A method for dynamic surface trajectory tracking control of a robotic arm system with preset performance.

Abstract

This application provides a preset performance dynamic surface trajectory tracking control method for a robotic arm system. The method includes the following steps: Step 1, analyzing the total kinetic and potential energy of the robotic arm system using the Lagrange formula, and solving for the control torque using a pneumatic artificial muscle mechanics model and rotational laws to establish a dynamic model of the robotic arm system; Step 2, designing a fixed-time disturbance observer to monitor the total disturbance of the system in real time, addressing the strong nonlinearity and joint coupling and frictional disturbance problems of the robotic arm system; Step 3, designing a preset performance fixed-time dynamic surface controller to compensate for the total disturbance of the system in real time, and designing a control law to enable the angular positions of each joint of the robotic arm system to quickly track the desired target trajectory signal within the allowable error range, thereby achieving the preset transient and steady-state performance constraints within a fixed time and ensuring that all signals of the multi-joint pneumatic robotic arm closed-loop system are bounded.

Description

Technical Field

[0001] This application relates to the field of robotic arm system technology, and in particular to a method for dynamic surface trajectory tracking control of a robotic arm system with preset performance. Background Technology

[0002] As a type of industrial robot, robotic arms have undergone continuous development along with advancements in robotics technology and are playing an increasingly important role in many application fields. Compared to traditional rigid robotic arms, multi-joint robotic arms driven by pneumatic artificial muscles occupy an important position in the industrial robot field due to their advantages such as lightweight, 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, such as unmodeled dynamics, load variations, time-varying parameters, air compressibility, and coupling and friction between joints, posing challenges to the precise trajectory tracking control of the robotic arm.

[0003] Therefore, numerous control strategies have been proposed for robotic arm control systems, such as PID control, sliding mode variable structure control, adaptive control, backstepping control, and active disturbance rejection control algorithms based on disturbance estimation and compensation. Among these, active disturbance rejection control algorithms based on disturbance estimation and compensation are popular due to their ability to significantly improve robotic arm control performance. Furthermore, most existing robotic arm system control methods only consider and guarantee the steady-state performance of the system, with less consideration given to transient performance such as overshoot and convergence speed. Improving the convergence speed of the robotic arm's trajectory tracking error helps increase its working efficiency, but a faster convergence speed may lead to a larger overshoot, and excessive overshoot may cause the system to go out of control in practical applications. Therefore, considering both the steady-state performance and the transient performance of the robotic arm system, such as error convergence speed and overshoot, is a valuable area of ​​research. Summary of the Invention

[0004] To overcome the shortcomings of existing methods, this invention provides a preset performance dynamic surface trajectory tracking control method for a pneumatic robotic arm system based on a fixed-time observer. First, the robotic arm system is modeled, and the control torques of each joint are calculated using the equivalent section coefficient of pneumatic artificial muscles and rotational laws, establishing a dynamic model of the robotic arm system. The strong nonlinearity of the robotic arm system, as well as coupling and friction between joints, are considered as total disturbances and introduced into the dynamic model. A fixed-time disturbance observer is used to estimate the total disturbances in the system, and a preset performance fixed-time dynamic surface controller is designed to compensate for the total disturbances in the pneumatic robotic arm system in real time. The controller designed using this method ensures that the robotic arm trajectory tracking error converges to a preset steady-state constraint range within a fixed time, exhibiting good steady-state and transient performance.

[0005] This application provides a method for dynamic surface trajectory tracking and control of a robotic arm system with preset performance, the method comprising the following steps:

[0006] Step 1: Analyze the total kinetic and potential energy of the robotic arm system using the Lagrange formula method, and solve for the control torque by combining the pneumatic artificial muscle mechanics model and rotation law to establish the dynamic model of the robotic arm system.

[0007] Step 2: To address the strong nonlinearity, joint coupling, and frictional disturbance issues in the robotic arm system, a fixed-time disturbance observer is designed to monitor the total disturbance of the system in real time.

[0008] Step 3: Design a preset performance fixed-time dynamic surface controller to compensate for the total disturbance of the system in real time. By designing a control law, the angular position of each joint of the robotic arm system can quickly track the desired target trajectory signal within the allowable error range, thereby achieving the preset transient and steady-state performance constraints within a fixed time and ensuring that all signals of the multi-joint pneumatic robotic arm closed-loop system are bounded.

[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 process of the fixed-time dynamic surface controller with preset performance. The design of the fixed-time disturbance observer is expressed as follows:

[0017]

[0018] In the formula, z is the estimated value of state x2. This is an estimate of the total disturbance d. The auxiliary variable is defined as e1 = x2 - z. k1>0, k2>0 are the observer gain parameters. ρ∈(0,1) is an adjustable parameter. According to the homogeneity theorem and the Lyapunov method, it can be proved that the designed perturbation observer is asymptotically stable in a fixed time, that is, the observation error of the perturbation observer can converge to the neighborhood near the origin in a fixed 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] Fixed-time preset performance function design is In the formula, h0, h ∞ These are the initial and final values ​​of the preset performance function, respectively, and they satisfy h0 > h. ∞ >0, using error transformation techniques, the equivalent transformation error of the performance constraint is derived as:

[0023] Fixed-time virtual control law is In the formula, p and q are positive odd numbers satisfying p>q, c 11 and c 12 Since it is a constant gain, a fixed-time filter is introduced to obtain the estimated signal and differential signal of the virtual control law in order to avoid the "term inflation" problem.

[0024]

[0025] In the formula, z2 represents the virtual control law. The estimated value, κ is the time constant of the fixed-time filter;

[0026] ② Define the second-order dynamic error surface: s2 = x2 - z2, and design the actual control law for the robotic arm system.

[0027] In the formula c 21 c 22 It is a constant gain; based on the Lyapunov stability criterion and the fixed-time stability theorem, by designing a suitable parameter c 11 c 12 c 21 c 22 κ enables the controller to gradually stabilize over a fixed time, meaning that the trajectory tracking error of the robotic arm system can reach the preset steady-state envelope within a fixed time.

[0028] This invention focuses on a pneumatic multi-joint robotic arm system, establishing a dynamic model of the robotic arm. The control objective is to ensure that the angular positions of each joint of the robotic arm can quickly track the desired target trajectory signal within an allowable error range. Considering the strong nonlinearity and disturbances such as coupling and friction between joints in the robotic arm system, these are treated as total disturbances and introduced into the dynamic model of the robotic arm system. A fixed-time disturbance observer is designed to estimate the total disturbance in the system in real time and compensate for it in the control law design, greatly improving the robustness of the robotic arm trajectory tracking control. In the controller design, a fixed-time preset performance function is introduced to constrain the performance of the robotic arm trajectory tracking error. An error transformation technique is applied to transform the constrained trajectory tracking error into an unconstrained transformed error. Combined with a fixed-time dynamic surface controller, this ensures that the trajectory tracking error of the robotic arm system can converge to the preset steady-state range within a fixed time, thereby enabling the transient and steady-state performance of the robotic arm trajectory tracking control process to simultaneously meet the preset performance requirements. Attached Figure Description

[0029] The accompanying drawings illustrate, by way of example and not limitation, the various embodiments discussed herein.

[0030] Figure 1 This is a simplified structural diagram of the pneumatic multi-joint robotic arm of the present invention;

[0031] Figure 2 This is a schematic diagram of the fixed-time dynamic surface control principle based on a fixed-time disturbance observer, according to the present invention. Detailed Implementation

[0032] 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.

[0033] 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.

[0034] 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.

[0035] The specific principles and steps of a preset performance dynamic surface trajectory tracking control method for a robotic arm system according to an embodiment of this application include:

[0036] Step 1: First, analyze the total kinetic energy and total potential energy of the robotic arm system using the Lagrange formula method. Then, combine the pneumatic artificial muscle mechanics model and rotation law to solve for the control torque and establish the dynamic model of the robotic arm system.

[0037] Step 2: Considering the strong nonlinearity of the robotic arm system and the disturbances such as coupling and friction between joints, design a fixed-time disturbance observer to observe the total disturbance of the system in real time.

[0038] Step 3: Design a preset performance fixed-time dynamic surface controller to compensate for the total disturbance of the system in real time. By designing the control law, the angular position of each joint of the robotic arm system can quickly track the desired target trajectory signal within the allowable error range, thereby achieving the preset transient and steady-state performance constraints within a fixed time and ensuring that all signals of the multi-joint pneumatic robotic arm closed-loop system are bounded.

[0039] This invention provides a method for tracking and controlling a robotic arm system based on a fixed-time perturbation observer with preset performance and dynamic surface trajectory. A detailed implementation method is given using a three-joint robotic arm as an example, comprising the following steps:

[0040] Step 1: Establish the dynamic model of the robotic arm

[0041] like Figure 1The diagram shows the structure of a three-joint robotic arm. Based on the Lagrange formula, the total kinetic and potential energy of the robotic arm system are analyzed. 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:

[0042]

[0043] 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:

[0044]

[0045] 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.

[0046] Step 2: Design a fixed-time perturbation observer

[0047] like Figure 2 The block diagram illustrates the control principle of a robotic arm system based on a fixed-time dynamic surface control technique with preset performance using a fixed-time 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 process of the preset performance fixed-time dynamic surface controller. The designed fixed-time disturbance observer is represented as follows:

[0048]

[0049] In the formula, z is the estimated value of state x2. This is an estimate of the total disturbance d. The auxiliary variable is defined as e1 = x2 - z. k1>0, k2>0 are the observer gain parameters. ρ∈(0,1) is an adjustable parameter. According to the homogeneity theorem and the Lyapunov method, it can be proved that the designed perturbation observer is asymptotically stable in a fixed time, that is, the observation error of the perturbation observer can converge to the neighborhood near the origin in a fixed time.

[0050] Step 3: Design a dynamic surface controller with preset performance and fixed time.

[0051] This step involves designing a pre-defined performance fixed-time dynamic surface controller for the robotic arm system. Based on the disturbance observer's estimation of the total disturbance, a corresponding pre-defined performance dynamic surface controller is designed to compensate for the total disturbance in real time. The design process for the fixed-time dynamic surface controller is as follows:

[0052] ③ Define the first-order dynamic error surface: s1 = x1 - x d In the formula x d It is the target trajectory

[0053] Fixed-time preset performance function design is In the formula, h0, h ∞ These are the initial and final values ​​of the preset performance function, respectively, and they satisfy h0 > h. ∞ >0, using error transformation techniques, the equivalent transformation error of the performance constraint is derived as:

[0054] Fixed-time virtual control law is In the formula, p and q are positive odd numbers satisfying p>q, c 11 and c 12 Since it is a constant gain, a fixed-time filter is introduced to obtain the estimated signal and differential signal of the virtual control law in order to avoid the "term inflation" problem.

[0055]

[0056] In the formula, z2 represents the virtual control law. The estimated value, κ is the time constant of the fixed-time filter.

[0057] ④ Define the second-order dynamic error surface: s2 = x2 - z2, and design the actual control law for the robotic arm system.

[0058]

[0059] In the formula c 21 c 22 It is a constant gain. Based on the Lyapunov stability criterion and the fixed-time stability theorem, by designing a suitable parameter c... 11 c 12 c 21 c 22κ enables the controller to gradually stabilize over a fixed time, meaning that the trajectory tracking error of the robotic arm system can reach the preset steady-state envelope within a fixed time.

[0060] This invention focuses on a pneumatic multi-joint robotic arm system, establishing a dynamic model of the robotic arm. The control objective is to ensure that the angular positions of each joint of the robotic arm can quickly track the desired target trajectory signal within an allowable error range. Considering the strong nonlinearity and disturbances such as coupling and friction between joints in the robotic arm system, these are treated as total disturbances and introduced into the dynamic model of the robotic arm system. A fixed-time disturbance observer is designed to estimate the total disturbance in the system in real time and compensate for it in the control law design, greatly improving the robustness of the robotic arm trajectory tracking control. In the controller design, a fixed-time preset performance function is introduced to constrain the performance of the robotic arm trajectory tracking error. An error transformation technique is applied to transform the constrained trajectory tracking error into an unconstrained transformed error. Combined with a fixed-time dynamic surface controller, this ensures that the trajectory tracking error of the robotic arm system can converge to the preset steady-state range within a fixed time, thereby enabling the transient and steady-state performance of the robotic arm trajectory tracking control process to simultaneously meet the preset performance requirements.

[0061] 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 method for dynamic surface trajectory tracking control of a robotic arm system with preset performance, characterized in that, Includes the following steps: Step 1: Analyze the total kinetic and potential energy of the robotic arm system using the Lagrange formula method, and solve for the control torque by combining the pneumatic artificial muscle mechanics model and rotation law to establish the dynamic model of the robotic arm system. Step 2: To address the strong nonlinearity, joint coupling, and frictional disturbance issues in the robotic arm system, a fixed-time disturbance observer is designed to monitor the total disturbance of the system in real time. Step 3: Design a preset performance fixed-time dynamic surface controller to compensate for the total disturbance of the system in real time. By designing a control law, the angular positions of each joint of the robotic arm system can quickly track the desired target trajectory signal within the allowable error range, thereby achieving the preset transient and steady-state performance constraints within a fixed time and ensuring that all signals of the multi-joint pneumatic robotic arm closed-loop system are bounded. 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 Fixed-time preset performance function design is In the formula These are the initial and final values ​​of the preset performance function, respectively, and satisfy the following conditions: Using error transformation techniques, the transformation error equivalent to the performance constraint is derived as follows: Fixed-time virtual control law is In the formula and Positive odd numbers satisfy , and Since the gain is constant, a fixed-time filter is introduced to obtain the estimated and differential signals of the virtual control law in order to avoid term inflation. 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 , It is a constant gain; based on the Lyapunov stability criterion and the fixed-time stability theorem, by designing appropriate parameters... , , , and This allows the controller to gradually stabilize over a fixed time, meaning that the trajectory tracking error of the robotic arm system can reach the preset steady-state envelope within a fixed time.

2. The method for preset performance dynamic surface trajectory tracking control of a 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 method for preset performance dynamic surface trajectory tracking control of a robotic arm system according to claim 1, characterized in that, Step two specifically includes: 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 process of the fixed-time dynamic surface controller with preset performance. The design of the fixed-time disturbance observer is expressed as follows: In the formula It is a state The estimated value, It is the total disturbance The estimated value, defining auxiliary variables , , , It is the observer gain parameter. , , It is an adjustable parameter. According to the homogeneity theorem and the Lyapunov method, it can be proved that the designed perturbation observer is asymptotically stable in a fixed time, that is, the observation error of the perturbation observer can converge to the neighborhood near the origin in a fixed time.

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