A fixed-time dynamic surface trajectory tracking control method for a pneumatic robotic arm system

Abstract

This application provides a fixed-time dynamic surface trajectory tracking control method for a pneumatic robotic arm system. The method includes the following steps: Step 1, establishing a dynamic model of the robotic arm based on the Lagrange formula method of energy analysis; Step 2, addressing the unmodeled dynamics, load variations, time-varying parameters, and coupling and friction disturbances between joints of the robotic arm system, designing a finite-time disturbance observer to observe the total disturbance of the system, and compensating for the total disturbance in real time during the controller design process; Step 3, designing a fixed-time dynamic surface controller to compensate for the total disturbance of the system in real time, designing a feedback control law to enable the angular positions of each joint of the robotic arm system to track the given target trajectory signal, and designing a time-varying gain function to dynamically adjust the controller gain.

Description

Technical Field

[0001] This application relates to the field of robotic arm systems, and in particular to a fixed-time dynamic surface trajectory tracking control method for a pneumatic robotic arm system. Background Technology

[0002] With the rapid development of intelligent technology, robotics has made significant progress. As a type of industrial robot, the robotic arm has continuously evolved alongside advancements in robotics technology and has been widely applied in human production and daily life. Compared to traditional rigid robotic arms, pneumatically driven multi-joint robotic arms occupy an important position in the industrial robot field due to their advantages such as lightweight design, low power consumption, good compliance, strong environmental adaptability, and high power-to-weight ratio. However, multi-joint pneumatic robotic arm systems are susceptible to disturbances during control, including unmodeled dynamics, load variations, time-varying parameters, air compressibility, and coupling and friction between joints, posing challenges to precise trajectory tracking control. To address the trajectory tracking problem of multi-joint pneumatic robotic arm systems under disturbance, scholars both domestically and internationally have proposed numerous advanced control strategies.

[0003] In general, trajectory tracking control algorithms for robotic arm systems can be broadly categorized into two types: model-free PID, fuzzy algorithms, and neural network algorithms, and model-based sliding mode control, adaptive control, and dynamic surface control. Among these, dynamic surface control is popular due to its more structured and systematic design process and is widely used in the design of control strategies for nonlinear systems. Furthermore, it is worth noting that in highly complex and variable real-world application scenarios, the control performance of robotic arm systems is always affected by disturbances. Traditional control methods often struggle to achieve good control performance across a wide operating range. Therefore, designing active disturbance rejection control algorithms based on disturbance estimation and compensation is of great significance for further improving the control performance of robotic arm systems. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies, this invention provides a fixed-time dynamic surface trajectory tracking control method for a robotic arm system based on a disturbance observer. First, the pneumatic robotic arm system is modeled, and considering the mechanical model of a pneumatic artificial muscle, the control torques of each joint of the robotic arm are solved using rotational laws to establish a dynamic model of the robotic arm system. Unmodeled dynamics, time-varying parameters, air compressibility, and coupling and friction between joints are considered as total disturbances and introduced into the robotic arm dynamic model. A disturbance observer is used to estimate the total disturbances in the system, and a fixed-time dynamic surface controller is designed to compensate for these disturbances in real time. To improve the response speed and trajectory tracking control accuracy of the robotic arm system, a time-varying gain function is introduced into the controller design to dynamically adjust the controller gain. Using the controller designed with this method, the pneumatic robotic arm system can achieve higher control accuracy and better robustness and disturbance rejection.

[0005] This application provides a fixed-time dynamic surface trajectory tracking control method for a pneumatic robotic arm system, including the following steps:

[0006] Step 1: Establish a dynamic model of the robotic arm based on the Lagrange formula method of energy analysis;

[0007] Step 2: To address the issues of unmodeled dynamics, load variations, time-varying parameters, and coupling and frictional disturbances between joints in the robotic arm system, a finite-time disturbance observer is designed to observe the total disturbance of the system, and the total disturbance is compensated for in real time during the controller design process.

[0008] Step 3: Design a fixed-time dynamic surface controller to compensate for the total disturbance of the system in real time. Design a feedback control law so that the angular position of each joint of the robotic arm system can track the given target trajectory signal. Design a time-varying gain function to dynamically adjust the controller gain.

[0009] In some embodiments, step one specifically includes:

[0010] Based on the analysis of the total kinetic and potential energy of the robotic arm system using the Lagrange formula, and combined with the mechanical model of the pneumatic artificial muscle, the control torques of each joint of the robotic arm are solved using the rotational law, resulting in the following dynamic model of the robotic arm system:

[0011]

[0012] In the formula, θ = [θ1, θ2, θ3] T Here, θ1, θ2, and θ3 represent the deflection angles of joint one, joint two, and joint three, respectively, and M(θ) is the inertia matrix of the robotic arm. This is the matrix of centrifugal and Coriolis forces of the robotic arm, G(θ) is the gravity vector, and τ is the centrifugal and Coriolis force matrices of the robotic arm. d It is an unknown perturbation vector, τ=[τ1,τ2,τ3] T This is the control torque vector, where τ1, τ2, and τ3 represent the control torques of joint one, joint two, and joint three, respectively; define x1 = θ, x = [x1, x2] T It is a state vector. The state-space expression of the robotic arm system is as follows:

[0013]

[0014] In the formula, f(x) = -M(x1) -1 (C(x1,x2)+G(x1)), g(x)=M(x1) -1 b, d = M(x1) -1 τ d y is the system output, d is the total disturbance of the robotic arm system, and b is the equivalent cross-sectional coefficient of the pneumatic artificial muscle.

[0015] In some embodiments, step two specifically includes:

[0016] Considering the total disturbance d in the robotic arm system, a disturbance observer is designed to estimate the total disturbance in real time and compensate for it during the design of the fixed-time dynamic surface controller. The finite-time disturbance observer is represented as follows:

[0017]

[0018] In the formula, z is the estimated value of the system state x2. This is the estimated value of the total disturbance d. The auxiliary variables σ = x² - z, m ∈ [1 / 2, 1), n ​​= 2m - 1, k1 > 0, k2 > 0 are the observer gain parameters, sig(σ). m =|σ| m sign(σ) defines the observation error of the perturbation observer. The estimated error vector is By designing appropriate observer gains k1 and k2, the designed perturbation observer can be made asymptotically stable in finite time, that is, the observation error of the perturbation observer can converge to the region near the origin in finite time.

[0019] In some embodiments, step three specifically includes:

[0020] The design process of a fixed-time dynamic surface controller is as follows:

[0021] ① Define the first-order dynamic error surface: s1 = x1 - x d In the formula x d It is the target trajectory

[0022] Design a fixed-time virtual control law:

[0023] In the formula, p and q are positive odd numbers satisfying p>q, φ 11 and φ 12 It is a time-varying gain, designed as follows:

[0024]

[0025] In the formula, α1, β1, ∈ 11 ,∈ 12 Since the parameters are positive, to avoid the "term explosion" problem and achieve fixed-time stability, the following fixed-time filter is designed.

[0026]

[0027] In the formula, z2 represents the virtual control law. The estimated value, where T is the time constant of the fixed-time filter;

[0028] ② Define the second-order dynamic error surface: s2 = x2 - z2, and design the actual control law for the robotic arm system.

[0029] In the formula φ 21 and φ 22 It is a time-varying gain, designed as follows:

[0030]

[0031] In the formula, α2, β2, ∈ 21 ,∈ 22 Since these are positive parameters, according to the Lyapunov stability criterion and the fixed-time stability theorem, by designing appropriate parameters α1, β1, α2, and β2, the controller can achieve asymptotic stability over a fixed time. That is, the trajectory tracking error s1 of the robotic arm system can converge to the neighborhood of the origin within a fixed time, and the convergence time is independent of the initial state information of the system.

[0032] This invention focuses on a pneumatic multi-joint robotic arm system. A dynamic model of the robotic arm is established, with the control objective being the rapid and high-precision tracking of the desired target trajectory signal by the joint angles of each joint. Considering the unmodeled dynamics, load variations, time-varying parameters, air compressibility, and coupling and friction between joints, disturbances in the robotic arm system are introduced as total disturbances into the dynamic model. A fixed-time dynamic surface control method based on a disturbance observer is designed and applied to the robotic arm system. The disturbance observer estimates the total disturbance in the pneumatic multi-joint robotic arm system, and the fixed-time dynamic surface controller compensates for this total disturbance in real time. This ensures that the output joint angles of the robotic arm system accurately track the desired target trajectory signal, thereby improving the robustness and disturbance rejection of the robotic arm system. Furthermore, a time-varying gain function is introduced into the controller design to dynamically adjust the controller gain, thereby improving the response speed and trajectory tracking control accuracy of the robotic arm system. Attached Figure Description

[0033] The accompanying drawings illustrate, by way of example and not limitation, the various embodiments discussed herein.

[0034] Figure 1 This is a simplified structural diagram of the pneumatic multi-joint robotic arm of the present invention;

[0035] Figure 2 This is a schematic diagram of the fixed-time dynamic surface control method based on a disturbance observer according to the present invention. Detailed Implementation

[0036] In order to gain a more detailed understanding of the features and technical content of the embodiments of this application, the implementation of the embodiments of this application will be described in detail below with reference to the accompanying drawings. The accompanying drawings are for reference and illustration only and are not intended to limit the embodiments of this application.

[0037] In the embodiments described in this application, it should be noted that, unless otherwise stated and limited, the term "connection" should be interpreted broadly. For example, it can be an electrical connection, or a connection between two internal components. It can be a direct connection or an indirect connection through an intermediate medium. Those skilled in the art can understand the specific meaning of the above term according to the specific circumstances.

[0038] It should be noted that the terms "first," "second," and "third" used in the embodiments of this application are merely used to distinguish similar objects and do not represent a specific ordering of objects. It is understood that "first," "second," and "third" can be interchanged in a specific order or sequence where permitted. It should be understood that the objects distinguished by "first," "second," and "third" can be interchanged where appropriate so that the embodiments of this application described herein can be implemented in an order other than those illustrated or described herein.

[0039] This application provides a fixed-time dynamic surface trajectory tracking and control method for a pneumatic robotic arm system, the specific principle and steps of which are as follows:

[0040] Step one: First, establish the dynamic model of the robotic arm based on the Lagrange formula method of energy analysis.

[0041] Step two: Considering the unmodeled dynamics, load changes, time-varying parameters, and coupling and friction between joints of the robotic arm system, design a finite-time disturbance observer to observe the total disturbance of the system, and compensate for the total disturbance in real time during the controller design process.

[0042] Step 3: Design a fixed-time dynamic surface controller to compensate for the total disturbance of the system in real time. By designing a feedback control law, the angular position of each joint of the robotic arm system can quickly and accurately track the given target trajectory signal. Furthermore, a time-varying gain function with the properties of "large error, small gain; small error, large gain" is designed to dynamically adjust the controller gain in order to obtain transient performance with fast response and small overshoot.

[0043] This invention provides a fixed-time dynamic surface trajectory tracking control method for a pneumatic robotic arm system based on a disturbance observer. A detailed implementation method is given using a three-joint robotic arm as an example, comprising the following steps:

[0044] Step 1: Establish the dynamic model of the robotic arm

[0045] like Figure 1 The diagram shown is a simplified schematic of a pneumatic three-joint robotic arm. The total kinetic and potential energy of the robotic arm system are analyzed using the Lagrange formula. Combining this with the mechanical model of a pneumatic artificial muscle, the control torques of each joint of the robotic arm are solved using the rotational law, resulting in the following dynamic model of the robotic arm system:

[0046]

[0047] In the formula, θ = [θ1, θ2, θ3] T Here, θ1, θ2, and θ3 represent the deflection angles of joint one, joint two, and joint three, respectively, and M(θ) is the inertia matrix of the robotic arm. This is the matrix of centrifugal and Coriolis forces of the robotic arm, G(θ) is the gravity vector, and τ is the centrifugal and Coriolis force matrices of the robotic arm. d It is an unknown perturbation vector, τ=[τ1,τ2,τ3] T This is the control torque vector, where τ1, τ2, and τ3 represent the control torques of joint one, joint two, and joint three, respectively. Define x1 = θ. x = [x1, x2] T If it is a state vector, then the state-space expression of the robotic arm system is as follows:

[0048]

[0049] In the formula, f(x) = -M(x1) -1 (C(x1,x2)+G(x1)), g(x)=M(x1) -1 b, d = M(x1) -1 τ d y is the system output, d is the total disturbance of the robotic arm system, and b is the equivalent cross-sectional coefficient of the pneumatic artificial muscle.

[0050] Step 2: Design a finite-time perturbation observer

[0051] like Figure 2 The block diagram illustrates the control principle of a robotic arm system based on a fixed-time dynamic surface control technique using a disturbance observer. Considering the total disturbance *d* in the robotic arm system, a disturbance observer is designed to estimate the total disturbance in real time and compensate for it during the design of the fixed-time dynamic surface controller. The designed finite-time disturbance observer is represented as follows:

[0052]

[0053] In the formula, z is the estimated value of the system state x2. This is the estimated value of the total disturbance d. The auxiliary variables σ = x² - z, m ∈ [1 / 2, 1), n ​​= 2m - 1, k1 > 0, k2 > 0 are the observer gain parameters, sig(σ). m =|σ| m sign(σ) defines the observation error of the perturbation observer. The estimated error vector is By designing appropriate observer gains k1 and k2, the designed perturbation observer can be made asymptotically stable in finite time, that is, the observation error of the perturbation observer can converge to the region near the origin in finite time.

[0054] Step 3: Design a fixed-time dynamic surface controller

[0055] This step involves designing a dynamic surface controller for the robotic arm system. Based on the disturbance observer's estimation of the total disturbance, a corresponding fixed-time dynamic surface controller is designed to compensate for the total disturbance in real time. Simultaneously, a time-varying gain function is introduced into the controller design to dynamically adjust the controller's gain, thereby improving the robotic arm system's response speed and trajectory tracking control accuracy. The design process of the fixed-time dynamic surface controller is as follows:

[0056] ① Define the first-order dynamic error surface: s1 = x1 - x d In the formula x d It is the target trajectory

[0057] Design a fixed-time virtual control law:

[0058] In the formula, p and q are positive odd numbers satisfying p>q, φ 11 and φ 12 It is a time-varying gain, designed as follows:

[0059]

[0060] In the formula, α1, β1, ∈ 11 ,∈ 12 Since the parameters are positive, to avoid the "term explosion" problem and achieve fixed-time stability, the following fixed-time filter is designed.

[0061]

[0062] In the formula, z2 represents the virtual control law. The estimated value is T, where T is the time constant of the fixed-time filter.

[0063] ② Define the second-order dynamic error surface: s2 = x2 - z2, and design the actual control law for the robotic arm system.

[0064] In the formula φ 21 and φ 22 It is a time-varying gain, designed as follows:

[0065]

[0066] In the formula, α2, β2, ∈ 21 ,∈ 22 These are positive parameters. According to the Lyapunov stability criterion and the fixed-time stability theorem, by designing appropriate parameters α1, β1, α2, and β2, the controller can achieve asymptotic stability over a fixed time. That is, the trajectory tracking error s1 of the robotic arm system can converge to the neighborhood of the origin within a fixed time, and the convergence time is independent of the initial state information of the system.

[0067] The above description is merely a preferred embodiment of this application and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of disclosure in this application is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the above-described concept. For example, the above features may be formed by substituting the above features with (but not limited to) technical features with similar functions disclosed in this application.

Claims

1. A fixed-time dynamic surface trajectory tracking control method for a pneumatic robotic arm system, characterized in that, Includes the following steps: Step 1: Establish a dynamic model of the robotic arm based on the Lagrange formula method of energy analysis; Step 2: To address the issues of unmodeled dynamics, load variations, time-varying parameters, and coupling and frictional disturbances between joints in the robotic arm system, a finite-time disturbance observer is designed to observe the total disturbance of the system, and the total disturbance is compensated for in real time during the controller design process. Step 3: Design a fixed-time dynamic surface controller to compensate for the total disturbance of the system in real time. A feedback control law is designed to ensure that the angular positions of each joint of the robotic arm system can track the given target trajectory signal. A time-varying gain function is designed to dynamically adjust the controller gain. Specifically, this includes: The design process of a fixed-time dynamic surface controller is as follows: ① Define the first-order dynamic error surface: In the formula It is the target trajectory Design a fixed-time virtual control law: In the formula and Positive odd numbers satisfy , and It is a time-varying gain, designed as follows: ， In the formula , , , Since the parameters are positive, to avoid the term explosion problem and achieve fixed-time stability, the following fixed-time filter is designed. In the formula Represents virtual control law The estimated value, It is the time constant of a fixed-time filter; ② Define the second-order dynamic error surface: Design the actual control law of the robotic arm system In the formula and It is a time-varying gain, designed as follows: ， In the formula , , , Since these are positive parameters, based on the Lyapunov stability criterion and the fixed-time stability theorem, appropriate parameters can be designed. , , , This allows the controller to gradually stabilize over a fixed time, thus reducing the trajectory tracking error of the robotic arm system. It can converge to the neighborhood of the origin within a fixed time, and the convergence time is independent of the initial state information of the system.

2. The fixed-time dynamic surface trajectory tracking control method for the pneumatic robotic arm system according to claim 1, characterized in that, Step one specifically includes: Based on the analysis of the total kinetic and potential energy of the robotic arm system using the Lagrange formula, and combined with the mechanical model of the pneumatic artificial muscle, the control torques of each joint of the robotic arm are solved using the rotational law, resulting in the following dynamic model of the robotic arm system: In the formula It is an angle vector. , and These represent the deflection angles of joint one, joint two, and joint three, respectively. It is the inertia matrix of the robotic arm. It is the matrix of centrifugal force and Coriolis force of the robotic arm. It is the gravity vector. It is an unknown perturbation vector. It is the control torque vector. , and These represent the control torques of joint one, joint two, and joint three, respectively; Definition , , It is a state vector. The state-space expression of the robotic arm system is as follows: In the formula , , , It is system output. It is the total disturbance of the robotic arm system. It is the equivalent cross-sectional coefficient of pneumatic artificial muscle.

3. The fixed-time dynamic surface trajectory tracking control method for the pneumatic robotic arm system according to claim 1, characterized in that, Step two specifically includes: Considering the total disturbance in the robotic arm system A disturbance observer is designed to estimate the total disturbance in real time and compensate for it during the design of the fixed-time dynamic surface controller. The finite-time disturbance observer is represented as follows: In the formula System status The estimated value, It is the total disturbance The estimated value, auxiliary variables , , , , It is the observer gain parameter. Define the observation error of the perturbation observer. = The estimated error vector is By designing appropriate observer gain , This allows the designed perturbation observer to be asymptotically stable in finite time, meaning that the observation error of the perturbation observer can converge to the region near the origin in a finite time.

Images