Real-time fault monitoring and self-adaptive fixed time fault-tolerant control method for mechanical arm

Abstract

The invention discloses a real-time fault monitoring and self-adaptive fixed-time fault-tolerant control method for a mechanical arm, is applied to the field of mechanical arms, and aims to solve the problem that most of the current control methods for the mechanical arm are difficult to adapt to the strict requirements on high precision, quick response and high reliability in a complex application scene. And the severity of the fault of the actuator cannot be quantified in real time. According to the method, firstly, a novel dynamic regulation variable is designed for approaching an actuator LOE fault coefficient and compensating the influence of a fault on system operation, and secondly, an adaptive rate is designed for quickly and accurately estimating the unknown nonlinear upper bound of the system within a fixed time. On the basis of the designed dynamic adjustment variable and the self-adaption rate, a self-adaption fixed time fault-tolerant controller is designed in combination with the sliding mode control technology, and real-time fault monitoring and rapid high-precision fault-tolerant control of the mechanical arm can be achieved through the method.

Description

Technical Field

[0001] The present invention belongs to the field of robotic arms, and particularly relates to a robotic arm control technology. Background Art

[0002] As a core equipment for automation and intelligent manufacturing, robotic arms have relatively rich application bases and broad application prospects in various fields. In intelligent manufacturing, robotic arms can achieve high-precision and high-efficiency automated production, and have become a key tool for ensuring product quality and reducing labor costs; in the medical field, robotic arm-assisted surgeries can improve surgical precision, reduce trauma, and promote the development of minimally invasive surgeries; in the field of service robots, robotic arms can provide home services, rehabilitation training, etc., effectively improving the quality of human life; in addition, robotic arms also play important roles in high-risk tasks such as space exploration and underwater operations. Whether it is which application scenario, stability, rapid response, and high-precision control are crucial for robotic arms. However, robotic arms still face many technical challenges in practical applications, among which non-linearity and actuator failures are two key problems. Since the robotic arm system is inherently non-linear, its dynamic model contains complex coupling relationships and non-linear characteristics, such as joint friction and parameter variations, and usually it is difficult or impossible to accurately model. These non-linear factors will affect the control accuracy and stability of the system. In addition, actuators are the core components of robotic arms, and their failures will lead to a decline in the performance of robotic arms or even complete failure. However, when robotic arms perform high-load or high-frequency motion tasks, they often face the risk of actuator failures, such as Loss of Effectiveness (LOE) failures, offset failures, or jamming failures, etc. These failures will seriously threaten the stability and safety of the system. Therefore, fault tolerance is also indispensable for robotic arms, especially for scenarios with high safety requirements.

[0003] Traditional robotic arm control technologies, including PID control, robust control, and model-based control strategies, exhibit certain limitations in practical applications. These methods are insufficient in terms of anti-interference ability, fault tolerance, flexibility, and dynamic response speed. They usually rely on accurate system models and preset control parameters. Therefore, when faced with unknown nonlinearities or model uncertainties, they often perform poorly. When an actuator fails, these methods also lack the ability to adaptively adjust, which may lead to a significant decline in system performance or even complete failure. In high-dynamic task scenarios, traditional control methods are difficult to meet the requirements of both fast response and high-precision control. In addition, to achieve a faster convergence speed or higher control precision, these methods usually adopt high-gain feedback, but this easily causes system oscillations, affects the smoothness of the robotic arm's movement, and increases energy consumption. Moreover, implementing real-time monitoring of the health status and degree of actuator faults of the robotic arm can enable operators to promptly understand the working conditions of the system and make corresponding arrangements. For example, when an operator discovers that an actuator of the robotic arm has a relatively serious fault and the application scenario has high safety requirements, the equipment should be stopped in a timely manner and the actuator should be repaired or replaced; if the operator discovers that the actuator of the robotic arm only has a minor fault, and the fault tolerance ability of the controller can basically compensate for its impact, and the application scenario has low safety requirements, in order to save time costs, the equipment can continue to run for a certain period of time. Many commonly used fault detection methods can only determine whether an actuator has failed, but cannot quantify the severity of the fault, and there are also deficiencies in real-time performance. Moreover, methods based on machine learning or deep learning require a large amount of fault data for training, which is also difficult to obtain in practical applications.

[0004] In summary, most current robotic arm control methods are difficult to meet the strict requirements of high precision, fast response, and strong reliability in complex application scenarios, and cannot quantify the severity of actuator faults in real time. Therefore, there is an urgent need for a new type of robotic arm fault monitoring and fault-tolerant control method that can ensure fast and high-precision motion control for systems with unknown nonlinearities, real-time feedback of the health status and degree of actuator faults, and achieve fast fault tolerance after the system experiences actuator LOE faults, ensuring the stability and reliability of the robotic arm. Summary of the Invention

[0005] To solve the above technical problems, the present invention proposes a real-time fault monitoring and adaptive fixed-time fault-tolerant control method for robotic arms, which can achieve real-time fault monitoring and fast and high-precision fault-tolerant control of robotic arms.

[0006] The technical solution adopted by the present invention is: a real-time fault monitoring and adaptive fixed-time fault-tolerant control method for robotic arms, including:

[0007] S1: Construct the Lagrangian dynamics model of an n - degree - of - freedom robotic arm considering the existence of partial actuator failure faults and unknown non - linearities;

[0008] S2: Construct a non - singular fixed - time terminal sliding mode surface;

[0009] S3: Based on the sliding mode surface constructed in step S2, design an adaptation rate to estimate the upper bound of the unknown non - linearities of the robotic arm within a fixed time;

[0010] S4: Design a dynamic adjustment variable to approximate the partial actuator failure fault coefficient;

[0011] S5: Based on the Lagrangian dynamics model of the robotic arm in step S1, the adaptation rate in step S3, and the dynamic adjustment variable in step S4, combined with the sliding mode surface constructed in step S2, design an adaptive fixed - time fault - tolerant controller;

[0012] S6: Use the adaptive fixed - time fault - tolerant controller designed in step S5 to control the robotic arm with unknown non - linearities and potential partial actuator failure faults.

[0013] Advantages of the present invention: First of all, the present invention designs a novel dynamic adjustment variable to approximate the actuator LOE fault coefficient and compensate for the impact of the fault on the system operation. At the same time, the health status and fault degree of each actuator in the robotic arm can be monitored in real time through this variable. Secondly, the present invention designs an adaptation rate to quickly and accurately estimate the upper bound of the unknown non - linearities of the system within a fixed time, thereby ensuring the robustness of the controller while avoiding the use of overestimated control gains and the corresponding chattering phenomenon. Based on the designed dynamic adjustment variable and adaptation rate, the present invention combines the sliding - mode control technology to design an adaptive fixed - time fault - tolerant controller, ensuring that the robotic arm with actuator LOE faults and unknown non - linearities can achieve high - precision motion control within a fixed time independent of the initial state of the system, and without any prior information related to unknown non - linearities and actuator faults. The method of the present invention has the following advantages:

[0014] (1) The present invention designs a dynamic adjustment variable to approximate the actuator LOE fault coefficient of the robotic arm and compensate for the impact of the fault on the system. Through this variable, the failure degree of the actuator can be monitored in real time, enabling the operator to timely understand the health status of the equipment.

[0015] (2) The present invention designs an adaptation rate to estimate the upper bound of the unknown non - linearities in the system, thereby ensuring the robustness of the controller while avoiding the use of overestimated control gains and the corresponding chattering phenomenon, and reducing the energy consumption to a certain extent.

[0016] (3) The fault-tolerant controller designed in the present invention can ensure that the robotic arm with unknown nonlinearity and actuator LOE faults achieves high-precision following motion control within a fixed time independent of the initial state of the system, and does not require any prior information about the unknown nonlinearity and actuator faults. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 is a flowchart of the method of the present invention;

[0018] Figure 2 is a block diagram of the architecture of the method of the present invention;

[0019] Figure 3 is a schematic structural diagram of the robotic arm in an embodiment of the present invention;

[0020] Figure 4 is the result of the following motion of the robotic arm affected by faults and the result of the dynamic adjustment variable approaching the actuator failure coefficient in an embodiment of the present invention;

[0021] wherein, (a) is the actual trajectory and the ideal trajectory of joints Q1-Q3, (b) is the position following error of joints Q1-Q3, and (c) is the approximation result of the actuator failure coefficient of joints Q1-Q3;

[0022] Figure 5 is the result of the following motion of the robotic arm under different initial system states in an embodiment of the present invention;

[0023] wherein, (a) is the trajectories of joints Q1-Q3 under different initial states, and (b) is the following errors of joints Q1-Q3 under different initial states. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0024] To facilitate the understanding of the technical content of the present invention by those skilled in the art, the content of the present invention will be further explained below with reference to the accompanying drawings.

[0025] The present invention aims to design a fault monitoring and fault-tolerant control method to ensure that a robotic arm with unknown nonlinearities and actuator LOE faults can still achieve fast and high-precision motion control, and the tracking error can converge to a neighborhood of zero within a fixed time independent of the initial state of the system. A key innovation of the present invention is to design a dynamically adjustable variable to approximate the actuator LOE fault coefficient. This variable is not only used in the design of the controller to compensate for the adverse effects of faults on the system operation, but also through this variable, the failure degree of each actuator in the robotic arm can be monitored in real time, enabling the operator to timely understand the health status of the equipment and make subsequent reasonable work arrangements, which is very valuable in practical applications. Another prominent feature of the present invention is to design an adaptation rate to estimate the upper bound of the unknown nonlinearities in the system, thereby ensuring the robustness of the controller, and enabling the controller not to use overestimated gains and avoid the chattering phenomenon easily caused by high gains. Based on the designed dynamically adjustable variable and adaptation rate, the present invention combines fixed-time control and sliding-mode control techniques to design an adaptive fixed-time fault-tolerant controller to achieve the above objectives.

[0026] As Figure 1 shown, a real-time fault monitoring and adaptive fixed-time fault-tolerant control method for a robotic arm proposed by the present invention includes the following steps:

[0027] S1: Construct the Lagrangian dynamics model of an n-degree-of-freedom robotic arm considering the existence of actuator LOE faults and unknown nonlinearities.

[0028] When not considering the above adverse factors, the basic Lagrangian dynamics model of an n-degree-of-freedom robotic arm is:

[0029]

[0030] where respectively represent the angular position, velocity, and acceleration vectors of the joints, represents the set of real numbers, is the inertia matrix and is positive definite and symmetric, is the centripetal force and Coriolis force matrix, G(q) is the gravity vector, is the driving torque.

[0031] During the motion of the robotic arm, it is generally affected by friction, and it is usually difficult or impossible to accurately model its nonlinear characteristics. In addition, due to the complex dynamic equations involved in the robotic arm, the model parameters are usually difficult to accurately obtain, or the physical parameters change in real time during the operation of the system, which will lead to the introduction of model uncertainties. The present invention collectively refers to these adverse factors as unknown nonlinearities and constructs the following dynamic model of the robotic arm considering the existence of actuator LOE faults and unknown nonlinearities:

[0032]

[0033] where is the friction force vector, and Δ is the model uncertainty, which can be expressed as where △M, △C, and △G are the differences between the corresponding true matrix and nominal matrix. Define and assume that d(t) and its derivative are bounded, that is, there exist and where and are unknown constants. In addition, Ω = diag{ρ i}(i = 1,…,n) is the actuator LOE fault coefficient matrix, and there is ρ i ≤ ρ i ≤ 1, ρ i ∈(0,1]. If ρ i = 1, it indicates that the i-th joint actuator of the robotic arm has no fault, and the smaller ρ i , the more serious the fault.

[0034] Those skilled in the art should note that the true matrix and the nominal matrix have the same form here, but only the parameters in the matrix are different. The parameters of the true matrix are the true physical parameters of the system, which are unknown, and those in the nominal matrix are nominal parameters, which are known, generally provided by the product manufacturer or measured by oneself.

[0035] S2: Construct the following non-singular fixed-time terminal sliding mode surface:

[0036]

[0037] where s = [s1,…,s n T , 0 < α < 1, β > 1, e = [e1,…,e n T = q d - q is the following error vector of the joint angle position, q d is the ideal joint angle position vector, and it is assumed that q d and its derivative and are all bounded, that is, there exist and where, ||·|| represents taking the norm, and are both unknown constants. The function S α (e) = [s α (e1),…,s α (e n )] T , where s​​α (e i ) is defined as:

[0038]

[0039] where δ is a very small constant, which takes the value of 0.01 in this embodiment and can be valued as needed in practical applications. The function Sig β (e) = [sig β (e1), …, sig β (e n )] T , and sig β (e i ) = sgn(e i )|e i | β . V1 and V2 are two positive definite parameter matrices, and the specific definitions are as follows:

[0040]

[0041] Taking the derivative of the sliding mode surface s and substituting the model (2) into it, we can obtain:

[0042]

[0043] where Θ = M -1 (F + Δ) = M -1 d(t) is defined as the unknown nonlinearity of the system, and Θ = [θ1, …, θ n )] T , and H β (e) is the matrix form of the derivative of the function Sig β (e), and F α (e) is the matrix form of the derivative of the function S α (e), and the specific definitions are as follows:

[0044]

[0045] where

[0046]

[0047] S3: Design the adaptation rate for estimating the upper bound r i of the unknown nonlinearity θ i , where θ i is the i-th element of the vector Θ in Equation (6), the upper bound r i is an unknown constant, and |θ i | ≤ r i . The adaptation rate Designed as:

[0048]

[0049] Where and are positive design parameters, and q > 1. Define the estimation error as

[0050] S4: Design a dynamic adjustment variable to approximate the actuator LOE fault coefficient ρ i . First, design the following nominal controller:

[0051]

[0052] Where tanh(s) = [tanh(s1), …, tanh(s n )] T , and 0 < p < 1. A s and B s are the following two positive definite parameter matrices:

[0053]

[0054] Where, and are both positive design parameters, Define the inverse of matrix M as:

[0055]

[0056] is the element in matrix M -1 , i = 1, 2, …, n, j = 1, 2, …, m;

[0057] The dynamic adjustment variable is designed as follows:

[0058]

[0059] Where and are positive design parameters. Define the approximation error as Now, conduct a theoretical analysis on the convergence of the error : Define the following Lyapunov function:

[0060]

[0061] According to Equation (6), we can obtain:

[0062]

[0063] Taking the derivative with respect to V1 and substituting Equation (15), we get:

[0064]

[0065] According to -r i tanh(s i )s i ≤0.2785r i -r i |s i | and r i ≥|θ i |, Equation (16) can be written as:

[0066]

[0067] Through some operations, the following inequality holds:

[0068]

[0069]

[0070] Where Substituting (18) and (19) into (17) and combining with Young's inequality, we get:

[0071]

[0072] Where

[0073]

[0074] According to Lyapunov stability theorem, the function V1 is ultimately uniformly bounded (UUB), that is, V1(t) ≤ η1 / χ, t → ∞. Thus, the sliding mode surface s i and the approximation error of the dynamic adjustment variable are both bounded, and Correspondingly, it can be known that E j and (i, j = 1, …, n) are both bounded. Therefore, there exists an unknown constant such that is an unknown constant, used to represent the existence of an unknown constant satisfies That is This variable is bounded.

[0075] S5: Based on the manipulator dynamic model, adaptive rate and dynamic adjustment variable obtained from the foregoing steps, combined with the sliding mode control technology, design the following adaptive fixed-time fault-tolerant controller:

[0076]

[0077] where the definition of E is shown in Equation (10). is the estimated value of matrix Ω, and the estimation error is defined as The architecture block diagram of the method of the present invention is as shown in Figure 2 Figure [figure number]. Now, the convergence of the tracking motion error of the robotic arm using the designed controller (22) and the estimation error of the adaptive rate r i is analyzed theoretically: Define the following Lyapunov function:

[0078]

[0079] Taking the derivative of V2 gives:

[0080]

[0081] Substituting into Equation (24) gives:

[0082]

[0083] Through some operations, the following inequality holds:

[0084]

[0085] Substituting Equation (26) and Equation (18) into Equation (25) and combining with Young's inequality gives:

[0086]

[0087] where Rearranging Equation (27) gives:

[0088]

[0089] where

[0090]

[0091] According to Lemma 1, V2 will converge to the following region:

[0092]

[0093] where 0 < φ < 1, and the convergence time T1 satisfies:

[0094]

[0095] Lemma 1 is defined as:

[0096] For the system If there exists a continuously radially unbounded function satisfying:

[0097]

[0098] wherein, it means that the independent variable x of the function V(x) is an n-dimensional constant, and the value of the function V(x) is a constant greater than or equal to 0, is the set of real numbers greater than 0, a > 0, b > 0, 0 < α < 1, β > 1, and 0 < η < ∞ are all constants. Then the origin of the system is actually fixed-time stable, and

[0099]

[0100] where 0 < φ < 1 is a constant. The convergence time satisfies:

[0101]

[0102] According to the definition of the function V2, it can be known that the sliding mode surface s i and the adaptive rate estimation error k i will converge to the neighborhood of the corresponding zero within the time . Define the following constants:

[0103]

[0104] It can be known that when t ≥ T1, |s i | ≤ ε i and e i s i ≤ ε i |e i |. If |e i | ≥ δ at this stage, then s i satisfies:

[0105]

[0106] Multiply both sides of Equation (36) by e i to obtain:

[0107]

[0108] Define the Lyapunov function Take its derivative and substitute Equation (37) to obtain:

[0109]

[0110] According to Lemma 1, the tracking error e i will converge to the following region:

[0111]

[0112] where \(0 \lt \lambda \lt 1\). The convergence time satisfies:

[0113]

[0114] Based on the above analysis, for any initial system state, the following motion error \(e\) of the robotic arm i converges to and the time satisfies \(T \leq T\) m = \(T\) max1 + \(T\) max2 . Here, the time upper limit \(T\) m is the most conservative estimate, and the actual convergence time may be much smaller than \(T\) m .

[0115] S6: Use the designed adaptive fixed-time fault-tolerant controller to control the robotic arm with unknown non-linearity and potential actuator LOE faults.

[0116] The present invention conducts relevant experimental verifications with the QArm robotic arm of Quanser Inc. as the control object. This robotic arm consists of 4 joints (Q1 - Q4), and its structural schematic diagram is as shown in Figure 3 . The present invention only considers the motions of joints Q1, Q2, and Q3, and their ideal angular motion trajectories are all set to \(0.5\sin(2\pi t / 15)\) rad. Additionally, at the 12th second, the efficiency coefficient of the Q1 actuator is set to \(\rho_1 = 0.4\) to simulate the actuator LOE fault, at the 26th second, a fault with \(\rho_2 = 0.6\) is set for the Q2 actuator, and at the 35th second, a fault with \(\rho_3 = 0.75\) is set for the Q3 actuator.

[0117] As shown in Figure 4 , (a) shows the actual trajectories and ideal trajectories of joints Q1 - Q3, as shown in Figure 4 , (b) shows the position following errors of joints Q1 - Q3, and as shown in Figure 4 , (c) shows the approximation results of the actuator failure coefficients of joints Q1 - Q3. It can be seen that during the overall operation of the robotic arm, joints Q1 - Q3 always maintain a high-precision following effect. After the actuator LOE faults occur in sequence at the joints, the corresponding dynamic adjustment variables promptly approximate the failure coefficients, thereby compensating for the impact of the faults on the system operation, achieving fast and effective fault tolerance, and ensuring that the motion control effect of the robotic arm after the actuator fails is almost the same as when no fault occurs. Additionally, by monitoring the dynamic adjustment variable , the failure states of the Q1 - Q3 actuators can be understood in real time, which is also reflected in the schematic diagram of Figure 2 , as shown in Figure 2As shown, in this embodiment, three joints are taken as examples for illustration. In actual applications, more joints may be designed. Therefore Figure 2 the ellipsis in it can also show the actuator failure coefficients of more joints.

[0118] In addition, the present invention has conducted relevant experiments when the robotic arm joints Q1-Q3 are in three different initial positions to verify that the control effect of the designed method is independent of the system initial state. Among them, the initial state 1 is Q1(0) = Q2(0) = Q3(0) = 0.1 rad; the initial state 2 is Q1(0) = Q2(0) = Q3(0) = 0 rad; the initial state 3 is Q1(0) = Q2(0) = Q3(0) = -0.1 rad. As Figure 5 (a) in it is the trajectories of joints Q1-Q3 under different initial states. As Figure 5 (b) in it is the tracking errors of joints Q1-Q3 under different initial states. It can be seen that even under different system initial states, joints Q1-Q3 can converge to a consistent tracking effect within a fixed time. The time marked in the figure is the upper bound of time T calculated when λ = φ = 0.9 is selected m .

[0119] Those of ordinary skill in the art will realize that the embodiments described herein are for helping the reader understand the principles of the present invention and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. For those skilled in the art, various changes and modifications can be made to the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the scope of the claims of the present invention.

Claims

1. A real-time fault monitoring and adaptive fixed-time fault-tolerant control method for a robotic arm, characterized in that Including: S1: Construct the Lagrangian dynamics model of an n-degree-of-freedom robotic arm considering the presence of actuator partial failure faults and unknown non-linearities; S2: Construct a non-singular fixed-time terminal sliding mode surface; S3: Based on the sliding mode surface constructed in step S2, design an adaptation rate to estimate the upper bound of the unknown non-linearities of the robotic arm within a fixed time; S4: Design a dynamic adjustment variable to approximate the actuator partial failure fault coefficient; S5: Based on the Lagrangian dynamics model of the robotic arm in step S1, the adaptation rate in step S3, and the dynamic adjustment variable in step S4, combined with the sliding mode surface constructed in step S2, design an adaptive fixed-time fault-tolerant controller; S6: Use the adaptive fixed-time fault-tolerant controller designed in step S5 to control the robotic arm with unknown non-linearities and potential actuator partial failure faults.

2. A real-time fault monitoring and adaptive fixed-time fault-tolerant control method for a robotic arm according to claim 1, characterized in that, The Lagrangian dynamics model of the robotic arm described in step S1 is expressed as: where, is the friction force vector, q, represent the angular position, velocity, and acceleration vectors of the joints respectively, and Δ is the model uncertainty, expressed as △M is the difference between the true inertia matrix and the nominal inertia matrix, △C is the difference between the true centripetal and Coriolis force matrices and the nominal centripetal and Coriolis force matrices, and △G is the difference between the true gravity vector and the nominal gravity vector.

3. A real-time fault monitoring and adaptive fixed-time fault-tolerant control method for a robotic arm according to claim 2, characterized in that, The sliding mode surface constructed in step S2 is expressed as: where \(s = [s_1,\ldots,s n T \), \(s i \) represents the \(i\)-th sliding surface, \(0 < \alpha < 1\), \(\beta>1\), \(e\) is the tracking error vector of the joint angle position, \(e = [e_1,\ldots,e n T = q d - q\), \(q d \) is the ideal joint angle position vector, \(V_1\) and \(V_2\) are two positive definite diagonal parameter matrices, \(S α (e)=[s α (e_1),\ldots,s α (e n )] T \), \(\delta\) is a very small constant, \(Sig β (e)=[sig β (e_1),\ldots,sig β (e n )] T \), and \(sig β (e i )=\text{sgn}(e i )|e i | β \).​​​ 4. A real-time fault monitoring and adaptive fixed-time fault-tolerant control method for a robotic arm according to claim 3, characterized in that The adaptation rate in step S3 is expressed as: wherein, and are positive design parameters, and q > 1, r i is the upper bound of the unknown non-linearity of the robotic arm, is the adaptation rate, is the derivative of.

5. A real-time fault monitoring and adaptive fixed-time fault-tolerant control method for a robotic arm according to claim 4, characterized in that The dynamic adjustment variable in step S4 is expressed as: Among them, and are positive design parameters, ρ i is the actuator partial failure fault coefficient, is the dynamic adjustment variable, is 's derivative, E i is the nominal controller corresponding to the i-th joint.

6. A real-time fault monitoring and adaptive fixed-time fault-tolerant control method for a robotic arm according to claim 5, characterized in that The nominal controller is expressed as: where tanh(s) = [tanh(s1), …, tanh(s n )] T , and 0 < p < 1, A s and B s are positive definite diagonal parameter matrices.

7. A real-time fault monitoring and adaptive fixed-time fault-tolerant control method for a robotic arm according to claim 6, characterized in that The adaptive fixed-time fault-tolerant controller in step S5 is expressed as: where E is the nominal controller,

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