximation error of the evaluation network;

[0141] Since it is difficult to directly and accurately obtain the evaluation network weight and approximation error, the following true approximate expressions are adopted:

[0142]

[0143] where, and represent the cost function and the estimated value of the evaluation network respectively, and is obtained from the definition of the weight estimate value;

[0144] Substituting Equation (14) into Equation (8), the error function of the evaluation network is obtained as:

[0145]

[0146] where,

[0147] It can be seen from Equation (8) that H = 0, from which we can get:

[0148]

[0149] Performing normalization on obtains Equation (17):

[0150]

[0151] where,

[0152] Step 5.2. Design a weight parameter update rule with finite-time convergence based on this error function The expression of

[0153]

[0154] In Equation (18), 0 < γ < 1; Λ is the learning rate matrix, and both P and Λ are positive definite matrices and P > 0.

[0155] Step 5.3. Based on the weight parameter update rule with finite-time convergence in Step 5.2, establish an adaptive event-triggered optimal control model The expression of

[0156]

[0157] In Equation (19), is the estimated value of W in the evaluation network. c

[0158] Step 5.4: Calculate the dynamic event trigger condition as follows:

[0159] Step 5.4.1: Define a dynamic variable Design the filtering dynamic equation for this variable as:

[0160]

[0161] In the equation, l is a positive filtering constant, and λ min (E) represents the minimum eigenvalue of matrix E;

[0162] Step 5.4.2: Design the dynamic event trigger condition based on the filtering dynamic equation as:

[0163]

[0164] where θ ∈ (0, 1) is a design parameter.

[0165] Step 6: Combine the dynamic event trigger condition with the adaptive event-triggered optimal control model obtained in Step 5. While achieving the finite-time convergent robot motion constraint tracking control, the convergence event also satisfies the following conditions:

[0166]

[0167] where

[0168] The tracking control method for the motion-limited robot system with finite-time convergence provided in this embodiment transforms the tracking control problem with motion constraints into an optimal control problem of an unconstrained auxiliary error system by introducing a nonlinear bijective mapping function. An adaptive dynamic programming algorithm based on dynamic event triggering is proposed. By designing a new weight parameter update rule in the adaptive dynamic learning optimal control framework, the convergence speed of the evaluation network parameters is accelerated, enabling the robot system to achieve the finite-time trajectory tracking control goal while satisfying the motion constraint conditions.

[0169] ​It will be understood that the present invention is described by way of some embodiments, and those skilled in the art will know that various changes or equivalent substitutions can be made to these features and embodiments without departing from the spirit and scope of the present invention. Additionally, under the teachings of the present invention, these features and embodiments can be modified to adapt to specific circumstances and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application belong to the scope protected by the present invention.

Claims

1. A tracking control method for a motion-constrained robot system with finite-time convergence, characterized in that: The following steps are involved: Step 1: Based on the dynamic model of the robot system and the system state constraints, a tracking error system with constraints is established; The specific steps include: Step 1.1: The dynamic model of the robot system is: Among them, H(q), and g(q) represent the inertia matrix, Coriolis force matrix and gravity vector respectively; q, They represent the joint state vector, velocity vector and acceleration vector of the robot system respectively; u represents the control input of the system; Define the motion tracking error as z1 = qq d and Among them, q d and Represent the target joint motion angle and angular velocity respectively; according to the motion tracking errors z1 and z2, the state variable x and the expected state x d , get the tracking error state Wherein, Τ represents transpose; Step 1.2: Establish the tracking error dynamics model based on the tracking error state z and the dynamics model formula In the formula, Where I represents the identity matrix, H represents the inertia matrix in the system dynamics model, C represents the Coriolis force matrix in the system dynamics model, g represents the gravity vector, and H -1 is the inverse matrix of H; Define the robot joint motion constraint range as i=1,2,...,2n; where n represents the robot's degree of freedom, and Indicates the upper and lower bounds of the robot's motion state; Step 2: By introducing a nonlinear bidirectional mapping function, the motion control problem of the tracking error system with constraints is equivalently transformed to obtain the optimal control problem of the unconstrained auxiliary error tracking system; the specific steps include: Step 2.1: Introduce the Barrier transformation equation and find the inverse function z of the equation i : The introduced Barrier conversion equation is: The inverse function z of the barrier transformation equation i for: Among them, χ i is the error state after transformation, It is a bidirectional reversible mapping function, a>1 is a positive constant; Step 2.2: Based on the tracking error dynamics model and the inverse function z i , construct an unconstrained auxiliary error tracking system, and the expression of the unconstrained tracking error system is as follows: in, Step 3: Construct a performance cost function based on the unconstrained auxiliary error tracking system, and then use the Hamilton function and Bellman optimality principle to solve the cost function to obtain the optimal control model for the unconstrained auxiliary error tracking system; the cost function used to evaluate its performance is as follows: Where R(χ,u)=χ T Eχ+u T Gu is the utility function, E and G are given positive definite matrices, and τ represents the variable of the integral function; Solving the cost function J(χ,u) yields the optimal control problem model u for the unconstrained auxiliary error tracking system * (χ) is as follows: Among them, G -1 is the inverse of a given positive definite matrix G, is the partial derivative of the optimal cost function; Step 4: Introduce an event trigger mechanism into the optimal control model obtained in step 3 to obtain an event triggered optimal control model; the specific steps include: Step 4.1: Define a monotonically increasing set of event triggering times The system state at the time of event triggering is recorded as Set up a continuous function to evaluate the error between the state at the sampling moment and the current state Θ i ≤t<Θ i+1 ; Step 4.2: Obtain the corresponding event-triggered optimal control model based on the continuous function The expression is: Step 5: Use the evaluation network to approximate the Bellman error to design a weight parameter update rule that converges in finite time, and use this to establish an adaptive event-triggered optimal control model; given a dynamic variable, design a dynamic event trigger condition based on the dynamic variable; the specific steps include: Step 5.1: Use the evaluation network to approximate the optimal cost function online to obtain the updated error function of the unconstrained auxiliary error tracking system The evaluation network is used to approximate the cost function, which is approximately: Among them, W c represents the evaluation network weight, ψ(χ) is the activation function, and ζ(χ) represents the evaluation network approximation error; Its true approximate expression is: in, and Represent the estimated values ​​of the cost function and the evaluation network respectively, Obtained by the weight estimate definition; According to the true expression and Hamilton function, the updated error function of the unconstrained auxiliary error tracking system is obtained in, Step 5.2: Design a weight parameter update rule that converges in finite time based on the error function The expression is: in, 0<γ<1; Λ is the learning rate matrix, P and Λ are both positive definite matrices and P>0; Step 5.3: Based on the weight parameter update rule with finite time convergence in step 5.2, an adaptive event-triggered optimal control model is established. The model is as follows: in, To evaluate the network W c The estimated value of Step 5.4: Design the dynamic event triggering conditions as follows: Step 5.4.1 Define a dynamic variable The filter dynamic equation designed for this variable is: Where l is a positive filter constant, λ min (E) represents the minimum eigenvalue of matrix E; Step 5.4.2: Design the dynamic event triggering conditions based on the filtering dynamic equation: in, is the design parameter; step 6, applying the dynamic event triggering condition and the adaptive event triggering optimal control model obtained in step 5 to the robot motion constraint tracking control to achieve the tracking control of the motion constrained robot system with finite time convergence; while achieving the robot motion constraint tracking control with finite time convergence, the convergence event also satisfies the following conditions: among them β=2λ min (Λ), κ∈(0,1).

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