An adaptive neural network fault-tolerant cooperative control method for a multi-flexible arm system

Abstract

The present invention discloses an adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system. The method comprises the following steps: constructing a dynamic model of N flexible arms that takes into account actuator failures and time-varying parameter uncertainties; selecting an external reference signal as a virtual leader, connecting the N flexible arms as followers into a multi-flexible arm system, and constructing an auxiliary function for the multi-flexible arm system to achieve collaborative control; constructing a Lyapunov function based on the dynamic model and the auxiliary function; constructing an adaptive parameter update law and an adaptive neural network fault-tolerant collaborative controller; and applying an adaptive neural network to the multi-flexible arm system to achieve fault-tolerant collaborative control. The present invention can effectively suppress vibrations of the multi-flexible arm system, mitigate the effects of actuator failures and time-varying parameter uncertainties on control performance, and enable the motion trajectory of the multi-flexible arm system to track the virtual leader, achieving fault-tolerant collaborative control.

Description

Technical Field

[0001] The present invention relates to the technical fields of adaptive neural network control, vibration control, fault-tolerant control and multi-agent collaborative control, and in particular to an adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system. Background Art

[0002] Flexible arms have the characteristics of light weight, low energy consumption, and fast response, and can be widely used in robotics, mechanical engineering, aerospace and other fields. In the study of flexible arms, the Euler Bernoulli beam is usually used as the basic model. However, due to the influence of external disturbances and internal flexibility, the flexible arm based on the Euler Bernoulli beam model will inevitably continue to produce elastic deformation, which in turn causes the flexible arm connecting rod to vibrate for a long time. These non-negligible elastic vibrations not only reduce the working performance of the flexible arm, but also affect its material wear and service life. In addition, vibrations may also cause the parameters of the flexible arm to change continuously, thereby affecting its control accuracy. Therefore, how to achieve vibration control of the flexible arm under the influence of changing parameters is an urgent problem to be solved.

[0003] Traditional flexible arm control methods often assume that the arm's actuators are in normal working order. However, as a critical mechanical component that requires frequent operation in engineering projects, flexible arms are prone to actuator failure in harsh operating environments. A failure in the actuator, the controller's actuator, can adversely affect control accuracy and even cause accidents. Furthermore, if every actuator failure requires immediate attention, it would incur significant financial and labor costs.

[0004] Currently, most research on flexible arm control methods focuses on simply adjusting or tracking the joint angles of a single flexible arm. However, this control approach limits the application of flexible arms, preventing them from fully leveraging their advantages in situations where multiple individuals need to collaborate. Therefore, achieving coordinated control of multiple flexible arm systems is an urgent problem. Summary of the Invention

[0005] The purpose of the present invention is to solve the above-mentioned defects in the prior art and to provide an adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system.

[0006] The purpose of the present invention can be achieved by taking the following technical solutions:

[0007] An adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system, the adaptive neural network fault-tolerant collaborative control method comprising the following steps:

[0008] S1. Based on the dynamic characteristics of N flexible arms, a dynamic model is constructed that takes into account actuator failures and time-varying parameter uncertainties.

[0009] S2. Select an external reference signal as a virtual leader, connect N flexible arms as followers and form a multi-flexible arm system according to the undirected and interconnected rule, and construct an auxiliary function of the multi-flexible arm system to achieve collaborative control;

[0010] S3. constructing a Lyapunov function based on the dynamic model and the auxiliary function;

[0011] S4. Based on the Lyapunov function, combined with the adaptive radial basis function neural network approximation method and the robust adaptive parameter estimation technology, an adaptive parameter update law and an adaptive neural network fault-tolerant cooperative controller for the multi-flexible arm system are constructed;

[0012] S5. Based on the adaptive neural network fault-tolerant collaborative controller, an adaptive neural network is applied to the multi-flexible arm system to realize fault-tolerant collaborative control.

[0013] Furthermore, the dynamic characteristics of the N flexible arms in step S1 include the kinetic energy, potential energy, and virtual work done by the non-conservative force on the flexible arms. Substituting the kinetic energy, potential energy, and virtual work into the Hamiltonian principle, the dynamic model of the flexible arms is obtained as follows:

[0014] Where subscript j is the number of the j-th flexible arm, and the parameter or variable with subscript j represents that the parameter or variable belongs to the j-th flexible arm, j∈{1,2,...,N}, N is the total number of flexible arms; s is the spatial position variable, t is the time variable, L represents the same length of all flexible arm links; r j (s, t) represents the vibration displacement of the j-th flexible arm at position s and time t; z j (s, t) is the displacement of the j-th flexible arm at position s and time t, defined as z j (s,t)=sΘ j (t)+r j (s,t), where Θ j (t) is the joint angle of the jth flexible arm; represent the mass per unit length and bending stiffness of the j-th flexible arm link, respectively, and are defined as: Where ρ and Ξ are the rated mass per unit length and rated bending stiffness, which are the same for all flexible arm links, and Δ ρj (t), Δ Ξj (t) are the time-varying uncertainty of the unit length mass and bending stiffness of the j-th flexible arm link, and Δ ρj (t), Δ Ξj (t)Assumptions need to be met: are Δ ρj (t), Δ Ξj The upper bound of (t); z jtt (s,t),r jssss (s,t) are z j The second-order partial derivative of (s,t) with respect to time t and r j The fourth-order partial derivative of (s,t) with respect to position s;

[0015] The boundary conditions of the j-th flexible arm are:

[0016] In the formula are the hub inertia and end load mass of the j-th flexible arm, respectively, and are defined as: I h , m are the rated hub inertia and rated end load mass, which are the same for all flexible arms, Δ Ij (t), Δ mj (t) are respectively the time-varying uncertainty of the hub inertia and the end load mass of the j-th flexible arm, and Δ Ij (t), Δ mj (t)Assumptions to be met: They are Δ Ij (t), Δ mj The upper bound of (t); F 1j (t) is the torque output by the torque actuator at the j-th flexible arm hub, F 2j (t) is the control force output by the force actuator at the end of the j-th flexible arm; η 1j (t), η 2j (t) are the unknown time-varying boundary disturbances acting on the hub and the end, respectively, and satisfy are the boundary perturbations η 1j (t), η 2j The upper bound of (t); represents the angular acceleration of the jth flexible arm joint angle; r js (0,t),r jss (0, t) represent the first-order position partial derivative and the second-order position partial derivative of the vibration offset of the j-th flexible arm at position s = 0 and time t, respectively; r jss (L,t),r jsss (L, t) represent the second-order position partial derivative and the third-order position partial derivative of the vibration offset of the j-th flexible arm at position s = L and time t, respectively;

[0017] Considering that the actuator of the flexible arm may suffer from partial failure and offset failure, the actuator failure is defined as follows:

[0018] Wherein, the subscript i∈{1,2} represents the number of the controller and actuator of the flexible arm. The actuator and controller with the same number of each flexible arm are installed together. The subscript i=1 represents the variable of the torque actuator or torque controller at the hub of the flexible arm, and the subscript i=2 represents the variable of the force actuator or force controller at the end of the flexible arm; Σ ij (t) is the actuator fault term of the i-th actuator of the j-th flexible arm, defined as Among them, τ ij (t) is the i-th controller of the j-th flexible arm, α ij (t) represents the time-varying effective coefficient of the i-th actuator of the j-th flexible arm, represents the time-varying bias coefficient of the i-th actuator of the j-th flexible arm, and the actuator fault parameter α ij (t), Need to meet: α ij (t)∈(0,1], Β ij yes The upper bound of .

[0019] Note that the above time-varying parameter Δ ρj (t), Δ Ξj (t), Δ Ij (t), Δ mj (t), actuator fault parameter α ij (t), and the boundary perturbation η 1j (t), η 2j (t) is assumed to be unknown and bounded in the control method. The above dynamic model constructed based on the Hamiltonian principle is a high-order partial differential equation related to both time and spatial position. The boundary conditions are a set of ordinary differential equations related only to time. The construction of the dynamic model and boundary conditions is to find the relationship between the state variables of the flexible arm and the output joint angle and vibration offset, thereby providing a theoretical basis for the subsequent construction of auxiliary functions for the multi-flexible arm system.

[0020] The force actuator and torque actuator of the above-mentioned flexible arm can be collectively referred to as actuators. The control force and torque are generated by the output of the force actuator and torque actuator respectively. Actuators with the same number are installed together with the controller, and the input of the actuator is the controller with the same number. If there is no actuator failure, the input of the actuator is equal to the output. Once an actuator failure occurs, the input and output of the actuator will be inconsistent.

[0021] Furthermore, in step S2, the external reference signal is selected as a virtual leader, and the N flexible arms are connected as followers to form a multi-flexible arm system according to the undirected and interconnected rule. The auxiliary function of the multi-flexible arm system is constructed to achieve collaborative control as follows:

[0022] Select the external reference signal as the motion trajectory of a virtual leader, and number the virtual leader as 0; define the trajectory of the virtual leader as: Θ0(t) = Θ d ,r0(s,t)=r d , where Θ d is the desired joint angle of the N flexible arms, and the desired joint angle Θ d Angular velocity and angular acceleration are all 0, r d is the expected vibration offset of the N flexible arms. To suppress the vibration of the N flexible arms, it is necessary to define the expected vibration offset as r d =0;

[0023] N flexible arms are used as followers and connected into a multi-flexible arm system according to the undirected and interconnected rule. The undirected connection rule means that two adjacent flexible arms can obtain signals from each other, and the interconnected connection rule means that there is at least one signal transmission path between any two flexible arms.

[0024] The collaborative error of the jth flexible arm in the multi-flexible arm system is defined as:

[0025]

[0026]

[0027]

[0028] Where, They are the joint angle coordination error, vibration offset coordination error, and displacement coordination error of the j-th flexible arm respectively; Θ n (t), r n (s,t),z n (s, t) represent the joint angle, vibration offset, and displacement of the nth flexible arm, respectively. n∈{1,2,...,N} is defined for the convenience of summation operation, and the value of n is consistent with j; a jn The element with the subscript (j,n) in the adjacency matrix A, where the subscripts j and n refer to the numbers of the flexible arms, is the adjacency matrix A = {a jn}∈R N×N It is a non-negative matrix describing the connection relationship of N flexible arms in a multi-flexible arm system, which is defined as: if the j-th flexible arm can obtain the n-th flexible arm signal and j≠n, then a jn=1, otherwise, a jn =0;a j0 are the diagonal elements of the diagonal matrix Δ, and a j0 Located in the jth row and jth column of the diagonal matrix Δ, the diagonal matrix Δ=diag{a j0}∈R N×N It is a non-negative matrix describing the connection relationship between the multi-flexible arm system and the virtual leader, which is defined as: if the j-th flexible arm in the multi-flexible arm system can obtain the signal of the virtual leader, then a j0 =1, otherwise, a j0 =0; In the control method, it is necessary to assume that at least one flexible arm in the multi-flexible arm system can obtain the signal of the virtual leader;

[0029] According to the above definition of collaborative error, the vector form of collaborative error is:

[0030]

[0031]

[0032]

[0033] Where, are the joint angle coordination error vector, vibration offset coordination error vector, and displacement coordination error vector in the multi-flexible arm system, respectively, and are defined as follows: Θ(t), r(s,t), and z(s,t) are the joint angle vector, vibration offset vector, and displacement vector in the multi-flexible arm system, respectively, and are defined as follows: Θ(t) = [Θ j (t)] T ∈R N×1 ,r(s,t)=[r j (s,t)] T ∈R N×1 ,z(s,t)=[z j (s,t)] T ∈R N×1 ; 1 N is an N-dimensional all-1 vector, defined as: 1 N =[1,1,...,1] T ∈R N×1 ; M is a positive definite communication matrix that describes the connection relationship between N flexible arms in the multi-flexible arm system and the connection relationship between the multi-flexible arm system and the virtual leader, defined as M = L f +Δ, where L f is the Laplace matrix, defined as L f ={l jn}∈R N×N , ljn It's L f For elements with subscript (j,n), when j≠n, there is l jn =-a jn ,otherwise,

[0034] Based on the above definition of coordination error, the auxiliary function of the multi-flexible arm system is constructed as follows:

[0035]

[0036]

[0037] Where, κ 1j (t),κ 2j (t) are the first auxiliary function and the second auxiliary function for realizing the coordinated control of the j-th flexible arm in the multi-flexible arm system; are the first-order position partial derivative and the third-order position partial derivative of the vibration offset coordination error of the j-th flexible arm at position s = L and time t, respectively; is the first-order time partial derivative of the displacement coordination error of the j-th flexible arm at position s = L and time t; is the first-order time partial derivative of the joint angle coordination error of the j-th flexible arm at position s = L and time t;

[0038] The auxiliary function κ of the above multi-flexible arm system 1j (t),κ 2j (t) and the coordinated error vector The definition of the auxiliary function κ 1j (t),κ 2j The vector form of (t) is:

[0039] κ1(t)=[κ 1j (t)] T =M[r s (L,t)+z t (L,t)-r sss (L,t)]∈R N×1 ,

[0040]

[0041] Where r s (L,t),r sss (L,t) are The vector form is defined as: s (L,t)=[r js (L,t)] T ∈R N×1 , r sss(L,t)=[r jsss (L,t)] T ∈R N×1 ;z t (L,t) is The vector form of z is defined as: t (L,t)=[z t (L,t)] T ∈R N×1 ; is the joint angular velocity vector of the multi-flexible arm system, defined as in, is the joint angular velocity of the jth flexible arm in the multi-flexible arm system.

[0042] The aforementioned collaborative error includes the state variables of the flexible arms in the multi-flexible-arm system, as well as the connections between the N flexible arms in the system and between the multi-flexible-arm system and the virtual leader. Therefore, the auxiliary function constructed based on the collaborative error is the foundation for achieving collaborative control and an important component of the subsequent construction of a Lyapunov function that reflects the total energy of the multi-flexible-arm system. Furthermore, the causal relationship between the auxiliary function and collaborative control is that if the auxiliary function can converge to near zero under the influence of the subsequently constructed controller, then the motion trajectory of the multi-flexible system can also converge to near the virtual leader, thus achieving collaborative control.

[0043] Furthermore, the process of constructing the Lyapunov function based on the dynamic model and the auxiliary function in step S3 is as follows:

[0044] Based on the dynamic model and auxiliary functions, the Lyapunov function is constructed as:

[0045] P(t)=P1(t)+P2(t)+P3(t)+P4(t),

[0046] in

[0047]

[0048]

[0049]

[0050] Where, π 1j , π 2j , π 3j are the first energy coefficient, second energy coefficient, and third energy coefficient of the j-th flexible arm in the multi-flexible arm system to achieve energy constraint; μ 1j 、μ 2jare the first control parameter and the second control parameter for realizing coordinated control of the j-th flexible arm in the multi-flexible arm system respectively; are the first weight estimation error vector and the second weight estimation error vector for adaptive neural network fault-tolerant control of the j-th flexible arm in the multi-flexible arm system, respectively, and are defined as: in, are the first ideal weight vector and the second ideal weight vector of the j-th flexible arm, respectively. are the first weight vector estimation value and the second ideal weight vector estimation value of the j-th flexible arm respectively; The first parameter estimation error and the second parameter estimation error of the j-th flexible arm in the multi-flexible arm system are respectively the estimation boundary perturbation and the upper bound of the neural network approximation error, which are defined as: in, are the first parameter estimation value and the second parameter estimation value of the j-th flexible arm, respectively. are the first unknown parameter and the second unknown parameter of the j-th flexible arm respectively and are defined as: are the upper bounds of the first neural network approximation error and the second neural network approximation error of the j-th flexible arm respectively; They are respectively the first weight estimation adjustment coefficient and the second weight estimation adjustment coefficient of the j-th flexible arm in the multi-flexible arm system to realize the neural network weight estimation; They are the first parameter estimation adjustment coefficient and the second parameter estimation adjustment coefficient for unknown parameter estimation of the j-th flexible arm in the multi-flexible arm system; are the vibration offset coordination errors of the jth flexible arm at position s and time t. The first-order partial derivative and second-order partial derivative of .

[0051] The Lyapunov function above includes the dynamic model of the N flexible arms in the multi-flexible arm system, the state variables in the boundary conditions, and the error variables in the parameter estimation, which can reflect the overall energy of the closed-loop multi-flexible arm system. The purpose of constructing this Lyapunov function is to ensure that the subsequent construction of the adaptive neural network fault-tolerant collaborative controller is based on the overall energy attenuation of the multi-flexible arm system that can be reflected by the Lyapunov function.

[0052] Furthermore, in step S4, based on the Lyapunov function, combined with the adaptive radial basis function neural network approximation method and the robust adaptive parameter estimation technology, the process of constructing the adaptive parameter update law and the adaptive neural network fault-tolerant cooperative controller of the multi-flexible arm system is as follows:

[0053] Adaptive radial basis function neural network approximation method is used to solve the problem of time-varying parameter uncertainty Δρj (t), Δ Ξj (t), Δ Ij (t), Δ mj (t) and actuator fault parameter α ij (t), The unknown terms are approximated as follows:

[0054] Where S 1j (Z 1j ), S 2j (Z 2j ) are the first radial basis function and the second radial basis function of the adaptive neural network fault-tolerant control of the j-th flexible arm in the multi-flexible arm system, respectively. The radial basis function S(Z) is defined in the radial basis neural network as: S(Z) = exp(-||ZC|| 2 / b 2 ), Z is the input vector of the radial basis neural network, C is the center of the input vector variation range, b is the width of the input vector variation range, and the symbol exp(*) is equivalent to e * , the symbol ||*|| is equivalent to taking the norm of the vector *; therefore, Z 1j , Z 2j are the first input vector and the second input vector of the j-th flexible arm, respectively, and are defined as:

[0055]

[0056]

[0057] Among them, sgn[*] is the sign function, r jssst (L, t) is the vibration displacement r of the jth flexible arm at position s = L j (L, t) Simultaneously find the first-order partial derivative with respect to time t and the third-order partial derivative with respect to position s; r jst (L, t) is the vibration displacement r of the jth flexible arm at position s = L j (L, t) calculates the first-order partial derivative with respect to time t and the first-order partial derivative with respect to position s at the same time; δ 1j (Z 1j ),δ 2j (Z 2j ) are the first neural network approximation error and the second neural network approximation error of the adaptive neural network fault-tolerant control of the j-th flexible arm in the multi-flexible arm system, and they satisfy:

[0058] The ideal weight vector of the neural network is estimated by using robust adaptive parameter estimation technology. and unknown parameters To make an estimate, the following adaptive parameter update law for the multi-flexible arm system is constructed:

[0059]

[0060]

[0061] Where g 1j 、g 2j 、g 3j 、g 4j are the first robust parameter, the second robust parameter, the third robust parameter, and the fourth robust parameter for ensuring the robustness of the parameter estimation value of the j-th flexible arm in the multi-flexible arm system;

[0062] The first-order derivative of the Lyapunov function with respect to time t is obtained. Based on the Lyapunov stability theory, combined with the above-mentioned adaptive radial basis function neural network approximation method and robust adaptive parameter estimation technology, the following adaptive neural network fault-tolerant cooperative controller for the multi-flexible arm system is finally constructed:

[0063]

[0064]

[0065] Where, τ 1j (t), τ 2j (t) are the first controller and the second controller of the j-th flexible arm in the multi-flexible arm system; w 1j 、.w 2j . are respectively the third control parameter and the fourth control parameter for realizing coordinated control of the j-th flexible arm in the multi-flexible arm system.

[0066] In the above adaptive neural network fault-tolerant cooperative controller, the actuator fault parameter α ij (t), and time-varying parameter uncertainty Δ ρj (t), Δ Ξj (t), Δ Ij (t), Δ mj The adverse effects of (t) on the control performance of the multi-flexible arm system are compensated by the adaptive radial basis function neural network approximation method, and the boundary disturbance η 1j (t), η 2j The upper bound of (t) and the neural network approximation error δ 1j (Z 1j ),δ 2j (Z 2j ) Estimated by the robust adaptive parameter estimation technology, the present invention can alleviate the impact of actuator failure and time-varying parameter uncertainty on control performance, and can also handle boundary disturbances and neural network approximation errors.

[0067] Note that the state signals in the above-mentioned adaptive parameter update law and adaptive neural network fault-tolerant cooperative controller can be obtained by sensor sampling or finite difference method. After obtaining the state signal, it needs to be calculated in the computing unit of the flexible arm controller before the actuator can be updated, thereby applying fault-tolerant cooperative control to the multi-flexible arm system.

[0068] Furthermore, the process of constructing the adaptive parameter update law and the adaptive neural network fault-tolerant cooperative controller of the multi-flexible arm system also includes a step of verifying the stability of the multi-flexible arm system under the action of the adaptive neural network fault-tolerant cooperative controller, and the process is as follows:

[0069] By constraining the energy coefficient π in the Lyapunov function 1j , π 2j , π 3j , ensuring the positive definiteness of the Lyapunov function;

[0070] Taking the first derivative of the Lyapunov function with respect to time t;

[0071] Lyapunov bounded stability theory is applied to verify the stability of the multi-flexible arm system under the action of the adaptive neural network fault-tolerant cooperative controller.

[0072] The above steps of verifying the stability of the closed-loop multi-flexible arm system are aimed at verifying the rationality of the adaptive neural network fault-tolerant collaborative controller, because a reasonable adaptive neural network fault-tolerant collaborative controller should be able to attenuate the overall energy of the multi-flexible arm system. According to the Lyapunov bounded stability theory, when the state variables or error variables of the closed-loop multi-flexible arm system are bounded and stable, it can be seen that the overall energy of the multi-flexible arm system is attenuated, and it is further concluded that the adaptive neural network fault-tolerant collaborative controller is reasonable.

[0073] Furthermore, in step S5, based on the adaptive neural network fault-tolerant cooperative controller, the process of applying the adaptive neural network to the multi-flexible arm system to achieve fault-tolerant cooperative control is as follows:

[0074] After N flexible arms are connected to form a multi-flexible arm system according to the rule of undirected and interconnected, for the j-th flexible arm in the multi-flexible arm system, the process of applying the adaptive neural network to realize fault-tolerant cooperative control specifically refers to calculating the adaptive parameter update law in the computing unit of the i-th controller of the j-th flexible arm. and adaptive neural network fault-tolerant cooperative controller τ ij(t), where the controller τ ij (t) contains adaptive parameter estimates and adaptive neural network estimates

[0075] Controller τ ij (t) After the calculation is completed, the i-th actuator of the j-th flexible arm is updated. As the control time increases, the actuator continuously outputs control force or torque, which affects the joint angle Θ j (t) and vibration offset r j (s, t) is continuously adjusted, and the multi-flexible arm system's trajectory ultimately tracks the virtual leader. Even if actuator failures and time-varying parameter uncertainties cause the multi-flexible arm system's trajectory to deviate from the virtual leader, the adaptive neural network can quickly self-adjust to compensate for the actuator failure and time-varying parameter uncertainty, and the multi-flexible arm system's deviated trajectory will be re-tracked to the virtual leader, achieving fault-tolerant collaborative control.

[0076] Furthermore, the adaptive neural network fault-tolerant collaborative controller can achieve vibration suppression of the multi-flexible arm system under the influence of actuator failure and time-varying parameter uncertainty, allowing the motion trajectory of the multi-flexible arm system to track the virtual leader, and ultimately achieve the adaptive neural network fault-tolerant collaborative control effect of the multi-flexible arm system.

[0077] The present invention has the following advantages and effects compared to the prior art:

[0078] (1) Compared with the traditional single flexible arm control method, the adaptive neural network fault-tolerant collaborative control method of a multi-flexible arm system proposed in the present invention can not only achieve vibration suppression of the multi-flexible arm system, but also enable the motion trajectory of the multi-flexible arm system to track the virtual leader to achieve collaborative control.

[0079] (2) In the control method proposed in the present invention, actuator failures and time-varying parameter uncertainties are compensated by the adaptive radial basis function neural network approximation method, and the upper bounds of boundary disturbances and neural network approximation errors are estimated by the robust adaptive parameter estimation technology. Therefore, the present invention can alleviate the impact of actuator failures and time-varying parameter uncertainties on control performance.

[0080] (3) In the control method proposed in the present invention, each flexible arm of the multi-flexible arm system includes two types of boundary controllers: one type of boundary controller is installed at the wheel hub to achieve joint angle coordinated control, and the controller signal mainly comes from the difference between its own signal and the signal of the adjacent flexible arm. In particular, the flexible arm connected to the virtual leader can also obtain the signal of the virtual leader (i.e., the external reference signal); the other type of boundary controller is installed at the end to achieve vibration suppression, and the control signal also comes from the difference between its own signal and the signal of the adjacent flexible arm. By selecting appropriate controller parameters, the adaptive parameter update law and radial basis function neural network in the controller can quickly converge to alleviate the adverse effects of actuator failure and time-varying parameter uncertainty, and thus the multi-flexible arm system can quickly track the virtual leader and achieve vibration suppression. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of this application. The exemplary 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. In the drawings:

[0082] Figure 1 This is a flow chart of an adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system disclosed in the present invention;

[0083] Figure 2 This is a schematic diagram of a typical flexible arm structure in Example 1 of the present invention;

[0084] Figure 3 This is an example diagram of the connection topology of the multi-flexible arm system in Example 1 of the present invention;

[0085] Figure 4 2 is a schematic diagram of the simulation results of the vibration offset of the multi-flexible arm system without control in Example 2 of the present invention;

[0086] Figure 5 2 is a schematic diagram of the simulation results of the vibration offset of the multi-flexible arm system under the adaptive neural network fault-tolerant collaborative control method in Example 2 of the present invention;

[0087] Figure 6 It is a schematic diagram of the joint angle simulation results of the multi-flexible arm system under the adaptive neural network fault-tolerant collaborative control method in Example 2 of the present invention. DETAILED DESCRIPTION

[0088] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts shall fall within the scope of protection of the present invention.

[0089] Example 1

[0090] Figure 1 : is a flow chart of an adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system disclosed in Example 1, which specifically includes the following steps:

[0091] S1. Based on the dynamic characteristics of N flexible arms, a dynamic model is constructed that takes into account actuator failures and time-varying parameter uncertainties.

[0092] Figure 2 The figure shows a typical single-link flexible arm based on the Euler Bernoulli beam model. The subscript j is the number of the j-th flexible arm. The parameter or variable with the subscript j represents that the parameter or variable belongs to the j-th flexible arm, j∈{1,2,...,N}, and N is the total number of flexible arms. Figure 2 The circular pattern on the left represents a wheel hub that can rotate on a plane and is equipped with a torque F. 1j The torque actuator of (t) may be subjected to the same direction torque disturbance η 1j (t), the thick line in the middle represents the connecting rod with a length of L, and the small ball on the right connected to the connecting rod represents the mass of The end load is subject to the control force F output by the force actuator 2j (t) and the same direction force disturbance η 2j (t) The coordinate system SoR represents the inertial coordinate system with the hub center o as the origin and the horizontal direction as the S axis; the coordinate system sor represents the self-coordinate system with the hub center o as the origin and the tangent direction of the connecting rod starting point as the s axis; therefore, r j (s, t) is the vibration offset of the j-th flexible arm at position s∈[0,L] in its own coordinate system sor and time t∈[0,∞), Θ j (t) is the joint angle of the hub rotation of the j-th flexible arm at position s in the coordinate system SoR at time t, z j (s,t)=sΘ j (t)+r j (s,t) is the displacement of the connecting rod at position s and time t in the SoR coordinate system.

[0093] The dynamic model of the j-th flexible arm is:

[0094] Where, s is the spatial position variable, t is the time variable, L represents the same length of all flexible arm links; r j (s, t) represents the vibration offset of the jth flexible arm at position s and time t; represent the mass per unit length and bending stiffness of the j-th flexible arm link, respectively, and are defined as: Where ρ and Ξ are the rated mass per unit length and rated bending stiffness, which are the same for all flexible arm links, and Δ ρj (t), Δ Ξj (t) are the time-varying uncertainty of the unit length mass and bending stiffness of the j-th flexible arm link, and Δ ρj (t), Δ Ξj (t)Assumptions need to be met: are Δ ρj (t), Δ Ξj The upper bound of (t); z jtt (s,t),r jssss (s,t) are z j The second-order partial derivative of (s,t) with respect to time t, r j The fourth-order partial derivative of (s,t) with respect to position s.

[0095] The boundary conditions of the j-th flexible arm are:

[0096] In the formula are the hub inertia and end load mass of the j-th flexible arm, respectively, and are defined as: I h , m are the rated hub inertia and rated end load mass, which are the same for all flexible arms, Δ Ij (t), Δ mj (t) are respectively the time-varying uncertainty of the hub inertia and the end load mass of the j-th flexible arm, and Δ Ij (t), Δ mj (t)Assumptions to be met: They are Δ Ij (t), Δ mj The upper bound of (t); F 1j (t) is the torque output by the torque actuator at the jth flexible arm hub (s = 0), F 2j (t) is the control force output by the force actuator at the end of the j-th flexible arm (s = L); η 1j (t), η2j (t) are the boundary disturbances acting on the hub and the end, respectively, and satisfy are the boundary perturbations η 1j (t), η 2j The upper bound of (t); represents the angular acceleration of the jth flexible arm joint angle; r js (0,t),r jss (0, t) represent the first-order position partial derivative and the second-order position partial derivative of the vibration offset of the j-th flexible arm at position s = 0 at time t; r jss (L,t),r jsss (L, t) represent the second-order position partial derivative and the third-order position partial derivative of the vibration offset of the j-th flexible arm at position s = L and time t, respectively.

[0097] Considering that the actuator of the flexible arm may suffer from partial failure and offset failure, the actuator failure is defined as follows:

[0098] Wherein, the subscript i∈{1,2} represents the number of the controller and actuator of the flexible arm. The actuator and controller with the same number of each flexible arm are installed together. The subscript i=1 represents the variable of the torque actuator or torque controller at the hub of the flexible arm, and the subscript i=2 represents the variable of the force actuator or force controller at the end of the flexible arm; Σ ij (t) is the actuator fault term of the i-th actuator of the j-th flexible arm, defined as Among them, τ ij (t) is the i-th controller of the j-th flexible arm, α ij (t) represents the time-varying effective coefficient of the i-th actuator of the j-th flexible arm, represents the time-varying bias coefficient of the i-th actuator of the j-th flexible arm, and the actuator fault parameter α ij (t), Need to meet: α ij (t)∈(0,1], Β ij yes The upper bound of .

[0099] Note that the above time-varying parameter Δ ρj (t), Δ Ξj (t), Δ Ij (t), Δ mj (t), actuator fault parameter α ij (t), and the boundary perturbation η 1j (t), η 2j(t) is assumed to be unknown and bounded in the control method. The above dynamic model constructed based on the Hamiltonian principle is a high-order partial differential equation related to both time and spatial position. The boundary conditions are a set of ordinary differential equations related only to time. Force actuators and torque actuators can be collectively referred to as actuators. The control force and torque are generated by the output of the force actuator and torque actuator respectively. Actuators with the same number are installed together with the controller, and the input of the actuator is the controller with the same number. If there is no actuator failure, the input of the actuator is equal to the output. Once an actuator failure occurs, the input and output of the actuator will be inconsistent.

[0100] S2. Select an external reference signal as a virtual leader, connect N flexible arms as followers and form a multi-flexible arm system according to the undirected and interconnected rule, and construct an auxiliary function of the multi-flexible arm system to achieve collaborative control.

[0101] Select the external reference signal as the motion trajectory of a virtual leader, and number the virtual leader as 0; define the trajectory of the virtual leader as: Θ0(t) = Θ d ,r0(s,t)=r d , where Θ d is the desired joint angle of the N flexible arms, and the desired joint angle Θ d Angular velocity and angular acceleration are all 0, r d is the expected vibration offset of the N flexible arms. To suppress the vibration of the N flexible arms, it is necessary to define the expected vibration offset as r d =0.

[0102] N flexible arms are used as followers and connected into a multi-flexible arm system according to the undirected and interconnected rules. The undirected connection rule means that two adjacent connected flexible arms can obtain signals from each other, and the interconnected connection rule means that there is at least one signal transmission path between any two flexible arms.

[0103] like Figure 3 The figure shows an example of the connection topology of a multi-flexible arm system composed of four flexible arms. In the figure, the one-way arrow indicates that the flexible arm pointed to by the arrow tip can obtain signals from the flexible arm where the arrow tail is located, and the two-way arrow indicates that two flexible arms can exchange signals with each other. As you can see, Figure 3 One virtual leader (external reference signal) is numbered 0, and four flexible arms as followers are numbered 1 to 4. The connection relationship between the four flexible arms as followers satisfies the undirected and connected rules. In addition, only the flexible arms numbered 2 and 4 can obtain signals from the virtual leader 0.

[0104] The collaborative error of the jth flexible arm in the multi-flexible arm system is defined as:

[0105]

[0106]

[0107]

[0108] Where, They are the joint angle coordination error, vibration offset coordination error, and displacement coordination error of the j-th flexible arm respectively; Θ n (t), r n (s,t),z n (s, t) represent the joint angle, vibration offset, and displacement of the nth flexible arm, respectively. n∈{1,2,...,N} is defined for the convenience of summation operation, and the value of n is consistent with j; a jn The element with the subscript (j,n) in the adjacency matrix A, where the subscripts j and n refer to the numbers of the flexible arms, is the adjacency matrix A = {a jn}∈R N×N It is a non-negative matrix describing the connection relationship of N flexible arms in a multi-flexible arm system, which is defined as: if the j-th flexible arm can obtain the n-th flexible arm signal and j≠n, then a jn =1, otherwise, a jn =0;a j0 are the diagonal elements of the diagonal matrix Δ, and a j0 Located in the jth row and jth column of the diagonal matrix Δ, the diagonal matrix Δ=diag{a j0}∈R N×N It is a non-negative matrix describing the connection relationship between the multi-flexible arm system and the virtual leader, which is defined as: if the j-th flexible arm in the multi-flexible arm system can obtain the signal of the virtual leader, then a j0 =1, otherwise, a j0 =0; In the control method, it is necessary to assume that at least one flexible arm in the multi-flexible arm system can obtain the signal of the virtual leader.

[0109] According to the above definition of collaborative error, the vector form of collaborative error is:

[0110]

[0111]

[0112]

[0113] Where, They are the joint angle coordination error vector, vibration offset coordination error vector, and displacement coordination error vector in the multi-flexible arm system, respectively, and are defined as: Θ(t), r(s,t), and z(s,t) are the joint angle vector, vibration offset vector, and displacement vector in the multi-flexible arm system, respectively, and are defined as: Θ(t) = [Θ j (t)] T ∈R N×1 ,r(s,t)=[r j (s,t)] T ∈R N×1 ,z(s,t)=[z j (s,t)] T ∈R N×1 ; 1 N is an N-dimensional all-1 vector, defined as: 1 N =[1,1,...,1] T ∈R N×1 ; M is a positive definite communication matrix that describes the connection relationship between N flexible arms in the multi-flexible arm system and the connection relationship between the multi-flexible arm system and the virtual leader, defined as M = L f +Δ, where L f is the Laplace matrix, defined as L f ={l jn}∈R N×N , l jn It's L f For elements with subscript (j,n), when j≠n, there is l jn =-a jn ,otherwise,

[0114] Based on the above definition of coordination error, the auxiliary function of constructing the multi-flexible arm system is as follows:

[0115]

[0116]

[0117] Where, κ 1j (t),κ 2j (t) are the first and second auxiliary functions for realizing coordinated control of the j-th flexible arm in the multi-flexible arm system; are the first-order and third-order position partial derivatives of the vibration offset coordination error of the j-th flexible arm at position s = L and time t, respectively; is the first-order time partial derivative of the displacement coordination error of the j-th flexible arm at position s = L and time t; is the first-order time partial derivative of the joint angle coordination error of the j-th flexible arm at position s=L and time t.

[0118] The auxiliary function κ of the above multi-flexible arm system1j (t),κ 2j (t) and the coordinated error vector The definition of the auxiliary function κ 1j (t),κ 2j The vector form of (t) is:

[0119] κ1(t)=[κ 1j (t)] T =M[r s (L,t)+z t (L,t)-r sss (L,t)]∈R N×1 ,

[0120]

[0121] Where r s (L,t),r sss (L,t) are The vector form is defined as: s (L,t)=[r js (L,t)] T ∈R N×1 , r sss (L,t)=[r jsss (L,t)] T ∈R N×1 ;z t (L,t) is The vector form of z is defined as: t (L,t)=[z t (L,t)] T ∈R N×1 ; is the joint angular velocity vector of the multi-flexible arm system, defined as in, is the joint angular velocity of the jth flexible arm in the multi-flexible arm system.

[0122] The above-mentioned collaborative error includes the state variables of the flexible arms in the multi-flexible-arm system, and also implies the connection relationship between the N flexible arms in the multi-flexible-arm system and the connection relationship between the multi-flexible-arm system and the virtual leader. Therefore, the auxiliary function constructed based on the collaborative error is the basis for realizing collaborative control, and is also an important component of the subsequent construction of the Lyapunov function that can reflect the total energy of the multi-flexible-arm system.

[0123] S3. Based on the dynamic model and auxiliary functions, construct the Lyapunov function.

[0124] Based on the dynamic model and auxiliary functions, the Lyapunov function is constructed as:

[0125] P(t)=P1(t)+P2(t)+P3(t)+P4(t),

[0126] in

[0127]

[0128]

[0129]

[0130] Where, π 1j , π 2j , π 3j are the first, second and third energy coefficients for the j-th flexible arm in the multi-flexible arm system to achieve energy constraint; μ 1j 、μ 2j are the first and second control parameters for achieving coordinated control of the j-th flexible arm in the multi-flexible arm system; are the first and second weight estimation error vectors for adaptive neural network fault-tolerant control of the j-th flexible arm in the multi-flexible arm system, respectively, and are defined as: in, are the first and second ideal weight vectors of the j-th flexible arm, are the first and second ideal weight vector estimates of the j-th flexible arm, respectively; are the first and second parameter estimation errors for the j-th flexible arm in the multi-flexible arm system, which are the estimated boundary perturbation and the upper bound of the neural network approximation error, respectively, and are defined as: in, are the first and second parameter estimates of the j-th flexible arm, are the first unknown parameter and the second unknown parameter of the j-th flexible arm respectively and are defined as: are the upper bounds of the first and second neural network approximation errors of the j-th flexible arm, respectively; are the first and second weight estimation adjustment coefficients for the j-th flexible arm in the multi-flexible arm system to realize the neural network weight estimation; are the first and second parameter estimation adjustment coefficients for unknown parameter estimation of the j-th flexible arm in the multi-flexible arm system; are the vibration offset coordination errors of the jth flexible arm at position s and time t. The first and second order position partial derivatives of .

[0131] The above Lyapunov function includes the dynamic model of the N flexible arms in the multi-flexible arm system and the state variables in the boundary conditions, as well as the error variables of parameter estimation, which can reflect the overall energy situation of the multi-flexible arm system. The subsequent construction of the adaptive neural network fault-tolerant collaborative controller needs to be based on the ability to attenuate the overall energy of the multi-flexible arm system.

[0132] S4. Based on the Lyapunov function, combined with the adaptive radial basis function neural network approximation method and robust adaptive parameter estimation technology, an adaptive parameter update law and an adaptive neural network fault-tolerant collaborative controller for the multi-flexible arm system are constructed.

[0133] Adaptive radial basis function neural network approximation method is used to solve the problem of time-varying parameter uncertainty Δ ρj (t), Δ Ξj (t), Δ Ij (t), Δ mj (t) and actuator fault parameter α ij (t), The unknown terms are approximated as follows:

[0134] Where S 1j (Z 1j ), S 2j (Z 2j ) are the first and second radial basis functions for adaptive neural network fault-tolerant control of the j-th flexible arm in the multi-flexible arm system, respectively. The radial basis function S(Z) is defined in the radial basis neural network as: S(Z) = exp(-||ZC|| 2 / b 2 ), Z is the input vector of the radial basis neural network, C is the center of the input vector variation range, b is the width of the input vector variation range, and the symbol exp(*) is equivalent to e * , the symbol ||*|| is equivalent to taking the norm of the vector *; therefore, Z 1j , Z 2j are the first and second input vectors of the j-th flexible arm, respectively, and are defined as:

[0135]

[0136]

[0137] Among them, sgn[*] is the sign function, r jssst (L, t) is the vibration displacement r of the jth flexible arm at position s = L j (L, t) Simultaneously find the first-order partial derivative with respect to time t and the third-order partial derivative with respect to position s; r jst(L, t) is the vibration displacement r of the jth flexible arm at position s = L j (L, t) calculates the first-order partial derivative with respect to time t and the first-order partial derivative with respect to position s at the same time; δ 1j (Z 1j ),δ 2j (Z 2j ) are the first and second neural network approximation errors of the adaptive neural network fault-tolerant control of the j-th flexible arm in the multi-flexible arm system, and they satisfy: The ideal weight vector of the neural network is estimated by using robust adaptive parameter estimation technology. and unknown parameters To make an estimate, the following adaptive parameter update law for the multi-flexible arm system is constructed:

[0138]

[0139]

[0140] Where g 1j 、g 2j 、g 3j 、g 4j They are the first robust parameter, second robust parameter, third robust parameter and fourth robust parameter for ensuring the robustness of parameter estimation of the j-th flexible arm in the multi-flexible arm system.

[0141] The first-order derivative of the Lyapunov function with respect to time t is obtained. Based on the Lyapunov stability theory, combined with the above-mentioned adaptive radial basis function neural network approximation method and robust adaptive parameter estimation technology, the following adaptive neural network fault-tolerant cooperative controller for the multi-flexible arm system is finally constructed:

[0142]

[0143]

[0144] Where, τ 1j (t), τ 2j (t) are the first controller and the second controller of the j-th flexible arm in the multi-flexible arm system; w 1j 、w 2j are the third control parameter and the fourth control parameter for realizing coordinated control of the j-th flexible arm in the multi-flexible arm system respectively.

[0145] In the above adaptive neural network fault-tolerant cooperative controller, the actuator fault parameter α ij (t), and time-varying parameter uncertainty Δ ρj (t), Δ Ξj (t), Δ Ij (t), Δmj The adverse effects of (t) on the control performance of the multi-flexible arm system are compensated by the adaptive radial basis function neural network approximation method, and the boundary disturbance η 1j (t), η 2j The upper bound of (t) and the neural network approximation error δ 1j (Z 1j ),δ 2j (Z 2j ) The robust adaptive parameter estimation technique can mitigate the impact of actuator failures and time-varying parameter uncertainty on control performance, while also handling boundary disturbances and neural network approximation errors. Furthermore, the state signals in the adaptive parameter update law and the adaptive neural network fault-tolerant collaborative controller can be obtained using sensor sampling or finite difference calculations.

[0146] Note that the process of constructing the adaptive parameter update law and adaptive neural network fault-tolerant cooperative controller of the multi-flexible arm system is actually based on the fact that the adaptive neural network fault-tolerant cooperative controller can attenuate the total energy of the multi-flexible arms. Therefore, it is also necessary to verify the stability of the multi-flexible arm system under the action of the adaptive neural network fault-tolerant cooperative controller to illustrate the rationality of the adaptive neural network fault-tolerant cooperative controller. The specific process is as follows:

[0147] (1) By constraining the energy coefficient π in the Lyapunov function 1j , π 2j , π 3j , to ensure the positive definiteness of the Lyapunov function, we first introduce the following inequality lemma:

[0148] Lemma 1: For N-dimensional vector x,y∈R N×1 ,have

[0149] For a symmetric positive definite matrix M∈R N×N , and its maximum eigenvalue λ max (M) and the minimum eigenvalue λ min (M), the inequality holds Among them, the symbol ||*||2 represents the 2-norm of vector *, and the symbol λ min (*), λ max (*) represents the minimum eigenvalue and maximum eigenvalue of the matrix *.

[0150] Lemma 2: For z(s,t), if z(0,t)=z s (0,t)=0,s∈[0,L],t∈[0,∞), then where z s(s,t),z ss (s,t) are the first-order partial derivative and second-order partial derivative of z(s,t) with respect to s respectively.

[0151] Using the above inequality Lemmas 1 and 2, we can get the Lyapunov function P(t) Scaling, we can get

[0152] Where, The symbol min{*,...,*} represents the minimum value of the elements in the set {*,...,*}, and the symbol max{*,...,*} represents the maximum value of the elements in the set {*,...,*}.

[0153] Adding P3(t) and P4(t) in the Lyapunov function to both sides of the inequality (1-χ1)P1(t)≤P2(t)+P1(t)≤(1+χ1)P1(t), we can obtain: χ2[P1(t)+P3(t)+P4(t)]≤P(t)≤χ3[P1(t)+P3(t)+P4(t)] where χ2=min{1-χ1,1}, χ3=max{1+χ1,1}, and the constraint π 1j , π 2j , π 3j The positive definiteness of the Lyapunov function can be guaranteed by making 1-χ1>0.

[0154] (2) Find the first-order derivative of the Lyapunov function with respect to time t, as follows:

[0155] Find the first derivative of the Lyapunov function P(t) with respect to time t And by scaling up the inequality Lemmas 1 and 2, we can get:

[0156]

[0157] Where θ 1j ,θ 2j are the first scaling factor and the second scaling factor respectively, ξ=χ4 / χ3,

[0158] (3) Applying Lyapunov bounded stability theory, the stability of the multi-flexible arm system under the action of the adaptive neural network fault-tolerant cooperative controller is verified as follows:

[0159] When the inequality is satisfied, we can get ξ>0, at this time, for the above Multiply both sides of the inequality by e ξt , and integrate from 0 to t, and we can get:

[0160]

[0161] Combined with the definition of the Lyapunov function, we can finally get:

[0162]

[0163]

[0164] According to the above results, the collaborative error It will eventually converge to a neighborhood near 0. Applying Lyapunov bounded stability theory, it can be obtained that the multi-flexible arm system is uniformly ultimately bounded and stable under the action of the adaptive neural network fault-tolerant cooperative controller.

[0165] The above stability verification results show that the overall energy of the multi-flexible arm system reflected by the Lyapunov function is attenuated under the action of the adaptive neural network fault-tolerant cooperative controller. Therefore, the adaptive neural network fault-tolerant cooperative controller constructed in this step is reasonable.

[0166] S5. Based on the adaptive neural network fault-tolerant cooperative controller, an adaptive neural network is applied to the multi-flexible arm system to realize fault-tolerant cooperative control.

[0167] After N flexible arms are connected to form a multi-flexible arm system according to the rule of undirected and interconnected, for the j-th flexible arm in the multi-flexible arm system, the process of applying the adaptive neural network to realize fault-tolerant cooperative control specifically refers to calculating the adaptive parameter update law in the computing unit of the i-th controller of the j-th flexible arm. and adaptive neural network fault-tolerant cooperative controller τ ij (t), after the calculation is completed, the i-th actuator of the j-th flexible arm is updated. As the control time increases, the actuator continuously outputs control force or torque, which affects the joint angle Θ j (t) and vibration offset r j (s, t) is continuously adjusted, and finally the motion trajectory of the multi-flexible arm system tracks the virtual leader to achieve fault-tolerant collaborative control.

[0168] In summary, this embodiment provides an adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system, including: constructing a dynamic model that takes into account actuator failures and time-varying parameter uncertainties based on the dynamic characteristics of N flexible arms; selecting an external reference signal as a virtual leader, and connecting N flexible arms as followers into a multi-flexible arm system, and constructing an auxiliary function for the multi-flexible arm system; constructing a Lyapunov function based on the dynamic model and the auxiliary function; constructing an adaptive parameter update law and an adaptive neural network fault-tolerant collaborative controller based on the Lyapunov function; and applying an adaptive neural network to the multi-flexible arm system based on the adaptive neural network fault-tolerant collaborative controller to achieve fault-tolerant collaborative control. The present invention can not only achieve vibration suppression of the multi-flexible arm system, but also enable the motion trajectory of the multi-flexible arm system to track the virtual leader (i.e., the external reference signal). Good control effects can be achieved by selecting appropriate control parameters and adjustment parameters.

[0169] Example 2

[0170] Based on the adaptive neural network fault-tolerant cooperative control method in Example 1, the MATLBA simulation software is used to perform digital simulation on the multi-flexible arm system to further verify the effectiveness of the adaptive neural network fault-tolerant cooperative control method proposed in the present invention. Therefore, this embodiment is based on the various steps of the adaptive neural network fault-tolerant cooperative control method disclosed in Example 1. Figure 3 The connection topology of the multi-flexible arm system is shown in Table 1, Table 2, Table 3, and Table 4. The multi-flexible arm system is digitally simulated.

[0171] Table 1. System simulation parameters of the multi-flexible arm system (j=1, 2, 3, 4)

[0172]

[0173] Table 2. Simulated values ​​of boundary perturbations of the jth flexible arm in a multi-flexible arm system (j = 1, 2, 3, 4)

[0174] parameter Numerical <![CDATA[η 1j (t)]]> 0.01×j×[sin(t)+sin(2t)]N <![CDATA[η 2j (t)]]> 0.1×(5-j)×[cos(t)+cos(2t)]Nm

[0175] Table 3. Simulated values ​​of actuator failure coefficients for the multi-flexible arm system (j = 1, 2, 3, 4)

[0176]

[0177] The time-varying effective coefficient α in Table 3 1j (t), α 2j (t) takes the value of 1.0 in the time interval t∈[0,2), and the time-varying bias coefficient The value is 0 in the time intervals t∈[0,2) and t∈(4,10].

[0178] Table 4. Control parameters and adjustment parameters of the jth flexible arm in the multi-flexible arm system (j = 1, 2, 3, 4)

[0179]

[0180] In Table 4, Θ0(t) represents the joint angle trajectory of the virtual leader. In the multi-flexible arm system of this embodiment, only the flexible arms numbered 2 and 4 can obtain Θ0(t).

[0181] This embodiment uses Figure 3 As the connection topology of the multi-flexible arm system composed of 4 flexible arms in the digital simulation, the matrix describing the connection relationship between the flexible arms in the multi-flexible arm system and between the multi-flexible arm system and the virtual leader is

[0182] Figure 4 、 Figure 5 、 Figure 6 These are all simulation result diagrams of this embodiment, and the total simulation time is 10s. Figure 4 As shown, when no control is applied (i.e. F 1j (t) = F 2j (t) = 0), the vibration offset r of the four flexible arms in the multi-flexible arm system j Schematic diagram of simulation results for (s,t)(j=1,2,3,4). Figure 5 As shown in the figure, after applying the adaptive neural network fault-tolerant cooperative control method to the multi-flexible arm system, the vibration offset r of the four flexible arms is j (s,t) Schematic diagram of simulation results. Figure 4 and Figure 5 By comparison, it is found that when there is no control, the multi-flexible arm system vibrates continuously and its vibration offset is large. However, after applying the adaptive neural network fault-tolerant collaborative control method to the multi-flexible arm system, even in the presence of time-varying parameter uncertainties and actuator failures described in Tables 2 and 3, the vibration offsets in the multi-flexible arm system are adjusted to near 0, that is, vibration suppression of the multi-flexible arm system is achieved. Figure 6 As shown, in the adaptive neural network fault-tolerant cooperative controller and Figure 3 The joint angle trajectory Θ of the four flexible arms in the multi-flexible arm system under the action of the connection topology shown j (t), and even with the time-varying parameter uncertainties and actuator failures described in Tables 2 and 3, the joint angle trajectories of the four flexible arms in the multi-flexible arm system can track the joint angle trajectory of the virtual leader Θ0(t) = 0.6 rad, that is, fault-tolerant collaborative control of the joint angles is achieved.

[0183] From the above simulation results, it can be seen that the present invention can effectively suppress the vibration of the multi-flexible arm system, alleviate the impact of time-varying parameter uncertainty and actuator failure on the control performance, and the motion trajectory of the multi-flexible arm system can track the virtual leader, ultimately achieving the adaptive neural network fault-tolerant collaborative control effect.

[0184] The above embodiments are preferred implementation modes of the present invention, but the implementation modes of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention should be considered as equivalent replacement methods and are included in the scope of protection of the present invention.

Claims

1. An adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system, characterized in that: The adaptive neural network fault-tolerant collaborative control method comprises the following steps: S1. Based on the dynamic characteristics of N flexible arms, a dynamic model is constructed that takes into account actuator failures and time-varying parameter uncertainties. S2. Select an external reference signal as a virtual leader, connect N flexible arms as followers and form a multi-flexible arm system according to the undirected and interconnected rule, and construct an auxiliary function of the multi-flexible arm system to achieve collaborative control; S3. constructing a Lyapunov function based on the dynamic model and the auxiliary function; S4. Based on the Lyapunov function, combined with the adaptive radial basis function neural network approximation method and the robust adaptive parameter estimation technology, an adaptive parameter update law and an adaptive neural network fault-tolerant collaborative controller for the multi-flexible arm system are constructed. The adaptive neural network fault-tolerant collaborative controller for the multi-flexible arm system is as follows: Where, τ 1j (t), τ 2j (t) are the first controller and the second controller of the j-th flexible arm in the multi-flexible arm system; w 1j 、w 2j are the third and fourth control parameters for the coordinated control of the j-th flexible arm in the multi-flexible arm system, κ 1j (t),κ 2j (t) are the first auxiliary function and the second auxiliary function for the j-th flexible arm in the multi-flexible arm system to achieve coordinated control, is the joint angle coordination error of the j-th flexible arm at position s = L and time t, is the first-order time partial derivative of the joint angle coordination error of the j-th flexible arm at position s = L and time t, are the first parameter estimation value and the second parameter estimation value of the j-th flexible arm, S 1j (Z 1j ), S 2j (Z 2j ) are the first radial basis function and the second radial basis function of the adaptive neural network fault-tolerant control of the j-th flexible arm in the multi-flexible arm system, respectively. 1j , Z 2j are the first input vector and the second input vector of the j-th flexible arm, μ 1j 、μ 2j are the first control parameter and the second control parameter for the j-th flexible arm in the multi-flexible arm system to achieve coordinated control, are the first weight vector estimate and the second ideal weight vector estimate of the j-th flexible arm, respectively, and sgn[*] is the sign function; S5. Based on the adaptive neural network fault-tolerant collaborative controller, an adaptive neural network is applied to the multi-flexible arm system to realize fault-tolerant collaborative control.

2. The adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system according to claim 1 is characterized in that: The dynamic characteristics of the N flexible arms in step S1 include the kinetic energy, potential energy, and virtual work done by the non-conservative force on the flexible arms. Substituting the kinetic energy, potential energy, and virtual work into the Hamiltonian principle, the dynamic model of the flexible arms is obtained as follows: Where subscript j is the number of the j-th flexible arm, and the parameter or variable with subscript j represents that the parameter or variable belongs to the j-th flexible arm, j∈{1,2,...,N}, N is the total number of flexible arms; s is the spatial position variable, t is the time variable, L represents the same length of all flexible arm links; r j (s, t) represents the vibration displacement of the j-th flexible arm at position s and time t; z j (s, t) is the displacement of the j-th flexible arm at position s and time t, defined as z j (s,t)=sΘ j (t)+r j (s,t), where Θ j (t) is the joint angle of the jth flexible arm; represent the mass per unit length and bending stiffness of the j-th flexible arm link, respectively, and are defined as: Where ρ and Ξ are the rated mass per unit length and rated bending stiffness, which are the same for all flexible arm links, and Δ ρj (t), Δ Ξj (t) are the time-varying uncertainty of the unit length mass and bending stiffness of the j-th flexible arm link, and Δ ρj (t), Δ Ξj (t)Assumptions need to be met: are Δ ρj (t), Δ Ξj The upper bound of (t); z jtt (s,t),r jssss (s,t) are z j The second-order partial derivative of (s,t) with respect to time t and r j The fourth-order partial derivative of (s,t) with respect to position s; The boundary conditions of the j-th flexible arm are: In the formula are the hub inertia and end load mass of the j-th flexible arm, respectively, and are defined as: I h , m are the rated hub inertia and rated end load mass, which are the same for all flexible arms, Δ Ij (t), Δ mj (t) are respectively the time-varying uncertainty of the hub inertia and the end load mass of the j-th flexible arm, and Δ Ij (t), Δ mj (t)Assumptions to be met: They are Δ Ij (t), Δ mj The upper bound of (t); F 1j (t) is the torque output by the torque actuator at the j-th flexible arm hub, F 2j (t) is the control force output by the force actuator at the end of the j-th flexible arm; η 1j (t), η 2j (t) are the unknown time-varying boundary disturbances acting on the hub and the end, respectively, and satisfy are the boundary perturbations η 1j (t), η 2j The upper bound of (t); represents the angular acceleration of the jth flexible arm joint angle; r js (0,t),r jss (0, t) represent the first-order position partial derivative and the second-order position partial derivative of the vibration offset of the j-th flexible arm at position s = 0 and time t, respectively; r jss (L,t),r jsss (L, t) represent the second-order position partial derivative and the third-order position partial derivative of the vibration offset of the j-th flexible arm at position s = L and time t, respectively; Considering that the actuator of the flexible arm may suffer from partial failure and offset failure, the actuator failure is defined as follows: Wherein, the subscript i∈{1,2} represents the number of the controller and actuator of the flexible arm. The actuator and controller with the same number of each flexible arm are installed together. The subscript i=1 represents the variable of the torque actuator or torque controller at the hub of the flexible arm, and the subscript i=2 represents the variable of the force actuator or force controller at the end of the flexible arm; Σ ij (t) is the actuator fault term of the i-th actuator of the j-th flexible arm, defined as Among them, τ ij (t) is the i-th controller of the j-th flexible arm, α ij (t) represents the time-varying effective coefficient of the i-th actuator of the j-th flexible arm, represents the time-varying bias coefficient of the i-th actuator of the j-th flexible arm, and the actuator fault parameter α ij (t), Need to meet: α ij (t)∈(0,1], Β ij yes The upper bound of .

3. The adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system according to claim 2 is characterized in that: In step S2, an external reference signal is selected as a virtual leader, N flexible arms are connected as followers and are formed into a multi-flexible arm system according to the undirected and interconnected rule. The auxiliary function of the multi-flexible arm system is constructed to achieve collaborative control as follows: Select the external reference signal as the motion trajectory of a virtual leader, and number the virtual leader as 0; define the trajectory of the virtual leader as: Θ0(t) = Θ d ,r0(s,t)=r d , where Θ d is the desired joint angle of the N flexible arms, and the desired joint angle Θ d Angular velocity and angular acceleration are all 0, r d is the expected vibration offset of the N flexible arms. To suppress the vibration of the N flexible arms, it is necessary to define the expected vibration offset as r d =0; N flexible arms are used as followers and connected into a multi-flexible arm system according to the undirected and interconnected rule. The undirected connection rule means that two adjacent flexible arms can obtain signals from each other, and the interconnected connection rule means that there is at least one signal transmission path between any two flexible arms. The collaborative error of the jth flexible arm in the multi-flexible arm system is defined as: Where, They are the joint angle coordination error, vibration offset coordination error, and displacement coordination error of the j-th flexible arm respectively; Θ n (t), r n (s,t),z n (s, t) represent the joint angle, vibration offset, and displacement of the nth flexible arm, respectively. n∈{1,2,...,N} is defined for the convenience of summation operation, and the value of n is consistent with j; a jn The element with the subscript (j,n) in the adjacency matrix A, where the subscripts j and n refer to the numbers of the flexible arms, is the adjacency matrix A = {a jn }∈R N×N It is a non-negative matrix describing the connection relationship of N flexible arms in a multi-flexible arm system, which is defined as: if the j-th flexible arm can obtain the n-th flexible arm signal and j≠n, then a jn =1, otherwise, a jn =0;a j0 are the diagonal elements of the diagonal matrix Δ, and a j0 Located in the jth row and jth column of the diagonal matrix Δ, the diagonal matrix Δ=diag{a j0 }∈R N×N It is a non-negative matrix describing the connection relationship between the multi-flexible arm system and the virtual leader, which is defined as: if the j-th flexible arm in the multi-flexible arm system can obtain the signal of the virtual leader, then a j0 =1, otherwise, a j0 =0; In the control method, it is necessary to assume that at least one flexible arm in the multi-flexible arm system can obtain the signal of the virtual leader; According to the above definition of collaborative error, the vector form of collaborative error is: Where, are the joint angle coordination error vector, vibration offset coordination error vector, and displacement coordination error vector in the multi-flexible arm system, respectively, and are defined as follows: Θ(t), r(s,t), and z(s,t) are the joint angle vector, vibration offset vector, and displacement vector in the multi-flexible arm system, respectively, and are defined as follows: Θ(t) = [Θ j (t)] T ∈R N×1 , r(s,t)=[r j (s,t)] T ∈R N×1 ,z(s,t)=[z j (s,t)] T ∈R N×1 ; 1 N is an N-dimensional all-1 vector, defined as: 1 N =[1,1,...,1] T ∈R N×1 ; M is a positive definite communication matrix that describes the connection relationship between N flexible arms in the multi-flexible arm system and the connection relationship between the multi-flexible arm system and the virtual leader, defined as M = L f +Δ, where L f is the Laplace matrix, defined as L f ={l jn }∈R N×N , l jn It's L f For elements with subscript (j,n), when j≠n, there is l jn =-a jn ,otherwise, Based on the above definition of coordination error, the auxiliary function of the multi-flexible arm system is constructed as follows: Where, κ 1j (t),κ 2j (t) are the first auxiliary function and the second auxiliary function for realizing the coordinated control of the j-th flexible arm in the multi-flexible arm system; are the first-order position partial derivative and the third-order position partial derivative of the vibration offset coordination error of the j-th flexible arm at position s = L and time t, respectively; is the first-order time partial derivative of the displacement coordination error of the j-th flexible arm at position s = L and time t; is the first-order time partial derivative of the joint angle coordination error of the j-th flexible arm at position s = L and time t; The auxiliary function κ of the above multi-flexible arm system 1j (t),κ 2j (t) and the coordinated error vector The definition of the auxiliary function κ 1j (t),κ 2j The vector form of (t) is: κ1(t)=[κ 1j (t)] T =M[r s (L,t)+z t (L,t)-r sss (L,t)]∈R N×1 , Where r s (L,t),r sss (L,t) are The vector form is defined as: s (L,t)=[r js (L,t)] T ∈R N×1 , r sss (L,t)=[r jsss (L,t)] T ∈R N×1 ;z t (L,t) is z jt The vector form of (L, t) is defined as: t (L,t)=[z t (L,t)] T ∈R N×1 ; is the joint angular velocity vector of the multi-flexible arm system, defined as in, is the joint angular velocity of the jth flexible arm in the multi-flexible arm system.

4. The adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system according to claim 3 is characterized in that: In step S3, the process of constructing the Lyapunov function based on the dynamic model and the auxiliary function is as follows: Based on the dynamic model and auxiliary functions, the Lyapunov function is constructed as: P(t)=P1(t)+P2(t)+P3(t)+P4(t), in Where, π 1j , π 2j , π 3j are the first energy coefficient, second energy coefficient, and third energy coefficient of the j-th flexible arm in the multi-flexible arm system to achieve energy constraint; μ 1j 、μ 2j are the first control parameter and the second control parameter for realizing coordinated control of the j-th flexible arm in the multi-flexible arm system respectively; are the first weight estimation error vector and the second weight estimation error vector for adaptive neural network fault-tolerant control of the j-th flexible arm in the multi-flexible arm system, respectively, and are defined as: in, are the first ideal weight vector and the second ideal weight vector of the j-th flexible arm, respectively. are the first weight vector estimation value and the second ideal weight vector estimation value of the j-th flexible arm respectively; The first parameter estimation error and the second parameter estimation error of the j-th flexible arm in the multi-flexible arm system are respectively the estimation boundary perturbation and the upper bound of the neural network approximation error, which are defined as: in, are the first parameter estimation value and the second parameter estimation value of the j-th flexible arm, respectively. are the first unknown parameter and the second unknown parameter of the j-th flexible arm respectively and are defined as: are the upper bounds of the first neural network approximation error and the second neural network approximation error of the j-th flexible arm respectively; They are respectively the first weight estimation adjustment coefficient and the second weight estimation adjustment coefficient of the j-th flexible arm in the multi-flexible arm system to realize the neural network weight estimation; They are the first parameter estimation adjustment coefficient and the second parameter estimation adjustment coefficient for unknown parameter estimation of the j-th flexible arm in the multi-flexible arm system; are the vibration offset coordination errors of the jth flexible arm at position s and time t. The first-order partial derivative and second-order partial derivative of .

5. The adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system according to claim 4 is characterized in that: In step S4, based on the Lyapunov function, combined with the adaptive radial basis function neural network approximation method and the robust adaptive parameter estimation technology, the process of constructing the adaptive parameter update law and the adaptive neural network fault-tolerant cooperative controller of the multi-flexible arm system is as follows: Adaptive radial basis function neural network approximation method is used to solve the problem of time-varying parameter uncertainty Δ ρj (t), Δ Ξj (t), Δ Ij (t), Δ mj (t) and actuator fault parameter α ij (t), The unknown terms are approximated as follows: Where S 1j (Z 1j ), S 2j (Z 2j ) are the first radial basis function and the second radial basis function of the adaptive neural network fault-tolerant control of the j-th flexible arm in the multi-flexible arm system, respectively. The radial basis function S(Z) is defined in the radial basis neural network as: S(Z) = exp(-||ZC|| 2 / b 2 ), Z is the input vector of the radial basis neural network, C is the center of the input vector variation range, b is the width of the input vector variation range, and the symbol exp(*) is equivalent to e * , the symbol ||*|| is equivalent to taking the norm of the vector *; therefore, Z 1j , Z 2j are the first input vector and the second input vector of the j-th flexible arm, respectively, and are defined as: Among them, sgn[*] is the sign function, r jssst (L, t) is the vibration displacement r of the jth flexible arm at position s = L j (L, t) Simultaneously find the first-order partial derivative with respect to time t and the third-order partial derivative with respect to position s; r jst (L, t) is the vibration displacement r of the jth flexible arm at position s = L j (L, t) calculates the first-order partial derivative with respect to time t and the first-order partial derivative with respect to position s at the same time; δ 1j (Z 1j ),δ 2j (Z 2j ) are the first neural network approximation error and the second neural network approximation error of the adaptive neural network fault-tolerant control of the j-th flexible arm in the multi-flexible arm system, and they satisfy: The ideal weight vector of the neural network is estimated by using robust adaptive parameter estimation technology. and unknown parameters To make an estimate, the following adaptive parameter update law for the multi-flexible arm system is constructed: Where g 1j 、g 2j 、g 3j 、g 4j are the first robust parameter, the second robust parameter, the third robust parameter, and the fourth robust parameter for ensuring the robustness of the parameter estimation value of the j-th flexible arm in the multi-flexible arm system; The first-order derivative of the Lyapunov function with respect to time t is obtained. Based on the Lyapunov stability theory, combined with the above-mentioned adaptive radial basis function neural network approximation method and robust adaptive parameter estimation technology, an adaptive neural network fault-tolerant collaborative controller for the multi-flexible arm system is finally constructed.

6. The adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system according to claim 5, characterized in that: The process of constructing the adaptive parameter update law and the adaptive neural network fault-tolerant cooperative controller of the multi-flexible arm system also includes a step of verifying the stability of the multi-flexible arm system under the action of the adaptive neural network fault-tolerant cooperative controller, and the process is as follows: By constraining the energy coefficient π in the Lyapunov function 1j , π 2j , π 3j , ensuring the positive definiteness of the Lyapunov function; Taking the first derivative of the Lyapunov function with respect to time t; Lyapunov bounded stability theory is applied to verify the stability of the multi-flexible arm system under the action of the adaptive neural network fault-tolerant cooperative controller.

7. The adaptive neural network fault-tolerant collaborative control method for a multi-flexible arm system according to claim 6 is characterized in that: In step S5, the process of applying the adaptive neural network to the multi-flexible arm system to realize fault-tolerant collaborative control based on the adaptive neural network fault-tolerant collaborative controller is as follows: The adaptive parameter update law is calculated in the computing unit of the i-th controller of the j-th flexible arm. and adaptive neural network fault-tolerant cooperative controller τ ij (t), after the calculation is completed, the i-th actuator of the j-th flexible arm is updated. As the control time increases, the actuator continuously outputs control force or torque, which affects the joint angle Θ j (t) and vibration offset r j (s, t) is continuously adjusted, and finally the motion trajectory of the multi-flexible arm system tracks the virtual leader to achieve fault-tolerant collaborative control.

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