An angle tracking control method for a flexible robotic arm based on input hysteresis

Abstract

The present invention relates to the technical field of flexible manipulator control and discloses an angle tracking control method for a flexible manipulator based on input hysteresis. The method comprises the following steps: S1, establishing a dynamic model of a system; S2, constructing a controller τ(t) based on a hysteresis inverse operator; S3, constructing a Lyapunov function W(t) and analyzing the stability of the flexible manipulator system with input hysteresis; S4, performing numerical simulation on the system using a MATLAB platform; and S5, verifying the stability of the system. An inverse compensation boundary control method is designed for a single-link flexible manipulator system based on Bouc-Wen type input hysteresis, so that the system can achieve vibration suppression and angle tracking while offsetting the influence of actuator input hysteresis.

Description

Technical Field

[0001] The present invention relates to the technical field of flexible robotic arm control, and in particular to an angle tracking control method of a flexible robotic arm based on input hysteresis. Background Art

[0002] Flexible robotic arms are made of flexible structures, which are lighter than rigid structures. They have high flexibility and adaptability, and can improve workspace utilization. They have attracted widespread attention in fields such as aerospace. However, due to the strong coupling of the structure, the structure is prone to deformation, which affects the operation accuracy. Therefore, it is of great significance to adopt appropriate and effective control methods to achieve vibration suppression and angle tracking of flexible robotic arms.

[0003] As a typical infinite-dimensional distributed parameter system, the flexible structure of the flexible manipulator is represented by an infinite number of modes, which makes many existing mature methods for traditional rigid systems unable to be directly applied. The control methods mainly include passive control and active control. Passive control is called passive control, which is achieved by adding damping shock absorbers to the main structure to dissipate energy to achieve the purpose of vibration control. However, passive control is limited, especially in the face of complex working environments and high working precision requirements. Therefore, we choose to use active control to suppress system vibration. Active control is also called active control. The boundary control method is a control method belonging to the category of active control. Compared with the truncated model control and distributed control methods, the boundary control method is easy to design and implement, and can avoid the spillover effect caused by ignoring high-frequency modes.

[0004] Hysteresis, as one of the input nonlinearities of actuators, is unavoidable in practical systems. The Bouc–Wen hysteresis model is commonly used to describe nonlinear hysteresis systems. It was introduced by Bouc and extended by Wen, who demonstrated its versatility by generating various hysteresis patterns. The model can analytically capture a range of hysteresis periodic shapes that match the behavior of various hysteresis systems. Hysteresis can significantly degrade system performance, cause undesirable oscillations, and even lead to system instability. Therefore, a controller is needed in this field that can offset the effects of input hysteresis, thereby improving the accuracy, precision, response speed, and safety of flexible robotic arm control. Summary of the Invention

[0005] The purpose of the present invention is to provide an angle tracking control method for a flexible robotic arm based on input hysteresis, which solves the problems mentioned in the above background technology. Its characteristic is that a reasonable hysteresis compensation controller is designed using an inverse compensation operator to effectively offset the influence of hysteresis, so that the flexible robotic arm system can suppress vibration and achieve angle tracking performance.

[0006] To achieve the above object, the present invention provides the following technical solution: an angle tracking control method for a flexible robotic arm based on input hysteresis, comprising the following steps:

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

[0008] S2, construct a controller τ(t) based on the hysteresis inverse operator;

[0009] S3. Construct the Lyapunov function W(t) and analyze the stability of the flexible manipulator system with input hysteresis.

[0010] S4, numerical simulation of the system using MATLAB platform;

[0011] S5. Verify the stability of the system.

[0012] Preferably, in step S1, for the convenience of expression, some formulas of the system dynamics model are simplified as follows:

[0013] (·)=(·)(y,t), ,

[0014] (·)0=·(0,t),(·) s = (s, t); the kinetic energy of the single-link flexible manipulator system is expressed as:

[0015]

[0016] Where y∈[0,s] represents the spatial variable of the single-link manipulator, t∈[0,∞) represents the time variable, s is the length of the flexible manipulator, m represents the uniform mass per unit length of the flexible manipulator, I represents the moment of inertia of the motor of the flexible manipulator, and the absolute displacement z(y,t) of the manipulator in the XOY coordinate system is defined as z(y,t=p(y,t)+yθ(t), where p(y,t) represents the elastic deformation of the flexible manipulator at position y and time t in the XOY coordinate system, and θ(t) represents the rotation angle position of the motor of the flexible manipulator. The potential energy of the single-link flexible manipulator system is expressed as:

[0017]

[0018] Where T and EI represent the tension and bending stiffness of the flexible manipulator, respectively. The virtual work done by damping and external disturbance is:

[0019]

[0020] Where c is the damping coefficient of the flexible manipulator, δ is the variational sign, and the virtual work of the controller on the flexible manipulator system is:

[0021] δW a (t)=τ(t)δz'(0,t),

[0022] Where τ(t) is the input control signal of the flexible manipulator system, and the total virtual work on the flexible manipulator system can be obtained as:

[0023] δW s (t) = δW a (t)+δW b (t),

[0024] The kinetic energy of the system E k (t), potential energy E P (t), total virtual work δW s Substituting Hamilton principle, the dynamic model of the flexible robotic arm system is as follows:

[0025]

[0026] p(0,t)=p'(0,t)=p″(s,t)=0, (2)

[0027] EIp”'(s,t)=Tp'(s,t), (3)

[0028]

[0029] Preferably, in step S2, the expression of the Bouc-Wen type input hysteresis is expressed as:

[0030] τ(t)=H(v)=μκv+(1-μ)κζ=μ1v+μ2ζ (5)

[0031] Where 1>μ>0, is the stiffness ratio, κ is a parameter related to the nonlinear pseudo-frequency of the system, parameters μ1 and μ2 have the same sign, ζ is an auxiliary variable, its change depends on the input v and its derivative The change in , which is generated by the following nonlinear first-order differential equation:

[0032]

[0033] Where β>|χ|, n≥1. β and χ describe the shape and amplitude of the hysteresis, respectively, and n affects the smoothness of the transition from the initial slope to the asymptotic slope. The definition is as follows:

[0034]

[0035] The controls for the intended design are:

[0036]

[0037] in, ζ1(t0)=0,

[0038] τ(t0)=u(t0)

[0039]

[0040] The boundary control law is:

[0041]

[0042] in, β1,β2,β θ is the gain parameter.

[0043] Preferably, in step S3, the Lyapunov function W(t) of the single-link flexible manipulator system is defined as:

[0044] W(t)=W1(t)+W2(t)+W3(t),

[0045] in,

[0046] represents the energy term;

[0047] represents the error term;

[0048] represents the coupling term;

[0049] Among them, z e (y,t)=p(y,t)+y[θ(t)-θ r ], verify the positive definiteness and boundedness of the Lyapunov function W(t), and then verify The negative definiteness of , it is concluded that the system is asymptotically stable.

[0050] Preferably, in step S4, a single-link flexible manipulator system with Bouc-Wen type input hysteresis is numerically simulated on a MATLAB platform, the simulation results are analyzed and it is determined whether the control effect meets the requirements. If it does not meet the requirements, the gain parameters β1, β2, β θ ; If the requirements are met, then end.

[0051] Preferably, in step S5, verifying the stability of the system includes the following two steps:

[0052] (1) Verify the positive definiteness and boundedness of the Lyapunov function W(t);

[0053] (2) Verification Negative definiteness.

[0054] Preferably, the method for verifying the positive definiteness and boundedness of the Lyapunov function W(t) is as follows: It is easy to obtain,

[0055] W1(t)>0,W2(t)>0, and by deduction we can conclude that |W3(t)|≤αS(t), α1=β2max{m(1+s),ms 2},in

[0056] We can further obtain: W(t)=W1(t)+W2(t)+W3(t)>0; α2S(t)≤W1(t)+W2(t)≤α3S(t), α2=min(m,T,EI,I,β θ ),α3=max(m,T,EI,I,β θ ); further, we can obtain 0≤λ1S(t)≤W(t)≤λ2S(t),λ1=α2-α1,λ2=α3+α1, that is, the positive definiteness and boundedness of the Lyapunov function W(t) are verified.

[0057] Preferably, the verification Negative definiteness:

[0058] The first derivative of W1(t) with respect to time

[0059]

[0060] Substitute equation (1) into equation (a), integrate the second half of equation (a), and then simplify to obtain

[0061]

[0062] Substituting equations (2) and (3) into equation (b) and simplifying them, we obtain:

[0063]

[0064] The first derivative of W2(t) with respect to time

[0065]

[0066] Substituting equation (4) into equation (d) yields:

[0067]

[0068] The first-order derivative of W3(t) with respect to time is:

[0069]

[0070] Substituting equation (1) into (f) and integrating it, we can simplify it to get:

[0071]

[0072] Substituting equations (c), (e), and (g) into W(t)=W1(t)+W2(t)+W3(t) yields:

[0073]

[0074] Among them, δ1 is a positive constant, and the parameters β1, β2, β θ The adjustment is to satisfy the following inequality:

[0075]

[0076]

[0077]

[0078]

[0079] λ3=2min(σ1,σ2,σ3,σ4β2EI)>0 (n)

[0080] Right now The negative definiteness of is verified. From formula (i), by solving the integral and simplifying, we can get: From the error term, we can get: Therefore, the elastic deformation of the system can converge to the origin, and the angle can also be tracked to the desired angle, proving that the system can have better vibration suppression and angle tracking performance based on input hysteresis nonlinearity under the control strategy proposed in the present invention.

[0081] The present invention provides an angle tracking control method for a flexible robotic arm based on input hysteresis. The angle tracking control method for a flexible robotic arm based on input hysteresis has the following beneficial effects:

[0082] (1) This angle tracking control method for a flexible manipulator based on input hysteresis designs an inverse compensation boundary control method for a single-link flexible manipulator system based on Bouc-Wen type input hysteresis, which achieves vibration suppression and angle tracking on the basis of offsetting the influence of actuator input hysteresis;

[0083] (2) The angle tracking control method of the flexible robotic arm based on input hysteresis adopts a boundary control design method. Compared with the existing distributed parameter control method, the control method of the present invention only needs to install controllers and sensors at the boundary position, requires fewer sensors and controllers, has low cost, and has broad prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 Schematic diagram of the flow of the flexible manipulator angle tracking control method based on input hysteresis of the present invention;

[0085] Figure 2 A schematic diagram of a mathematical model of a flexible robotic arm system according to an embodiment of the flexible robotic arm angle tracking control method based on input hysteresis of the present invention;

[0086] Figure 3 is the elastic deformation p(y, t) of the flexible manipulator without control in the present invention;

[0087] Figure 4 is the displacement z(y,t) of the flexible manipulator without control in the present invention;

[0088] Figure 5 is the angular position θ(t) of the flexible manipulator without control in the present invention;

[0089] Figure 6 is the elastic deformation p(y, t) of the flexible manipulator after the control of the present invention is applied;

[0090] Figure 7 is the displacement z(y, t) of the flexible manipulator after the control of the present invention is applied;

[0091] Figure 8 is the angular position θ(t) of the flexible manipulator hub after the present invention applies control (the expected angle is 15°);

[0092] Figure 9 is the angular position θ(t) of the flexible manipulator hub after the control of the present invention is applied (the expected angle is 30°);

[0093] Figure 10 The ideal input of the present invention directly acts on the actual control τ(t) of the hysteresis system;

[0094] Figure 11 is the elastic deformation p(y, t) under the ideal input of the present invention directly acting on the hysteresis system;

[0095] Figure 12 is the angular position θ(t) under the ideal input of the present invention directly acting on the hysteresis system (the desired angle is 15°);

[0096] Figure 13 The actual control τ(t) designed for the present invention;

[0097] Figure 14 The ideal boundary control law u(t) designed for the present invention. DETAILED DESCRIPTION

[0098] like Figure 1-14 As shown, the present invention provides a technical solution: an angle tracking control method of a flexible robotic arm based on input hysteresis, comprising the following steps:

[0099] S1. Establish the system's dynamic model. For the sake of convenience, some formulas of the system's dynamic model are simplified as follows:

[0100] (·)=(·)(y,t), (·)0=·(0,t),(·) s =·(s,t);

[0101] The kinetic energy of the single-link flexible robotic arm system is expressed as:

[0102]

[0103] Where y∈[0,s] represents the spatial variable of the single-link manipulator, t∈[0,∞) represents the time variable, s is the length of the flexible manipulator, m represents the uniform mass per unit length of the flexible manipulator, I represents the moment of inertia of the motor of the flexible manipulator, and the absolute displacement z(y,t) of the manipulator in the XOY coordinate system is defined as z(y,t)=p(y,t)+yθ(t), where p(y,t) represents the elastic deformation of the flexible manipulator at position y and time t in the XOY coordinate system, and θ(t) represents the rotation angle position of the motor of the flexible manipulator. The potential energy of the single-link flexible manipulator system is expressed as:

[0104]

[0105] Where T and EI represent the tension and bending stiffness of the flexible manipulator, respectively. The virtual work done by damping and external disturbance is:

[0106]

[0107] Where c is the damping coefficient of the flexible manipulator, δ is the variational sign, and the virtual work of the controller on the flexible manipulator system is:

[0108] δW a (t)=τ(t)δz'(0,t),

[0109] Where τ(t) is the input control signal of the flexible manipulator system, and the total virtual work on the flexible manipulator system can be obtained as:

[0110] δW s (t) = δW a (t)+δW b (t),

[0111] The kinetic energy of the system Ek (t), potential energy E P (t), total virtual work δW S Substituting Hamilton principle, the dynamic model of the flexible robotic arm system is as follows:

[0112]

[0113] p(0,t)=p'(0,t)=p″(s,t)=0, (2)

[0114] EIp”'(s,t)=Tp'(s,t), (3)

[0115]

[0116] S2. Construct a controller τ(t) based on the hysteresis inverse operator. The expression of Bouc-Wen type input hysteresis is expressed as:

[0117] τ(t)=H(v)=μκv+(1-μ)kζ=μ1v+μ2ζ (5)

[0118] Where 1>μ>0, is the stiffness ratio, κ is a parameter related to the nonlinear pseudo-frequency of the system, parameters μ1 and μ2 have the same sign, ζ is an auxiliary variable, its change depends on the input v and its derivative The change in , which is generated by the following nonlinear first-order differential equation:

[0119]

[0120] Where β>|χ|, n≥1. β and χ describe the shape and amplitude of the hysteresis, respectively, and n affects the smoothness of the transition from the initial slope to the asymptotic slope. The definition is as follows:

[0121]

[0122] The controls for the intended design are:

[0123]

[0124] in, ζ1(t0)=0,

[0125] τ(t0)=u(t0)

[0126]

[0127] The boundary control law is:

[0128]

[0129] in, β1,β2,β θ is the gain parameter;

[0130] S3. Construct the Lyapunov function W(t) and analyze the stability of the flexible manipulator system with input hysteresis. The Lyapunov function W(t) of the single-link flexible manipulator system is defined as:

[0131] W(t)=W1(t)+W2+t)+W3(t),

[0132] in,

[0133] represents the energy term;

[0134] represents the error term;

[0135] represents the coupling term;

[0136] Among them, z e (y,t)=p(y,t)+y[θ(t)-θ r ], verify the positive definiteness and boundedness of the Lyapunov function W(t), and then verify The negative definiteness of , it is concluded that the system is asymptotically stable;

[0137] S4. Use the MATLAB platform to perform numerical simulation on the system. Use the MATLAB platform to perform numerical simulation on the single-link flexible manipulator system with Bouc-Wen type input hysteresis. Analyze the simulation results and judge whether the control effect meets the requirements. If it does not meet the requirements, re-modify the gain parameters β1, β2, β θ ; If the requirements are met, then end;

[0138] S5. Verify the stability of the system. Verifying the stability of the system includes the following two steps:

[0139] (1) Verify the positive definiteness and boundedness of the Lyapunov function W(t) as follows: It is easy to obtain that W1(t)>0, W2(t)>0, and by deduction, it can be concluded that |W3(t)|≤αS(t), α1=β2max{m(1+s),ms 2},in

[0140] We can further obtain: W(t)=W1(t)+W2(t)+W3(t)>0; α2S(t)≤W1(t)+W2(t)≤α3S(t), α2=min(m,T,EI,I,β θ ),α3=max(m,T,EI,I,β θ ); furthermore, we can obtain 0≤λ1S(t)≤W(t)≤λ2S(t),λ1=α2-α1,λ2=α3+α1, that is, the positive definiteness and boundedness of the Lyapunov function W(t) are verified;

[0141] (2) Verification Negative definiteness, W1(t) takes the first-order derivative with respect to time

[0142]

[0143] Substitute equation (1) into equation (a), integrate the second half of equation (a), and then simplify to obtain

[0144]

[0145] Substituting equations (2) and (3) into equation (b) and simplifying them, we obtain:

[0146]

[0147] The first derivative of W2(t) with respect to time

[0148]

[0149] Substituting equation (4) into equation (d) yields:

[0150]

[0151] The first-order derivative of W3(t) with respect to time is:

[0152]

[0153] Substituting equation (1) into (f) and integrating it, we can simplify it to get:

[0154]

[0155] Substituting equations (c), (e), and (g) into W(t)=W1(t)+W2(t)+W3(t) yields:

[0156]

[0157] Among them, δ1 is a positive constant, and the parameters β1, β2, β θ The adjustment is to satisfy the following inequality:

[0158]

[0159]

[0160]

[0161]

[0162] λ3=2min(σ1,σ2,σ3,σ4β2EI)>0 (n)

[0163] Right now The negative definiteness of is verified. From formula (i), by solving the integral simplification, we can get: From the error term, we can get: Therefore, the elastic deformation of the system can converge to the origin, and the angle can also be tracked to the desired angle, proving that the system can have better vibration suppression and angle tracking performance based on input hysteresis nonlinearity under the control strategy proposed in the present invention.

[0164] This input hysteresis-based angle tracking control method for a flexible manipulator analyzes the dynamics of a single-link manipulator system and designs a compensation boundary controller for the flexible manipulator system using the hysteresis inverse operator. The Lyapunov function W(t) is then constructed to analyze the stability of the flexible manipulator under control. The system's motion is numerically simulated using the MATLAB platform. The control system's design parameters are adjusted based on the simulation results to achieve better control. This demonstrates that the proposed control strategy, based on input hysteresis nonlinearity, can achieve superior vibration suppression and angle tracking performance.

Claims

1. An angle tracking control method for a flexible manipulator based on input hysteresis, characterized in that: The following steps are involved: S1. Establish the dynamic model of the system; S2. Construct a controller based on the hysteresis inverse operator ; S3. Construct Lyapunov function and analyze the stability of a flexible robotic arm system with input hysteresis; S4, numerical simulation of the system using MATLAB platform; S5. Verify the stability of the system; In step S1, for the convenience of expression, some formulas of the system dynamics model are simplified as follows: , , , , , , , ； The kinetic energy of the single-link flexible robotic arm system is expressed as: in, represents the spatial variables of the single-link robotic arm, represents the time variable, s is the length of the flexible robotic arm, represents the uniform mass per unit length of the flexible manipulator, Represents the rotational inertia of the flexible robotic arm motor and the absolute displacement of the robotic arm in the XOY coordinates Defined as , It represents the position of the flexible robotic arm in the XOY coordinate system. y , time t The elastic deformation, represents the motor rotation angle position of the flexible robotic arm. The potential energy of the single-link flexible robotic arm system is expressed as: ， Where T and EI represent the tension and bending stiffness of the flexible manipulator, respectively. The virtual work done by damping and external disturbance is: ， in, is the damping coefficient of the flexible manipulator, is the variational symbol, and the virtual work done by the controller on the flexible manipulator system is: ， in, is the input control signal of the flexible manipulator system. The total virtual work performed on the flexible manipulator system can be obtained as: ， The kinetic energy of the system , potential energy Total virtual work Substituting Hamilton principle, the dynamic model of the flexible robotic arm system is as follows: m ,(1) ,(2) ,(3) ，（4） In step S2, the expression of the Bouc-Wen type input hysteresis is expressed as: （5） in , is the stiffness ratio, Is a parameter related to the nonlinear pseudo-frequency of the system, parameter and have the same sign, is an auxiliary variable that changes depending on the input and its derivatives The change in , which is generated by the following nonlinear first-order differential equation: , (6) in, describe the shape and amplitude of the hysteresis, respectively, n affects the smoothness of the transition from the initial slope to the asymptotic slope, The definition is as follows: The controls for the intended design are: (7) in, , , (8) The boundary control law is: （9） in, , is the gain parameter; In step S3, the Lyapunov function of the single-link flexible manipulator system is defined as follows: for: ， in, represents the energy term; , represents the error term; , represents the coupling term; in, ], verify the Lyapunov function The positivity and boundedness of The negative definiteness of , it is concluded that the system is asymptotically stable.

2. The angle tracking control method of a flexible manipulator based on input hysteresis according to claim 1, characterized in that: In step S4, a single-link flexible manipulator system with Bouc-Wen type input hysteresis is numerically simulated on the MATLAB platform, the simulation results are analyzed and it is determined whether the control effect meets the requirements. If it does not meet the requirements, the gain parameters of the hysteresis inverse boundary controller are modified. ; If the requirements are met, then end.

3. The angle tracking control method of a flexible manipulator based on input hysteresis according to claim 1, characterized in that: In step S5, verifying the stability of the system includes the following two steps: (1) Verify the Lyapunov function The positivity and boundedness of (2) Verification Negative definiteness.

4. The angle tracking control method of a flexible manipulator based on input hysteresis according to claim 3, characterized in that: Verify the Lyapunov function The positive definiteness and boundedness of , the method is as follows: It is easy to get, , and by deduction we can get , ,in Further we can get: ; , ), ); further , , , that is, the Lyapunov function The positivity and boundedness of are verified.

5. The angle tracking control method of a flexible manipulator based on input hysteresis according to claim 3, characterized in that: verify Negative definiteness: First derivative with respect to time （10） Substitute equation (1) into equation (10), integrate the second half of equation (10), and then simplify to obtain （11） Substituting equations (2) and (3) into (11) and simplifying them, we obtain: （12） First derivative with respect to time （13） Substituting formula (4) into (13), we obtain: （14） Take the first derivative with respect to time: （15） Substituting equation (1) into (14) and integrating it, we can obtain: （16） Substitute equations (12), (14), and (16) into get: (17) in, is a positive constant, the parameter The adjustment is to satisfy the following inequality: (18) = (19) (20) (21) = , , , (22) Right now The negative positivity is verified.

Images