Mechanical arm anti-interference and fault-tolerant control method based on fixed time expansion state observer

Abstract

The invention discloses a mechanical arm anti-interference and fault-tolerant control method based on a fixed time expansion state observer, is applied to the technical field of mechanical arm control, and aims to overcome the defects of most existing mechanical arm anti-interference and fault-tolerant control methods in the aspects of model dependence, real-time performance, robustness and the like. And the requirements of high precision, fast response and strong reliability in a complex application scene are difficult to meet. According to the method, a novel FxTESO is designed to estimate the joint angular velocity of the mechanical arm and lumped uncertainty formed by an actuator fault component, external disturbance and unknown nonlinearity in the system, the self-adaption rate is designed to be used for estimating a boundary related to the FxTESO error of the observer, and finally, the self-adaption rate is used for estimating the joint angular velocity of the mechanical arm and the lumped uncertainty formed by the actuator fault component, the external disturbance and the unknown nonlinearity. On the basis of the work, a self-adaptive fixed-time fault-tolerant anti-interference controller is designed. By adopting the control method disclosed by the invention, the quick and high-precision anti-interference and fault-tolerant control of the mechanical arm can be realized.

Description

Technical Field

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

[0002] Nowadays, robotic arms have made important contributions in many fields such as industrial manufacturing, healthcare, and aerospace. Their control performance directly determines the overall efficiency and reliability of the system. Therefore, high-precision control, strong reliability, and fast response speed are crucial for robotic arms. At the same time, they also need to have anti-interference and fault tolerance capabilities to cope with various challenges under different working conditions. Due to the complex structure and dynamic working environment, robotic arms are often affected by unknown non-linearities and external disturbances. The unknown non-linearities are mainly reflected in the complex non-linear terms included in the dynamic model of the robotic arm, such as friction, gravity, and centripetal force. These non-linear characteristics are usually difficult to accurately model. In addition, there are also differences between the nominal parameters and the actual parameters of the robotic arm, as well as model mismatches caused by changes in system parameters due to changes in working conditions. In addition, robotic arms are often used to perform tasks with heavy loads or high-frequency movements, which easily lead to actuator failures, such as Loss of effectiveness (LOE) failures, bias failures, or jamming failures, etc., affecting the stability and safety of the system.

[0003] Traditional robotic arm control methods, such as PID control, robust control, and model-based control, etc., have certain limitations in dealing with the above problems. For example, they have insufficient anti-interference ability, limited fault tolerance ability, excessive conservatism, and slow response speed, etc. These methods usually rely on accurate models and fixed control parameters, and it is difficult to effectively cope with unknown disturbances and model uncertainties. When the actuator fails, it is often impossible to adjust the control strategy, resulting in a decline in system performance or even failure. In a high-dynamic environment, it is also difficult for traditional methods to simultaneously meet the requirements of fast response and high-precision control. In addition, these methods usually require high-gain feedback to obtain a faster convergence speed or higher control precision, which easily causes system oscillations, affecting the motion smoothness and energy consumption efficiency of the robotic arm.

[0004] The Extended State Observer (ESO) is a very effective system state and disturbance estimation technique. It can not only estimate system state variables such as position and velocity, but also integrate various uncertain factors of the system into an "extended state" (also known as "lumped uncertainty") and estimate it, and then achieve robust control through feedback compensation. The ESO does not require an accurate system model, has strong anti-interference ability, simple structure and is easy to implement. Usually, only an observer gain matrix needs to be designed. However, the convergence speed of the traditional ESO usually depends on the selection of the observer gain. In a high-dynamic disturbance or fast-changing working environment, it may not be able to estimate the system state and lumped uncertainty in time and accurately, and there are certain limitations. Fixed-time control is a relatively advanced control strategy that can achieve the state convergence of the system within a fixed time, and the convergence time does not depend on the initial state of the system. This characteristic makes fixed-time control very suitable for application scenarios that require fast response.

[0005] In summary, most of the existing anti-disturbance and fault-tolerant control methods for robotic arms have certain defects in terms of model dependence, real-time performance and robustness, etc., and it is difficult to meet the high-precision, fast-response and strong reliability requirements in complex application scenarios. Therefore, there is an urgent need for a new type of robotic arm control method that can achieve fast and high-precision anti-disturbance control in the presence of unknown nonlinearities and external disturbances in the system, and achieve fast fault tolerance after the occurrence of actuator LOE faults, ensuring the overall performance and reliability of the robotic arm. Combining the ESO and the fixed-time control strategy can give full play to the advantages of both, providing effective ideas and solutions for the high-precision fast anti-disturbance and fault-tolerant control problems of robotic arms, and having important theoretical significance and practical application value. However, the existing ESO needs to meet the condition that its coefficient matrix is a Hurwitz matrix. Therefore, when adjusting parameters, the existing fixed-time ESO needs to jointly adjust the parameters to meet this condition, resulting in more parameters to be adjusted and complex debugging. Summary of the Invention

[0006] To solve the above technical problems, the present invention proposes a robotic arm anti-disturbance and fault-tolerant control method based on a fixed-time extended state observer to achieve fast and high-precision anti-disturbance and fault-tolerant control of the robotic arm.

[0007] The technical solution adopted by the present invention is: a robotic arm anti-disturbance and fault-tolerant control method based on a fixed-time extended state observer, including:

[0008] S1: Construct a Lagrangian dynamics model of an n-degree-of-freedom robotic arm considering the presence of actuator LOE faults, external disturbances and unknown nonlinearities;

[0009] S2: Based on the Lagrangian dynamics model of the constructed robotic arm, design a fixed-time extended state observer to estimate the joint angular velocity and the lumped uncertainty composed of actuator fault components, external disturbances, and unknown non-linearities in the system;

[0010] S3: Construct a sliding surface;

[0011] S4: Based on the constructed sliding surface, design an adaptation rate to estimate a bound related to the error of the fixed-time extended state observer;

[0012] S5: Based on the robotic arm dynamics model, observer FxTESO, sliding surface, and adaptation rate in the foregoing steps, design an adaptive fixed-time fault-tolerant disturbance rejection controller;

[0013] S6: Use the designed adaptive fixed-time fault-tolerant disturbance rejection controller to control the robotic arm with external disturbances, unknown non-linearities, and potential actuator LOE faults.

[0014] Advantages of the present invention: First, the present invention designs a novel FxTESO (Fixed-time extended state observer) to estimate the joint angular velocity of the robotic arm and the lumped uncertainty composed of actuator fault components, external disturbances, and unknown non-linearities in the system, and its estimation error can be reduced by increasing the bandwidth of the observer. Second, an adaptation rate is designed to estimate a bound related to the error of the observer FxTESO, which not only improves the robustness of the controller but also avoids using overestimated control gains. Finally, based on the above work, an adaptive fixed-time fault-tolerant disturbance rejection controller is designed, which can ensure that even when the robotic arm has actuator LOE faults, external disturbances, and unknown non-linearities, its motion control can still achieve practical fixed-time stability without any information about the lumped uncertainty in the system. The method of the present invention has the following advantages:

[0015] (1) The fixed-time extended state observer FxTESO designed by the present invention can accurately estimate the joint angular velocity of the robotic arm and the lumped uncertainty of the system, and the estimation error can converge to a neighborhood of zero within a fixed time independent of the initial state of the system, and the estimation accuracy can be improved by increasing the observer bandwidth ω0.

[0016] (2) The adaptive fixed-time fault-tolerant disturbance rejection controller designed in the present invention can ensure that for a robotic arm with unknown nonlinearities, external disturbances, and actuator LOE faults, the tracking error of each joint angle can converge to a neighborhood of zero within a fixed time independent of the initial state of the system. Additionally, the designed adaptation rate is used to estimate a bound related to the FxTESO error, which not only improves the robustness of the controller but also avoids using overestimated control gains, thereby enhancing the stability of the system.

[0017] (3) The joint angular velocity signal of the robotic arm used in the controller designed in the present invention is estimated by FxTESO rather than obtained by differentiating the joint angle position signal measured by the sensor, thus effectively reducing the influence of measurement noise on the control performance. Additionally, the designed controller does not require any prior information about the lumped uncertainties of the system. Description of the Drawings

[0018] Figure 1 is a flowchart of the method of the present invention;

[0019] Figure 2 is a block diagram of the architecture of the method of the present invention;

[0020] Figure 3 is a schematic structural diagram of the robotic arm in the embodiment of the present invention;

[0021] Figure 4 is the estimation result of the lumped uncertainties related to joints Q2 and Q3 by FxTESO in the embodiment of the present invention;

[0022] Figure 5 is the tracking motion result of joints Q2 and Q3 of the robotic arm under the influence of disturbances and faults in the embodiment of the present invention;

[0023] Figure 6 is the tracking motion result of joints Q2 and Q3 of the robotic arm under different initial system states in the embodiment of the present invention. Detailed Embodiment

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

[0025] The method of the present invention aims to design a control strategy. In the case where the robotic arm system faces problems such as unknown non - linearity, external disturbances, and actuator LOE faults in practical applications, it can still ensure the stability of the system and high - precision motion control. Moreover, the motion error of the robotic arm can converge to a neighborhood of zero within a fixed time independent of the initial state of the system. A key innovation and prominent feature of the method of the present invention is the design of a fixed - time extended state observer FxTESO to estimate the joint angular velocity of the robotic arm and the lumped uncertainty composed of unknown non - linearity, external disturbances, and actuator fault components. The designed FxTESO has high estimation accuracy, a simple structure, and the estimation error can converge to a neighborhood of zero within a fixed time. Different from most observers and estimators, FxTESO not only estimates the lumped uncertainty of the system to compensate for the effects of disturbances and faults, etc., but also estimates the state variables of the system, that is, the joint angle position and velocity of the robotic arm, thereby reducing the influence of sensor noise on the control performance (if the joint angular velocity is obtained by differentiating the joint angle position signal measured by the sensor, the measurement noise will be further amplified). In addition, the present invention designs an adaptation rate to estimate a bound related to the error of FxTESO, thereby improving the robustness of the controller while avoiding the use of over - estimated control gains, ensuring the stability of the system and reducing energy consumption to a certain extent, which is also a feature of the present invention. Combining the designed observer FxTESO and the adaptation rate above, the present invention designs an adaptive fixed - time fault - tolerant disturbance - rejection controller within the framework of the sliding - mode control technology to achieve the above - mentioned control objectives.

[0026] As Figure 1 shown, a disturbance - rejection and fault - tolerant control method for a robotic arm based on a fixed - time extended state observer 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, external disturbances, and unknown non - linearity.

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

[0029]

[0030] where are respectively the angular position, velocity, and acceleration of the robotic arm joints, is the driving torque, is the inertia matrix, is the centripetal force and Coriolis force matrix, and G(q) is the gravity vector.

[0031] The Lagrangian dynamics model of an n - degree - of - freedom robotic arm considering actuator LOE faults, external disturbances, and unknown non - linearity is constructed as:

[0032]

[0033] where is the friction force vector, Λ = diag{λ i} is the actuator LOE fault coefficient matrix, and λ i ∈(0,1], τ d is the external disturbance, and Δ is the model uncertainty, which can be expressed as where △M, △C, and △G are the uncertainty parts of the model, which are the inertia matrix uncertainty, the centripetal force and Coriolis force matrix uncertainty, and the gravity vector uncertainty, respectively. Multiply both sides of Equation (2) by M -1 (q), and after rearrangement, we can obtain:

[0034]

[0035] where I n is the n-order identity matrix, d(t) = [d1(t), …, d n (t)] T is the lumped uncertainty, which includes the actuator fault component, the external disturbance, and the unknown nonlinearity. The superscript T represents the transpose. In Equation (3), G(q), , and Δ are generally difficult to obtain or cannot be obtained, and we collectively refer to these as unknown nonlinearities. In addition, assume that the lumped uncertainty d(t) and its derivative in model (3) are bounded, that is, there exist and where and are unknown constants.

[0036] S2: Based on the constructed manipulator dynamics model, design a fixed-time extended state observer FxTESO to estimate the joint angular velocity and the lumped uncertainty in the system composed of the actuator fault component, the external disturbance, and the unknown nonlinearity.

[0037] First, define the lumped uncertainty d(t) as an extended state variable x3, and assume that the derivative of x3 with respect to time is where γ = [γ1, …, γ n T is an unknown function. Then, the state variables of the manipulator can be written accordingly as x1 represents the joint angle, and x2 represents the joint angular velocity. The fixed-time extended state observer FxTESO is designed in the following form:

[0038] ​

[0039] where (j = 1, …, 3) are the state estimated values, and the estimation error is defined as gj is the designed gain, and ω0 is the bandwidth of the FxTESO. The function Sig a (x)=[sig a (x1), …, sig a (x n )] T , where sig a (x)=sgn(x)|x| a , where the superscript a = α j or β j or b in Equation (31), α j = j(α - 1)+1, β j = 1 / α+(j - 1)(α - 1), and α ∈ (1 - ε1, 1), where ε1 > 0 is a small constant; x1, ..., x n here represent the elements of the vector

[0040] The theoretical analysis of the estimation error convergence of the designed FxTESO is carried out. According to Equations (3) and (4), the estimation error dynamics of the FxTESO are as follows:

[0041]

[0042] where and ξ j =[ξ j,1 , …, ξ j,n T . First, consider the following subsystem:

[0043]

[0044] where

[0045]

[0046] Define v i =[ξ 1,i , …, ξ 3,i T (i = 1, …, n). If α = 1, then the error subsystem (6) can be written as where the matrix A = [-g1, 1, 0; -g2, 0, 1; -g3, 0, 0], and the symbol represents the Kronecker product. If the matrix A is designed as a Hurwitz matrix, then there exists a Lyapunov equation PA + A T ​​​P = -Q, where P and Q are positive definite matrices, and P is symmetric. Construct the following Lyapunov function:

[0047]

[0048] where μ = α1α2α3. According to Definition 1, the function V1(α, ξ) is homogeneous of order l1 = 2 / μ with respect to the weight r = (1, α, 2α - 1); the Lie derivative of V1(α, ξ) along S α (ξ) is homogeneous of order l2 = 2 / μ + α - 1 with respect to the weight r. According to Lemma 1, there exists such that the following inequality holds:

[0049]

[0050] where λ min (·) and λ max (·) are the minimum eigenvalue and the maximum eigenvalue of the matrix respectively, and l2 / l1 = 1 + μ(α - 1) / 2 < 1.

[0051] Definition 1 is as follows:

[0052] If a function satisfies for all a > 0, then the function S is homogeneous of degree d with respect to the weight , denotes the set of real numbers, denotes the n-dimensional real vector space, denotes the n-dimensional vector space whose all components are positive real numbers. If the i-th (1 ≤ i ≤ n) element of the vector field v all satisfies then v is homogeneous of degree d with respect to r.

[0053] Lemma 1 is as follows:

[0054] If the continuous functions Y1(x) > 0 and Y2(x) > 0, are homogeneous of order l1 and l2 with respect to r respectively, and l1 > 0, l2 > 0. Then the following expression holds:

[0055]

[0056] x is an arbitrary independent variable;

[0057] where

[0058] (11)

[0059] g is the independent variable when Y1(g) = 1;​

[0060] Then consider the following subsystem:

[0061]

[0062] where

[0063]

[0064] the Lie derivative of V1(α, ξ) along S β (ξ) is homogeneous of order l3 = 2 / μ + 1 / α - 1 with respect to the weight r. Thus, there exists such that:

[0065]

[0066] where l3 / l1 = 1 + μ(1 / α - 1) / 2 > 1.

[0067] Finally, combining the above analysis on and and considering the existence of the term in (5), the derivative of the Lyapunov function V1(α, ξ) along the FxTESO estimation error dynamics (5) can be expressed as:

[0068]

[0069] where, it is homogeneous of order l4 = 2 / μ - 2α + 1 with respect to r, and the following inequality holds:

[0070]

[0071] where, from Eqs. (9), (14) and (15), it can be obtained that:

[0072]

[0073] where,

[0074] By introducing an arbitrary constant δ (0 < δ < 1), Eq. (17) can be written in the following two forms:

[0075] Form 1:

[0076]

[0077] where 0 < δ < 1 is a constant. Obviously, if then there is According to Lemma 2, it can be obtained that: ​​

[0078]

[0079] The convergence time satisfies:

[0080]

[0081] Form 2:

[0082]

[0083] If then According to Lemma 2, it can be obtained that:

[0084]

[0085] The convergence time satisfies:

[0086]

[0087] Combining the above Form 1 and Form 2, it can be obtained that the estimation error of FxTESO will converge to the following region:

[0088]

[0089] The convergence time satisfies:

[0090]

[0091] It can be seen that T max1 is a fixed-time constant independent of the initial state of the system. According to Equation (24), it can be known that where is the i-th element of the FxTESO estimation error vector

[0092] Lemma 2 is:

[0093] For the system If there exists a continuous radially unbounded function satisfying:

[0094]

[0095] where indicates 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; represents the set of real numbers greater than 0, and a>0, b>0, 0<α<1, and β>1 are all constants. Then the origin of the system is fixed-time stable, and the convergence time satisfies:

[0096] ​

[0097] If the following inequality holds:

[0098]

[0099] where \(0 < \eta < \infty\). Then the system has an actually fixed-time stable origin, and

[0100]

[0101] where \(0 < \varphi < 1\) is a constant. The convergence time satisfies:

[0102]

[0103] S3: Construct a sliding mode surface for the subsequent design of the adaptive rate and the fixed-time fault-tolerant disturbance rejection controller. At the point where the tracking error of the manipulator joint angle approaches 0, the derivative of this sliding mode surface still exists and no singularity problem will occur. The form of the sliding mode surface is constructed as follows:

[0104]

[0105] where \(s\) is an \(n\)-dimensional vector, i.e., \(s = [s_1, \ldots, s n T , \(0 < a < 1\) and \(b > 1\) are two design parameters, \(q d \) is the ideal joint angle trajectory, is the first derivative of \(q d , \(e = q d - q\) is the tracking error and \(e = [e_1, \ldots, e n T , \(e_1, \ldots, e n \) respectively correspond to the angle trajectory tracking errors of the 1st to the \(n\)th joints respectively, \(U a (e) = [u a (e_1), \ldots, u a (e n )] T , and there is:

[0106]

[0107] where \(\sigma > 0\) is a small constant, which is taken as 0.001 in this embodiment and can be taken according to needs in practical applications. The function \(u a (e i )\) is to prevent the singularity problem when the joint angle tracking error \(e i \to 0\). In addition, \(H_1\) and \(H_2\) in Equation (31) are the following two positive definite parameter matrices:

[0108] ​​​

[0109] Taking the derivative of \(s\) in Equation (31) gives:

[0110]

[0111] and \(\varPsi\) is an \(n\)-dimensional vector \(\varPsi=[\psi_1,\cdots,\psi\) n T , \(R\) b (e) and \(F\) a (e) are two matrices as follows:

[0112]

[0113] where

[0114]

[0115] S4: Based on the constructed sliding mode surface, design an adaptation rate to estimate a bound related to the FxTESO error, which can not only improve the robustness of the controller but also avoid overestimating the control gain. The said bound refers to the upper bound \(k\) i , where \(\psi\) i is the \(i\)-th element of the vector \(\varPsi\) in Equation (34), is the \(i\)-th element of the estimation error vector of FxTESO, and the upper bound \(k\) i is an unknown constant. The adaptation rate i for estimating \(k\) is designed as:

[0116]

[0117] where \(a\) k,i and \(b\) k,i are positive design parameters, and the estimation error is defined as

[0118] S5: Based on the manipulator dynamic model, the observer FxTESO, the sliding mode surface, and the adaptation rate obtained from the foregoing steps, design an adaptive fixed-time fault-tolerant disturbance rejection controller for the manipulator with external disturbances, unknown nonlinearities, and potential actuator LOE faults to ensure that the joint angle \(q\) of the manipulator follows the ideal trajectory \(q\) d with high precision, and the tracking error converges to a neighborhood of zero within a fixed time independent of the initial state of the system. The controller is designed as:

[0119]

[0120] where​ and o i is a small constant of the design, and the value can be 0.1, 0.05, 0.02, etc. A suitable value is selected during the actual system debugging. 0 < m < 1, l > 1, and the symbol ⊙ represents the Hadamard product. Matrix A s and B s are as follows:

[0121]

[0122] where A s and B s are two positive definite parameter matrices of the design. The architecture block diagram of the method of the present invention is as Figure 2 shown.

[0123] The convergence of the following motion error of the manipulator using the designed controller and the estimation error of the adaptation rate k i is theoretically analyzed. For the stage of t ≥ T1, the following Lyapunov function is constructed:

[0124]

[0125] Taking the derivative of it, we can get:

[0126]

[0127] According to -k i tanh(s i / o i )s i ≤0.2785k i o i -k i |s i |, Equation (41) can be written as:

[0128]

[0129] According to some operations and lemmas, the following inequality can be obtained:

[0130]

[0131] Substituting Equation (43) into Equation (42), we can get:

[0132]

[0133] Equation (44) can be sorted out as:

[0134]

[0135] where

[0136]

[0137] Further arranging Equation (45) gives:

[0138]

[0139] where According to Lemma 2, the sliding mode surface s i and the estimation error of the adaptive rate will converge to the following region:

[0140]

[0141] where The convergence time T2 satisfies:

[0142]

[0143] Define a constant After the sliding mode surface s i converges into D2, there are and If |e i |≥σ, then there is:

[0144]

[0145] Multiply Equation (50) by e i to get:

[0146]

[0147] Construct a Lyapunov function Take its derivative and substitute Equation (51) to get:

[0148]

[0149] According to Lemma 2, the joint angle following error e of the robotic arm i will converge to the following region:

[0150]

[0151] The convergence time T3 satisfies:

[0152]

[0153] where 0 < υ < 1. From the above analysis, it can be seen that for any initial system state, the total time for the following error e i to converge into D3 satisfies T ≤ T m = T max1 + T max2 + Tmax3 The time upper limit T here m is the most conservative estimate, and the actual convergence time may be much smaller than T m .

[0154] S6: Use the designed adaptive fixed-time fault-tolerant disturbance rejection controller to control the robotic arm with external disturbances, unknown non-linearities, and potential actuator LOE faults.

[0155] The present invention conducts relevant experimental verifications with the QArm robotic arm of Quanser Company as the control object. This robotic arm consists of 4 joints (Q1 - Q4), and its structural schematic diagram is as Figure 3 shown. The present invention only considers the movements of joints Q2 and Q3, and their ideal angle trajectories are both set to sin(2πt / 15) rad. Additionally, an external disturbance signal τ d,2 = 1.06sin(πt) N·m is given to joint Q2 between 20 - 30 s; an external disturbance signal τ d,3 = 0.53sin(πt) N·m is given to joint Q3 between 40 - 50 s; the efficiency coefficient λ2 of joint Q2 is set to λ2 = 0.5 at the 60th s to simulate the actuator LOE fault; the efficiency coefficient λ3 of joint Q3 is set to λ3 = 0.6 at the 70th s.

[0156] Figure 4 In (a) of Figure 4 is the lumped uncertainty estimation result of joint Q2, -1 (q)(Λ - I n )τ term is no longer 0, which will cause the amplitude of d(t) to increase significantly. From Figure 4 it can be seen that the amplitude of the corresponding lumped uncertainty estimated by FxTESO also increases rapidly, thereby compensating for the influence of the fault in a timely manner, demonstrating the effectiveness of the designed observer FxTESO.

[0157] From Figure 5 (a) of Figure 5 it can be seen that joints Q2 and Q3 both maintain a very high-precision following effect during the overall movement of the robotic arm. From

[0158] To verify that the control effect of the designed method is independent of the initial state of the system, the present invention conducted relevant experimental verifications when the robotic arm joints Q2 and Q3 were in three different initial positions. Among them, the initial state 1 was Q2(0) = Q3(0) = 0.1 rad; the initial state 2 was Q2(0) = Q3(0) = 0 rad; the initial state 3 was Q2(0) = Q3(0) = -0.1 rad. From Figure 6 It can be seen that even under different initial states of the system, the joints Q2 and Q3 can converge to a consistent following effect, and the convergence time is independent of the initial state of the system, which once again demonstrates the superior performance of the method of the present invention.

[0159] In summary, compared with the existing ESO, the present invention introduces the observer bandwidth ω0. In this way, on the premise of ensuring that the coefficient matrix of the fixed-time ESO (referred to as FxTESO designed by the present invention) is a fixed Hurwitz matrix, only by adjusting the bandwidth ω0 can the observer performance be changed. The larger the ω0 is designed, the higher the observer accuracy, which significantly simplifies the parameter design and reduces the debugging difficulty.

[0160] In addition, the present invention also designs an adaptation rate (see Equation (37)) to estimate a bound k related to the error of FxTESO i , thereby reducing the influence of the FxTESO estimation error on the system performance, and while improving the robustness of the controller, avoiding the use of overestimated control gains.

[0161] Those of ordinary skill in the art will realize that the embodiments described herein are to assist the reader in understanding the principles of the present invention and should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. For those skilled in the art, the present invention can have various changes and modifications. Any modifications, equivalent replacements, improvements, 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 disturbance rejection and fault tolerance control method for a robotic arm based on a fixed-time extended state observer, characterized in that, Including: S1: Construct the Lagrangian dynamics model of an n-degree-of-freedom robotic arm considering actuator LOE faults, external disturbances, and unknown non-linearities; S2: Based on the constructed Lagrangian dynamics model of the robotic arm, design a fixed-time extended state observer to estimate the joint angular velocity and the lumped uncertainty in the system composed of actuator fault components, external disturbances, and unknown non-linearities; S3: Construct a sliding mode surface; S4: Based on the constructed sliding mode surface, design an adaptation rate to estimate a bound related to the error of the fixed-time extended state observer; S5: Based on the robotic arm dynamics model, fixed-time extended state observer, sliding mode surface, and adaptation rate in the foregoing steps, design an adaptive fixed-time fault-tolerant disturbance rejection controller; S6: Use the designed adaptive fixed-time fault-tolerant disturbance rejection controller to control the robotic arm with external disturbances, unknown non-linearities, and potential actuator LOE faults.

2. A disturbance rejection and fault tolerance control method for a robotic arm based on a fixed-time extended state observer according to claim 1, characterized in that The Lagrangian dynamics model of the robotic arm described in step S1 is expressed as: where q, are the angular position, velocity, and acceleration of the robotic arm joints, respectively, is the friction force vector, Λ is the actuator LOE fault coefficient matrix, τ d is the external disturbance, and Δ is the model uncertainty, expressed as △M, △C, and △G are the uncertain parts of the model; Pair After sorting, we get: where, I n is an n - order identity matrix, d(t) is the lumped uncertainty, and d(t) includes actuator fault components, external disturbances, and unknown non - linearities.

3. A disturbance rejection and fault tolerance control method for a robotic arm based on a fixed-time extended state observer according to claim 2, characterized in that, The fixed-time extended state observer in step S2 is specifically: Define the lumped uncertainty \(d(t)\) as an extended state variable \(x_3\), and assume that the derivative of \(x_3\) with respect to time is where \(\gamma = [\gamma_1,\ldots,\gamma n T is an unknown function;​ Then, the state variables of the robotic arm are written accordingly The fixed-time extended state observer FxTESO is designed in the following form: where, is the state estimate value, is the derivative of, j = 1, …, 3, and the estimation error is defined as g j is the designed gain, ω0 is the bandwidth of the fixed-time extended state observer; the function Sig a (x) = [sig a (x1), …, sig a (x n )] T , where sig a (x) = sgn(x)|x| a , α j = j(α - 1) + 1, β j = 1 / α + (j - 1)(α - 1), and α ∈ (1 - ε1, 1), where ε1 > 0 is a small constant.

4. A disturbance rejection and fault tolerance control method for a robotic arm based on a fixed-time extended state observer according to claim 3, characterized in that The representation form of the sliding mode surface in step S3 is: where s is an n-dimensional vector, i.e., s = [s1, …, s n T , s i represents the i-th sliding mode surface, i = 1, 2, …, n, a and b are two design parameters, 0 < a < 1, b > 1, q d is the ideal joint angle trajectory, e is the tracking error, e = q d - q, U a (e) = [u a (e1), …, u a (e n )] T , ​ 5. A disturbance rejection and fault tolerance control method for a robotic arm based on a fixed-time extended state observer according to claim 4, characterized in that, The adaptation rate designed in step S4 is expressed as: where k i is a bound related to the fixed-time extended state observer error, is the adaptation rate for estimating k i , o i is a small designed constant, a k,i and b k,i are positive design parameters, l > 1.

6. A disturbance rejection and fault tolerance control method for a robotic arm based on a fixed-time extended state observer according to claim 5, characterized in that, The adaptive fixed-time fault-tolerant disturbance rejection controller in step S5 is expressed as: wherein, and o i is a small constant of the design, 0 < m < 1, and the symbol ⊙ represents the Hadamard product, In addition, A s and B s are positive definite diagonal parameter matrices.

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