Fixed-Time Synchronization Control Method and System for a Manipulator System under Actuator Faults

Abstract

The present invention discloses a fixed-time synchronization control method and system for a robotic arm system under actuator faults. The method includes: constructing a fixed-time sliding mode observer according to the state information and control input information of the robotic arm system itself; using the fixed-time sliding mode observer to estimate the total disturbance, where the total disturbance includes the uncertainty of the dynamic model parameters and external disturbances; constructing a sliding mode variable based on a neural network sign function; based on the sliding mode variable, the total disturbance, and the dynamic model of the robotic arm system, considering actuator faults, and designing a fixed-time synchronization controller according to the Lyapunov stability principle, such that both the system error and the derivative of the system error can converge simultaneously, and the upper bound of the settling time is independent of the initial value; verifying the stability of the fixed-time synchronization controller, and if the verification is passed, applying the fixed-time synchronization controller and the sliding mode variable based on the neural network sign function to the robotic arm system.

Description

Technical Field

[0001] The present invention relates to the technical field of manipulator control, and particularly to a fixed-time synchronization control method and system for a manipulator system under actuator faults. Background Art

[0002] At present, the trajectory control problem of manipulator systems has attracted wide attention due to its extensive application in modern industry. In actual engineering systems, robot systems have high requirements for reliability, stability, and safety. Therefore, it is very necessary to handle disturbances, uncertainties, unknown frictions, and even actuator faults. In addition, due to the suddenness and uncertainty of actuator faults, the scale of the faults is difficult to predict, and actuator faults can be regarded as the most critical problem in the system. The lack of fault-tolerant control ability will lead to a serious decline in performance and system instability, resulting in the failure of control tasks. Fortunately, with the unremitting efforts of researchers, several fault-tolerant control schemes have been proposed to mitigate the impact of actuator faults and external disturbances on the performance of manipulator systems.

[0003] Fault-tolerant control systems are divided into two methods: active and passive. For the passive fault-tolerant control method, robust control laws are designed to solve actuator faults, and there is no need to observe and estimate faults or reconstruct the controller. On the contrary, the active fault-tolerant control scheme aims to identify faults and then use the obtained diagnostic information to reconstruct the controller. Compared with passive fault-tolerant control, when actuator faults are accurately estimated, active fault-tolerant control exhibits better performance. Therefore, currently, researchers have studied some active fault-tolerant control schemes for the reconstruction of actuator faults. It should be emphasized that most fault reconstruction methods cannot achieve reconstruction within a fixed time. In actual engineering, the failure to implement fault reconstruction as soon as possible may lead to a decline in system performance and even system instability. Therefore, further research is needed on fault-tolerant control considering dynamic uncertainties and disturbances.

[0004] In addition, in the research of manipulator control systems, the tracking rate is also an important performance index. The emergence of finite-time control meets the requirements for the fast response performance of control systems. Although great progress has been made in fixed-time control theory, the convergence time of each state of the control system cannot be guaranteed to be the same. In some specific operations, the present invention hopes that each state of the system can reach the specified position at the same time, such as the grasping task of a manipulator, the synchronous formation of unmanned aerial vehicles, etc. The fixed-time synchronization control method was proposed to ensure the synchronous convergence of all state elements. However, currently, few studies focus on the fixed-time synchronization control of manipulator control systems with dynamic uncertainties and actuator faults. In addition, when the system state is far from equilibrium, the convergence rate of fixed-time synchronization control also needs to be further improved. Summary of the Invention

[0005] To solve the deficiencies of the prior art, the present invention provides a fixed-time synchronization control method and system for a robotic arm system under actuator faults; the present invention is applicable to an uncertain robotic arm control system with actuator faults and external disturbances. In the face of actuator faults, most fault reconstruction methods cannot achieve rapid reconstruction within a fixed time. In practical engineering, the failure to implement fault reconstruction as soon as possible may lead to a decline in system performance and even system instability. The present invention proposes a fast fixed-time synchronization control method based on observer technology, which can rapidly reconstruct faults within a fixed time. In the face of the problem that the convergence time of each state of the control system cannot be guaranteed to be the same, the present invention proposes a control strategy to enable all system states to synchronously converge within a certain time.

[0006] On the one hand, a fixed-time synchronization control method for a robotic arm system under actuator faults is provided, including:

[0007] Establish a dynamic model of the robotic arm system. The dynamic model takes into account that there are n joints in the robotic arm system, and also takes into account the uncertainties, external disturbances, and actuator faults existing in the robotic arm system;

[0008] Construct a fixed-time sliding mode observer based on the state information and control input information of the robotic arm system itself; use the fixed-time sliding mode observer to estimate the total disturbance; the total disturbance includes the parameter uncertainties of the dynamic model and external disturbances;

[0009] Construct a sliding mode variable based on the neural network sign function; based on the sliding mode variable, the total disturbance, and the dynamic model of the robotic arm system, considering actuator faults, design a fixed-time synchronization controller according to the Lyapunov stability principle, so that both the system error and the derivative of the system error can converge simultaneously, and the upper bound of the stable time is independent of the initial value;

[0010] Verify the stability of the fixed-time synchronization controller. If the verification is passed, apply the fixed-time synchronization controller and the sliding mode variable based on the neural network sign function to the robotic arm system.

[0011] On the other hand, a fixed-time synchronization control system for a robotic arm system under actuator faults is provided, including:

[0012] A building module, which is configured to: establish a dynamic model of the robotic arm system. The dynamic model takes into account that there are n joints in the robotic arm system, and also takes into account the uncertainties, external disturbances, and actuator faults existing in the robotic arm system;

[0013] An estimation module, configured to: construct a fixed-time sliding mode observer according to the state information and control input information of the robotic arm system itself; use the fixed-time sliding mode observer to estimate the total disturbance; the total disturbance includes: the dynamic model parameter uncertainty and external disturbance;

[0014] A design module, configured to: construct a sliding mode variable based on a neural network sign function; based on the sliding mode variable, the total disturbance, and the dynamic model of the robotic arm system, considering actuator faults, design a fixed-time synchronization controller according to the Lyapunov stability principle, so that both the system error and the derivative of the system error can converge simultaneously, and the upper bound of the settling time is independent of the initial value;

[0015] A verification module, configured to: verify the stability of the fixed-time synchronization controller, and if the verification passes, apply the fixed-time synchronization controller and the sliding mode variable based on the neural network sign function to the robotic arm system.

[0016] On the other hand, an electronic device is further provided, including:

[0017] A memory for non-temporarily storing computer-readable instructions; and

[0018] A processor for running the computer-readable instructions,

[0019] wherein, when the computer-readable instructions are run by the processor, the method described in the first aspect above is executed.

[0020] On the other hand, a storage medium is further provided, which non-temporarily stores computer-readable instructions, wherein when the non-temporary computer-readable instructions are executed by a computer, the method described in the first aspect is executed.

[0021] On the other hand, a computer program product is further provided, including a computer program, and the computer program is used to implement the method described in the first aspect above when running on one or more processors.

[0022] The above technical solution has the following advantages or beneficial effects:

[0023] (1) In view of the fact that most fault reconstruction methods of the present invention cannot achieve reconstruction within a fixed time. In actual engineering, the failure to implement fault reconstruction as soon as possible may lead to a decline in system performance and even cause the problem of system instability. A fast fixed-time control method based on observer technology is designed, which can quickly reconstruct faults within a fixed time;

[0024] (2) For the majority of current research where the convergence time of each state in the control system cannot be guaranteed to be the same, the present invention proposes a new sliding mode variable based on the neural network sign function. Different from the traditional fixed-time sliding mode variable, the proposed sliding mode variable can not only ensure a fast convergence rate but also achieve synchronous convergence of system states.

[0025] (3) The present invention proposes a fast fixed-time synchronous control method applicable to robotic arm systems with uncertain dynamics, actuator faults, and external disturbances, where all system states converge synchronously within a certain time. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] The accompanying drawings forming a part of this specification are used to provide a further understanding of the present invention. The schematic embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention.

[0027] Figure 1 It is a flowchart of the method of the present invention.

[0028] Figure 2(a) shows the trajectory tracking q of the fast fixed-time synchronous control of the present invention.

[0029] Figure 2(b) shows the trajectory tracking of traditional fixed-time control.

[0030] Figure 3(a) shows the tracking error of the fast fixed-time synchronous control of the present invention.

[0031] Figure 3(b) shows the tracking error of traditional fixed-time control.

[0032] Figure 4(a) shows the average error of the fast fixed-time synchronous control of the present invention.

[0033] Figure 4(b) shows the average error of traditional fixed-time control.

[0034] Figure 5(a) shows the control input of the fast fixed-time synchronous control of the present invention.

[0035] Figure 5(b) shows the control input of traditional fixed-time control.

[0036] Figure 6(a) shows the ratio of the errors e1 and e2 of the fast fixed-time synchronous control of the present invention.

[0037] Figure 6(b) shows the ratio of the errors e1 and e2 of traditional fixed-time control.

[0038] Figure 7(a) shows the system state and its observed value of the first joint based on the fast fixed-time synchronous control of the present invention.

[0039] Figure 7(b) shows the system state and its observed value of the second joint based on the fast fixed-time synchronous control of the present invention.

[0040] Figure 8(a) shows the total system disturbance and its estimated value of the first joint based on the fast fixed-time synchronization control of the present invention;

[0041] Figure 8(b) shows the total system disturbance and its estimated value of the second joint based on the fast fixed-time synchronization control of the present invention.

[0042] Figure 9(a) shows the desired position trajectory q d Images of using the control strategies in Article 1, Article 2, and the control strategy of the present invention respectively.

[0043] Figure 9(b) shows the images of the joint position q using the control strategies in Article 1, Article 2, and the control strategy of the present invention respectively.

[0044] Figure 10(a) shows the average error of using the control strategy in Article 1 and the control strategy of the present invention.

[0045] Figure 10(b) shows the average error of using the control strategy in Article 2 and the control strategy of the present invention.

[0046] Figure 11(a) shows the control input of the robotic arm system of Article 1.

[0047] Figure 11(b) shows the control input of the robotic arm system of Article 2.

[0048] Figure 11(c) shows the control input of the robotic arm system of the present invention.

[0049] Figure 12(a) shows the system state and its estimated value of the first joint of the fast fixed-time synchronization control method of the present invention when a fault occurs.

[0050] Figure 12(b) shows the system state and its estimated value of the second joint of the fast fixed-time synchronization control method of the present invention when a fault occurs.

[0051] Figure 13(a) shows the total system disturbance and its estimated value of the first joint based on the fast fixed-time synchronization control of the present invention.

[0052] Figure 13(b) shows the total system disturbance and its estimated value of the second joint based on the fast fixed-time synchronization control of the present invention. Detailed implementation manners

[0053] It should be noted that the following detailed description is exemplary and is intended to provide further illustration of the present invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which the present invention belongs.

[0054] Glossary: Observer: A device or algorithm used in the field of automatic control to estimate the states of a system that cannot be directly measured. Specifically, an observer estimates the unmeasurable state signals of a system based on the measurable output signals of the system (such as sensor data). In practical engineering applications, the state variables of many systems cannot be directly measured, so an observer is needed to estimate these state variables.

[0055] Actuator failure: Refers to the phenomenon in a control system where the actuator fails to execute the control command properly due to various reasons.

[0056] Fixed-time synchronization: Fixed-time synchronization control means that all the errors of the system reach the stable state simultaneously within a fixed time, and this synchronization time does not depend on the initial state of the system.

[0057] Example 1: As Figure 1 shown, this example provides a fixed-time synchronization control method for a robotic arm system under actuator failure, including:

[0058] S101: Establish the dynamic model of the robotic arm system. The dynamic model takes into account that there are n joints in the robotic arm system, and also takes into account the uncertainties, external disturbances, and actuator failures existing in the robotic arm system.

[0059] S102: Construct a fixed-time sliding mode observer based on the state information and control input information of the robotic arm system itself; use the fixed-time sliding mode observer to estimate the total disturbance; the total disturbance includes the uncertainties of the dynamic model parameters and external disturbances.

[0060] S103: Construct a sliding mode variable based on the neural network sign function; based on the sliding mode variable, total disturbance, and dynamic model of the robotic arm system, considering actuator failure, design a fixed-time synchronization controller according to the Lyapunov stability principle, so that both the system error and the derivative of the system error can converge simultaneously, and the upper bound of the stable time is independent of the initial value.

[0061] S104: Verify the stability of the fixed-time synchronization controller. If the verification passes, apply the fixed-time synchronization controller and the sliding mode variable based on the neural network sign function to the robotic arm system.

[0062] Further, in step S101: Establish the dynamic model of the robotic arm system. The dynamic model takes into account that there are n joints in the robotic arm system, and also takes into account the uncertainties, external disturbances, and actuator failures existing in the robotic arm system. Specifically, it includes:

[0063]

[0064] Among them, denote the generalized position, velocity, and acceleration. Among them denotes the inertia matrix, denotes the matrix of Coriolis forces and centrifugal torques, is the action of gravity, is the frictional action, F = diag{f1, f2,..., f n} is the friction coefficient, f dis unknown external disturbances, denotes the applied torque input.

[0065] When an actuator fault occurs, the control torque u is modeled as:

[0066] u = u nom + u f (2)

[0067] where, denotes the commanded torque generated by the designed controller, denotes the vector of unknown actuator faults;

[0068] In addition, due to aging and wear, the parameters of the system cannot be accurately obtained in advance, and at the same time the frictional force and the disturbance f dis also cannot be estimated. Therefore, the dynamic model formula (1) of the robotic arm system is rewritten as:

[0069]

[0070] where, B n (q) = B(q) - ΔB(q), and G n (q) = G(q) - ΔG(q) and is the sum of uncertainty factors.

[0071] Furthermore, in step S102: According to the state information and control input information of the robotic arm system itself, a fixed-time sliding mode observer is constructed; using the fixed-time sliding mode observer, the total disturbance is estimated; the total disturbance includes: the dynamic model parameter uncertainty and external disturbances, specifically including: fixed-time means that starting from any initial condition, the system state will converge to the equilibrium point within a finite time, and this convergence time is consistent and bounded;

[0072] Introduce a state variable vector Then the dynamic model formula (3) is transformed into:

[0073]

[0074] where, τ = u nom and

[0075] Assume that the total disturbance d total (t) is a differentiable unknown signal, and for the time derivative of the total disturbance There exists a constant Satisfying

[0076] Next, a fixed-time sliding mode observer is constructed to estimate the total disturbance d total (t). First, define a new auxiliary variable As: Formula (5); where λ1 is a positive constant.

[0077] Considering the dynamic equation (4), the following results are obtained:

[0078]

[0079] Where Then, regard the total disturbance d total (t) in formula (6) as the extended state And formula (6) can be expressed as:

[0080]

[0081] Where h(t) is the first-order time derivative of d total (t).

[0082] Finally, the fixed-time sliding mode observer is designed as follows:

[0083]

[0084] Where And And Are the observer states of the system state And the disturbance Respectively, and the functions Φ1 and Φ2 are designed as follows:

[0085]

[0086] Where h1, g1, l1 and p1 are four positive odd integers, and satisfy h1 > g1 and l1 < p1, k i > 0 is the observer gain, i = 1, 2,..., 6, and

[0087] Combining formulas (7) and (8), the following dynamic equation of the observer error is obtained:

[0088]

[0089] Considering that the observer (8) proposed for the robotic arm system (3) ensures that the observer error (10) converges to zero within a finite or fixed time, provided that the following are satisfied The observer error can reach the origin after the steady-state time, and the steady-state time is:

[0090]

[0091] where, and (h1 + g1) / 2g1 > 1 and 0 < (l1 + p1) / 2p1 < 1.

[0092] It is inferred that will be reached within a fixed time, and the steady-state time can be estimated as:

[0093]

[0094] From the above analysis, it is concluded that the observer states and converge to zero after a finite / fixed time, and the observer law formula (8) approximates the total disturbance d total (t).

[0095] Furthermore, the step S103: Construct a sliding-mode variable based on the neural network sign function; based on the sliding-mode variable, the total disturbance, and the dynamic model of the robotic arm system, considering actuator faults, design a fixed-time synchronization controller according to the Lyapunov stability principle, such that both the system error and the derivative of the system error can converge simultaneously, and the upper bound of the settling time is independent of the initial value, specifically including:

[0096] S103-1: Construct a fixed-time synchronization stability system theory; S103-2: Based on the fixed-time synchronization stability system theory, construct a fixed-time synchronization control strategy.

[0097] Furthermore, the step S103-1: Construct a fixed-time synchronization stability system theory, including:

[0098] For the following nonlinear system:

[0099]

[0100] where, α0 > 0 and β0 > 0 are two scalars, h0 > 0, g0 > 0, l0 > 0, and p0 > 0 are odd integers and satisfy h0 > g0 and p0 > l0. Then the equilibrium point of the system is fixed-time stable, and the steady-state time is bounded and limited to:

[0101]

[0102] Further, in step S103-2: Based on the fixed-time synchronization stability system theory, a fixed-time synchronization control strategy is constructed, including:

[0103] Define the trajectory tracking error as e = q - q d , where q d is the desired trajectory.

[0104] Introduce the neural network sign function:

[0105] sig n (z) α = ||z|| α sign n (z) (15)

[0106] sig c (z) α = [|z1| α sign c (z1), …, |z n | α sign c (z n )] T (16)

[0107] where α is a positive constant. The neural network sign function sign n and the classical sign function sign c are defined as follows respectively:

[0108]

[0109] where

[0110]

[0111] where i = 1, 2, …, n.

[0112] Based on the introduced neural network sign function formula (15) and the fast fixed-time stable system formula (13), construct a new sliding mode variable s * , which is expressed as:

[0113]

[0114] where α1 and β1 are positive constants, h2, g2, l2 and p2 are positive odd numbers, and satisfy h2 > g2 and l2 < p2.

[0115] Then, the fixed-time synchronization controller is as follows:

[0116] u nom = u nom_1 + u nom_2 (21)

[0117]

[0118] where α2, β2, and κ are positive constants, h3, g3, l3, and p3 are positive odd numbers, satisfying h3 > g3 and l3 < p3, and the variables η1 and η2 satisfy:

[0119]

[0120] Due to the negative powers of η1 and η2 in the controller (23), when e = 0, a singularity problem will occur. To solve the singularity problem, the following modified sliding-mode variable is designed:

[0121]

[0122] where the forms of the variables and their time derivatives are:

[0123]

[0124] and δ > 0 is a very small constant. The variable s * is:

[0125]

[0126] And the constants μ1 and μ2 are designed respectively as:

[0127]

[0128] And the fractional powers ρ1 ∈ (1, 4), l2 / p2 ∈ (0.5, 1), where h2, g2, l2, and p2 are positive odd numbers. It should be emphasized that by designing the modified sliding-mode variable, the singularity problem can be eliminated, and the designed constants μ1 and μ2 ensure the continuity of the variables s c and . Based on the modified sliding-mode variable (26), the controller can be designed as:

[0129] u nom = u nom_1 + u nom_2 (32)

[0130]

[0131] By applying the newly designed modified sliding-mode variable formula (26) and the proposed controller formula (32), all states of the closed-loop system achieve fixed-time synchronous convergence, and the steady-state time is bounded and limited to:

[0132]

[0133] Further, in S104: verifying the stability of the fixed-time synchronization controller specifically includes:

[0134] Lyapunov candidate function V s is:

[0135] V s = s T s (36)

[0136] Derive V s with respect to time and use the controller (32) to obtain:

[0137]

[0138] If the variable s satisfies |s| > 1, then it can be obtained that When |s| ≤ 1 is satisfied, it can be obtained that It can be concluded that all states will reach the surface of s = 0 within a certain time, and the convergence time of s is bounded and limited to:

[0139]

[0140] When s = 0 is reached, it can be obtained that:

[0141]

[0142] Then, select another Lyapunov function as V e = e T e. Derive V e to obtain:

[0143]

[0144] If the variable s satisfies |s| > 1, then it can be obtained that When |s| ≤ 1 is satisfied, it can be obtained that It can be concluded that the states e and will converge to zero within a certain time:

[0145]

[0146] Assume that there are two error state variables e i and e j , and their convergence times are T i and T j , and T j > T i . In the state e i and ej Before reaching the origin, the derivative of

[0147]

[0148] we can obtain

[0149]

[0150] where c ij is a non - zero constant. Since the variable e i reaches equilibrium at time T i this results in:

[0151]

[0152] Based on the above analysis, we get e j (T i ) = 0. However, the convergence time of the variable e j is T j instead of T i , which contradicts the previous assumption. Therefore, all variables of the state e(t) will reach equilibrium synchronously.

[0153] The present invention makes the following three contributions to the research on uncertain manipulator systems: (1) The present invention proposes a fixed - time sliding - mode observer for realizing the reconstruction of actuator faults and system uncertainties, where the observer error converges to the origin after a finite / fixed time. (2) A new sliding - mode variable based on the neural - network sign function is proposed. Different from traditional fixed - time sliding - mode variables, the proposed sliding - mode variable can not only ensure a fast convergence rate but also achieve synchronous convergence of system states. (3) A fast fixed - time synchronous control method is proposed, which is applicable to manipulator systems with uncertain dynamics, actuator faults, and external disturbances, and all system states converge synchronously within a certain time.

[0154] In this example, the computer software MATLAB is used to verify the invented control method. In the following simulation, a two - joint manipulator system is considered. The first joint refers to the first rotating joint of the manipulator, which is the first movable part starting from the base of the manipulator. The second joint refers to the second rotating joint following the first joint and connecting to the end - effector of the manipulator. The steps in the simulation are as follows:

[0155] (1) Selection of some specific physical values of the manipulator system: b1 represents the mass of the first joint, and the value of b1 is 0.5 kg; b2 represents the mass of the second joint, and the value of b2 is 1.5 kg; b 10 represents the standard value of b1, and the value of b 10 is 0.4 kg; b20 Represents the standard value of b2, b 20 The value of is 1.2 kg; g represents the acceleration due to gravity, and the value of g is 9.81 m / s 2 ; r1 represents the length of the first joint, and the value of r1 is 1 m; r2 represents the length of the second joint, and the value of r2 is 0.8 m; j1 represents the inertia of the first joint, and the value of j1 is 5 kg·m 2 , j2 represents the inertia of the second joint; the value of j2 is 5 kg·m 2 .

[0156] (2) Dynamic model of the dual-joint robotic arm system: The dynamic model of the dual-joint robotic arm system with the position vector q = [q1, q2] T is represented by Equation (1), and the system matrix is as follows:

[0157]

[0158] where a2 = b2r1r2, a5 = (b1 + b2)r1 and a6 = b2r2. b 10 and b 20 represent the standard values of the parameters b1 and b2, respectively.

[0159] The initial value of the system state is: q(0) = [1.2, 1.5] T and

[0160] Friction force vector and external disturbance f dis are as follows:

[0161]

[0162] (3) Compared with traditional fixed-time control: To demonstrate the characteristics of fixed-time synchronization stability, the proposed fixed-time synchronization controller (32) and the sliding-mode variable using the neural network sign function (26) are applied to the robotic arm system. Then, for better comparison, a fixed-time control scheme using the traditional sign function is introduced and applied to the robotic arm system, as shown below. The design of the fixed-time sliding-mode variable for system (3) is as follows:

[0163]

[0164] where i = 1, 2,..., n, and while the control law is designed as follows:

[0165]

[0166] The observer gains of the proposed fixed-time synchronization controller are: k1 = 4, k2 = 5, k3 = 3.5, k4 = 4, k5 = 5, k6 = 2, λ1 = 0.5, λ2 = 2, h1 / g1 = 13 / 9, l1 / p1 = 7 / 13. Meanwhile, the control gains are selected as: α1 = 1, β1 = 1, α2 = 1, β2 = 1, h2 / g2 = 13 / 7, h3 / g3 = 13 / 7, l2 / p2 = 7 / 13, l3 / p3 = 7 / 13, δ = 0.001.

[0167] The selected desired trajectory is:

[0168]

[0169] Assume that the actuator works properly without bias torque, i.e., d i = 1 and where i = 1, 2.

[0170] In the following simulations, the comparison results between the proposed fast fixed-time synchronization control scheme and the traditional fixed-time control scheme are provided, as shown in Figs. 2(a) and 2(b).

[0171] Figs. 2(a) and 2(b), Figs. 3(a) and 3(b) show the trajectory tracking and tracking errors of the two control schemes, indicating that both the proposed fast fixed-time synchronization control scheme and the traditional fixed-time control scheme provide good tracking performance. Figs. 3(a) and 3(b) show the trajectory tracking errors e1 and e2 under the proposed fast fixed-time synchronization control scheme and the traditional fixed-time control scheme. In Fig. 3(a), it can be clearly seen that under the fast fixed-time synchronization control scheme of the present invention, the tracking errors e1 and e2 synchronously reach the origin at t = 2.5 s. While in Fig. 3(b), it can be seen that the errors e1 and e2 converge to zero at different time instants. Specifically, e1 reaches equilibrium at t = 2.5 s, while e2 converges to zero at t = 2.6 s. To make the comparison more obvious, Figs. 4(a) and 4(b) show the average error (e1(t) + e2(t)) / 2 when using two different control strategies. It can be clearly seen from Figs. 4(a) and 4(b) that when using the fast fixed-time synchronization control method of the present invention, the average error converges to zero faster. Therefore, the performance of the fast fixed-time synchronization control method of the present invention is better.

[0172] Figures 5(a) and 5(b) show the control curves under the fast fixed-time synchronization control scheme and the traditional fixed-time control scheme of the present invention. Figures 6(a) and 6(b) show the error ratio e1(t) / e2(t) under the fast fixed-time control scheme and the fixed-time control scheme of the present invention. It can be observed from Figure 6(a) that the proposed fast fixed-time control method ensures that this ratio remains unchanged before the state reaches the origin. For the proposed fast fixed-time control method, this verifies the conclusion proposed by the present invention. In Figure 6(b), it can be clearly seen that there are many jitters in the image, indicating that the performance of the proposed fast fixed-time synchronization control method is better. Figures 7(a) and 7(b) show the state and its observed value Figures 8(a) and 8(b) show the total system uncertainty (i.e.) and its observed value.

[0173] Comparison of the fast fixed-time control scheme of the present invention with the active fault-tolerant control in Article 1 and the nonsingular fast terminal sliding-mode control in Article 2:

[0174] Article 1 is "An Intelligent Actuator Fault Reconstruction Scheme for Robotic Manipulators" published by Bing Xiao et al. in IEEE Transactions on Cybernetics (vol. 48, no. 2, pp. 639-647); Article 2 is "Study of Nonsingular Fast Terminal Sliding-Mode Fault-Tolerant Control" published by Sendren Sheng-Dong Xu et al. in IEEE Transactions on Industrial Electronics (vol. 62, no. 5, pp. 3906-3913). The control schemes of Article 1 and Article 2 are as follows:

[0175] Control scheme of Article 1:

[0176]

[0177] where K>0, κ1>0, κ2>0, k d >0, k p >0 are parameters to be designed, ρ = δ1 / δ2, and δ1 and δ2 are positive odd numbers satisfying δ1 < δ2.

[0178] Control scheme of Article 2:

[0179]

[0180] where α > 0, β > 0, p and q are positive odd numbers, satisfying 1 < p / q < 2, and γ > p / q.

[0181] In the simulation test, the control gains of the proposed fast fixed-time synchronization control are the same as those in the above text. The control parameters of the scheme in Article 1 are K = 10, κ1 = 2, κ2 = 2, k p = 750, k d = 150 and ρ = 13 / 17. The control gains of the scheme in Article 2 are selected as α1 = 8, β1 = 6, γ = 9 / 5, p / q = 17 / 15, η = 10 and ν = 2. To simulate the influence of system faults, the actuator bias fault is considered. For the dual-link manipulator system, the bias torque is expressed as u f = [u f1 , u f2 T . For the first actuator, the bias fault is:

[0182]

[0183] For the second actuator, it is assumed that the fault is u f2 = -0.8 N·m, meaning there is a bias fault of -0.8 N·m. The selected desired trajectory is:

[0184]

[0185] The comparison results of the proposed fast fixed-time synchronization control, the control method in Article 1 and the control method in Article 2 are shown in Figs. 9(a) and 9(b). Figs. 9(a) and 9(b) show the trajectory tracking performance of the manipulator under uncertain dynamics, actuator faults and external disturbances, respectively, for the proposed fast fixed-time synchronization control, the control method in Article 1 and the control method in Article 2. It can be observed from these figures that the proposed fast fixed-time synchronization control, the control method in Article 1 and the control method in Article 2 can all successfully track the desired trajectory, which means that these three methods can compensate or estimate the total disturbance. However, by comparing the three algorithms, it can be found that the proposed fast fixed-time synchronization control method provides a faster tracking rate than the control method in Article 1 and the control method in Article 2.

[0186] ​For better comparison, Figs. 10(a) and 10(b) show the average tracking errors of the proposed fast fixed-time synchronization control, the control method of Article 1, and the control method of Article 2. From the results of Figs. 9(a) and 9(b), it can be seen that the fast fixed-time synchronization control scheme provides higher tracking accuracy and reaches the steady state faster than the control methods of Article 1 and Article 2. However, the control methods of Article 1 and Article 2 will exhibit lower tracking accuracy after a fault occurs. Therefore, the proposed scheme provides better control performance compared with the control methods of Article 1 and Article 2.

[0187] Figures 11(a) to 11(c) show the control inputs of the proposed fast fixed-time synchronization control, the control method of Article 1, and the control method of Article 2. From Figures 11(a) to 11(c) it can be seen that the fast fixed-time synchronization control scheme provides continuous control inputs, while the control methods of Article 1 and Article 2 exhibit chattering phenomena and large control torques. Figs. 12(a) and 12(b) show the states and their observations under the proposed fast fixed-time synchronization control scheme when a fault occurs. Figs. 13(a) and 13(b) show the total system disturbances (including uncertain dynamics, actuator faults, and external disturbances) and their observations under the proposed fast fixed-time synchronization control scheme when a fault occurs suddenly. The estimation process will be completed in a short time when a fault occurs suddenly. Figs. 12(a) and 12(b), Figs. 13(a) and 13(b) show that the proposed fixed-time sliding mode observer guarantees good estimation performance. Therefore, high-precision estimation can be obtained under the control method proposed in the present invention.

[0188] Embodiment 2: This embodiment provides a fixed-time synchronization control system for a robotic arm system under actuator faults, including: a construction module configured to construct a dynamic model of the robotic arm system. The dynamic model takes into account that there are n joints in the robotic arm system, as well as the uncertainties, external disturbances, and actuator faults existing in the robotic arm system; an estimation module configured to construct a fixed-time sliding mode observer based on the state information and control input information of the robotic arm system itself; and use the fixed-time sliding mode observer to estimate the total disturbance. The total disturbance includes the parameter uncertainties of the dynamic model and external disturbances; a design module configured to construct a sliding mode variable based on a neural network sign function; based on the sliding mode variable, the total disturbance, and the dynamic model of the robotic arm system, considering actuator faults, and according to the Lyapunov stability principle, design a fixed-time synchronization controller such that both the system error and the derivative of the system error can converge simultaneously, and the upper bound of the settling time is independent of the initial value; a verification module configured to verify the stability of the fixed-time synchronization controller. If the verification is passed, the fixed-time synchronization controller and the sliding mode variable based on the neural network sign function are applied to the robotic arm system.

[0189] It should be noted here that the above construction module, estimation module, design module, and verification module correspond to steps S101 to S104 in Embodiment 1. The examples and application scenarios implemented by the above modules and the corresponding steps are the same, but are not limited to the content disclosed in Embodiment 1 above. It should be noted that the above modules, as part of the system, can be executed in a computer system such as a set of computer-executable instructions.

[0190] In the above embodiments, the descriptions of each embodiment have their own emphases. For parts not detailed in a certain embodiment, reference can be made to the relevant descriptions of other embodiments.

[0191] The proposed system can be implemented in other ways. For example, the system embodiments described above are merely illustrative. For example, the above module division is only a logical function division. In actual implementation, there can be other division methods. For example, multiple modules can be combined or integrated into another system, or some features can be ignored or not executed.

[0192] Embodiment 3

[0193] This embodiment also provides an electronic device, including: one or more processors, one or more memories, and one or more computer programs; wherein, the processor is connected to the memory, and the above one or more computer programs are stored in the memory. When the electronic device runs, the processor executes the one or more computer programs stored in the memory so that the electronic device executes the method described in Embodiment 1 above.

[0194] It should be understood that in this embodiment, the processor may be a central processing unit (CPU), and the processor may also be other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), off-the-shelf programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor may be a microprocessor or any conventional processor, etc.

[0195] The memory may include a read-only memory and a random access memory, and provide instructions and data to the processor. A part of the memory may also include a non-volatile random access memory. For example, the memory may also store information about the device type.

[0196] In the implementation process, each step of the above method may be completed by the integrated logic circuit in the hardware of the processor or the instructions in the form of software.

[0197] The method in Embodiment 1 may be directly embodied as being executed and completed by the hardware processor, or executed and completed by the combination of the hardware and software modules in the processor. The software module may be located in a mature storage medium in the art such as a random access memory, a flash memory, a read-only memory, a programmable read-only memory, or an electrically erasable programmable memory, a register, etc. This storage medium is located in the memory, and the processor reads the information in the memory and combines its hardware to complete the steps of the above method. To avoid repetition, it will not be described in detail here.

[0198] Those of ordinary skill in the art can realize that the units and algorithm steps of each example described in combination with this embodiment can be implemented by electronic hardware or the combination of computer software and electronic hardware. Whether these functions are executed in a hardware or software manner depends on the specific application and design constraints of the technical solution. Professional technicians can use different methods to implement the described functions for each specific application, but such implementation should not be considered to exceed the scope of the present invention.

[0199] Embodiment 4

[0200] This embodiment also provides a computer-readable storage medium for storing computer instructions. When the computer instructions are executed by the processor, the method described in Embodiment 1 is completed.

[0201] The above are only the preferred embodiments of the present invention and are not used to limit the present invention. For those skilled in the art, the present invention may have various changes and modifications. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. Fixed-time synchronization control method for a robotic arm system under actuator faults, characterized by comprising: Establish a dynamic model of the robotic arm system, which, considering there are n joints in the robotic arm system, also takes into account the uncertainties, external disturbances, and actuator faults existing in the robotic arm system; Construct a fixed-time sliding mode observer based on the state information and control input information of the robotic arm system itself; Use the fixed-time sliding mode observer to estimate the total disturbance; The total disturbance includes the uncertainties of the dynamic model parameters and external disturbances; Construct a sliding mode variable based on the neural network sign function; based on the sliding mode variable, total disturbance, and dynamic model of the robotic arm system, considering actuator faults, design a fixed-time synchronization controller according to the Lyapunov stability principle, such that both the system error and the derivative of the system error can converge simultaneously, and the upper bound of the stabilization time is independent of the initial value; Verify the stability of the fixed-time synchronization controller. If the verification passes, then apply the fixed-time synchronization controller and the sliding mode variable based on the neural network sign function to the robotic arm system.

2. The fixed-time synchronization control method for a robotic arm system under actuator faults as described in claim 1, characterized in that, Establish a dynamic model of the robotic arm system, which, considering there are n joints in the robotic arm system, also takes into account the uncertainties, external disturbances, and actuator faults existing in the robotic arm system. Specifically, it includes: wherein, represents the generalized position, velocity and acceleration; wherein represents the inertia matrix, represents the matrix of Coriolis forces and centrifugal torques, is the action of gravity, is the friction action, F = diag{f1, f2,..., f n} is the friction coefficient, f dis is the unknown external disturbance, represents the applied torque input; When actuator faults occur, the control torque u is modeled as: u = u nom + u f (2) wherein, represents the commanded torque generated by the designed controller, represents the vector of unknown actuator faults; The dynamic model formula (1) of the robotic arm system is rewritten as: Among them, B n (q) = B(q) - ΔB(q), and G n (q) = G(q) - ΔG(q) and is the sum of uncertainty factors.

3. The fixed-time synchronization control method for the robotic arm system under actuator failure as described in claim 2, characterized in that, Construct a fixed-time sliding mode observer based on the state information and control input information of the robotic arm system itself; Use the fixed-time sliding mode observer to estimate the total disturbance; The total disturbance includes the uncertainties of the dynamic model parameters and external disturbances. Specifically, fixed time means that starting from any initial condition, the system state will converge to the equilibrium point within a finite time, and this convergence time is consistent and bounded; Introduce a state variable vector Then the dynamic model formula (3) is transformed into: Among them, τ = u nom and Assume the total disturbance d total (t) is a differentiable unknown signal, and for the time derivative of the total disturbance There exists a constant Satisfying Next, a fixed-time sliding mode observer is constructed to estimate the total disturbance d total (t); First, a new auxiliary variable is defined as: Equation (5); where λ1 is a positive constant; Considering the dynamic equation (4), the following results are obtained: Among them Then, regard the total disturbance d total (t) in formula (6) as the extended state And formula (6) is expressed as: where h(t) is the first time derivative of d total (t); Finally, the fixed-time sliding mode observer is designed as follows: where and and are the observer states of the system state and the disturbance respectively. The functions Φ1 and Φ2 are designed as follows: where h1, g1, l1, and p1 are four positive odd integers, satisfying h1 > g1 and l1 < p1, and k i > 0 is the observer gain, i = 1, 2,..., 6, and Combining formulas (7) and (8), the following dynamic equation of the observer error is obtained: Considering that the observer (8) proposed by the robotic arm system formula (3) ensures that the observer error (10) converges to zero within a finite or fixed time, provided that observer error reaches the origin after the steady-state time, and the steady-state time is: Among them, and (h1 + g1) / 2g1 > 1 and 0 < (l1 + p1) / 2p1 < 1; Inferred will be reached within a fixed time, and the steady-state time is estimated to be: Based on the above analysis, the conclusion is drawn that the observer states and converge to zero after a finite / fixed time, and the observer law formula (8) approximates the total disturbance d total (t) within a fixed time.

4. The fixed-time synchronization control method for a robotic arm system under actuator faults according to claim 3, wherein Construct a sliding mode variable based on the neural network sign function; based on the sliding mode variable, total disturbance, and dynamic model of the robotic arm system, considering actuator faults, design a fixed-time synchronization controller according to the Lyapunov stability principle, such that both the system error and the derivative of the system error can converge simultaneously, and the upper bound of the stabilization time is independent of the initial value. Specifically, it includes: Construct a fixed-time synchronization stability system theory; Based on the fixed-time synchronization stability system theory, construct a fixed-time synchronization control strategy.

5. The fixed-time synchronization control method for a robotic arm system under actuator faults as described in claim 4, characterized in that, Construct a fixed-time synchronization stability system theory, including: For the following nonlinear system: where, α0>0 and β0>0 are two scalars, h0>0, g0>0, l0>0 and p0>0 are odd integers and satisfy h0>g0 and p0>l0, then the equilibrium point of the system is fixed-time stable, and the steady-state time is bounded and restricted to:

6. The fixed-time synchronization control method for the robotic arm system under actuator failure according to claim 5, characterized in that, Based on the fixed-time synchronization stability system theory, construct a fixed-time synchronization control strategy, including: Define the trajectory tracking error as \(e = q_d - q\), where \(q_d\) is the desired trajectory; d where \(q_d\) d is the desired trajectory; Introduce the neural network sign function: sig n (z) α = ||z|| α sign n (z) (15) sig c (z) α =[|z1| α sign c (z1),…,|z n | α sign c (z n )] T (16) where α is a positive constant; the neural network sign function sign n and the classical sign function sign c are defined as follows respectively: Among them where i = 1, 2,..., n; Based on the introduced neural network sign function formula (15) and the fast fixed-time stable system formula (13), construct a new sliding mode variable s * , which is expressed as: where α1 and β1 are positive constants, h2, g2, l2, and p2 are positive odd numbers, and satisfy h2 > g2 and l2 < p2; Then, the fixed-time synchronization controller is as follows: u nom = u nom_1 + u nom_2 (21) where α2, β2, and κ are positive constants, h3, g3, l3, and p3 are positive odd numbers, and satisfy h3 > g3 and l3 < p3, and the variables η1 and η2 satisfy: Since there are negative powers of η1 and η2 in the controller formula (23), when e = 0, a singularity problem occurs; to solve the singularity problem, the following modified sliding mode variable is designed: Among them, the forms of the variables and their time derivatives are as follows: and δ > 0 is a very small constant; the variable s * is defined as: And the constants μ1 and μ2 are designed respectively as: And the fractional powers ρ1 ∈ (1, 4), l2 / p2 ∈ (0.5, 1), where h2, g2, l2 and p2 are positive odd numbers; Based on the modified sliding mode variable (26), the controller is designed as: u nom = u nom_1 + u nom_2 (32) By applying the newly designed modified sliding mode variable formula (26) and the proposed controller formula (32), all states of the closed-loop system achieve fixed-time synchronous convergence, and the steady-state time is bounded and limited to:

7. The fixed-time synchronization control method for the robotic arm system under actuator failure as described in claim 6, characterized in that, Verify the stability of the fixed-time synchronous controller, specifically including: Lyapunov candidate function V s is as follows: V s = s T s (36) For V s Taking the derivative with respect to time and using the controller (32), we obtain: If the variable s satisfies |s| > 1, then we get When |s| ≤ 1 is satisfied, we get It is concluded that all states will reach the surface of s = 0 within a certain time, and the convergence time of s is bounded and limited to: When s = 0 is reached, it is obtained that: Then, select another Lyapunov function as V e = e T e; Take the derivative of V e to obtain: If the variable s satisfies |s| > 1, then obtain When |s| ≤ 1 is satisfied, obtain Derive state e and will converge to zero within a certain time: Suppose there exist two error state variables e i and e j , whose convergence times are T i and T j , respectively, and T j >T i ; before the states e i and e j reach the origin, 's derivative satisfies: It is obtained that: where c ij is a non-zero constant, and since the variable e i reaches equilibrium at time T i this results in: Based on the above analysis, e is obtained j (T i ) = 0; However, the convergence time of variable e j is T j instead of T i , which contradicts the previous assumption; Therefore, all variables of state e(t) will reach equilibrium synchronously.

8. A fixed-time synchronous control system for a robotic arm system under actuator faults, characterized by comprising: A building module, which is configured to: build a dynamic model of the robotic arm system. The dynamic model takes into account that there are n joints in the robotic arm system, and also takes into account the uncertainties, external disturbances and actuator faults existing in the robotic arm system; An estimation module, which is configured to: construct a fixed-time sliding mode observer according to the state information and control input information of the robotic arm system itself; Adopt a fixed-time sliding mode observer to estimate the total disturbance; The total disturbance includes: the dynamic model parameter uncertainties and external disturbances; A design module, which is configured to: construct a sliding mode variable based on a neural network sign function; based on the sliding mode variable, the total disturbance and the dynamic model of the robotic arm system, considering actuator faults, design a fixed-time synchronous controller according to the Lyapunov stability principle, so that both the system error and the derivative of the system error can converge simultaneously, and the upper bound of the stable time is independent of the initial value; A verification module, which is configured to: verify the stability of the fixed-time synchronous controller. If the verification is passed, apply the fixed-time synchronous controller and the sliding mode variable based on the neural network sign function to the robotic arm system.

9. An electronic device, characterized by comprising: A memory for non-temporarily storing computer-readable instructions; And A processor for running the computer-readable instructions, Wherein, when the computer-readable instructions are run by the processor, the method described in any one of claims 1-7 above is executed.

10. A storage medium, characterized in that, Non-temporarily storing computer-readable instructions, wherein when the non-temporarily computer-readable instructions are executed by a computer, the method described in any one of claims 1-7 is executed.

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