A mechanical arm trajectory tracking control method and system based on backstepping sliding mode active disturbance rejection

Abstract

The application discloses a kind of based on anti-step sliding mode self-disturbance mechanical arm trajectory tracking control method and system, belong to robot control technical field.The system includes: sliding mode tracking differentiator, for the desired trajectory arrangement transition process and extract high-quality differential signal;Linear extended state observer, for real-time estimation system state and total disturbance, improve observation speed and noise suppression ability;Anti-step sliding mode controller, for generating control torque and ensuring the stability of closed-loop system and tracking desired trajectory in finite time.The application integrates anti-step method and sliding mode control into the framework of self-disturbance, overcomes the defects of slow convergence speed, stability proof difficulty and other defects of traditional self-disturbance control, significantly improves the trajectory tracking accuracy and robustness of mechanical arm in complex disturbance environment, suitable for high-precision operation scenarios such as industrial robot, medical robot.

Description

Technical Field

[0001] This invention belongs to the field of robot control technology, specifically relating to a robotic arm trajectory tracking control method and system based on backstepping sliding mode self-disturbance rejection, applicable to industrial robots, medical robots, service robots and other applications requiring high-precision trajectory tracking. Technical Background

[0002] With the development of industrial automation and intelligent manufacturing, robotic arms are widely used in welding, assembly, and material handling. A robotic arm is a complex, nonlinear, strongly coupled, and time-varying system, and in actual operation it is often affected by uncertainties such as modeling errors, joint friction, load changes, and external environmental disturbances. Therefore, achieving high-precision trajectory tracking control has always been a research hotspot.

[0003] Active Disturbance Rejection Control (ADRC), proposed by researcher Han Jingqing, treats all internal and external disturbances as a total disturbance and uses an Extended State Observer (ESO) for real-time estimation and compensation. It boasts advantages such as independence from precise models and strong anti-interference capabilities. However, traditional ADRC also has some shortcomings: 1) Slow convergence speed: Based on linear error feedback, traditional ADRC can only guarantee asymptotic convergence, theoretically requiring an infinite amount of time to completely eliminate the error; 2) Difficulty in proving stability, especially for nonlinear systems, lacking a systematic Lyapunov analysis method; 3) Complex parameter tuning, particularly key parameters such as observer bandwidth, which require manual adjustment and are difficult to adapt to changing environments.

[0004] In recent years, scholars have attempted to combine sliding mode control, backstepping, neural networks, and ADRC. Sliding mode control has the advantages of strong robustness and fast response, but it suffers from chattering problems; backstepping can ensure system stability through recursive design, but it requires model information. Therefore, how to combine the speed of sliding mode control with the systematic nature of backstepping, while improving the observer structure and parameter self-tuning capability, is a technical problem that urgently needs to be solved in this field. Summary of the Invention

[0005] The present invention aims to overcome the shortcomings of the prior art and provide a robotic arm trajectory tracking control method and system based on backstepping sliding mode active disturbance rejection, so as to solve the problems of slow convergence, difficulty in proving stability and limited observer performance of traditional active disturbance rejection control.

[0006] To address the aforementioned problems, this invention proposes a robotic arm trajectory tracking control method and system based on backstepping sliding mode active disturbance rejection (ADDR). By real-time estimation and compensation of internal and external disturbances, the motion accuracy and robustness of the robotic arm are improved. Specifically, addressing the issues of slow convergence, difficulty in proving stability, and limited observer performance in traditional ADDR systems, the method includes: 1. Designing a sliding mode tracking differentiator to arrange the transition process of the desired trajectory and extract the smooth tracking signal and its derivative signal, overcoming the noise sensitivity of traditional tracking differentiators. 2. Designing a linearly extended state observer to observe unknown disturbances in the robotic arm system in real time and compensate for their impact. 3. Designing a backstepping sliding mode controller, constructing a Lyapunov function based on the backstepping method, introducing a sliding surface design control law to ensure finite-time convergence, weakening chattering through a fast power-law approach, and using feedforward compensation based on the total disturbance estimated by the observer.

[0007] This invention provides a robotic arm trajectory tracking control method and system based on backstepping sliding mode self-disturbance rejection, comprising the following steps:

[0008] S1. Construct a dynamic model of the robotic arm based on the Lagrange method.

[0009] S2, Sliding mode tracking differentiator: used to arrange the transition process of the desired trajectory and extract the smooth tracking signal and its differential signal, overcoming the shortcomings of traditional tracking differentiators that are sensitive to noise;

[0010] S3, Linear Extended State Observer: Real-time observation of unknown disturbances in the robotic arm system, compensating for the impact of disturbances on the robotic arm system;

[0011] S4. Backstepping sliding mode controller: Based on the backstepping method, Lyapunov functions are constructed, and a sliding surface is introduced to design the control law to achieve accurate tracking of the desired trajectory in finite time. Chattering is weakened by a fast power-law approach, and feedforward compensation is performed using the total disturbance estimated by the observer.

[0012] S5 integrates the control algorithms of modules S1 to S4 to achieve precise control of the robotic arm.

[0013] In step S1, the method for establishing the dynamic model of the robotic arm is as follows:

[0014] The Lagrangian method is used for dynamic modeling of a six-DOF robotic arm. The Lagrangian method starts from the overall system and establishes the dynamic model of the six-DOF robotic arm:

[0015]

[0016] in, , representing the joint position angle, joint angular velocity, and joint angular acceleration vector of the robotic arm, respectively. It is a symmetric positive definite inertial matrix. The centrifugal Coriolis force matrix of the robotic arm. The matrix representing the gravity terms of the robotic arm. Indicates external disturbance. The input vector representing the torque of the robotic arm;

[0017] Considering the uncertainty of the dynamic parameters of the robotic arm, i.e. , , ,in , , This represents the nominal value of the robotic arm's dynamics system. , , This represents the parameter uncertainty of the dynamic system; therefore, the above dynamic model is rewritten as follows: ,in, ;

[0018] Rewrite the model in state-space form. In the formula, Uncertainty of lumped parameters Represented as ;

[0019] Furthermore, in step S2, the basic mathematical model of the tracking differentiator is:

[0020]

[0021] in, To track the input signal, yes The derivative of the sigmoid function. The improved expression for the sigmoid function is: .in, It is a small constant that controls the degree of smoothness.

[0022] Furthermore, in step S3, for a nonlinear uncertain controlled object, its linearly extended state observer is established as follows:

[0023] Based on Luenberger state observer theory, a set of linear equations for a third-order linear extended state observer can be constructed under the active disturbance rejection control paradigm:

[0024]

[0025] in, Estimate the gain for the controller. To control the input, for The observed values, As input to the linearly extended state observer, its state variables The expanded state variables will be tracked separately. Specifically, It is used in active disturbance rejection control to capture unknown disturbances in the system, such as external interference. The robotic arm is an extended state variable in the modeling part.

[0026] After simplification, the observer equation can be obtained:

[0027]

[0028] Parameters in the formula To determine the gain of the observer, the poles of the observer are placed using the bandwidth method. The relationship between gain and bandwidth can be obtained using the following formula.

[0029]

[0030] available .

[0031] Furthermore, in step S4, the algorithm design for constructing the backstepping sliding mode controller is as follows:

[0032] To achieve accurate tracking of the desired trajectory within a finite time, a backstepping sliding mode control was designed. First, the tracking error was defined. , For reference only.

[0033] Define the first Lyapunov function: Differentiate: .

[0034] Design virtual control quantity ,in .

[0035] Define a second error variable:

[0036] Substituting the above equation into... :

[0037]

[0038] Design the sliding surface:

[0039] Differentiating, we get:

[0040]

[0041] Define the enhanced Lyapunov function:

[0042]

[0043] Differentiate:

[0044]

[0045]

[0046] The control law is designed as follows:

[0047]

[0048] in, These are the parameters of the sliding surface. It is feedback gain. It is a power coefficient. , It is a sign function, a lumped disturbance. By observer Alternative compensation.

[0049] The present invention also provides a robotic arm trajectory tracking control system based on backstepping sliding mode active disturbance rejection, comprising: at least one memory for storing computer-executable instructions; and at least one processor for executing the computer-executable instructions to implement the robotic arm trajectory tracking backstepping sliding mode active disturbance rejection control method described above. Attached Figure Description

[0050] Figure 1 This is a flowchart illustrating the robotic arm trajectory tracking control method of the present invention;

[0051] Figure 2 This is a block diagram of the overall system structure of the present invention;

[0052] Figure 3 Schematic diagram of a sliding mode tracking differentiator;

[0053] Figure 4 This is a simulation diagram of the robotic arm of the present invention;

[0054] Figure 5 These are the observations of the joint angles by the observer;

[0055] Figure 6 This is a simulation trajectory tracking effect diagram of the method of the present invention applied to a six-degree-of-freedom robotic arm;

[0056] Figure 7 This is a graph showing the tracking error of the present invention under external disturbances;

[0057] Figure 8 This is a comparison chart of tracking errors between the present invention and other methods. Detailed Implementation

[0058] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0059] Example 1: Running the control algorithm proposed in this invention, the system includes the following modules:

[0060] Sliding mode tracking differentiator: software implementation, input is the desired trajectory The output is a smooth tracking signal. and differential signal .

[0061] Linear extended state observer: based on joint angle feedback and control quantity Real-time estimation of angular velocity Total disturbance .

[0062] Reverse sliding mode controller: based on and estimated value Calculate joint control torque .

[0063] Example 2: Dynamics Modeling of a Robotic Arm

[0064] A six-DOF robotic arm dynamic model is established using the Lagrange method. The dynamic model of a single-joint robotic arm is as follows:

[0065]

[0066] In the formula, To control torque; The moment of inertia of the joint wheel; This refers to the joint damping coefficient; For unknown external disturbances; This represents the angle through which the joint wheel rotates; Let be the angular velocity of the joint wheel. This is considered as the unknown total disturbance of the system. In this embodiment, a pulse disturbance is added: amplitude 0.1, pulse width 0.1s, phase delay 5.5s, simulating sudden load changes in an industrial scenario. The sum of the original total disturbance and the offset caused by the introduction of control parameters is denoted as the new total disturbance. Select state variables , here defined Then the state-space expression of the system can be obtained as:

[0067]

[0068] in Let be the rate of change of the disturbance, which is assumed to be bounded.

[0069] Example 3: Sliding Mode Tracking Differentiator Design

[0070] To avoid noise amplification by traditional differentiators, a sliding mode tracking differentiator is designed as follows. Consider a second-order system:

[0071]

[0072] Define the sliding surface:

[0073]

[0074] The improved sigmoid function is used instead of the sign function to suppress chattering, and the control law is designed as follows:

[0075]

[0076] Consider Lyapunov functions Differentiating, we get:

[0077]

[0078] because And when hour, This indicates that the system is asymptotically stable and will eventually converge to the boundary layer. .

[0079] Example 4: Design of a Linearly Extended State Observer

[0080] To perform real-time observation of unknown disturbances in the robotic arm system and compensate for the impact of disturbances on the system's performance, a linear extended state observer is established as follows:

[0081] Based on Luenberger state observer theory, a set of linear equations for a third-order linear extended state observer can be constructed under the active disturbance rejection control paradigm:

[0082]

[0083] in, To control the amplification factor, For system control variables, for The observed values, As input to the linearly extended state observer, its state variables The expanded state variables will be tracked separately. Specifically, It is an extended state variable used in the extended state observer to capture unknown disturbances in the system, such as external disturbances or extended state variables of the unmodeled parts of the robotic arm.

[0084] After simplification, the observer equation can be obtained:

[0085]

[0086] Parameters in the formula To determine the gain of the observer, the poles of the observer are placed using the bandwidth method. The relationship between gain and bandwidth can be obtained using the following formula.

[0087]

[0088] available .

[0089] Example 5: Design of a Backstepping Sliding Mode Controller

[0090] To achieve finite-time convergence of control, a backstepping sliding mode control was designed. First, the tracking error was defined. , For reference only.

[0091] Define the first Lyapunov function: Differentiate: .

[0092] Design virtual control quantity , in .

[0093] Define a second error variable:

[0094] Substituting the above equation into... :

[0095]

[0096] Design the sliding surface:

[0097] Differentiating, we get:

[0098]

[0099] Define the enhanced Lyapunov function:

[0100]

[0101] Differentiate:

[0102]

[0103]

[0104] The control law is designed as follows:

[0105]

[0106] in: These are the parameters of the sliding surface. It is feedback gain. It is a power coefficient. , It is a sign function. Since traditional sign functions can lead to strong chattering, a fast power-approaching law is used to balance response speed and suppress chattering.

[0107] Stability analysis:

[0108] Construct a Lyapunov function for the entire system:

[0109]

[0110] It is clearly positive definite. Taking its derivative:

[0111]

[0112] therefore To ensure Write it in matrix form: in, By selecting appropriate parameters, it becomes possible to achieve the desired result. It can be guaranteed Thus, the system asymptotically stabilizes.

[0113] once The system error is then simplified to There exists a constant. , so that: .therefore: .

[0114] This proves that the system is exponentially stable and converges exponentially to zero. To prove that it converges to the sliding surface in finite time, when the system state is near the sliding surface, the final convergence characteristic is determined by... Dominant. Construct a framework about Lyapunov function Its derivative satisfies: This allows us to prove that the surface converges to the sliding surface in a finite amount of time.

[0115] Example 6: This example is an implementation of the method according to Examples 1-5.

[0116] 1. Simulation system construction:

[0117] A simulation system was built in the Simulink environment. The mass and inertia data of the six-degree-of-freedom robotic arm used in this experiment are shown in Table 1.

[0118] Table 1 Robotic Arm Parameters

[0119] quality lxx lyy Lzz J1 0.4159 0.0002861 0.0003323 0.0003752 J2 2.4617 0.0017 0.0212 0.0203 J3 0.9768 0.0009 0.0026 0.0030 J4 0.7819 0.0058 0.0058 0.0002 J5 0.3930 0.0008106 0.0001453 0.0007495 J6 0.1062 0.00002732 0.00002732 0.00002726

[0120] 2. Backstepping sliding mode self-disturbance rejection motion control:

[0121] The designed backstepping sliding mode active disturbance rejection controller controls the six joints of the robotic arm, and the tracking performance of the angular displacement output of the six joints relative to the expected value is analyzed. In the simulation, the expected function is set to radians. The perturbation was set to an impulse perturbation with an amplitude of 0.1, a pulse width of 0.1, and a phase delay of 5.5 s. The simulation step size of the system was uniformly set to 0.0001 s, and the bandwidth settings for the linear expansion state observers of the six joints are shown in Table 2.

[0122] Table 2 Bandwidth Settings

[0123] parameter Setting value parameter Setting value 33 60 33 68 53 150

[0124] The specific parameters of the reverse sliding mode controller of the present invention are shown in Table 3.

[0125] Table 3 Controller Parameter Settings

[0126] J1 7.5 7.35 0.8 19 25 0.8 J2 4.8 9 0.4 29 39 0.08 J3 48 12 0.005 29 45 0.01 J4 170 7 0.8 25 32 0.08 J5 230 8 0.002 29 39 0.2 J6 240 7.35 0.8 13 18 0.08

[0127] The above parameters are a preferred embodiment of the present invention. Those skilled in the art can adjust these parameters within a reasonable range based on the physical characteristics and performance indicators of the actual robotic arm. For example, increasing... It can speed up the convergence of the observer, but may also amplify noise; increase and It can accelerate the convergence of the sliding surface, but if it is too large, it will cause control signal chattering.

[0128] Substituting the above parameters into the controller and robotic arm dynamics model of this invention, the simulation results are obtained: the position tracking curves and position tracking error curves of each joint of the six-DOF robotic arm with the backstepping sliding mode controller of this invention are as follows: Figure 6 and Figure 7 As shown.

[0129] pass Figure 6 and Figure 7 It can be seen that the robotic arm trajectory tracking control method based on backstepping sliding mode self-disturbance rejection proposed in this invention still has good tracking performance under conditions of uncertain system parameters and disturbances. Figure 8 It can be seen that the robotic arm trajectory tracking control method based on backstepping sliding mode self-disturbance rejection proposed in this invention has better tracking performance compared with PID and ADRC.

[0130] The present invention provides a robotic arm trajectory tracking control system based on backstepping sliding mode self-disturbance rejection, comprising a memory and a processor, wherein the memory stores a computer program, and the processor executes the program to implement the steps in the above method embodiments.

[0131] The above embodiments are merely illustrative of the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to the preferred embodiments above, those skilled in the art should understand that modifications or equivalent substitutions to the technical solutions of the present invention should not depart from the spirit and scope of the present invention. Those skilled in the art can also make other changes within the spirit of the present invention and use them in the design of the present invention, as long as they do not deviate from the technical effects of the present invention. These changes made according to the spirit of the present invention should all be included within the scope of protection claimed by the present invention.

Claims

1. A method and system for tracking and controlling the trajectory of a robotic arm based on backstepping sliding mode self-disturbance rejection, characterized in that, Includes the following steps: S1. Construct a dynamic model of the robotic arm using the Lagrange method; S2. Design a sliding mode tracking differentiator module: to receive the desired trajectory signal, generate a smooth tracking signal and its first-order differential signal through the sliding mode algorithm, and use the differential signal as a velocity feedforward to avoid unsmooth or discontinuous phenomena in the system input; S3. Build a linear extended state observer: to observe unknown disturbances in the robotic arm system in real time and compensate for the impact of disturbances on the robotic arm system. S4. Construct a backstepping sliding mode controller module: This module receives the tracking signal and differential signal output by the sliding mode tracking differentiator, as well as the state estimate output by the linearly extended state observer. Based on the backstepping method, it constructs a Lyapunov function, designs the sliding surface and a fast power-law approach, generates control torque, and achieves precise tracking of the desired trajectory in finite time. S5. The backstepping sliding mode controller module receives the output signals from the sliding mode tracking differentiator module and the linear expansion state observer module to generate control torque and achieve precise control of the joint angle displacement of the robotic arm.

2. The robotic arm trajectory tracking control method based on backstepping sliding mode self-disturbance rejection as described in claim 1, characterized in that, Its single-joint robotic arm dynamics model is as follows: In the formula, , , These are joint angle, angular velocity, and angular acceleration, respectively. It is a symmetric positive definite inertial matrix. The centrifugal Coriolis force matrix of the robotic arm. The matrix representing the gravity terms of the robotic arm. Indicates external disturbance. The control torque represents the torque of the robotic arm. The model is standardized as follows: in The total disturbance includes model uncertainties and external disturbances. This is the nominal value of the control gain; To control the input.

3. The robotic arm trajectory tracking control method based on backstepping sliding mode self-disturbance rejection as described in claim 1, characterized in that, The sliding mode tracking differentiator is designed as follows: Define the sliding surface: In the formula, It is a small constant that controls the degree of smoothness; To track the velocity factor of the differentiator, The larger the value, the faster the tracking speed; This is the steepness parameter for the sigmoid function, used to control how steep the sigmoid function is. To avoid the denominator being zero; To track the input tracking signal of the differentiator; for The differential signal.

4. The robotic arm trajectory tracking control method based on backstepping sliding mode self-disturbance rejection as described in claim 1, characterized in that, Unknown nonlinear terms in the control system As the state variables of the extended state observer, a third-order linear extended state observer is designed as follows: In the formula, This refers to the actual angle of the robotic arm joint. These are the estimated values ​​for joint angle, angular velocity, and total disturbance, respectively. To determine the gain of the observer, the poles of the observer are placed using the bandwidth method. Where can be obtained . For the observer bandwidth, For controller gain estimation.

5. The robotic arm trajectory tracking control method based on backstepping sliding mode self-disturbance rejection as described in claim 1, characterized in that, The reverse-stepping sliding mode controller is designed as follows: First, define the tracking error. , For the desired trajectory, For the joint angles of the robotic arm, design virtual control variables. ,definition , For the angular velocity of the robotic arm joints, the sliding surface The control law is: in, These are the parameters of the sliding surface. It is feedback gain. It is a power coefficient. , It is a sign function. Wherein, the lumped disturbance... Perturbation estimate output by the linear extended state observer Perform feedforward compensation.

6. A method and system for tracking and controlling the trajectory of a robotic arm based on backstepping sliding mode self-disturbance rejection, characterized in that, Includes the following steps: Step 1: Establish a dynamic model of the robotic arm, use the Lagrange method to obtain the inertia matrix, Coriolis matrix and gravity term of the multi-degree-of-freedom robotic arm, and rewrite it into a second-order integral series form suitable for active disturbance rejection control. Step 2: Design a sliding mode tracking differentiator for the desired trajectory. Arrange the transition process to generate a smooth tracking signal. and its differential signal ; Step 3: Design a linear expansion state observer based on the actual joint angles. and control input Real-time estimation of angular velocity Total disturbance ; Step 4: Design a backstepping sliding mode controller based on tracking error and sliding surface, combined with disturbance estimation. Calculate the control torque ; Step 5: Apply the calculated control torque to the robotic arm joints, collect joint angle feedback, form closed-loop control, and repeat steps 2 to 4 to achieve high-precision trajectory tracking.

7. The method according to any one of claims 1 to 6, characterized in that, The robotic arm is a six-degree-of-freedom serial robotic arm.

8. The method according to claim 6, characterized in that, In step 4, the backstepping sliding mode control law adopts a fast power-law approach. The system's exponential stability and finite-time convergence are proven through Lyapunov functions, and the disturbance observations are fully feedforward compensated to the control law.

9. A robotic arm trajectory tracking control system based on backstepping sliding mode self-disturbance rejection, characterized in that, include: At least one memory for storing computer-executable instructions; At least one processor is configured to execute the computer-executable instructions to implement the method of any one of claims 1 to 8.

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