A manipulator trajectory tracking control method based on a nonlinear tracking differentiator

Abstract

The present invention discloses a manipulator trajectory tracking control method based on a nonlinear tracking differentiator. This method designs a nonlinear tracking differentiator, which can not only quickly track the signal output by the manipulator and filter noise, but also estimate the unknown state in the manipulator system. The method includes the following steps: S1: Design a nonlinear tracking differentiator by combining the inverse hyperbolic sine function and the terminal attractor function; S2: For the manipulator dynamic model, use an RBF neural network to approximate the uncertainty of the manipulator and design a sliding mode controller with a robust factor; S3: Use the nonlinear tracking differentiator to estimate the unknown state in the manipulator, track the position signal output by the manipulator and reconstruct its speed signal, and feedback it to the sliding mode controller. The present invention realizes the manipulator trajectory tracking control through the designed nonlinear tracking differentiator combined with the sliding mode controller, and has the characteristics of fast response and noise filtering.

Description

Technical Field

[0001] The present invention relates to the field of manipulator control, and particularly to a manipulator trajectory tracking control method based on a nonlinear tracking differentiator. Background Art

[0002] With the continuous development of technology, manipulators are more and more widely used in industrial automation and are more common in actual industrial sites such as national defense, chemical industry, and medical treatment. Its emergence has provided a lot of convenience for human life. It saves a large amount of labor for enterprises, reduces some cost expenses, and also avoids the operation errors that will inevitably occur during the production process by manual labor. It also ensures the personal safety of workers in high-risk workplaces. People have carried out extensive research on the tracking control problem of manipulators. However, since the manipulator system is a complex nonlinear system, there are modeling errors and uncertain disturbances in its actual industrial application sites, which are all uncertain factors affecting the tracking accuracy of the system. The powerful learning ability of the neural network algorithm is used to approximate the uncertain part in the manipulator system, reducing the dependence on the mathematical model of the system. For a complex nonlinear system, sliding mode control is an excellent control method that is widely used, but it will inevitably appear chattering phenomenon during the control process. Summary of the Invention

[0003] The purpose of this invention patent is to provide a manipulator trajectory tracking control method based on a nonlinear tracking differentiator to solve one or more of the problems raised in the above background art.

[0004] To achieve the above purpose, this invention patent provides the following technical solutions: A manipulator trajectory tracking control method based on a nonlinear tracking differentiator, characterized by including the following steps:

[0005] S1: Design a nonlinear tracking differentiator by combining the inverse hyperbolic sine function and the terminal attractor function;

[0006] S2: For the manipulator dynamic model, use the RBF neural network to approximate the uncertainty of the manipulator and design a sliding mode controller with a robust factor;

[0007] S3: Use the nonlinear tracking differentiator to estimate the unknown state in the manipulator, track the position signal output by the manipulator and reconstruct its speed signal, and feedback it to the sliding mode controller to achieve the effect of controlling the manipulator.

[0008] Preferably, the step S1 includes the following steps:

[0009] S1-1: Design the nonlinear tracking differentiator based on the inverse hyperbolic sine function and the terminal attractor function:

[0010] Equation (1)

[0011] In the said Equation (1), is the input signal, is the tracking signal of, is the differential signal of; is the speed factor, The larger, the faster the tracking speed, but chatter will occur when it is too large; is an adjustable parameter, The influence on the system is the same as that of i.e., When increasing, it will accelerate the tracking speed of the system; is also an adjustable parameter, When increasing, the noise suppression ability is enhanced, but the tracking speed will be weakened; , and are both odd numbers, is the terminal attractor parameter, The role of is the same as that of i.e., The larger, the stronger the noise suppression ability;

[0012] S1-2. Construct a Lyapunov function and prove the stability of the said nonlinear tracking differentiator according to the Barbashin-Krasovskii theorem:

[0013] First, transform the form of Equation (1), and the transformed form is:

[0014] Equation (2)

[0015] Determine the stability of Equation (1), that is, determine the stability of Equation (2), and construct the Lyapunov function as:

[0016] Equation (3)

[0017] In the said Equation (3), the inverse hyperbolic sine function is an odd function, then is also an odd function, so is always greater than zero. And when is, . Therefore, In the neighborhood of (0,0):

[0018]

[0019] From the above properties of, it can be known that the first-order derivative function of Equation (3) always holds; if and only if when ; that is, in equation (2), there are no other points except the origin that can satisfy ; equation (2) is asymptotically stable at the origin (0, 0); equation (1) is also asymptotically stable at the origin (0, 0).

[0020] Preferably, the dynamic model of the robotic arm in step S2 is:

[0021] Equation (4)

[0022] In the above equation (4), is the joint variable of the robotic arm, that is, the angular displacement of the joint; is the positive definite symmetric inertia matrix of the robotic arm; is the centrifugal force and Coriolis force term of the robotic arm; is the gravity term of the robotic arm; is the friction term of the robotic arm; is the external bounded input disturbance of the robotic arm; is the control torque;

[0023] Step S2 includes the following steps:

[0024] S2-1. Define the sliding mode function for the tracking error , where is the desired position command, ; Use the RBF neural network algorithm to approximate the unknown uncertainties in the robotic arm system, and obtain after approximation. Design the robust factor for the error term after neural network approximation and the external disturbance term of the system, where is the critical value of the neural network approximation error, and is the critical value of the external disturbance of the robotic arm.

[0025] S2-2. Design the sliding mode controller:

[0026] Equation (5)

[0027] In the above equation (5), is the function after the RBF neural network approximates the unknown uncertainties in the robotic arm; is the sliding mode control term, is the sliding mode function, is the sliding mode control coefficient; is the robust factor, is the critical value of the approximation error of the neural network, is the critical value of the external disturbance of the robotic arm.

[0028] Preferably, the nonlinear tracking differentiator in step S3 can estimate the unknown state of the system. The position signal output by the robotic arm is input into the tracking differentiator, the noise is filtered to output a tracking signal and the speed signal is reconstructed, and then the output tracking signal and speed signal are fed back to the sliding mode controller. The specific expressions are as follows:

[0029] Equation (6)

[0030] In the said Equation (6), is the actual position signal output by the th joint of the robotic arm; is the estimated signal of ; is the speed signal of the th joint after reconstruction; is the intermediate state variable.

[0031] Compared with the prior art, the beneficial effects of the present invention are:

[0032] (1) The present invention designs a new nonlinear tracking differentiator using the inverse hyperbolic sine function and the terminal attractor function, which has the advantages of faster response speed, better smoothness in the transition process, and better filtering performance compared with the traditional tracking differentiator;

[0033] (2) The present invention fully considers the influence of external disturbances, structural uncertainties, dynamics, etc. of the robotic arm on the tracking control, and the RBF neural network algorithm can reduce the dependence on the system mathematical model and approximate the unknown and uncertain parts of the system;

[0034] (3) Compared with the traditional sliding mode control method, the present invention designs a robust factor, which improves the robust performance of the system and the anti-interference ability to the outside world;

[0035] (4) The present invention considers the situation where the system state is unknown due to complex external conditions in the actual industrial site, and uses the designed nonlinear tracking differentiator to estimate and reconstruct the unmeasurable unknown state of the robotic arm;

[0036] (5) The present invention has a simple structure and is easier to implement in practical applications. It has fast response, good smoothness, and good filtering performance for the robotic arm tracking control system. Description of the Drawings

[0037] Figure 1 is the schematic diagram of the present invention;

[0038] Figure 2 It is the structural diagram of a nonlinear tracking differentiator;

[0039] Figure 3 It is the structural diagram of a sliding mode controller;

[0040] Figure 4 It is the simulation result diagram of the position signals of the two joints in the specific implementation manner of the present invention;

[0041] Figure 5 It is the simulation result diagram of the speed signals of the two joints in the specific example of the present invention;

[0042] Figure 6 It is the simulation result diagram of the control torque in the specific example of the present invention;

[0043] Figure 7 It is the simulation comparison diagram before and after the approximation of the RBF neural network in the specific example of the present invention. Specific implementation manner

[0044] The present invention will be described in detail below in conjunction with the accompanying drawings of the specification, but the solution of the present invention is not limited thereto.

[0045] In a specific embodiment of the present invention, taking the robotic arm with two degrees of freedom as an example, the present invention will be further described below in conjunction with the accompanying drawings and specific embodiments:

[0046] As Figure 1 shown, a robotic arm trajectory tracking control method based on a nonlinear tracking differentiator includes the following steps:

[0047] Step S1-1, designing a nonlinear tracking differentiator based on the inverse hyperbolic sine function and the terminal attractor function as:

[0048] Equation (1)

[0049] In the said Equation (1), is the input signal, is 's tracking signal, is 's differential signal; is the speed factor, The larger it is, the faster the tracking speed, but excessive value will cause chattering; is an adjustable parameter, The influence on the system is the same as that of , that is, When it increases, it will accelerate the tracking speed of the system; is also an adjustable parameter, When it increases, the noise suppression ability is also enhanced, but the tracking speed will be weakened; , and are all odd numbers, is the terminal attractor parameter, has the same effect as , that is the larger it is, the stronger the ability to suppress noise;

[0050] Step S1-2, construct the Lyapunov function and prove the stability of the designed nonlinear tracking differentiator according to the Barbashin-Krasovskii theorem.

[0051] Construct the Lyapunov function and prove the stability of the nonlinear tracking differentiator according to the Barbashin-Krasovskii theorem:

[0052] First, transform the form of Equation (1), and the transformed form is:

[0053] Equation (2)

[0054] Determine the stability of Equation (1), that is, determine the stability of Equation (2), and construct the Lyapunov function as:

[0055] Equation (3)

[0056] The inverse hyperbolic sine function in Equation (3) is an odd function, so is also an odd function, and thus is always greater than zero. And when , . Therefore, in the neighborhood of (0,0):

[0057]

[0058] From the above properties, it can be seen that the first derivative function of Equation (3) always holds; when and only when , ; that is, in Equation (2), there are no other points except the origin that can satisfy ; Equation (2) is asymptotically stable at the origin (0,0); Equation (1) is also asymptotically stable at the origin (0,0).

[0059] Step S2, the robotic arms proposed in this paper are all n-degree-of-freedom, that is, multi-degree-of-freedom robotic arms.

[0060] Then when the robotic arm has two degrees of freedom and all are revolute joints, its specific dynamic model is as follows:

[0061] Equation (7)

[0062] Equation (8)

[0063] Equation (9)

[0064] Among them, the specific parameters in Equations (7) to (9) are as follows: , , , , . In the formula, and are the masses of the robotic arm link 1 and link 2 respectively, and are the arm lengths of link 1 and link 2 respectively, is the acceleration due to gravity.

[0065] Step S2-1, define the tracking error , where is the desired position command. Design the sliding mode function , where . At this time, we can get:

[0066] Equation (10)

[0067] The following is the specific calculation process:

[0068]

[0069] Let the unknown and uncertain function be , where . Use the RBF neural network algorithm to approximate the function to obtain . For the error term after the neural network approximation and the external disturbance term of the system, design the robust factor , where is the critical value of the neural network approximation error, is the critical value of the external disturbance of the robotic arm.

[0070] The symbols in the above formulas are consistent with those in the dynamic model of the robotic arm. Specifically: such as , and in the formula respectively represent , and in the dynamic model.

[0071] Step S2-2, design a sliding mode controller based on the RBF neural network with a robust factor:

[0072] Equation (5)

[0073] In Equation (5), is the coefficient matrix of the sliding mode control term, and the diagonal matrix .

[0074] Step S3: Estimate the unknown state of the robotic arm using the designed new type of nonlinear tracking differentiator, and its specific form is as follows:

[0075] Equation (11)

[0076] In Equation (11), is the actual position signal output by the th joint of the robotic arm; is the estimated signal; is the reconstructed th joint speed signal; is the intermediate state variable.

[0077] That is, input the position signal output by the robotic arm into the tracking differentiator, filter the noise to output the tracking signal and reconstruct to obtain the speed signal , and then feedback the output signal to the sliding mode controller.

[0078] In this example, the structure diagram of the nonlinear tracking differentiator is as shown in Figure 2 , and the structure diagram of the sliding mode controller is as shown in Figure 3 .

[0079] To verify the tracking control ability of the robotic arm under time-varying disturbances for the proposed method, use MATLAB / Simulink to perform simulation verification on the specific embodiment. Assume that the structural parameters of the robotic arm are , let the initial state of the system be , and the initial weights of the network are 0. Assume that the position commands of the two joints are , , the friction term of the robotic arm is set to , and the external disturbance signal is . Adjust the controller parameters to , , , and the parameters of the nonlinear tracking differentiator are . Figure 4Shown are the angular position signals of two joints during the movement of the robotic arm. The simulation results verify that the control method can enable the angular position signals of each joint to quickly track the desired signals, improving the tracking performance of the robotic arm.

[0080] Figure 5 Shown are the angular velocity signals of two joints during the movement of the robotic arm, indicating that the tracking differentiator can effectively suppress noise and reasonably extract the differential signals to be transmitted to the controller under the interference of random noise.

[0081] Figure 6 Shown is the change of the control torque during the movement of the robotic arm. The curve in the figure can quickly reach a steady state, indicating that the control method effectively reduces the chattering in the sliding mode control.

[0082] Figure 7 Shown is the unknown non-linear function before and after the approximation by the RBF neural network 。

[0083] Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A manipulator trajectory tracking control method based on a nonlinear tracking differentiator, characterized in that, It includes the following steps: S1: Design a nonlinear tracking differentiator; S2: Design a sliding mode controller with a robust factor; S3: Use the nonlinear tracking differentiator to estimate the unknown state in the robotic arm, track the position signal output by the robotic arm and reconstruct its speed signal, and feedback the output tracking signal and speed signal to the sliding mode controller to achieve the effect of tracking the trajectory of the robotic arm; The step S1 includes the following steps: S1-1: Design a nonlinear tracking differentiator based on the inverse hyperbolic sine function and the terminal attractor function: Formula (1) In the formula (1), is the input signal, is 's tracking signal, is 's differential signal; is the speed factor; is an adjustable parameter; is also an adjustable parameter; , and are both odd numbers, is the terminal attractor parameter; S1-2: Construct a Lyapunov function and prove the stability of the nonlinear tracking differentiator according to the Barbashin-Krasovskii theorem; In the step S2, it includes: S2-1: Set the dynamic model of the robotic arm as Formula (4) In the formula (4), is the joint variable of the robotic arm, i.e., the angular displacement of the joint; is the positive definite symmetric inertia matrix of the robotic arm; is the centrifugal force and Coriolis force terms of the robotic arm; is the gravity term of the robotic arm; is the friction term of the robotic arm; is the external bounded input disturbance of the robotic arm; is the control torque; S2-2： Define the tracking error , where is the desired position command Design a sliding mode function , where , Let the unknown and uncertain function be , where ; After approximating the function using the RBF neural network algorithm, we obtain ; For the error term after neural network approximation and the external disturbance term of the system, design the robust factor , where is the critical value of the neural network approximation error, is the critical value of the external disturbance of the robotic arm; In the formula, , and respectively represent , and in the dynamic model; S2-3: Design the sliding mode controller equation Formula (5) In the formula (5), is the function after the RBF neural network approximates the unknown and uncertain terms in the manipulator; is the sliding mode control term, is the sliding mode function, is the sliding mode control coefficient; is the robustness factor, is the critical value of the neural network approximation error, is the critical value of the external disturbance of the manipulator.

2. The method for controlling the trajectory tracking of a robotic arm based on a nonlinear tracking differentiator according to claim 1, wherein The expression of the step S3 is as follows: Formula (6) In the said formula (6), is the actual position signal output by the th joint of the robotic arm; is the estimated signal; is the velocity signal of the th joint after reconstruction; is the intermediate state variable.

3. A manipulator trajectory tracking control method based on a nonlinear tracking differentiator according to claim 2, characterized in that, Let be the actual position signal output by the th joint of the robotic arm. The position signal output by the robotic arm is input into the tracking differentiator to filter out noise and output the tracking signal and reconstruct the velocity signal . Then, the output signal is fed back to the sliding mode controller to achieve the effect of tracking the trajectory of the robotic arm.

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