A mechanical arm finite time control method based on non-singular robust control

Abstract

The application relates to the technical field of mechanical arm motion control, and discloses a mechanical arm finite time control method based on a non-singular robust control, which comprises the following steps: a dynamics model with uncertain terms of an N-joint mechanical arm is established; a trajectory tracking error is defined, a state space equation and a control target are established; a non-singular robust controller is designed in combination with the robust control; initial states and final states of each joint of the mechanical arm are determined, and an expected trajectory is obtained; then a closed loop system is formed by combining the real-time obtained trajectory tracking error, joint variables, derivatives of the joint variables and the non-singular robust controller, so that real-time control of each joint is realized. The motion control law is improved on the basis of the robust control, singular value problems in the motion process of the system are avoided, and the motion control law has a better convergence speed than general finite time non-singular robust control; meanwhile, the uncertainty of self parameters and external disturbances are considered, and the motion control law has a good performance in the stability of the system.

Description

Technical Field

[0001] This invention belongs to the field of robotic arm motion control technology, specifically relating to a finite-time control method for robotic arms based on non-singular robust control. Background Technology

[0002] With the urgent need for improved operational precision and optimized production capacity in industrial manufacturing, the control technology of multi-joint robotic arms has attracted widespread attention. For example, multi-joint robotic arms are used in automotive body painting and component assembly, electronic component assembly, and minimally invasive surgical assistance systems. For the motion control of robotic arms, a set of expected trajectories is often given, and it is hoped that each joint of the robotic arm will move along the given trajectory under the control law. However, due to the inherent uncertainties in the robotic arm's parameters and external disturbances during movement, there is often a certain error between the actual trajectory and the expected trajectory. To ensure the control precision of the robotic arm, we hope that the error can converge to zero within a finite time.

[0003] Currently, the main control methods for robotic arms include PID control, robust control, fuzzy control, and finite-time robust control. PID control is widely used due to its simple design, but general PID control struggles to simultaneously achieve good response speed and error convergence, and its handling of external disturbances is not ideal, failing to meet the needs of high-precision control. Robust control offers advantages such as fast convergence speed, high accuracy, and no overshoot, enabling good convergence of system errors; however, it cannot guarantee that the convergence time of the system error falls within an ideal range. Finite-time robust control can ensure that the system error converges within a finite time, but it often suffers from singular values, which can reduce system safety during actual operation.

[0004] Therefore, non-singular finite-time robust control has emerged. However, even though non-singular finite-time robust control can avoid the problem of singular values, general non-singular finite-time robust control methods usually only consider external disturbances to the system without further discussion of internal parameter uncertainties. At the same time, there is still a lot of room for improvement in terms of convergence speed. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide a finite-time control method for a robotic arm based on non-singular robust control, so as to solve the problem of finite-time robust control of the system under the condition that the system has its own uncertainty and complex external disturbances.

[0006] To address the aforementioned technical problems, this invention provides a finite-time control method for a robotic arm based on non-singular robust control, comprising the following process:

[0007] S1. Establish a dynamic model of a robotic arm with N joints and uncertainties;

[0008] S2. Define the trajectory tracking error, establish the state-space equation and control objective;

[0009] S3. Design a non-singular robust controller by combining robust control;

[0010] S4. Determine the initial and final states of each joint of the robotic arm, and obtain the expected trajectory δ using polynomial trajectory programming. d Then, the trajectory tracking error σ, joint variables δ, and derivatives of the joint variables are obtained in real time by the sensors on the robotic arm. A closed-loop system is formed by the non-singular robust controller, enabling real-time control of each joint of the robotic arm by the non-singular robust controller.

[0011] As an improvement to the finite-time control method for robotic arms based on non-singular robust control, this invention provides the following:

[0012] The specific process of establishing the dynamic model of the robotic arm with N joints with uncertain terms in step S1 is as follows:

[0013] Step S1.1: The dynamic model of the N-joint robotic arm is as follows:

[0014]

[0015] Where δ represents the joint variable of the robotic arm, in radians. Let I(δ) be the derivative of the joint variable δ; I(δ) is the inertia matrix composed of the robot arm's own parameters and the joint variables. These are the Coriolis force and centrifugal force terms, G(δ) is the gravity term, and Ξ is the torque; the robot arm's own parameters include mass, length, and moment of inertia;

[0016] Step S1.2: Considering the internal uncertainties of the system, the nominal dynamic model is:

[0017]

[0018] Among them, I0(δ), G0(δ) is the nominal parameter matrix. This is the error term between the nominal and actual parameters, and it also includes external interference to the system.

[0019] Where d(t) represents the external disturbance.

[0020] As a further improvement to the finite-time control method for a robotic arm based on non-singular robust control of the present invention:

[0021] The trajectory tracking error is: σ = δ - δ d , where δ d It is the given expected trajectory.

[0022] As a further improvement to the finite-time control method for a robotic arm based on non-singular robust control of the present invention:

[0023] The state-space equation is:

[0024]

[0025] Where x1 = σ, u = Ξ;

[0026] Based on the assumptions of the robotic arm's dynamic parameters:

[0027]

[0028] Where a0, a1, and a0 are all positive real numbers, and ||·|| represents the Euclidean norm.

[0029] As a further improvement to the finite-time control method for a robotic arm based on non-singular robust control of the present invention:

[0030] The control objective is: under conditions of parameter uncertainty and external disturbances in the system, the trajectory tracking error σ of the robotic arm can converge to 0 within a finite time, thereby ensuring that the actual motion trajectory δ and the expected trajectory δ are consistent. d Consistent.

[0031] As a further improvement to the finite-time control method for a robotic arm based on non-singular robust control of the present invention:

[0032] The non-singular robust control function is:

[0033]

[0034] Where, x 11 ,x 12 ,...,x 1n Let C be the trajectory tracking error corresponding to each joint, which is a diagonal matrix with all terms being positive real numbers, i.e., C = diag(c1,...,c n ); p and q are both positive odd numbers, and p>q.

[0035] As a further improvement to the finite-time control method for a robotic arm based on non-singular robust control of the present invention:

[0036] The non-singular robust controller is:

[0037]

[0038]

[0039] Where, x 21 ,x 22 ,...,x 2n x2 corresponds to each joint.

[0040] The beneficial effects of this invention are mainly reflected in:

[0041] 1. This invention adds a new control term to the robust control, and the system error converges within a finite time.

[0042] 2. The motion control law designed in this invention does not contain switching terms, resulting in smaller control signal jitter and effectively improving the control accuracy of the system.

[0043] 3. The motion control law designed in this invention is an adjustment based on robust control, which can avoid the singular value problem of the system during motion and has a better convergence speed than general finite-time non-singular robust control.

[0044] 4. The motion control law designed in this invention takes into account both the uncertainty of its own parameters and the complex external disturbances to the system, and has good performance in terms of system stability. Attached Figure Description

[0045] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0046] Figure 1 This is a flowchart illustrating a finite-time control method for a robotic arm based on non-singular robust control according to the present invention.

[0047] Figure 2 This is a comparison diagram of the motion trajectory of joint 1 of the robotic arm under the condition of uncertainty of the robotic arm itself and external disturbance, and the expected trajectory.

[0048] Figure 3 This is a comparison diagram of the motion trajectory of joint 2 of the robotic arm under the condition that the robotic arm has its own uncertainties and external disturbances, and the expected trajectory. Detailed Implementation

[0049] The present invention will be further described below with reference to specific embodiments, but the scope of protection of the present invention is not limited thereto:

[0050] Example 1: A finite-time control method for a robotic arm based on non-singular robust control. The robotic arm mainly consists of a multi-joint linkage robotic arm connected to a hydraulic system. Sensors are installed in both the multi-joint linkage robotic arm and the hydraulic system. The sensors measure the joint variables of the robotic arm and transmit these variables to the non-singular robust controller to control the motion trajectory of the robotic arm. Under conditions of parameter uncertainty and external disturbances in the system, the method ensures that the trajectory tracking error of the robotic arm converges to zero within a finite time, thereby guaranteeing that the actual motion trajectory is consistent with the expected trajectory. The steps of the finite-time control method for a robotic arm based on non-singular robust control are as follows: Figure 1 As shown, the details are as follows:

[0051] Step S1: Establish a nominal dynamic model of an N-joint robotic arm with uncertainties, including:

[0052] Step S1.1: Dynamic model of N-joint robotic arm

[0053] The joint variable of the robotic arm is δ, and its unit is radians. Let δ be the derivative of the joint variable; the robot arm's own parameters include mass, length, and moment of inertia. Define an inertia matrix I(δ) composed of the robot arm's own parameters and the joint variables, and the expression for the system's kinetic energy can be obtained. According to the Lagrange equation:

[0054]

[0055] Where U(δ) is the potential energy of the system, The difference between kinetic and potential energy, (1) can be further written in the standard form of dynamics as follows:

[0056]

[0057] in, G(δ) is the Coriolis force term and the centrifugal force term, G(δ) is the gravity term, and Ξ is the torque.

[0058] Step S1.2: Consider the internal uncertainties of the system

[0059] For a system, the control law is often designed using given nominal parameters. There will be a certain error between the nominal and actual parameters, which leads to uncertainty within the system. If the impact of these errors is ignored, the system may deviate from its original trajectory under the control law, with the error iteratively increasing. This not only affects the control accuracy but also the system's stability. Furthermore, the robotic arm may be subject to external disturbances d(t) during its movement; therefore, these unknown errors need to be considered when deriving the dynamic model. Define the nominal parameter matrix I0(δ). G0(δ), by replacing equation (2), we obtain the nominal dynamic model as follows:

[0060]

[0061] in, This is the error term between the nominal and actual parameters, and it also includes external disturbances to the system. It is a bounded column vector.

[0062] Step S2: Define the trajectory tracking error, establish the state-space equations of the system, and give the control objective.

[0063] Step S2.1: State-space equations of the system

[0064] The trajectory tracking error can be defined as σ = δ - δ d , where δ d It is the given expected trajectory.

[0065] Let x1 = σ, The following state-space equations exist:

[0066]

[0067] in, u = Ξ.

[0068] Based on the assumptions of the robotic arm's dynamic parameters:

[0069]

[0070] Where a0, a1, and a0 are all positive real numbers, and ||·|| represents the Euclidean norm.

[0071] S2.2: System control objective

[0072] Based on the above description, the motion control objective of the multi-joint robotic arm is: under conditions of parameter uncertainty and external disturbances in the system, the trajectory tracking error σ of the robotic arm can converge to 0 within a finite time, thereby ensuring that the actual motion trajectory δ and the expected trajectory δ are consistent. d Consistent.

[0073] Step S3: Design a non-singular robust controller by combining robust control.

[0074] Step S3.1: Design non-singular robust control function

[0075]

[0076] Where, x 11 ,x 12 ,...,x 1nLet C be the trajectory tracking error corresponding to each joint, which is a diagonal matrix with all terms being positive real numbers, i.e., C = diag(c1,...,c n ); p and q are both positive odd numbers, and p>q.

[0077] Step S3.2 Design of Non-Singular Robust Controller

[0078]

[0079]

[0080] Where, x 21 ,x 22 ,...,x 2n x2 corresponds to each joint.

[0081] Step 4: Verify the stability of the system using the designed non-singular robust controller.

[0082] Consider the following energy function:

[0083]

[0084] Differentiating V(γ) gives:

[0085]

[0086] in, Let represent the Hadamard product; to simplify the subsequent formula derivation, let The Hadamard product in the substitution formula.

[0087] Substituting (4) into (10) yields:

[0088]

[0089] (11) can be further transformed into the following form:

[0090]

[0091] in, Therefore Since the value is negative definite, the system is stable in finite time. That is, the tracking error σ of the robotic arm can be stabilized in a finite time t. co The system converges to zero and maintains good robustness under external disturbances and uncertainties in its own parameters. The convergence time t is [not specified]. co The following constraints must be satisfied:

[0092]

[0093] Step S5: Non-singular robust control of the robotic arm control system under uncertainties

[0094] A finite-time stability control method for robotic arms based on non-singular robust control is applied to robotic arm control.

[0095] First, determine the initial and final states of each joint of the robotic arm. The initial state includes the initial joint variables δ of each joint. bi (i = 1, 2, ..., n), the derivatives of the initial joint variables (i = 1, 2, ..., n); the final state includes the final joint variables δ of each joint. ei (i = 1, 2, ..., n), the derivative of the final joint variable (i = 1, 2, ..., n); then the expected trajectory δ is obtained through polynomial trajectory programming. d ;

[0096] Secondly, the sensors on the robotic arm provide real-time feedback on the trajectory tracking error σ, joint variables δ, and the derivatives of the joint variables. Together with the aforementioned non-singular robust controller, a closed-loop system is formed. The system uses the physical quantities fed back above as inputs to the controller described in equations (7) and (8); trajectory tracking error σ, joint variable δ, and derivative of the joint variable. As input to the system, the non-singular robust controller will construct a non-singular robust control function according to the equation described by equation (6). This function will then be used as a further input to the next step of the non-singular robust controller output expression. Thus, when the robotic arm system, as the controlled object, receives a control signal that changes with the error σ and the joint variable δ, it will adjust the joint variable δ at the next moment. This process is repeated to form negative feedback and gradually correct the system error, so that the system error gradually decreases until it becomes zero. Finally, the system error converges to 0, and the movement trajectory of the robotic arm is consistent with the expected trajectory.

[0097] The physical quantities specified in the motion control of the robotic arm are as follows:

[0098] In the robotic arm, all physical quantities such as mass and joint length are defined by the parameters listed in the instruction manual or label as actual physical parameters; nominal parameters used in the controller design are described using subscripts, such as m. 10 l 10 These represent the nominal mass and nominal length of joint 1, respectively. Unless otherwise specified, the actual parameters are assumed to be the same as the nominal parameters.

[0099] The expected trajectory function shows a gradual increase in the image, eventually leveling off. Therefore, continuous functions with this property can be considered as expected trajectories in simulation experiments.

[0100] The external influence on the system is selected as a persistent disturbance d(t). This disturbance does not disappear with time and changes with time. It exists throughout the motion of the system.

[0101] Experiment 1

[0102] The finite-time stability control method for a robotic arm based on non-singular robust control, as described in Example 1, was used to conduct simulation experiments to verify the control of a two-joint robotic arm. This validated the effectiveness and feasibility of the finite-time stability control method for the robotic arm based on non-singular robust control. The simulation results are as follows: Figures 2 to 3 .

[0103] The parameters of the robotic arm dynamics model used in the simulation experiment are as follows:

[0104]

[0105] Where, δ=[δ1 δ2] T , representing the joint variables of joint 1 and joint 2 of the robotic arm, which are obtained in real time by sensors on the robotic arm. The expressions of each item in the matrix in (14) are as follows:

[0106]

[0107]

[0108]

[0109] In a dual-joint robotic arm, the masses of joint 1 and joint 2 are m1 = 0.5 kg and m2 = 1.5 kg, respectively, and the nominal mass is m 10 =0.4kg, m 20 =1.6kg; the lengths of joint 1 and joint 2 are l1 = 1.0m and l2 = 0.8m respectively; the moments of inertia of joint 1 and joint 2 are J1 = 5.0kg·m respectively. 2 J2 = 5.0 kg·m 2 The acceleration due to gravity is taken as 9.8 m / s². 2 .

[0110] The expected trajectories of joints 1 and 2 in a dual-joint robotic arm are δ d1 =2-1.5e -t +0.5e -4t (rad), δ d2 =2+1.2e -t -0.4e -4t(rad); External disturbances d1 = 2sin(t) + 0.5sin(200πt) (N·m), d2 = cos(2t) + 0.5sin(200πt) (N·m); The initial state of the system is δ1 = 1.5rad, δ2 = 1.5rad,

[0111] In the designed control law, p = 5, q = 3; a0 = 8, a1 = 8, a2 = 10; and the positive real diagonal matrix C = diag(1,1).

[0112] Figure 2 and Figure 3 The image shows a comparison between the actual motion trajectories of each joint of the robotic arm under conditions of inherent parameter uncertainties and external disturbances, and the given desired trajectory. The comparison trajectories in the image represent the motion trajectories of a typical finite-time non-singular robust control system. The information reflected in the image indicates that the novel finite-time stable control method for robotic arms based on non-singular robust control described in this invention maintains good robustness when facing internal parameter errors and external disturbances. Furthermore, the system error gradually converges to zero within a relatively short time, exhibiting a better convergence speed than typical finite-time non-singular robust control.

[0113] Finally, it should be noted that the above examples are merely some specific embodiments of the present invention. Obviously, the present invention is not limited to the above embodiments and many variations are possible. All variations that can be directly derived or conceived by those skilled in the art from the disclosure of the present invention should be considered within the scope of protection of the present invention.

Claims

1. A finite-time control method for a robotic arm based on non-singular robust control, characterized in that, The process includes: S1. Establish a dynamic model of a robotic arm with N joints and uncertainties. The specific process is as follows: Step S1.1: The dynamic model of the N-joint robotic arm is as follows: （2） in, These are the joint variables of the robotic arm, measured in radians. Represented as joint variables The derivative; The inertia matrix is ​​composed of the robot arm's own parameters and joint variables. These are the Coriolis force term and the centrifugal force term. It is a gravity term. It is torque; the robot arm's own parameters include mass, length, and moment of inertia; Step S1.2: Considering the internal uncertainties of the system, the nominal dynamic model is: （3） in, , , For the nominal parameter matrix, This is the error term between the nominal and actual parameters, and it also includes external interference to the system. ,in External disturbances; S2. Define the trajectory tracking error, establish the state-space equation and control objective; The trajectory tracking error is: ,in, It is the given expected trajectory; The state-space equation is: （4） in, , , , ; Based on the assumptions of the robotic arm's dynamic parameters: （5） in, , , All are positive real numbers. Denotes the Euclidean norm; The control objective is to minimize the trajectory tracking error of the robotic arm under conditions of parameter uncertainty and external disturbances in the system. It can converge to 0 within a finite time, thus ensuring the actual motion trajectory. in line with expected trajectory Consistent; S3. Design a non-singular robust controller by combining robust control; The non-singular robust control function is: （6） in, The trajectory tracking error corresponding to each joint, It is a diagonal matrix in which all terms are positive real numbers, i.e. ; , All are positive odd numbers, and ; S4. Determine the initial and final states of each joint of the robotic arm, and obtain the expected trajectory through polynomial trajectory programming. Then, the trajectory tracking error obtained in real time by the sensors on the robotic arm... Joint variables Derivatives of joint variables A closed-loop system is formed by the non-singular robust controller, enabling real-time control of each joint of the robotic arm by the non-singular robust controller.

2. The finite-time control method for a robotic arm based on non-singular robust control according to claim 1, characterized in that: The non-singular robust controller is: （7） （8） in, For each joint .

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