Boundary control methods, devices, and systems for flexible robotic arms with input gaps

Abstract

This invention relates to the field of automatic control technology, and particularly to a boundary control method, device, and system for a flexible robotic arm with input gap. The invention establishes a dynamic model of the system, constructs an adaptive inverse controller, builds a Lyapunov function, and analyzes the stability of the flexible robotic arm system with input gap. MATLAB is used to perform digital simulation of the system, and the simulation results are analyzed to determine whether the control effect meets the requirements. This boundary control method, device, and system for a flexible robotic arm with input gap utilizes a smooth inverse compensation operator to design a reasonable adaptive compensation controller, effectively mitigating the influence of the gap, enabling the flexible robotic arm system to reach a stable bounded state and achieve tracking performance. Furthermore, it avoids input chatter problems caused by using non-smooth inverse compensation operators, extending the service life of the robotic arm.

Description

Technical Field

[0001] This invention relates to the field of automatic control technology, specifically to a boundary control method, apparatus, and system for a flexible robotic arm with input gaps. Background Technology

[0002] Due to their advantages of low energy consumption, portability, and high flexibility, flexible robotic arms are widely used in various fields, including medical devices, industrial manufacturing, and agricultural production. However, the operating environment of flexible robotic arms faces unknown disturbances, inevitably causing deformation and vibration, which often leads to mechanical failures and affects the expected precise positioning. Therefore, adopting appropriate and effective control methods to achieve vibration suppression and trajectory tracking of flexible robotic arms is of great significance.

[0003] A flexible robotic arm is essentially an infinite-dimensional distributed parameter system composed of partial differential equations. If the controller is designed based on a discrete finite-dimensional system established by ordinary differential equations, spillover effects may occur. While distributed control is better at suppressing vibrations than modal control, it requires a large number of sensors and actuators, making it impractical for real-world applications. Boundary control can avoid spillover and requires fewer actuators and sensors during execution, making it considered the most effective control method.

[0004] Backlash, as one of the input nonlinearity problems of actuators, is unavoidable in practical systems. It may be caused by incomplete contact between gears in mechanical actuators and friction between control components. This nonlinearity can significantly degrade system performance, cause unwanted oscillations, and even lead to system instability. Summary of the Invention

[0005] The purpose of this invention is to provide a boundary control method, apparatus, and system for a flexible robotic arm with an input gap, so as to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention provides the following technical solution: a boundary control method for a flexible robotic arm with an input gap, comprising the following steps:

[0007] S1: Establish the dynamic model of the system.

[0008] S2: Construct an adaptive inverse controller τ(t).

[0009] S3: Construct the Lyapunov function χ(t) and analyze the stability of the flexible robotic arm system with input gaps. Verify the positive definiteness of the Lyapunov function χ(t), and conclude that the system is stable in the Lyapunov sense. Then verify... The negative definiteness of the system leads to the conclusion that the system is asymptotically stable.

[0010] S4: Use MATLAB to perform digital simulation of the system. Use MATLAB simulation software to perform digital simulation of the single-link flexible robotic arm system with input gap, analyze the simulation results and determine whether the control effect meets the requirements. If it does not meet the requirements, modify the gain parameters of the adaptive inverse controller. If it meets the requirements, then end the process.

[0011] Preferably, verifying the stability of the system in S3 includes verifying the positive definiteness of the Lyapunov function χ(t) and verifying... The negative qualitative property.

[0012] Preferably, in step S1, a dynamic model of the system is established. For ease of representation, some formulas are simplified as follows:

[0013] and

[0014] Boundary control device for a flexible robotic arm with input clearance, comprising a single-link robotic arm and MATLAB simulation software.

[0015] The boundary control system of a flexible manipulator with input gap includes the dynamic model of the single-link manipulator system under distributed disturbance, the design of the adaptive compensation controller of the flexible manipulator system using the dynamic model and smoothing gap inverse operator, the construction of Lyapunov function, the stability analysis of the flexible manipulator under control action, and the numerical simulation of the system motion state using MATLAB. The design parameters of the control system are adjusted according to the simulation results.

[0016] Compared with the prior art, the beneficial effects of the present invention are:

[0017] 1. The boundary control method, device and system for a flexible robotic arm with input gap is proposed. An adaptive gap compensation control method is designed for a single-link flexible robotic arm. A reasonable adaptive compensation controller is designed using a smooth inverse compensation operator to effectively reduce the influence of the gap, so that the flexible robotic arm system can reach a stable bounded state and achieve tracking performance. It can also avoid the input chatter problem caused by the use of a non-smooth inverse compensation operator and extend the service life of the robotic arm.

[0018] 2. The boundary control method, device and system for the flexible robotic arm with input gap uses the hyperbolic tangent function instead of the sign function to further reduce the problem of controller input chatter.

[0019] 3. The boundary control method, device, and system of the flexible robotic arm with input gap requires fewer sensors and controls, has low cost, and has broad prospects. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 This is a schematic diagram of the operation of the single-link robotic arm of the present invention;

[0022] Figure 2 This is a flowchart illustrating the implementation of the robust adaptive inverse control method for a flexible robotic arm with input gaps according to the present invention.

[0023] Figure 3 The elastic deformation w(s,t) of the uncontrolled flexible robotic arm of the present invention is given.

[0024] Figure 4 γ(s,t) represents the displacement of the uncontrolled flexible robotic arm of the present invention.

[0025] Figure 5 The angular position θ(t) of the uncontrolled flexible robotic arm of the present invention is given.

[0026] Figure 6 Let w(x,t) be the elastic deformation of the flexible robotic arm after the application of control according to the present invention.

[0027] Figure 7 Let γ(s,t) be the displacement of the flexible robotic arm after the application of control according to the present invention.

[0028] Figure 8 θ(t) represents the angular position of the hub of the flexible robotic arm after the application of control according to the present invention;

[0029] Figure 9 e(t) represents the angular position error of the hub of the flexible robotic arm after the application of control according to the present invention;

[0030] Figure 10 The actual control τ(t) of the design of this invention;

[0031] Figure 11 The control v(t) output by the gap inverse compensation operator of the present invention;

[0032] Figure 12 The boundary control law u of the present invention r (t). Detailed Implementation

[0033] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0034] Please see Figure 1-12 This invention provides a technical solution: a boundary control method for a flexible robotic arm with an input gap, comprising the following steps:

[0035] S1: Establish the dynamic model of the system. For ease of representation, some formulas are simplified as follows:

[0036]

[0037] and

[0038] The kinetic energy of a single-link flexible robotic arm system can be expressed as:

[0039]

[0040] Where s∈[0,L] represents the spatial variable of the single-link manipulator, t∈[0,∞) represents the time variable, L is the length of the flexible manipulator, ρ represents the uniform mass per unit length of the flexible manipulator, I represents the unit moment of inertia of the flexible manipulator, and the absolute displacement γ(s,t) of the manipulator in the XOY coordinate system is defined as γ(s,t)=w(s,t)+sθ(t), where w(s,t) represents the elastic deformation of the flexible manipulator at position s and time t in the XOY coordinate system, and θ(t) represents the rotation angle of the manipulator.

[0041] The potential energy of a single-link flexible robotic arm system can be expressed as:

[0042]

[0043] Where T and EI represent the tension and bending stiffness of the flexible robotic arm, respectively.

[0044] The virtual work done by damping and external disturbance is:

[0045]

[0046] Where f(s,t) is the distributed disturbance experienced by the robotic arm, c is the damping coefficient of the flexible robotic arm, and δ is the variational symbol.

[0047] The controller's virtual work on the flexible robotic arm system:

[0048] δWa (t)=τ(t)δγ'(0,t), where τ(t) is the controller.

[0049] The total virtual work done on the flexible robotic arm system can be obtained as follows:

[0050] δW s (t)=δW a (t)+δW b (t).

[0051] The system's kinetic energy E k (t), potential energy E P (t) Total virtual work δW s Substituting Hamilton's principle, the dynamic model of the flexible robotic arm system is obtained as follows:

[0052]

[0053]

[0054]

[0055] EIω”'(L,t)=Tω'(L,t) (3)

[0056]

[0057] S2: Construct an adaptive inverse controller τ(t).

[0058] The expression for the input gap is as follows:

[0059]

[0060] Where m>0, is the slope of the line, and B r >0 and B l <0 are constants, v(t) represents the expected control design, and τ(t) describes the actual input affected by the rebound. _ ), which means that τ(t) remains unchanged.

[0061] The expected design controls are:

[0062]

[0063] in,

[0064] σ is a positive number.

[0065] We define and

[0066] The compensation error can be derived from (5)-(6) as follows:

[0067]

[0068] The compensation error is:

[0069]

[0070] The boundary control law is:

[0071]

[0072] in, g,g1,η,g θ It is a gain parameter. Its value satisfies

[0073] The adaptive law is designed as follows: Where Λ>0.

[0074] S3: Construct the Lyapunov function χ(t) and analyze the stability of the flexible robotic arm system with input gaps. Verify the positive definiteness of the Lyapunov function χ(t), and conclude that the system is stable in the Lyapunov sense. Then verify... The negative definiteness of the system leads to the conclusion that the system is asymptotically stable.

[0075] Define the Lyapunov function χ(t) for a single-link flexible robotic arm system as follows:

[0076] χ(t)=χ1(t)+χ2(t)+χ3(t)+χ4(t)

[0077] in,

[0078]

[0079] This represents the energy term;

[0080] Indicates the error term;

[0081] Indicates coupling terms;

[0082] Indicates auxiliary items;

[0083] The positive definiteness of the Lyapunov function χ(t) can be verified as follows: It is easy to obtain...

[0084] χ1(t)>0, χ2(t)>0, χ4(t)>0, and by derivation and choosing appropriate parameter values, we can obtain...

[0085] |χ3(t)|≤α[χ1(t)+χ2(t)],

[0086]

[0087] Furthermore, we can obtain:

[0088] χ(t)=(1+α)[χ1(t)+χ2(t)+χ3(t)]+χ4(t)>χ1(t)+χ2(t)+χ3(t)]+χ4(t)>(1-α)[χ1(t)+χ2(t)+χ3(t)]+χ4(t)>0.

[0089] That is, the positive definiteness of the Lyapunov function χ(t) is verified.

[0090] verify The negative qualitative property is determined as follows:

[0091] Take the first derivative of χ1(t) with respect to time:

[0092]

[0093] Substituting equation (1) into equation (Ka), integrating the second half of equation (Ka), and then simplifying, we obtain...

[0094]

[0095] Substituting equations (2) and (3) into (Kb) and simplifying, we get...

[0096]

[0097] Take the first derivative of χ2(t) with respect to time.

[0098]

[0099] Substituting equation (4) into (Kd) yields:

[0100]

[0101] Take the first derivative of χ3(t) with respect to time:

[0102]

[0103] Substituting equation (1) into (Kf) and integrating, we can simplify to obtain:

[0104]

[0105] Take the first derivative of χ₄(t) with respect to time:

[0106]

[0107] Substituting equations (c)(e)(f) into χ(t)=χ1(t)+χ2(t)+χ3(t)+χ4(t), we get:

[0108]

[0109] in,

[0110] If φ1, φ2 > 0, then according to the adaptive projection mapping operator, there exists a constant ψ. m satisfaction

[0111] Right now The negative qualitative nature of the expression was verified.

[0112] From equation (Ki), by solving the integral and simplifying, we can obtain:

[0113]

[0114] Similarly, from the error term, we can obtain:

[0115]

[0116] in Let (Ki) be the coefficient of the minterm. This indicates that the elastic deformation and angular error of the system can converge to a very small range, demonstrating that the system has good vibration suppression and angle tracking performance.

[0117] S4: Use MATLAB to perform digital simulation of the system. Use MATLAB simulation software to perform digital simulation of the single-link flexible robotic arm system with input gap, analyze the simulation results and determine whether the control effect meets the requirements. If it does not meet the requirements, modify the gain parameters of the adaptive inverse controller. If it meets the requirements, then end the process.

[0118] S5: View and analyze the simulation effect of the flexible robotic arm.

[0119] Figure 3 and Figure 4 These are schematic diagrams of the vibration simulation and displacement variables of the system without control. As shown in the figures, when no control is applied, the robotic arm exhibits vibration (elastic deformation) and displacement variables at various points. Figure 6 and Figure 7 This is a simulation diagram of vibration and displacement when the control of this invention is enhanced. (See diagram below.) Figure 6 and Figure 7 As shown, the present invention is used to suppress the vibration of a flexible beam robotic arm with an input gap. After t = 1.5s, the amplitude of the flexible robotic arm tends to be relatively stable, and the amplitude is near the equilibrium position. Figure 5 and Figure 8This is a simulation diagram of the angle of the control system of this invention, with and without the addition of the present invention. For example... Figure 9 As shown, the present invention is used to suppress the angle tracking of a flexible robotic arm with an input gap, and the angle tracking performance of the flexible robotic arm is good.

[0120] S6: Determine whether the gain parameter needs to be adjusted based on the simulation results from step S5.

[0121] Based on the simulation results, determine whether the vibration, displacement, and angle of the flexible robotic arm meet the requirements. If they do not, readjust the gain parameters g,g1,η,g of the boundary controller. θ If the requirements are met, the process ends.

[0122] Boundary control devices for flexible robotic arms with input gaps, including single-link robotic arms and MATLAB simulation software.

[0123] For the boundary control system of a flexible manipulator with input gap, the dynamic model of the single-link manipulator system under distributed disturbance is included. The dynamic model and smoothing gap inverse operator are used to design an adaptive compensation controller for the flexible manipulator system. Then, a Lyapunov function is constructed to perform stability analysis on the flexible manipulator under control action. The system motion state is numerically simulated using MATLAB. The design parameters of the control system are adjusted according to the simulation results.

[0124] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A boundary control method for a flexible robotic arm with input gap, characterized in that, Includes the following steps: S1: Establish the dynamic model of the flexible robotic arm system; S2: Construct an adaptive inverse controller; specifically: The expression for the actual input gap is as follows: （5） in , is the slope of the straight line. and They are constants, Indicates the expected design control, The actual input gap of the flexible robotic arm system affected by rebound is described. represent No change; The expected design controls are: （6） in, , It is a positive number; definition and ; The compensation error can be derived from (5)-(6) as follows: （7） The boundary control law is: ； in, , It is a gain parameter. Its value satisfies ; The adaptive law is designed as follows: (9), among which, ; S3: Constructing Lyapunov functions The stability of the flexible robotic arm system with input gaps is analyzed, and the Lyapunov function is verified. The positive definiteness of the system is used to deduce that it is stable in the Lyapunov sense, and then verification is performed. The negative definiteness leads to the conclusion that the system is asymptotically stable; S4: Use MATLAB to perform digital simulation of the system. Use MATLAB simulation software to perform digital simulation of the single-link flexible robotic arm system with input gap, analyze the simulation results and determine whether the control effect meets the requirements. If it does not meet the requirements, modify the gain parameters of the adaptive inverse controller. If it meets the requirements, then end the process.

2. The boundary control method for a flexible robotic arm with an input gap according to claim 1, characterized in that: The stability verification of the system in S3 includes verifying the Lyapunov function. Positive definiteness and verification The negative qualitative property.

3. A boundary control device for a flexible robotic arm with an input gap, characterized in that: It includes a single-link robotic arm and MATLAB simulation software, wherein the single-link robotic arm is controlled by the boundary control method of the flexible robotic arm with input clearance as described in claim 1 or 2.

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