A mechanical arm tracking control method based on finite time tracking differentiator and disturbance observer

Abstract

The application discloses a mechanical arm tracking control method based on a finite time tracking differentiator and a disturbance observer, and comprises the following steps: combining inverse hyperbolic sine functions and terminal attractor functions to construct a finite time tracking differentiator; using the form of the finite time tracking differentiator, designing a new type of nonlinear disturbance observer; for a dynamics model of a rigid mechanical arm, combining an adaptive law to design an adaptive sliding mode backstepping controller; using the designed new type of nonlinear disturbance observer to estimate external disturbances and uncertain items of the system, and using the finite time tracking differentiator to respectively reconstruct unknown states of system output and derivative items of an estimated virtual control law, and feeding back position and velocity signals after reconstruction of the former to the adaptive sliding mode backstepping controller. The control method provided by the application can effectively realize tracking control of the mechanical arm, and has the advantages of fast response, high tracking precision and the like.

Description

Technical Field

[0001] This invention belongs to the field of tracking control technology, specifically relating to a robotic arm tracking control method based on a finite-time tracking differentiator and a disturbance observer. Background Technology

[0002] With the rapid development of mass production and the increasing market demand, robotic arms are playing an increasingly important role in various fields, leading to higher requirements for their tracking accuracy and speed. However, the complex characteristics of robotic arms, such as multiple inputs and outputs, parameter uncertainties, strong coupling, and external interference, make it difficult to accurately establish the system's dynamic model. This poses challenges to high-precision and high-speed tracking control. Therefore, researching a feasible control method is of great significance in the field of robotic arm tracking control.

[0003] Currently, commonly used control methods include: traditional PID control, adaptive control, sliding mode control, and optimal control. In actual industrial applications, the tracking control accuracy of robotic arm systems is often affected by uncertain external disturbances; moreover, in complex environments, the output signals of robotic arm systems are often difficult to measure, leading to unknown system states. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies, the present invention aims to provide a robotic arm tracking control method based on a finite-time tracking differentiator and a disturbance observer. The proposed disturbance observer estimates uncertain external disturbances in the robotic arm system, and the tracking differentiator reconstructs the unknown state of the robotic arm system and estimates the derivative terms of the virtual control law. An adaptive sliding mode inversion controller is constructed by combining adaptive law and sliding mode control methods to achieve high-precision tracking control of a multi-degree-of-freedom robotic arm. This allows for the estimation of uncertain external disturbances under complex and harsh measurement conditions, improving tracking speed and ensuring convergence accuracy.

[0005] To achieve the above objectives, the specific technical solution of the present invention is as follows:

[0006] A robotic arm tracking control method based on a finite-time tracking differentiator and a disturbance observer, characterized by the following steps:

[0007] Step S1: Combine the inverse hyperbolic sine function and the terminal attractor function to construct a finite-time tracking differentiator;

[0008] Step S2: Using the form of the finite-time tracking differentiator in step S1, design a novel nonlinear perturbation observer;

[0009] Step S3: For the dynamic model of the rigid manipulator, design an adaptive sliding mode inversion controller in combination with the adaptive law;

[0010] Step S4: Use the novel nonlinear disturbance observer to estimate the external disturbance and uncertainty of the system. The finite-time tracking differentiator reconstructs the unknown state of the system output and estimates the derivative of the virtual control law. The reconstructed position and velocity signals are then fed back to the adaptive sliding mode inversion controller.

[0011] Furthermore, the specific process of constructing the finite-time tracking differentiator in step S1 is as follows:

[0012]

[0013] In the formula, v(t) is the system input signal, x1(t) and x2(t) are the tracking signal and differential signal of v(t), respectively; R > 0 is the velocity factor, and a1, a2 > 0 are adjustable parameters. Both m and n are odd numbers, and b is the terminal attractor parameter.

[0014] Furthermore, the specific form of the novel nonlinear perturbation observer in step S2 is as follows:

[0015]

[0016] in,

[0017]

[0018] In the formula, x1 and x2 are state vectors, and Let x1, x2, and d be the estimators of the disturbance d, respectively; A1, A2, and Λ are the positive definite matrices to be designed. Let μ be the terminal attractor parameter, ξ > 0, and all be odd numbers; as ||Λ|| → ∞, and

[0019] Furthermore, the dynamic model of the rigid robotic arm in step S3 is as follows:

[0020]

[0021] In the formula, q∈R n Let M(q) be the joint variable of the robotic arm; M(q)∈R n×n , and G(q)∈R n Let R represent the positive definite symmetric inertia matrix, centrifugal force and Coriolis force terms, and gravity term in the robotic arm dynamics model, respectively; τ∈R n τ is the control torque of the robotic arm. d ∈R n For the uncertain disturbance outside the robotic arm;

[0022] Transform the dynamic model (4) into a differential equation form, and let the state variable x1 = q. Transform into the following form:

[0023]

[0024] Where f(x) = -M -1 (x1)[C(x1,x2)x2+G(x1)], g(x)=M -1 (x1), d=-M -1 (x1)τ d , u=τ; where f(x) and g(x) are both given nonlinear functions, and d and u represent the disturbance and control input, respectively.

[0025] Furthermore, the specific steps of step S3 include:

[0026] Step S31: Tracking error Define virtual control law Where x d For the desired position instruction, and To track the reconstructed state variables of the differentiator, k is the positive definite symmetric matrix to be designed; the sliding surface s = λe1 + e2 is designed, where the sliding coefficient matrix λ ∈ R. n×n >0; set estimated values ​​for the upper bound of the disturbance δ and the disturbance amount d. The adaptive law is chosen as follows: And γ > 0;

[0027] Step S32: Design an adaptive sliding mode inversion controller:

[0028]

[0029] In the formula, g(x) and f(x) are given nonlinear functions in the dynamic model of the robotic arm, respectively; s and λ are the sliding mode function and its coefficient matrix, respectively; e1 and e2 are defined error vectors; The derivative of the virtual control law α The estimated term is given by k, where k is the coefficient matrix and h is the parameter matrix of the controller. It is an estimator of the upper bound value δ of the disturbance;

[0030] Step S33: Construct the Lyapunov function and prove that the designed closed-loop system is exponentially stable; that is, the Lyapunov function is:

[0031]

[0032] Its first derivative:

[0033]

[0034] Let a0 = min{2||k||, 2||k-λ||, 2||h||}, then

[0035]

[0036] Therefore, this closed-loop system is exponentially convergent, meaning that all signals converge exponentially to 0.

[0037] Furthermore, the specific process of step S4 is as follows:

[0038] S41: Introduce intermediate variables σ = [σ1 σ2] T At this time, the virtual control law and its derivative It will be obtained from the following formula:

[0039]

[0040] Among them, R αi >0, a αi1 >0, a αi2 >0 and b αi For the parameter variables to be designed, m αi >n αi >0 and m αi n αi All are odd numbers, i = 1, 2;

[0041] S42: The position signal is obtained and the velocity signal is reconstructed by suppressing random disturbances using a finite-time tracking differentiator, as shown in the following form:

[0042]

[0043] In the formula, q i (t) represents the actual position signal output by the i-th joint of the robotic arm; For q i The estimated signal of (t); χ represents the velocity signal of the i-th joint after reconstruction. i (t) represents the intermediate state variable; R Fi >0, a Fi1 >0, a Fi2 >0 and These are the system parameters to be designed.

[0044] Compared with the prior art, the present invention has the following advantages:

[0045] (1) The present invention uses an inverse hyperbolic sine function and a terminal attractor function to construct a finite-time tracking differentiator. Compared with the traditional tracking differentiator, this tracking differentiator has a faster response speed, better transition stability, and better filtering performance.

[0046] (2) This invention fully considers the influence of uncertain external disturbances and structural uncertainties of the robotic arm system on tracking control. It uses the differential form of the proposed tracking differentiator to construct a novel nonlinear disturbance observer to estimate the uncertain disturbance terms of the robotic arm system.

[0047] (3) The present invention adopts a sliding mode inversion controller based on adaptive law. Adaptive law can improve the robustness and anti-interference ability of the system. Applying sliding mode control to the inversion controller has the characteristics of faster response speed and stronger robustness compared with the traditional inversion control method.

[0048] (4) The present invention takes into account the situation in actual industrial sites where the system state is unknown due to complex measurement conditions, and uses the constructed finite-time tracking differentiator to reconstruct the unknown state of the robotic arm.

[0049] (5) This invention is aimed at the robotic arm tracking control system under complex and harsh conditions. This control method has fast response, good stability and good filtering performance. Attached Figure Description

[0050] Figure 1 This is a system block diagram of the present invention;

[0051] Figure 2 This is a structural diagram of the novel nonlinear perturbation observer designed for this invention;

[0052] Figure 3 This is a structural diagram of the adaptive sliding mode inversion controller of the present invention;

[0053] Figure 4 This is a simulation diagram of the position, velocity, and disturbance signal tracking and error of joint one of the dual-joint rigid robotic arm in an embodiment of the present invention.

[0054] Figure 5 This is a simulation diagram showing the position, velocity, and disturbance signal tracking and error of joint two of the dual-joint rigid robotic arm in an embodiment of the present invention. Detailed Implementation

[0055] To enable those skilled in the art to better understand the technical solution of the present invention, the following description, in conjunction with the accompanying drawings, further illustrates a robotic arm tracking control method based on a finite-time tracking differentiator and a disturbance observer.

[0056] Taking a two-degree-of-freedom robotic arm as an example, the present invention will be further described below with reference to the accompanying drawings and specific embodiments:

[0057] like Figure 1 As shown, a disturbance tracking control method for a robotic arm based on a finite-time tracking differentiator and a disturbance observer includes the following steps:

[0058] Step S1: Construct a finite-time tracking differentiator. The finite-time tracking differentiator based on the inverse hyperbolic sine function and the terminal attractor function takes the following form:

[0059]

[0060] In the formula, v(t) is the system input signal, x1(t) and x2(t) are the tracking signal and the derivative signal of v(t), respectively; R > 0 is the velocity factor. When R increases, the system tracking speed will become faster, but if it is too large, it will produce chattering; a1, a2 > 0 are adjustable parameters. The effect of a1 is similar to that of R; increasing a2 will lead to a slower tracking speed, but the suppression capability will be enhanced. Both m and n are odd numbers, and b is the terminal attractor parameter, which has the same effect as a2.

[0061] Step S2: Construct a novel nonlinear perturbation observer using the differential form of the proposed tracking differentiator. The specific form of this novel nonlinear perturbation observer is as follows:

[0062]

[0063] in,

[0064]

[0065] In the formula, x1 and x2 are state vectors, and Let x1, x2, and d be the estimators of the perturbation, respectively; A1, A2, and Λ are positive definite matrices with design. And all are odd numbers) are terminal attractor parameters; when ||Λ||→∞, and

[0066] Step S3: For the dynamic model of the rigid robotic arm, an adaptive sliding mode inversion controller is designed using an adaptive law.

[0067] The rigid robotic arms proposed in this invention are all n-degree-of-freedom, i.e., multi-degree-of-freedom rigid robotic arms. The dynamic model of the n-degree-of-freedom rigid robotic arm is as follows:

[0068]

[0069] In the formula, q∈R nLet M(q) be the joint variable of the robotic arm; M(q)∈R n×n , and G(q)∈R n Let R represent the positive definite symmetric inertia matrix, centrifugal force and Coriolis force terms, and gravity term in the robotic arm dynamics model, respectively; τ∈R n τ is the control torque of the robotic arm. d ∈R n This refers to the uncertain disturbance outside the robotic arm.

[0070] For example, when the robotic arm has two degrees of freedom, both of which are rotational joints, its specific dynamic model is as follows:

[0071]

[0072]

[0073]

[0074] in, and In the formula, m1 and m2 are the masses of link 1 and link 2 of the robotic arm, respectively; l1 and l2 are the arm lengths of link 1 and link 2, respectively; I1 and I2 are the inertia matrices of link 1 and link 2 of the robotic arm, respectively; and g is the acceleration due to gravity.

[0075] Transform the dynamic model (4) into a differential equation form, and let the state variable x1 = q. Transform into the following form:

[0076]

[0077] Where f(x) = -M -1 (x1)[C(x1,x2)x2+G(x1)], g(x)=M -1 (x1), d=-M -1 (x1)τ d , u=τ. In the formula, f(x) and g(x) are both given nonlinear functions, and d and u represent the disturbance and control input, respectively.

[0078] Therefore, the specific process for designing an adaptive sliding mode inversion controller is as follows:

[0079] (1) Define tracking error Where x d For the desired position instruction, and To track the state variables after the differentiator is reconstructed, a virtual control law is designed. Where k is the positive definite symmetric matrix to be designed; the sliding surface s is defined as λe1+e2, where the sliding coefficient matrix λ∈R n×n>0; set estimated values ​​for the upper bound of the disturbance δ and the disturbance amount d. The adaptive law is chosen as follows: And γ > 0.

[0080] (2) Design of an adaptive sliding mode inversion controller:

[0081]

[0082] In the formula, g(x) and f(x) are given nonlinear functions in the dynamic model of the robotic arm, respectively; s and λ are the sliding mode function and its coefficient matrix, respectively; e1 and e2 are defined error vectors; Let be the derivative of the virtual control law α, and k be the coefficient matrix therein; h be the parameter matrix of the controller; It is an estimate of the upper bound value δ of the perturbation.

[0083] (3) Construct the Lyapunov function:

[0084]

[0085] Its first derivative

[0086]

[0087] Let a0 = min{2||k||, 2||k-λ||, 2||h||}, then

[0088]

[0089] Therefore, this closed-loop system is exponentially convergent, meaning that all signals converge exponentially to 0.

[0090] Step S4: Using a novel disturbance observer, estimate the external disturbances and uncertainties of the system. A finite-time tracking differentiator reconstructs the unknown state of the system output and estimates the derivative terms of the virtual control law. The position and velocity signals obtained from the reconstructed state are then fed back to the adaptive sliding mode inversion controller. Specifically, this includes the following steps:

[0091] (1) The inversion controller contains many repeated differential terms, leading to an "explosive term" problem. Estimating the derivative terms of the virtual control law using a finite-time tracking differentiator will effectively solve this problem. An intermediate variable σ = [σ1 σ2] is introduced. T At this point, the virtual control law and its derivative It will be obtained from the following formula:

[0092]

[0093] Among them, R αi >0, a αi1 >0, aαi2 >0 and And m αi n αi All numbers are odd, and i = 1, 2) are the parameter variables to be designed.

[0094] (2) The unknown state of the robot arm output is reconstructed using the finite-time tracking differentiator constructed in step S1. Its specific form is as follows:

[0095]

[0096] In the formula, q i (t) represents the actual position signal output by the i-th (i = 1, 2) joint of the robotic arm; For q i The estimated signal of (t); χ represents the velocity signal of the i-th joint after reconstruction. i (t) represents the intermediate state variable; R Fi >0, a Fi1 >0, a Fi2 >0 and These are the system parameters to be designed.

[0097] The position response q output by the robotic arm i (t) is input to a finite-time tracking differentiator to filter noise, track the position signal and reconstruct the velocity signal, and then feed it back to the sliding mode controller.

[0098] In this example, such as Figure 2 The diagram shown is a structural diagram of a novel nonlinear perturbation observer. Figure 3 The diagram shows the structure of the adaptive sliding mode inversion controller in the robotic arm tracking control system.

[0099] To verify the tracking and control capability of the robotic arm in complex environments with external disturbances, MATLAB / Simulink was used to perform simulations on a specific example, demonstrating the feasibility of the proposed disturbance observer. The structural parameters of the robotic arm were set as m1 = m2 = 4 kg, l1 = l2 = 0.5 m, and I1 = I2 = 0.2 kg·m. 2 And the maximum input torque of the two joints is u 1max =100, u 2max =45. Assume the position command q for the two joints. d =[q 1d q 2d ] T = [15sin(0.5t) 10sin(0.5t)] T The external interference signal is d = [2sin(q1) 4sin(q2)]. TThe controller parameters are adjusted to k = diag{2,2}, h = diag{2,2}, λ = diag{50,50}, and γ = 5. The parameters of the finite-time tracking differentiator IHSTD1, which reconstructs the virtual control law by differentiating the derivative, are R. α1 =R α1 =500, a α11 =a α21 =150, a α12 =a α22 =20, and b I1 =b I2 (n I1 =n I2 =1,m I1 =m I2 =15). The parameters of the finite-time tracking differentiator IHSTD2 in the feedback channel are R. F1 =R F2 =500, a F11 =a F21 =100, a F12 =a F22 =10 and b F1 =b F2 (n F1 =n F2 =1,m F1 =m F2 =21), and the parameters of the disturbance observer are set as Λ=diag{25,25}, A1=diag{5,5}, A2=diag{2,2},

[0100] like Figure 4 and Figure 5 The figure shows a simulation of the position, velocity, and disturbance signal tracking and error of the two joints during the motion of the dual-joint rigid manipulator in this example. It can be seen from the figure that although there is some overshoot in the initial tracking stage, the error can be quickly converged to zero. This indicates that the proposed control method can not only accurately track the position and velocity signals in a short time, but also accurately estimate external uncertain disturbances; it also confirms the disturbance estimation capability of the proposed nonlinear disturbance observer and the differential reconstruction capability of the finite-time tracking differentiator.

[0101] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A robotic arm tracking control method based on a finite-time tracking differentiator and a disturbance observer, characterized in that, Includes the following steps: Step S1: Combine the inverse hyperbolic sine function and the terminal attractor function to construct a finite-time tracking differentiator; The specific process of constructing a finite-time tracking differentiator is as follows: （1） In the formula, Input signals to the system, , They are respectively The tracking signal and the differential signal; For velocity factor, It is an adjustable parameter. , and All are odd numbers. For terminal attractor parameters; Step S2: Using the form of the finite-time tracking differentiator in Step S1, design a novel nonlinear perturbation observer; the specific form of the novel nonlinear perturbation observer is as follows: （2） in, （3） In the formula , Let be the state vector, and , , They are respectively , and disturbance The estimate; , and The matrix to be designed is a positive definite matrix; For terminal attractor parameters, And all are odd numbers; when hour, and ; Step S3: Based on the dynamic model of the rigid robotic arm, design an adaptive sliding mode inversion controller in combination with the adaptive law; Step S4: Utilize the newly designed nonlinear disturbance observer to estimate the external disturbances and uncertainties of the system. A finite-time tracking differentiator reconstructs the unknown state of the system output and estimates the derivative terms of the virtual control law, respectively. The reconstructed position and velocity signals are then fed back to the adaptive sliding mode inversion controller. The specific process is as follows: S41: Introducing intermediate variables At this time, the virtual control law and its derivative It will be obtained from the following formula: （10） in, , , and For the parameter variables to be designed, , and , All are odd numbers. ; S42: The position signal is obtained and the velocity signal is reconstructed by suppressing random disturbances using a finite-time tracking differentiator, as shown in the following form: （11） In the formula, For the first robotic arm The actual position signal output by each joint; for The estimated signal; For the reconstructed first The velocity signal of each joint; For intermediate state variables; , , as well as These are the system parameters to be designed.

2. The robotic arm tracking control method based on a finite-time tracking differentiator and a disturbance observer as described in claim 1, characterized in that, The dynamic model of the rigid robotic arm in step S3 is as follows: （4） In the formula For the joint variables of the robotic arm; , and These represent the positive definite symmetric inertia matrix, centrifugal force and Coriolis force terms, and gravity term in the robotic arm dynamics model, respectively. This refers to the control torque of the robotic arm; For the uncertain disturbance outside the robotic arm; Dynamic model Transform it into a differential equation form, let the state variables... , Transformed into the following form: （5） in , , , In the formula and All are given nonlinear functions. and These are represented as disturbance and control input, respectively.

3. The robotic arm tracking control method based on a finite-time tracking differentiator and a disturbance observer as described in claim 2, characterized in that, The specific steps of step S3 include: Step S31: Tracking error Define virtual control laws ,in For the desired position instruction, and To track the state variables after the differentiator is reconstructed, The positive definite symmetric matrix to be designed; design the sliding surface. The sliding mode coefficient matrix ; for the upper bound of the disturbance and disturbance quantity Set the estimated value The adaptive law is chosen as ,and ; Step S32: Design an adaptive sliding mode inversion controller: （6） In the formula, and These are the given nonlinear functions in the robotic arm's dynamics model; and These are the sliding mode function and its coefficient matrix, respectively; and The defined error vector; For virtual control laws derivative The estimated term, and This is the coefficient matrix; This is the parameter matrix of the controller; The upper bound of the perturbation The estimate; Step S33: Construct the Lyapunov function and prove that the designed closed-loop system is exponentially stable.

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