Fixed-time stabilization control method for single degree of freedom manipulator with full state constraints

Abstract

The application relates to the technical field of robots and discloses a fixed-time stable control method for a single-degree-of-freedom mechanical arm with full-state constraints, which comprises the following steps: modeling a single-degree-of-freedom mechanical arm to obtain a state equation; defining an error system and designing a first virtual control law alpha i,1 ; designing a relative threshold event trigger mechanism; designing a second virtual law alpha i,2 and an adaptive law; and based on a Matlab experiment platform, performing simulation experiments on the algorithm. The fixed-time stable control method provided by the application can realize the convergence of tracking errors within a fixed time under different initial states of a mechanical arm system, the convergence time is irrelevant to the initial state of the system, a higher convergence speed and better tracking precision can be obtained by selecting appropriate parameters, and the relative threshold event trigger mechanism arranged in the controller can reduce the update frequency of a control input signal under the premise of ensuring control precision.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of robots, in particular to a fixed-time stable control method for a single-degree-of-freedom robot arm with full state constraints. BACKGROUND

[0002] In recent years, as a frontier technology, robot technology has been widely studied in the industrial and academic fields. It is of great practical significance to study how to improve the control accuracy and convergence speed of the robot arm. In recent years, although many finite time control methods have appeared for robot systems, they can achieve fast and stable systems, but still cannot stabilize in a fixed time. It is of great significance to further study how to stabilize the system in a fixed time.

[0003] The existing finite time control method can realize the finite time stability of the system, but the convergence time of the system is related to the initial state of the system. The present application is based on the fixed-time stable control theory to design the controller, which can realize the fast finite time stability of the system, and the convergence time is independent of the initial state of the system.

[0004] Due to the working conditions or the inherent characteristics of the mechanical structure, the robot system often has full state constraint problems, such as: the joints of the robot arm can only move within a certain joint angle. The present application uses barrier Lyapunov function to design the controller, which can make the system stable in a fixed time, and meet the full state constraint requirements of the system.

[0005] Most of the existing technical solutions do not consider that the communication resources of the system are limited, and a large amount of communication resources is needed to maintain the stability of the system. The present application designs an event-triggered mechanism, which can reduce the update frequency of the control signal, and can alleviate the communication pressure of the system to a certain extent. SUMMARY

[0006] The present application aims to provide a fixed-time stable control method for a single-degree-of-freedom robot arm with full state constraints, to solve the problems raised in the background art.

[0007] To achieve the above purpose, the present application provides the following technical solutions:

[0008] A fixed-time stable control method for a single-degree-of-freedom robot arm with full state constraints, comprising the following steps:

[0009] S01, modeling the single-degree-of-freedom robot arm to obtain the state equation.

[0010] S02, defining the error system and designing the first virtual control law i,1 .

[0011] S03, designing a relative threshold event-triggered mechanism.

[0012] S04, design the second virtual control law i,2 and adaptive control law

[0013] S05, based on Matlab experimental platform, the algorithm is simulated.

[0014] Preferably, the S01 detailed steps are as follows:

[0015] The mathematical model of single degree of freedom manipulator is set as follows:

[0016]

[0017] Where x1, x2 represent joint angle and joint angle acceleration; The derivative of x1 is represented as The derivative of x2 is represented as y represents the system output, and u is the input torque. It is the uncertain part of the system model, J is the moment of inertia, B is the friction damping coefficient, m is the mass of the connecting rod, g is the acceleration of gravity, and l is the length of the connecting rod.

[0018] Preferably, the S02 detailed steps are as follows:

[0019] The error system is defined as follows:

[0020]

[0021] Where z1 is the tracking error, z2 is the virtual control error, α1 is the first virtual control law, y r The desired output signal is

[0022] The first virtual control law α1 is designed as follows:

[0023]

[0024] Where b1, c1, κ1 are all design parameters greater than zero, and p ∈ (0, 1).

[0025] Preferably, the S03 detailed steps are as follows:

[0026] In order to save communication resources, an event-triggered mechanism is designed. The definition of event-triggered control signal ω(t) is as follows:

[0027]

[0028] Only when the pre-designed trigger condition |m(t)| ≥ ρ|u(t)| + m1 is true, the control input signal u(t) will be updated, and its expression is as follows:

[0029]

[0030] where σ, m1, p, m2 are design parameters, and satisfy σ > 0, 0 < p < 1, m1 > 0, R represents a real number; a2 is a second virtual control law to be designed; represents a conversion error, and κ2 is a design parameter greater than zero; inf{·} represents the lower limit; k is an integer, t k is the kth triggering time, t k+1 is the (k+1)th triggering time; m(t) represents a measurement error, and m(t) = ω(t) - u(t).

[0031] Preferably, the S04 detailed steps are as follows:

[0032] Since there is an uncertain part in the system model, the fuzzy logic system is used to process the uncertain part in the system model, and the virtual control law a2 and the adaptive law Further, the step S04 specifically includes:

[0033] S041, a fuzzy logic system is used to process the uncertain combination part in the system model X represents an input vector, and X = [x1, x2] T ; an unknown positive parameter is introduced, where ||·|| represents a two-norm; and the parameter θ can be estimated by , i.e. is the estimated value of the parameter θ, then the final estimation error can be defined as Therefore, the expression of the approximation processing is as follows:

[0034]

[0035] where, is an ideal unknown weight vector, and represents ; Φ(X) is a base function vector, and Φ(X) = [Φ1(X), Φ2(X),..., Φ q (X)] T ; q is the number of fuzzy rules, and q > 1; μ(X) represents an approximation error, and satisfies is a constant greater than zero.

[0036] S042, according to the backstepping design method and the barrier Lyapunov function, the second virtual control law a2 and the adaptive law are designed by using the second error variable z2 and the above fuzzy logic system

[0037]

[0038]

[0039] where ||·|| denotes the two-norm, b2, c2, κ2, a, ε, μ, ζ are all positive design parameters, and p∈(0, 1).

[0040] Preferably, the S05 detailed steps are as follows:

[0041] To verify the effectiveness of the proposed method, the algorithm is simulated based on the Matlab experimental platform. The control target of the simulation experiment is to make the joint angular velocity x1 track the set trajectory signal y r = sin(2t). The parameters of the related system are as follows: moment of inertia J = 0.8, damping coefficient B = 1, and mgl = 10. It is assumed that the system has full state constraints: x1≤1.5, x2≤3.

[0042] The set values of the related parameters are as follows: b1 = c1 = 4, b2 = c2 = 4, κ1 = 0.5, κ2 = 2, μ = 1, a = 0.1, σ = 0.01, ρ = 0.1, m1 = 0.1, The initial estimated value is: The initial state of the system is: x1(0) = 0.2, x2(0) = 0. The sampling period of the simulation is 0.01s.

[0043] The fuzzy basis function is designed as follows:

[0044]

[0045] where i and j represent the labels, X represents an input vector, x i represents an element in the input vector X, represents a membership function, which is designed as follows:

[0046]

[0047] Compared with the prior art, the beneficial effects of the present application are:

[0048] 1. The fixed-time stable control method for the single-degree-of-freedom manipulator with full state constraints can realize the convergence of tracking error within a fixed time under different initial states of the manipulator system, and the convergence time is independent of the initial state of the system, and by selecting appropriate parameters, a high convergence speed and good tracking accuracy can be obtained.

[0049] 2. The fixed-time stable control method of the single degree of freedom manipulator with full state constraints, in order to reduce the communication burden between the manipulators, a relative threshold event trigger mechanism is designed in the controller, which can reduce the update frequency of the control input signal under the premise of ensuring the control accuracy.

[0050] 3. The fixed-time stable control method of the single degree of freedom manipulator with full state constraints, the controller is designed based on barrier Lyapunov function, which can make the system state always meet the full state constraint requirements of the system. BRIEF DESCRIPTION OF DRAWINGS

[0051] Figure 1 It is a schematic diagram of the principle of the fixed-time stable control method of the application;

[0052] Figure 2 It is a schematic diagram of the reference signal and system output of the fixed-time stable control method of the application;

[0053] Figure 3 It is a schematic diagram of the tracking error of different system initial states of the fixed-time stable control method of the application;

[0054] Figure 4 It is a schematic diagram of the system state x2 structure of the fixed-time stable control method of the application;

[0055] Figure 5 It is a schematic diagram of the control signal of the fixed-time stable control method of the application;

[0056] Figure 6 It is a schematic diagram of the trigger event time interval of the fixed-time stable control method of the application. DETAILED DESCRIPTION

[0057] Embodiment:

[0058] Reference Figures 1-4 The fixed-time stable control method of the single degree of freedom manipulator with full state constraints provided by the application comprises the following steps:

[0059] S01, modeling the single degree of freedom manipulator to obtain the state equation.

[0060] S02, define the error system, and design the first virtual control law i,1 .

[0061] S03, design a relative threshold event trigger mechanism.

[0062] S04, design the second virtual law i,2 And adaptive law

[0063] S05, based on the Matlab experimental platform, the algorithm is simulated.

[0064] S01 detailed steps as follows:

[0065] The mathematical model of single degree of freedom robot arm is set as follows:

[0066]

[0067] Where x1, x2 represents the joint angle and joint angle acceleration; The derivative of x1 is represented by x1; The derivative of x2 is represented by x2; y represents the system output, and u is the input torque; It is the uncertain part of the system model, J is the moment of inertia, B is the friction damping coefficient, m is the mass of the connecting rod, g is the acceleration of gravity, and l is the length of the connecting rod.

[0068] S02 detailed steps as follows:

[0069] The error system is defined as follows:

[0070]

[0071] Where z1 is the tracking error, z2 is the virtual control error, α1 is the first virtual control law, y r The desired output signal.

[0072] The first virtual control law α1 is designed as follows:

[0073]

[0074] Where b1, c1, κ1 are all greater than zero design parameters, p∈(0,1).

[0075] S03 detailed steps as follows:

[0076] In order to save communication resources, an event triggered mechanism is designed. The definition of event triggered control signal ω(t) is as follows:

[0077]

[0078] Only when the trigger condition |m(t)|≥ρ|u(t)|+m1 is true, the control input signal u(t) will be updated, and its expression is as follows:

[0079]

[0080] Where σ, m1, ρ, m2 are design parameters, and satisfy σ>0, 0<ρ<1, m1>0, R represents a real number; α2 is the second virtual control law to be designed; This indicates the conversion error, and κ2 is a design parameter greater than zero; inf{·} denotes the infimum, k is an integer, and t k For the k-th trigger time, t k+1 This is the (k+1)th trigger time; m(t) represents the measurement error, and m(t) = ω(t) - u(t).

[0081] The detailed steps for S04 are as follows:

[0082] Since the system model contains uncertainties, fuzzy logic is used to handle these uncertainties, and virtual control law α2 and adaptive law are designed based on the virtual control error z2. Furthermore, step S04 specifically includes:

[0083] S041. Use fuzzy logic systems to handle the uncertain combination components in the system model. X represents the input vector, and X = [x1, x2] T Introduce an unknown positive parameter. Where ||·|| represents the L2 norm; and the parameter θ can be obtained through... Estimate, i.e. Let θ be an estimator of the parameter, then the final estimation error can be defined as... Therefore, approximation processing The expression is as follows:

[0084]

[0085] in, Let be an ideal unknown weight vector, and express The transpose of ; Φ(X) is a basis function vector, and Φ(X) = [Φ1(X), Φ2(X), ..., Φ q (X)] T ; q is the number of fuzzy rules, and q>1; μ(X) represents the approximation error, and satisfies It is a constant greater than zero.

[0086] S042. Based on the backstepping design method and the barrier Lyapunov function, design the second virtual control law α2 and the adaptive law using the second error variable z2 and the above-mentioned fuzzy logic system.

[0087]

[0088]

[0089] where ||·|| denotes the two-norm, b2, c2, κ2, a, ε, μ, ζ are all positive design parameters, and p ∈ (0, 1).

[0090] The detailed steps of S05 are as follows:

[0091] To verify the effectiveness of the proposed method, the algorithm is simulated based on the Matlab experimental platform. The control target of the simulation experiment is to make the joint angular velocity x1 track the set trajectory signal y r = sin(2t). The parameters of the related system are as follows: moment of inertia J = 0.8, damping coefficient B = 1, and mgl = 10. It is assumed that the system has full state constraints: x1 ≤ 1.5 and x2 ≤ 3.

[0092] The set values of the related parameters are as follows: b1 = c1 = 4, b2 = c2 = 4, κ1 = 0.5, κ2 = 2, μ = 1, a = 0.1, σ = 0.01, ρ = 0.1, m1 = 0.1, The initial estimated value is as follows: The initial state of the system is x1(0) = 0.2 and x2(0) = 0. The sampling period of the simulation is 0.01 s.

[0093] The fuzzy basis function is designed as follows:

[0094]

[0095] where i and j represent the labels, X represents the input vector, x i represents an element in the input vector X, represents the membership function, which is designed as follows:

[0096]

[0097] The simulation results are shown in Figures 2-6 , and all signals are bounded. Figure 2 The expected output signal and the system output signal are depicted, and it can be seen from the figure that the system has good tracking accuracy and meets the state constraint requirements. Figure 3 The tracking error under different initial states of the system is expressed, and the error enters the error band of 5% after about 0.1 s, which indicates that the convergence speed is very fast. Figure 4 is the curve of the system state x2, and obviously, x2 meets the state constraint requirements. The control input signal of the system is shown in Figure 5 , the event-triggered control input w(t) is continuous and smooth, and the value of the control input u(t) exists in the w(t) curve. Figure 6The time interval between trigger events is expressed, and the minimum trigger time interval is 0.01s, obviously, there is no Zeno phenomenon. The event trigger statistics are shown in Table 1, and the total trigger times of the traditional time trigger control and the event trigger control proposed in the application are 2000 and 521 respectively. Therefore, the event trigger rate is 26.05%, which means that 73.95% of the communication resources are saved.

[0098]

[0099] Table 1 Trigger times comparison

[0100] The fixed time stable control method provided by the above-mentioned embodiment of the application can realize the convergence of tracking error within a fixed time under different initial states of the system, and the convergence time is irrelevant to the initial state of the system, and by selecting appropriate parameters, a higher convergence speed and better tracking accuracy can be obtained, and the relative threshold event trigger mechanism is arranged in the controller, so that the update frequency of the control input signal can be reduced under the premise of ensuring the control accuracy.

[0101] The above only describes the exemplary embodiments of the application and is not intended to limit the application, and any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the application shall be included in the protection scope of the application.

Claims

1. A fixed-time stabilization control method for a single degree of freedom manipulator with full state constraints, characterized in that: The method comprises the following steps: S01, modeling a single degree of freedom manipulator to obtain a state equation; S02, define error system and design first virtual control law a i,1 ; S03, designing a relative threshold event trigger mechanism; S04. Designing a second virtual law a i,2 and adaptive laws S05, performing simulation experiments on the algorithm based on a Matlab experimental platform; The S01 detailed steps are as follows: The mathematical model of the single degree of freedom manipulator is set as follows: where x1, x2 represent joint angle and joint angle acceleration; represents the derivative of x1; represents the derivative of x2; y represents system output, and u is input torque; is the system model uncertainty part, J is the moment of inertia, B is the friction damping coefficient, m is the link mass, g is the acceleration of gravity, and l is the link length; The S02 detailed steps are as follows: The error system is defined as follows: where z1 is the tracking error, z2 is the virtual control error, a1 is the first virtual control law, y r is the desired output signal, The first virtual control law α1 is designed as follows: Wherein, b1, c1, κ1 are all design parameters greater than zero p∈(0,1).

2. The fixed-time stable control method of single-input single-output manipulator with full state constraints according to claim 1, characterized in that: The S03 detailed steps are as follows: An event trigger mechanism is introduced to save communication resources, and the definition of the event trigger control signal ω(t) is as follows: Only when the pre-designed trigger condition |m(t)|≥ρ|u(t)|+m1 is true, the control input signal u(t) will be updated, and its expression is as follows: where σ, m1, p, m2 are design parameters, and satisfy σ > 0, 0 < p < 1, m1 > 0, R represents a real number; a2 is a second virtual control law to be designed; represents a conversion error, and k2 is a design parameter greater than zero; inf{·} represents the lower limit, k is an integer, t k is the kth triggering time, t k+1 is the k+1th triggering time; m(t) represents a measurement error, and m(t) = ω(t) - u(t).

3. The fixed-time stable control method of single-input single-output manipulator with full state constraints according to claim 1, characterized in that: The step S04 comprises the following steps: Because of the uncertain part of the system model, the fuzzy logic system is used to deal with the uncertain part of the system model, and the virtual control law a2 and the adaptive law are designed according to the virtual control error z2 The step S04 specifically comprises the following steps: S041、Adopting a fuzzy logic system to handle the uncertain combination part in system model X represents the input vector, and X = [x1, x2] T ; introduce an unknown positive parameter where ||·|| represents the two-norm; and the parameter θ can be estimated by , i.e. is the estimate of the parameter θ, then the final estimation error can be defined as Therefore, the expression of the approximation process is as follows: wherein is an ideal unknown weight vector, and denotes the transpose; Φ(X) is a basis function vector, and Φ(X) = [Φ1(X), Φ2(X),..., Φ q (X)] T ; q is the number of fuzzy rules, and q > 1; μ(X) represents an approximation error, and satisfies is a constant greater than zero; S042、According to the backstepping design method and barrier Lyapunov function, a second virtual control law α2 and an adaptive law are designed by using a second error variable z2 and the above fuzzy logic system Wherein, ||·|| represents the two norm, b2, c2, κ2, a, ε, μ, ζ are all design parameters greater than zero p∈(0,1).

4. The fixed-time stable control method of single-input single-output manipulator with full state constraints according to claim 1, characterized in that: The step S05 comprises the following steps: Based on Matlab experimental platform, the algorithm is simulated to verify the effectiveness of the proposed method: the control objective of the simulation experiment is to make the joint angular velocity x1 track the set trajectory signal y r = sin(2t), the parameters of the related system are as follows: moment of inertia J = 0.8, damping coefficient B = 1, mgl = 10, it is assumed that the system exists full state constraint: x1≤1.5, x2≤3; The set values of the relevant parameters are as follows: b1 = c1 = 4, b2 = c2 = 4, κ1 = 0.5, κ2 = 2, μ = 1, a = 0.1, σ = 0.01, p = 0.1, m1 = 0.1, The initial estimated value is: The initial state of the system is x1(0) = 0.2, x2(0) = 0, and the simulation sampling period is 0.01 s; The fuzzy basis function is designed as follows: where i and j denote indices, X denotes the input vector, x i denotes an element in the input vector X, denotes a membership function, which is designed as follows:

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