Preset time robust sliding mode control method for mechanical arm system

Abstract

A preset time robust sliding mode control method for a mechanical arm system comprises the steps that (1) under the condition that parameter perturbation, joint friction and external disturbance are considered, a dynamic model and a tracking error system model of the mechanical arm system are established; (2) a robust preset time controller is designed to ensure that the speed tracking error of the mechanical arm converges to zero within preset time; 3) designing a non-singular fast terminal sliding mode controller based on a preset time reaching law so as to ensure that a position tracking error can also converge to zero within the preset time; 4) the feasibility and effectiveness of the method are verified by using a Lyapunov stability analysis method; aiming at a mechanical arm system with parameter perturbation, joint friction and external disturbance, the invention develops a robust nonsingular fast terminal sliding mode control method in preset time so as to ensure that the mechanical arm system can accurately track an expected trajectory signal in the preset time, so that the tracking speed of a mechanical arm is improved, and the tracking precision is improved. And the tracking accuracy and safety of the mechanical arm are ensured.

Description

Technical Field

[0001] The present invention belongs to the technical field of manipulator system control, and specifically relates to a preset-time robust sliding mode control method for a manipulator system. Background Technique

[0003] Traditional manipulator control methods, such as PID control, computed torque control, etc., although they can meet the basic requirements in simple scenarios, their performance is often limited when facing complex nonlinear systems. As a strongly robust nonlinear control method, sliding mode control has received extensive attention in the field of manipulator control in recent years. Sliding mode control designs a sliding surface to make the system state converge to the sliding surface within a finite time and maintain a sliding motion on the sliding surface. However, traditional sliding mode control has a slow convergence speed and chattering phenomenon. To further improve the control performance of the manipulator system, preset-time control is introduced into the sliding mode control framework. Preset-time control can make the system state converge to the expected value within a preset time, and the convergence time is independent of the initial state of the system. This characteristic enables the manipulator system to complete high-precision control tasks within a specified time, and is especially suitable for application scenarios with strict time requirements, such as high-speed grasping, precision assembly, etc.

[0004] Therefore, aiming at the fact that the manipulator system is easily affected by external disturbances and model uncertainties, a preset-time based robust sliding mode control method is invented. This invention can not only make the manipulator system track the desired trajectory within the preset time, but also effectively suppress external disturbances and model uncertainties, thereby improving the overall control performance of the manipulator system. This is of great significance for enhancing the application ability of the manipulator in complex environments, and also provides new ideas for the research in related fields. Summary of the Invention

[0005] Aiming at the above problems existing in the prior art, the purpose of the present invention is to provide a preset-time based robust sliding mode control method applied to the trajectory tracking of a manipulator.

[0006] The present invention provides the following technical solutions: A preset-time robust sliding mode control method for a manipulator system, comprising the following steps: Step 1: First, establish the dynamic model of the manipulator system in a complex working environment, and then, based on the dynamics of the manipulator system, establish an error system for the manipulator to track the desired trajectory; its design process is as follows: 1.1) Describe the dynamic model of the manipulator system as: ; Wherein, , and are the position vector, velocity vector and acceleration vector of each joint of the robotic arm, represents the inertia matrix of the robotic arm system, represents the centrifugal force and the Coriolis matrix, is the gravity vector, and are the viscous friction coefficient matrix and the static friction vector, respectively. is the control torque of the robotic arm, represents the external disturbance, express dimensional column vector, express Matrix of 1.2) There are parameter perturbations in the modeling of the robotic arm. , and They are described as follows: ; in, , and Indicates the actual parameter value of the robotic arm system, , and Represents the parameter perturbation in the robotic arm system; 1.3) The dynamic model of the robotic arm system is rewritten as: ; in, is the concentrated uncertainty, which includes parameter perturbations, joint frictions and external disturbances; 1.4) Define the desired position and velocity tracking trajectory of the robotic arm system as and , then the position and velocity tracking errors of the robotic arm system are and , so we get , therefore, the error system of the robot arm tracking the desired trajectory is: ; in, , , Represents the concentrated uncertainty disturbance that the robotic arm system may encounter in practical applications; Step 2: Design a novel robust preset time controller, specifically: ; in, is a time-varying scaling function, and are respectively constants greater than zero, is a known computable function that makes assumptions about the centralized uncertainty, of, is an unknown constant greater than zero, represents the speed tracking error of the robotic arm joint ; Step 3: Design a nonsingular fast terminal sliding mode controller based on a preset time reaching law. The detailed process is as follows: 3.1) Design the nonsingular fast terminal sliding mode surface as: ; wherein, represents the th sliding mode surface, , , is a constant greater than 1, and are respectively constants greater than 0, represents the position tracking error of the robotic arm joint ; represents the speed tracking error of the robotic arm joint ; Derive the sliding mode variable , and we can get: ; wherein, , so an equivalent sliding mode controller is designed as: ; wherein, , , ; 3.2) Design the reaching rate based on the preset time as: ; wherein, , , , , and are respectively constants greater than zero; Finally, the designed nonsingular fast terminal sliding mode controller based on the preset time reaching law is: ; Step 4: Use the Lyapunov stability analysis method to verify the feasibility and effectiveness of the present invention. The specific proof process is as follows: 4.1) First, prove the effectiveness of the robust preset-time controller. Define the Lyapunov function as: ; where, ; Take the derivative of , and we can get: ; where, , , Using Young's inequality, we can obtain ; Substitute inequality (35) into inequality (34), and we can get ; According to the preset-time stability theory, we get can converge to zero at the preset time . Since , that is, can converge to zero at the preset time ; 4.2) Prove the effectiveness of the preset-time reaching law sliding mode controller. Under the action of the preset-time robust controller, the tracking error system of the robotic arm system becomes ; Define the Lyapunov function as: ; Take the derivative of , and we can get ; Substitute the sliding mode controller (32) based on the preset-time reaching law into equation (39), and we can get ; where, . According to the preset-time stability theory, we can obtain that the non-singular fast terminal sliding mode variable 0 can reach the sliding mode surface at the preset time , that is, . Since in 4.1), it has been proved that can converge to zero at the preset time , therefore, we can get can also converge to zero at the preset time .

[0007] Therefore, under the action of the preset-time robust sliding mode control, the robotic arm system can reach the preset time Tracking the desired trajectory verifies the effectiveness and feasibility of the present invention.

[0008] The design idea of the present invention is as follows: A preset-time robust sliding mode control method for a robotic arm system. 1) Considering parameter perturbation, joint friction, and external disturbance, the dynamic model of the uncertain robotic arm system and the tracking error system model are established; 2) A robust preset-time controller is designed to ensure that the velocity tracking error of the uncertain robotic arm converges to zero within a preset time; 3) A non-singular fast terminal sliding mode controller based on a preset-time reaching law is designed to ensure that the position tracking error can also converge to zero within a preset time; 4) The feasibility and effectiveness of the present invention are verified by using the Lyapunov stability analysis method, and it is also proved that the tracking error system of the robotic arm can converge to zero within a preset time. The present invention develops a preset-time robust non-singular fast terminal sliding mode control method for a robotic arm system with parameter perturbation, joint friction, and external disturbance to ensure that the robotic arm system can accurately track the desired trajectory signal within a preset time, not only improving the tracking speed of the robotic arm but also ensuring the accuracy and safety of the robotic arm tracking.

[0009] By adopting the above technologies, compared with the prior art, the beneficial effects of the present invention are as follows: The present invention is applied to a robotic arm system, and a robust sliding mode control method based on preset time is invented, enabling the robotic arm system to track the desired trajectory within a preset time. This can not only improve the working efficiency of the robotic arm system but also effectively suppress the influence of external disturbances and model uncertainties on the system, thereby achieving high-precision and fast-response control of the robotic arm system. This is of great significance for enhancing the application ability of the robotic arm in complex environments and also provides new ideas for research in related fields. Description of the Drawings

[0010] Figure 1 It is the system structure diagram of the robust sliding mode control method based on preset time in the embodiment of the present invention; Figure 2 It is the simple model of a two-link robotic arm in the embodiment of the present invention; Figure 3 It is the curve of the robotic arm joints 1 and 2 tracking the desired position response in the embodiment of the present invention; Figure 4 It is the curve of the robotic arm joints 1 and 2 tracking the desired velocity response in the embodiment of the present invention; Figure 5 It is the position tracking error response of the robotic arm joints 1 and 2 in the embodiment of the present invention; Figure 6 It is the velocity tracking error response of the robotic arm joints 1 and 2 in the embodiment of the present invention. Detailed implementation manners

[0011] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings of the specification and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0012] On the contrary, the present invention covers any alternatives, modifications, equivalent methods and solutions made within the spirit and scope of the present invention defined by the claims. Further, in order to enable the public to have a better understanding of the present invention, some specific details are described in detail in the following detailed description of the present invention. Those skilled in the art can also fully understand the present invention without the description of these details.

[0013] Please refer to Figure 1 , a preset time robust sliding mode control method for a robotic arm system, comprising the following steps: Step 1: First, establish the dynamic model of the robotic arm system in a complex working environment, and then, based on the dynamics of the robotic arm system, establish an error system for the robotic arm to track the desired trajectory; the design process is as follows: 1.1) Considering the influence of external disturbances, parameter perturbations and friction on the robotic arm, the dynamic model of the robotic arm is described as: ; Wherein, , and are respectively the position vector, velocity vector and acceleration vector of each joint of the robotic arm, represents the inertia matrix of the robotic arm system, represents the centrifugal force and Coriolis matrix, is the gravity vector, and are respectively the viscous friction coefficient matrix and the static friction vector, is the control torque of the robotic arm, represents the external disturbance, represents a column vector of dimension represents matrix; 1.2) Since there are parameter perturbations in the modeling of the robotic arm, therefore, , and are respectively described in the following forms: ; Wherein, , and Denote the actual parameter values of the robotic arm system, , and represent the parameter perturbations existing in the robotic arm system; 1.3) Combining Equation (42), the dynamic model of the robotic arm system can be re - described as: ; where, is the lumped uncertainty, which includes parameter perturbations, joint friction, and external disturbances; 1.4) For the convenience of controller design, it is necessary to construct the tracking error system of the robotic arm. Define the desired position and velocity tracking trajectories of the robotic arm system as and respectively. Then the position and velocity tracking errors of the robotic arm system are and respectively. Thus, we can obtain . Therefore, the error system for the robotic arm to track the desired trajectory is: ; where, , , represents the lumped uncertainty disturbance that the robotic arm system may encounter in practical applications; Step 2: To ensure that the robotic arm can accurately track the desired velocity trajectory within a preset time, a novel robust preset - time controller is designed as follows: ; where, is a time - varying scaling function, and are constants greater than zero respectively, is a known computable function, making an assumption about the lumped uncertainty , is an unknown constant greater than zero, represents the velocity tracking error of the th joint of the robotic arm; Step 3: To ensure that the robotic arm can accurately track the position trajectory signal within a preset time, a non - singular fast terminal sliding - mode controller based on the preset - time reaching law is designed. The detailed process is as follows: 3.1) Design the non - singular fast terminal sliding - mode surface as: ; where, represents the th sliding - mode surface, , , is a constant greater than 1, and are constants greater than 0, respectively, represents the position tracking error of the robotic arm joint ; represents the speed tracking error of the robotic arm joint ; Differentiating the sliding mode variable yields: ; wherein, , so, according to Equation (49), the equivalent sliding mode controller is designed as: ; wherein, , , ; 3.2) To ensure that the sliding mode variable can reach the sliding mode surface within a preset time, the reaching law based on the preset time is designed as: ; wherein, , , , , and are constants greater than zero, respectively; Finally, the non-singular fast terminal sliding mode controller based on the preset time reaching law is designed as: ; The structure diagram of the robust sliding mode control method based on the preset time is as shown in Figure 1 ; Step 4. Use the Lyapunov stability analysis method to verify the feasibility and effectiveness of the present invention. The specific proof process is as follows: 4.1) First, prove the effectiveness of the robust preset time controller. Define the Lyapunov function as: ; wherein, ; Differentiating yields: ; wherein, , , Using Young's inequality, we can obtain ; Substituting inequality (55) into inequality (54), we can obtain ; According to the preset-time stability theory, it is obtained that can converge to zero within the preset time Since , that is can converge to zero within the preset time ; 4.2) Prove the effectiveness of the preset-time reaching law sliding mode controller. Under the action of the preset-time robust controller, the tracking error system of the manipulator system becomes ; Define the Lyapunov function as: ; Taking the derivative of , we can get ; Substituting the sliding mode controller (52) based on the preset-time reaching law into equation (59), we can get ; where , according to the preset-time stability theory, the non-singular fast terminal sliding mode variable can reach the sliding mode surface within the preset time , that is , since in 4.1), it has been proved that can converge to zero within the preset time , therefore, it can be obtained that can also converge to zero within the preset time . Thus, under the action of the preset-time robust sliding mode control, the manipulator system can track the desired trajectory within the preset time , which verifies the effectiveness and feasibility of the present invention.

[0014] To further verify the effectiveness of the preset-time robust sliding mode control method of the present invention, we conducted a simulation experiment on the trajectory tracking of a two-link manipulator on the Matlab / Simulink platform. The specific process is as follows: The model of the two-link manipulator is as shown in Figure 2 . The dynamic model of the manipulator with external disturbances, joint friction, and parameter perturbations is described as: ; where ; where and are the joint position vectors of the robotic arm, and are the velocity vectors of the robotic arm joints, and their corresponding initial states are , and , and the dynamic friction coefficients of each joint of the robotic arm are , and the static friction coefficient is . In addition, the specific physical parameters are shown in Table 1. Further, the desired position trajectory is selected as: ; The external disturbance signal is set as: ; The parameters of the robust preset-time controller are set as , and , the preset time is set as , and the parameters of the nonsingular fast terminal sliding mode controller based on the preset-time reaching law are set as , , , , , , , ; Table 1 Related physical parameters of the robotic arm ; The specific simulation results are as shown in Figures 3 - 6 .

[0015] Figure 3 shows the curves of the position tracking responses of joints 1 and 2 of the robotic arm, Figure 4 shows the curves of the velocity tracking responses of joints 1 and 2 of the robotic arm, Figure 5 shows the curves of the position tracking errors of joints 1 and 2 of the robotic arm, Figure 6 shows the curves of the velocity tracking errors of joints 1 and 2 of the robotic arm. It can be clearly seen from Figures 3 to 6 that when the preset time is set as , joints 1 and 2 of the robotic arm can just track the desired position trajectory and velocity trajectory respectively at , which indicates that under the action of the preset-time robust sliding mode controller, the robotic arm can accurately and quickly track the desired trajectory, achieving satisfactory trajectory tracking performance. This also demonstrates the effectiveness and feasibility of the preset-time robust sliding mode control method of the present invention.

[0016] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A preset time robust sliding mode control method for a robotic arm system, characterized in that, Including the following steps: Step 1: First, establish the dynamic model of the robotic arm system. Then, based on the dynamic model of the robotic arm system, establish the error system for the robotic arm to track the desired trajectory; Step 2: Design a robust preset-time controller to ensure that the velocity tracking error of the robotic arm converges to zero within the preset time; Step 3: Design a non-singular fast terminal sliding mode controller based on the preset-time reaching law to ensure that the position tracking error can also converge to zero within the preset time; Step 4: Use the Lyapunov stability analysis method to verify the feasibility and effectiveness, and prove that the tracking error system of the robotic arm can converge to zero within the preset time.

2. The preset time robust sliding mode control method for a robotic arm system according to claim 1, wherein The specific content of Step 1 is as follows: 1.1) Describe the dynamic model of the robotic arm system as: ； Among them, , and are the position vector, velocity vector, and acceleration vector of each joint of the robotic arm respectively, represents the inertia matrix of the robotic arm system, represents the centrifugal force and Coriolis matrix, is the gravity vector, and are the viscous friction coefficient matrix and static friction vector respectively, is the control torque of the robotic arm, represents the external disturbance, represents a column vector of dimension represents a matrix of 1.2) There are parameter perturbations in the modeling of the robotic arm. Therefore, , and are respectively described in the following forms: ； Among them, , and represent the actual parameter values of the robotic arm system, , and represent the parameter perturbations existing in the robotic arm system; 1.3) Redescribe the dynamic model of the robotic arm system as: ； in, is the concentrated uncertainty, including parameter perturbations, joint frictions and external disturbances; 1.4) Define the desired position and velocity tracking trajectories of the robotic arm system as and , respectively. Then the position and velocity tracking errors of the robotic arm system are and , respectively. Thus, we obtain . Therefore, the error system for the robotic arm to track the desired trajectory is: ； Among them, , , represents the concentrated uncertain disturbances that the robotic arm system may encounter in practical applications.

3. A preset time robust sliding mode control method for a robotic arm system according to claim 1, characterized in that The specific form of the robust preset-time controller in Step 2 is: ； wherein, is a time-varying scaling function, and are respectively constants greater than zero, is a known computable function, making an assumption about the lumped uncertainty , is an unknown constant greater than zero, represents the velocity tracking error of the robotic arm joint .

4. A preset time robust sliding mode control method for a robotic arm system according to claim 1, characterized in that, The specific process of Step 3 is as follows: 3.1) Design the non-singular fast terminal sliding mode surface as: ； Among them, represents the th sliding mode surface, , , is a constant greater than 1, and are respectively constants greater than 0, represents the position tracking error of the robotic arm joint , represents the velocity tracking error of the robotic arm joint ; Derive the sliding mode variable to obtain: ； Among them, , so the designed equivalent sliding mode controller is: ； Among them, , , ; 3.2) Design the reaching rate based on the preset time as: ； Among them, , , , , and are constants greater than zero, respectively; The designed non-singular fast terminal sliding mode controller based on the preset-time reaching law is: 。 5. A preset time robust sliding mode control method for a robotic arm system according to claim 1, characterized in that, The specific process of Step 4 is as follows: 4.1) First, prove the effectiveness of the robust preset-time controller. Define the Lyapunov function as: ； Among them, ; Derive to obtain: ； Among them, , , Using Young's inequality, we get ； Substitute inequality (15) into inequality (14) to obtain ； According to the preset time stability theory, it is obtained that can converge to zero within the preset time because , that is can converge to zero within the preset time ; 4.2) Prove the effectiveness of the sliding mode controller based on the preset-time reaching law. Under the action of the preset-time robust controller, the tracking error system of the robotic arm system becomes ； Define the Lyapunov function as: ； Derive to obtain ； Substitute the sliding mode controller (12) based on the preset-time reaching law into equation (19) to get ； Among them, , according to the preset time stability theory, a non-singular fast terminal sliding mode variable can reach the sliding mode surface at the preset time , that is . Since in 4.1), it has been proved that can converge to zero at the preset time , therefore, it is obtained that can also converge to zero at the preset time .

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