A mechanical arm predetermined time non-singular fast terminal sliding mode control method
By designing a predetermined time non-singular fast terminal sliding mode control method, the problem of convergence time uncertainty of the robotic arm in complex environments is solved, realizing fast and accurate tracking within a predetermined time, improving the control accuracy and robustness of the robotic arm, and making it suitable for tasks such as high-speed sorting and precision welding.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- ZHEJIANG UNIV OF TECH
- Filing Date
- 2026-05-22
- Publication Date
- 2026-06-19
AI Technical Summary
Existing non-singular fast terminal sliding mode control methods are insufficient in terms of the determinism of convergence time, making it difficult to accurately control the operation of the robotic arm within a predetermined time. This is especially true in high-speed sorting and precision welding tasks where time accuracy is critical, leading to production process chaos and product quality degradation.
A non-singular fast terminal sliding mode control method for a robotic arm with a predetermined time is designed. By establishing a dynamic model that includes parameter perturbation, joint friction and external disturbance, a non-singular fast terminal sliding mode surface and an auxiliary controller with a predetermined time are constructed. Combined with Lyapunov stability analysis, the method ensures that the system can quickly and accurately track the desired trajectory within a predetermined time and suppresses the influence of external disturbances and model uncertainties.
It enables the robotic arm to track quickly and accurately within a predetermined time, improving the system's tracking accuracy and robustness, effectively suppressing the effects of external disturbances and parameter perturbations, and improving production efficiency and product quality.
Smart Images
Figure CN122231920A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of robotic arm motion control technology, specifically relating to a non-singular rapid terminal sliding mode control method for a robotic arm at a predetermined time. Background Technology
[0002] As a key component of modern industrial production and automation technology, robotic arms play an indispensable role in many fields such as automobile manufacturing, electronic assembly, logistics warehousing, and food processing due to their high flexibility and programmability. From simple material handling to complex parts processing and assembly, the application scope of robotic arms continues to expand, and their control precision, response speed, and reliability are directly related to production efficiency and product quality.
[0003] As industrial automation develops towards higher levels of intelligence and flexibility, increasingly stringent requirements are being placed on the control performance of robotic arms. On the one hand, the accelerated pace of production demands that robotic arms be able to complete various operations quickly and accurately to adapt to high-efficiency production processes. On the other hand, complex production environments and ever-changing task requirements present robotic arms with numerous challenges, such as the elastic deformation of mechanical structures, the nonlinear characteristics of transmission systems, the uncertainty of external loads, and various interference factors. These factors can all lead to a decrease in the control accuracy of robotic arms, or even system instability.
[0004] Traditional robotic arm control strategies, such as PID control, while offering advantages in terms of simple structure and ease of implementation, often fall short when facing complex and ever-changing dynamic environments and high-precision control requirements. Torque control, although improving control accuracy to some extent, demands high precision in the system model; deviations in model parameters or disturbances in the system significantly reduce control effectiveness. Sliding mode control, however, has garnered widespread attention as an effective nonlinear control method due to its robustness. By designing a sliding surface, it allows the system state to maintain sliding motion on the surface, thus providing strong suppression of uncertainties in system parameters, external disturbances, and unmodeled dynamics. However, traditional sliding mode control suffers from relatively slow convergence speed and significant chattering, which not only affects control accuracy but may also subject the robotic arm's mechanical structure to additional stress and wear, reducing its lifespan.
[0005] In recent years, non-singular fast terminal sliding mode control (NFM) has emerged, overcoming some of the shortcomings of traditional sliding mode control. NFM enables rapid convergence of the system state and avoids singularity issues, preventing instability near singular points. However, existing NFM methods still have limitations in the determinism of convergence time, which often depends on the initial state of the system. This makes it difficult to precisely plan and control the robotic arm's operation time in practical applications, especially in tasks requiring extremely high time accuracy, such as high-speed sorting and precision welding. This uncertainty can lead to production process disruptions and product quality degradation. Summary of the Invention
[0006] To address the aforementioned problems in the existing technology, the present invention aims to provide a non-singular fast terminal sliding mode control method for a robotic arm within a predetermined time, enabling the robotic arm to quickly and accurately track the desired trajectory within a predetermined time, effectively suppressing the influence of parameter perturbations, friction, and external disturbances, and improving the system's tracking accuracy, convergence speed, and control robustness.
[0007] This invention provides the following technical solution: a method for non-singular fast end-of-arm sliding mode control of a robotic arm at a predetermined time, comprising the following steps: 1) Establish a dynamic model of the robotic arm system including parameter perturbation, joint friction and external disturbance, and construct a tracking error system model based on the dynamic model of the robotic arm system; the tracking error system model includes position tracking error and velocity tracking error; 2) Based on the position tracking error and velocity tracking error in step 1), design a non-singular fast terminal sliding surface for a predetermined time; 3) Based on the predetermined time non-singular fast terminal sliding surface in step 2), design an auxiliary controller based on the predetermined time sliding mode reaching law, and construct a predetermined time non-singular fast terminal sliding mode controller by combining the predetermined time non-singular fast terminal sliding surface and the auxiliary controller. 4) Using the Lyapunov stability analysis method, it is proved that the tracking error system is stable within a predetermined time during both the sliding mode arrival and sliding phases, and the system stability time is determined. 5) Verify the feasibility and effectiveness of the control method on the robotic arm platform.
[0008] Furthermore, the specific process of 1) is as follows: 1.1) Establish a dynamic model of the multi-joint robotic arm system. The dynamic model considers the influence of parameter perturbation and includes the inertia matrix, centrifugal force and Coriolis matrix, gravity vector, friction term and external disturbance term. 1.2) Based on the non-singular properties of the dynamic model and the inertia matrix, the dynamic model is redescribed as a form containing concentrated uncertainty disturbances; 1.3) Define the desired position signal and desired velocity signal, calculate the deviation between the actual position and actual velocity of the robotic arm joint and the corresponding desired value, obtain the position tracking error and velocity tracking error respectively, and construct a tracking error system model containing concentrated uncertainty interference based on the model in step 1.2).
[0009] Furthermore, the specific process of 2) is as follows: Step 2.1: Based on the position tracking error and velocity tracking error, construct a non-singular fast terminal sliding surface for a predetermined time; the sliding surface includes a fractional power term of the position tracking error and an integral term of the position tracking error, and the sliding surface parameter configuration ensures that the sliding surface converges within the predetermined time and avoids singularities; Step 2.2: Differentiate the sliding surface of the non-singular fast terminal at the predetermined time to obtain the sliding surface derivative expression including the derivative of the tracking error; Step 2.3: Based on the tracking error system model and the sliding surface derivative expression, construct a predetermined time non-singular fast terminal sliding mode controller; the controller can drive the robotic arm system to track the desired trajectory within a predetermined time; it includes compensation terms related to the system model and gain terms for adjusting the approach speed.
[0010] Furthermore, the specific process of 3) is as follows: Step 3.1: To address the concentrated uncertainty disturbances in the robotic arm system, design an auxiliary controller based on a sliding mode reaching law with a predetermined time. Step 3.2: Based on the equivalent sliding mode controller and the auxiliary controller, a predetermined time non-singular fast terminal sliding mode controller is constructed; the predetermined time non-singular fast terminal sliding mode controller enables the robotic arm system to accurately track the desired trajectory within a predetermined time and is robust to parameter perturbations, joint friction and external disturbances.
[0011] Furthermore, in step 3.1, the control law of the auxiliary controller includes a sign function term and a fractional power term based on the sliding mode variable, and the parameter configuration of the auxiliary controller ensures that the system state reaches the sliding surface within a predetermined time.
[0012] Furthermore, in step 3.2 above, the predetermined time non-singular fast terminal sliding mode controller is a linear superposition of the equivalent sliding mode controller and the auxiliary controller, and the parameter configuration of the controller ensures that the robotic arm tracking error system is stable within the predetermined time.
[0013] Furthermore, the specific process of 4) is as follows: Step 4.1, Stability analysis of sliding mode arrival stage: Define the first Lyapunov function with the sliding mode variable as the state variable. By calculating its derivative and combining it with the expression of the non-singular fast terminal sliding mode controller within a predetermined time, prove that the sliding mode variable can converge to the sliding surface within the first predetermined time. Step 4.2, Stability analysis of sliding mode: After the system state reaches the sliding surface, a second Lyapunov function with tracking error as the state variable is defined. By analyzing the properties of its derivative, it is proved that the position tracking error and velocity tracking error can converge to zero in the second predetermined time. The total time for the tracking error system to reach stability is the sum of a first predetermined time and a second predetermined time, and the total time can be preset during system design.
[0014] Furthermore, in step 4.1, the first Lyapunov function is the squared L2 norm of the sliding mode variable, and the first predetermined time is determined by the auxiliary controller and preset parameters in the sliding surface.
[0015] Furthermore, in step 4.2, the second Lyapunov function is the squared norm of the position tracking error, and the second predetermined time is determined by preset parameters in the sliding surface.
[0016] By employing the above-described technology, the beneficial effects of the present invention compared to the prior art are as follows: This invention applies to robotic arm systems. By combining the advantages of pre-set time control, the robotic arm system can achieve non-singular fast terminal sliding mode control within a pre-defined time frame, ensuring rapid and accurate convergence of the system state while effectively suppressing the effects of external disturbances and model uncertainties. Through this control method, the robotic arm can complete various predetermined tasks with higher precision, faster speed, and stronger robustness in complex and ever-changing industrial environments, providing a more reliable and efficient solution for industrial automation production and promoting the in-depth application and development of robotic arm technology in a wider range of fields. Attached Figure Description
[0017] Figure 1 This is a schematic diagram of a non-singular fast terminal sliding mode control structure based on a predetermined time, as described in an embodiment of the present invention. Figure 2 This is a schematic diagram of the robotic arm platform in an embodiment of the present invention; Figure 3 This is a schematic diagram of the robotic arm joints 1 and 2 tracking the desired signal curve in an embodiment of the present invention; Figure 4 This is a schematic diagram of the tracking of the desired signal curve by robotic arm joints 3 and 4 in an embodiment of the present invention; Figure 5This is a schematic diagram of the tracking of the desired signal curve by robotic arm joints 5 and 6 in an embodiment of the present invention; Figure 6 This is a schematic diagram of the tracking error curves of robotic arm joints 1 and 2 in an embodiment of the present invention; Figure 7 This is a schematic diagram of the tracking error curves of robotic arm joints 3 and 4 in an embodiment of the present invention; Figure 8 This is a schematic diagram of the tracking error curves of robotic arm joints 5 and 6 in an embodiment of the present invention; Figure 9 This is a schematic diagram of the control torque curves of robotic arm joints 1 and 2 in an embodiment of the present invention; Figure 10 This is a schematic diagram of the control torque curves of robotic arm joints 3 and 4 in an embodiment of the present invention; Figure 11 This is a schematic diagram of the control torque curves of robotic arm joints 5 and 6 in an embodiment of the present invention. Detailed Implementation
[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.
[0019] Conversely, this invention encompasses any substitutions, modifications, equivalent methods, and solutions made within the spirit and scope of the invention as defined in the claims. Furthermore, to provide a better understanding of the invention, certain specific details are described in detail below. However, those skilled in the art will fully understand the invention even without these detailed descriptions.
[0020] Please see Figure 1 A method for sliding mode control of a robotic arm with a predetermined time non-singular fast end state includes the following steps: Step 1: First, establish the dynamic model of the robotic arm system and the error system for the robotic arm to track the desired trajectory; the specific process is as follows: 1.1) Considering the external disturbances, nonlinear joint friction, and unknown parameter perturbations that may occur during the movement of the robotic arm, the dynamic model of the multi-joint robotic arm system can be described as follows: (1); in, , and These represent the position vector, velocity vector, and acceleration vector of each joint of the robotic arm, respectively. This represents the inertia matrix of the robotic arm system. Representing centrifugal force and the Coriolis matrix, The force vector is the gravitational force. Due to parameter perturbations during the robotic arm modeling process, the relevant parameters of the robotic arm are... , and They can be described as follows: , and .in, , and These represent the true values of the model coefficients, , and These represent parameter perturbations that exist during the robotic arm modeling process. Represents the matrix of viscous friction coefficients. This represents the static friction vector between the joints of the robotic arm. The control torque for the robotic arm, This indicates an external disturbance.
[0021] 1.2) Based on step 1.1) and the inertia matrix Since it is non-singular, through calculation, the dynamic model of the robotic arm system can be redescribed as follows: (2); in, It is a set of uncertainties in the movement process of a robotic arm, including unknown parameter perturbations, nonlinear joint friction, and external disturbances.
[0022] 1.3) Define the desired position signal as The desired speed signal is Therefore, the position and velocity tracking errors are respectively and Based on step 1.2), the robotic arm tracking error system can be obtained as follows: (3); in, , , This represents the concentrated uncertainty interference encountered during the movement of the robotic arm.
[0023] Step 2: Based on the tracking error variable in Step 1.3), a predetermined-time non-singular fast terminal sliding surface and an equivalent sliding controller were designed; the specific process is as follows: 2.1) To ensure the robotic arm system has fast transient and tracking performance, a predetermined-time nonsingular fast end-effector sliding surface was designed as follows: (4); in, , and , , Represents a symbolic function. , , , , and All are normal numbers.
[0024] 2.2) Based on step 2.1), the non-singular fast terminal sliding surface at the predetermined time. Taking the derivative, we get: (5); in, , .
[0025] 2.3) Based on steps 1.3) and 2.2), the equivalent sliding mode controller is designed through derivation as follows: (6); in, , , , It is an arbitrarily small positive number.
[0026] Step 3: Based on the sliding surface in Step 2), an auxiliary controller based on the sliding mode reaching law at a predetermined time and a non-singular fast terminal sliding mode controller at a predetermined time are designed; the specific process is as follows: 3.1) To address the issues of parameter perturbation, joint friction, and external disturbances during the robotic arm's movement, an auxiliary controller based on a predetermined time sliding mode reaching law was designed as follows: (7); in, , , , Represents a symbolic function. , and , , , , All are normal numbers; here we assume concentrated uncertainty interference. , It is a normal constant.
[0027] 3.2) Based on steps 2.3) and 3.1), a predetermined-time non-singular fast termination sliding mode controller is constructed as follows: (8).
[0028] Step 4: Using the Lyapunov stability analysis method, it was proved that the tracking error system is stable for a predetermined time during both the sliding mode arrival and sliding phases. The specific proof process is as follows: 4.1) First, analyze the stability of the sliding mode arrival stage, and define the Lyapunov function as: (9); right Taking the derivative, we get: (10); Substituting equations (3) and (8) into equation (10), we get: (11); Based on the predetermined time stability theory, the sliding mode variable can be obtained. Able to be at the scheduled time Based on step 2.1), the following can be obtained from the non-singular fast terminal sliding surface that arrives at the predetermined time: (12); 4.2) Analyze the stability during the sliding phase of the sliding mode, and define the Lyapunov function as follows: (13); right Taking the derivative, we get: (14); According to the predetermined time stability theory, the tracking error variable can be stable within a predetermined time. The convergence reaches zero. Therefore, under the action of this invention, the robotic arm tracking error system is stable for a predetermined time, the settling time being: (15); This theoretically verifies the effectiveness and feasibility of the invention.
[0029] Step 5: From a practical perspective, the feasibility and effectiveness of the invention were verified on a robotic arm platform; the specific process is as follows: To further verify the effectiveness of the predetermined time non-singular fast terminal sliding mode control method of the present invention, we conducted a six-joint robotic arm trajectory tracking experiment on a robotic arm control platform, as shown in the figure. Figure 2 As shown, robotic arm joints 1 to 6 are respectively... This indicates that the initial values for the 6 joints are set to... The desired trajectory signal is set as follows: (16); External interference signal settings: (17); The relevant parameters of the scheduled time non-singular fast terminal sliding mode controller were set to... , , , , , , , , , and .
[0030] Specific experimental results are as follows: Figure 3-11 As shown: Figures 3 to 5 The results of the robotic arm's joints 1 to 6 tracking the desired position trajectory are shown. Figures 6 to 8 The tracking error curves for joints 1 to 6 of the robotic arm are shown. Figures 9 to 11 This is the control torque curve for joints 1 to 6 of the robotic arm. Figures 3 to 8 It can be clearly seen that all six joints of the robotic arm are operating at their predetermined times. The robot arm accurately tracked the desired position trajectory signal, which shows that under the action of the predetermined time non-singular fast terminal sliding mode controller, the robot arm can quickly and accurately track the desired trajectory, and has a good suppression effect on external disturbances, parameter perturbations and joint friction, and enables the robot arm to achieve satisfactory tracking effect. This shows that the invented predetermined time non-singular fast terminal sliding mode control method is effective and feasible.
[0031] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.
Claims
1. A method for sliding mode control of a robotic arm at a predetermined time non-singular rapid end point, characterized in that, Includes the following steps: 1) Establish a dynamic model of the robotic arm system including parameter perturbation, joint friction and external disturbance, and construct a tracking error system model based on the dynamic model of the robotic arm system; the tracking error system model includes position tracking error and velocity tracking error; 2) Based on the position tracking error and velocity tracking error in step 1), design a non-singular fast terminal sliding surface for a predetermined time; 3) Based on the predetermined time non-singular fast terminal sliding surface in step 2), design an auxiliary controller based on the predetermined time sliding mode reaching law, and construct a predetermined time non-singular fast terminal sliding mode controller by combining the predetermined time non-singular fast terminal sliding surface and the auxiliary controller. 4) Using the Lyapunov stability analysis method, it is proved that the tracking error system is stable within a predetermined time during both the sliding mode arrival and sliding phases, and the system stability time is determined. 5) Verify the feasibility and effectiveness of the control method on the robotic arm platform.
2. The method for non-singular rapid terminal sliding mode control of a robotic arm at a predetermined time according to claim 1, characterized in that, The specific process of 1) is as follows: 1) Establish a dynamic model of the multi-joint robotic arm system. The dynamic model considers the influence of parameter perturbation and includes the inertia matrix, centrifugal force and Coriolis matrix, gravity vector, friction term and external disturbance term. 2) Based on the non-singular properties of the dynamic model and the inertia matrix, the dynamic model is redescribed as a form containing concentrated uncertainty disturbances; 3) Define the desired position signal and desired velocity signal, calculate the deviation between the actual position and actual velocity of the robotic arm joint and the corresponding desired value, obtain the position tracking error and velocity tracking error respectively, and construct a tracking error system model containing concentrated uncertainty interference based on the model in step 1.2).
3. The method for non-singular rapid terminal sliding mode control of a robotic arm at a predetermined time according to claim 1, characterized in that, The specific process of 2) is as follows: Step 2.1: Based on the position tracking error and velocity tracking error, construct a non-singular fast terminal sliding surface for a predetermined time; the sliding surface includes a fractional power term of the position tracking error and an integral term of the position tracking error, and the sliding surface parameter configuration ensures that the sliding surface converges within the predetermined time and avoids singularities; Step 2.2: Differentiate the sliding surface of the non-singular fast terminal at the predetermined time to obtain the sliding surface derivative expression including the derivative of the tracking error; Step 2.3: Based on the tracking error system model and the sliding surface derivative expression, construct a non-singular fast terminal sliding mode controller with a predetermined time. The controller can drive the robotic arm system to track the desired trajectory within a predetermined time; it includes a compensation term related to the system model and a gain term for adjusting the approach speed.
4. The method for non-singular rapid terminal sliding mode control of a robotic arm at a predetermined time according to claim 3, characterized in that, The specific process of 3) is as follows: Step 3.1: To address the concentrated uncertainty disturbances in the robotic arm system, design an auxiliary controller based on a sliding mode reaching law with a predetermined time. Step 3.2: Based on the equivalent sliding mode controller and the auxiliary controller, a predetermined time non-singular fast terminal sliding mode controller is constructed; the predetermined time non-singular fast terminal sliding mode controller enables the robotic arm system to accurately track the desired trajectory within a predetermined time and is robust to parameter perturbations, joint friction and external disturbances.
5. The method for non-singular rapid terminal sliding mode control of a robotic arm at a predetermined time according to claim 4, characterized in that, In step 3.1, the control law of the auxiliary controller includes a sign function term and a fractional power term based on the sliding mode variable, and the parameter configuration of the auxiliary controller ensures that the system state reaches the sliding surface within a predetermined time.
6. The method for non-singular rapid terminal sliding mode control of a robotic arm at a predetermined time according to claim 4, characterized in that, In step 3.2 above, the predetermined time non-singular fast terminal sliding mode controller is a linear superposition of the equivalent sliding mode controller and the auxiliary controller, and the parameter configuration of the controller ensures that the robotic arm tracking error system is stable within the predetermined time.
7. The method for non-singular rapid terminal sliding mode control of a robotic arm at a predetermined time according to claim 1, characterized in that, The specific process of 4) is as follows: Step 4.1, Stability analysis of sliding mode arrival stage: Define the first Lyapunov function with the sliding mode variable as the state variable. By calculating its derivative and combining it with the expression of the non-singular fast terminal sliding mode controller within a predetermined time, prove that the sliding mode variable can converge to the sliding surface within the first predetermined time. Step 4.2, Stability analysis of sliding mode: After the system state reaches the sliding surface, a second Lyapunov function with tracking error as the state variable is defined. By analyzing the properties of its derivative, it is proved that the position tracking error and velocity tracking error can converge to zero in the second predetermined time. The total time for the tracking error system to reach stability is the sum of a first predetermined time and a second predetermined time, and the total time can be preset during system design.
8. The method for non-singular rapid terminal sliding mode control of a robotic arm at a predetermined time according to claim 7, characterized in that, In step 4.1, the first Lyapunov function is the squared L2 norm of the sliding mode variable, and the first predetermined time is determined by the auxiliary controller and the preset parameters in the sliding surface.
9. A method for non-singular rapid terminal sliding mode control of a robotic arm at a predetermined time, as described in claim 7, is characterized in that... In step 4.2, the second Lyapunov function is the squared norm of the position tracking error, and the second predetermined time is determined by preset parameters in the sliding surface.