A multi-flexible robot system iterative learning fault-tolerant consistency control method

Abstract

A multi-flexible manipulator system iterative learning fault-tolerant consensus control method comprises the following steps: S1, a directed communication topology is constructed, a flexible manipulator is taken as a follower, and a virtual leader is established; a mathematical model of the multi-flexible manipulator system is established based on the Hamilton principle; S2, a finite-time observer for estimating the angular displacement and angular velocity of the virtual leader is constructed for each follower of the multi-flexible manipulator system; S3, based on the virtual leader state estimated by the finite-time observer, the iterative learning law, the parameter adaptive law and the control signal of the multi-flexible manipulator system are designed in combination with the actuator fault model and the event-triggered rule based on the sampling data; the control method combines the estimated value of the finite-time observer to design the fault-tolerant consensus control based on the event-triggered mechanism of the sampling data, so that the system realizes the consensus control while saving the communication resources.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of flexible manipulator, in particular to a multi-flexible manipulator system iterative learning fault-tolerant consensus control method. BACKGROUND

[0002] As an important part of modern automation technology, manipulators have been widely used in industry, medicine, service and other fields under the impetus of scientific and technological progress and the development of advanced productive forces. Traditional rigid manipulators have high structural rigidity and limited movement path, and cannot meet the increasingly stringent requirements of lightness, precision and intelligence. In this context, flexible manipulators are widely studied due to their light weight, high flexibility, strong operability and ability to complete tasks in complex environments. Flexible manipulators not only can flexibly pass through narrow spaces, but also can avoid damage caused by collision through their flexible characteristics. However, in actual production activities, due to the complexity of the external environment and the flexibility of the flexible manipulator structure, the flexible manipulator is prone to vibration and deformation during operation, which affects the control accuracy of the system and even damages the structural integrity of the system.

[0003] Therefore, the prior art proposes to model the flexible manipulator using a fourth-order partial differential equation and proposes many control methods to achieve trajectory tracking while suppressing the vibration of the flexible manipulator, which can reduce the hardware loss of the flexible manipulator to a certain extent and prolong the service life. It should be noted that the existing control method usually only considers the trajectory tracking and vibration control of a single flexible manipulator system, and there is little research on the consensus control of multiple flexible manipulator systems. There are two technical problems in the actual application of multi-flexible manipulator systems: (1) Due to the limitations of system communication bandwidth and other factors, system communication delay, network congestion and other situations may occur, increasing the burden of network sensors; (2) The error of a single flexible manipulator accumulates during control, resulting in a decrease in the control accuracy of the entire system. SUMMARY

[0004] The purpose of the present application is to propose a multi-flexible manipulator system iterative learning fault-tolerant consensus control method, which combines the estimated value of the finite time observer to design a fault-tolerant consensus control based on the sampling data event trigger mechanism, to save communication resources while achieving consensus control of the system.

[0005] The technical solution adopted by the present application is: a multi-flexible manipulator system iterative learning fault-tolerant consensus control method, comprising the following steps:

[0006] S1, a directed communication topology is constructed, and the flexible manipulator is used as a follower, and a virtual leader is established for the follower;

[0007] establish a mathematical model of the multi-flexible manipulator system based on Hamilton principle;

[0008] S2, construct a finite-time observer for estimating the virtual leader angular displacement and angular velocity for each follower of the multi-flexible manipulator system;

[0009] S3, design an iterative learning law, a parameter adaptive law and a control signal of the multi-flexible manipulator system based on the virtual leader state estimated by the finite-time observer, an actuator fault model and an event-triggered rule based on sampling data;

[0010] S4, control the followers in the multi-flexible manipulator system based on the designed iterative learning law, parameter adaptive law and control signal, so that the angular position tracking error between the followers and the virtual leader and the elastic deformation generated by the followers themselves are bounded.

[0011] As a preferred solution, the mathematical model of the multi-flexible manipulator system is:

[0012]

[0013] The boundary conditions are:

[0014]

[0015] y i (0,t)=y′ i (0,t)=y″ i (L,t)=0

[0016] wherein x∈[0,L] represents the spatial position of the manipulator, L represents the length of the manipulator, i∈{1,2,...,N} represents the i-th follower manipulator, N is an integer greater than 2 and represents the maximum serial number of the follower manipulator; t represents the system running time, m represents the load mass of the manipulator; ρ, EI and J respectively represent the unit length mass, bending stiffness and hub inertia of the manipulator; θ i (t) represents the angular displacement of the i-th follower manipulator at time t; represents the second-order derivative of θ i (t) with respect to time; z i (x,t) represents the displacement of the i-th follower manipulator at the spatial position x, represents the second-order derivative of z i (x,t) with respect to time; z i (L,t) represents the displacement of the i-th follower manipulator at the spatial position L, represents the second-order derivative of z i (x,t) with respect to time at the spatial position L; y i(x, t) represents the elastic deformation of the i-th follower manipulator at spatial position x; y i (x, t) represents y i (x, t) represents the fourth order derivative of y i (0, t) represents the elastic deformation of the i-th follower manipulator at spatial position 0; y i (0, t) and y i (0, t) represents y i (0, t) represents the first order derivative of y i (L, t) represents the elastic deformation of the i-th follower manipulator at spatial position L; y i (L, t) and y i (L, t) represents y i (L, t) represents the second order derivative of y 1,i (L, t) and d 2,i (L, t) represents the disturbance of the i-th follower manipulator; u 1,i (L, t) and u 2,i (L, t) represents the control law of the i-th follower manipulator.

[0017] As a preferred solution, the directed communication topology is:

[0018] The communication topology of the multi-flexible manipulator system is described by a graph G = (P, Z), where the virtual leader and each manipulator are regarded as an agent, P = {0, 1, …, N} represents the set of nodes, represents the set of edges. It is assumed that the directed graph G has a directed spanning tree rooted at the leader. 0 represents the serial number of the virtual leader, and 1 to N represent the serial numbers of the follower flexible manipulators. The edge (i, j) ∈ Z represents that the j-th flexible manipulator can obtain information from the i-th flexible manipulator; the adjacency matrix A = [a ij ] ∈ R (N+1)×(N+1) , where i, j ∈ P and a ii = 0. If (j, i) ∈ Z, then a ij = 1, otherwise a ij = 0. The leader can only send information to the followers and cannot obtain information from the followers, i.e., a 0j = 0, j ∈ P. The Laplacian matrix of the graph G is where L1 ∈ R N×N represents the Laplacian matrix between the follower manipulators, and L2 ∈ R N represents the Laplacian matrix between the leader and the follower manipulators; L1 = [l ij ], and when i ≠ j, l ij = -aij ,otherwise

[0019] As a preferred option, the constructed finite-time observer is as follows:

[0020]

[0021] in, 0 < γ < 1 is a constant; Γ1(t)=[Γ 1,1 (t),...,Γ 1,N (t)] T Γ2(t)=[Γ 2,1 (t),...,Γ 2,N (t)] T ; and Let represent the finite-time observer estimates of the virtual leader's angular displacement and angular velocity, respectively, for the i-th follower flexible robotic arm. and They represent The first derivative with respect to time and The first derivative with respect to time; h1, h2, and ι are all constants, and ι > k, where k is the upper bound of the angular acceleration of the flexible robotic arm; θ0(t) represents the angular displacement of the virtual leader at time t. It represents the first derivative of θ0(t) with respect to time.

[0022] As a preferred option, the actuator failure model is as follows:

[0023]

[0024] Among them, w a,i (t) represents the control signal; Represented as an unknown bounded continuous signal; 0 < v a,i (t)<1, a=1,2, i=1,2,...,N.

[0025] As a preferred approach, the event triggering rules based on the sampled data are as follows:

[0026]

[0027] in, This is the control signal for the i-th following robotic arm that needs to be designed, where a = 1, 2; This represents the i-th control signal of the following robotic arm. The kth trigger time, where k is a positive integer; express The absolute value of ψ; 0 < ψ a,i <1 and μa,i 0 is the event trigger parameter of the i-th following robot arm; inf represents the infimum of the elements in the set; Y1 is a positive integer, and h is the sampling interval of the sampling data.

[0028] As a preferred solution, the designed iterative learning law is:

[0029]

[0030] where 0 < γ1 < 1, 0 < γ2 < 1; p a,i (t) is an iterative term designed for compensating r a,i (t) and r is an upper bound of the disturbance d a,i (t), and and respectively represent the first-order derivative of p a,i (t) with respect to time and the first-order derivative of r a,i (t) with respect to time, a = 1, 2, i = 1, 2,..., N; F represents the period of the disturbance d a,i (t). represents the first-order derivative of e 1,i (t) with respect to time;

[0031] As a preferred solution, the parameter adaptive law is:

[0032]

[0033] where and respectively represent the estimated values of o 1,i , o 2,i , λ 1,i and λ 2,i ,

[0034] q a,i = sup t≥0 v a,i (t), a = 1, 2, i = 1, 2,..., N; and respectively represent the first-order derivatives of and with respect to time; and are a part of the control signal to be designed; l1, l2, l3, l4, η1, η2, η3 and η4 are normal numbers.

[0035] As a preferred solution, the control signal is:

[0036]

[0037] wherein tanh represents hyperbolic tangent function; c1, c2, τ4, τ5 and ε are normal numbers; represents second order derivative with respect to time; y′ i (L,t) represents y i (x,t) first order derivative with respect to space at L.

[0038] As a preferred scheme, Lyapunov functions V(t) are established for stability analysis of the multi-flexible manipulator system, and the Lyapunov functions are respectively:

[0039] V(t) = V1(t) + V2(t) + V3(t) + V4(t) + V5(t)

[0040] In the formula, c1, c2, τ4, τ5 and ε are normal numbers.

[0041]

[0042] wherein dx and dκ represent differential operators; τ1, τ2 and τ3 are normal numbers. and respectively represent and estimation errors of y represents z i (x,t) first order derivative with respect to time; y′ i (x,t) and y″ i (x,t) respectively represent y i (x,t) first order derivative with respect to space and second order derivative with respect to space, a = 1, 2, i = 1, 2,..., N.

[0043] Compared with the prior art, the beneficial effects of the present application are:

[0044] 1. The multi-flexible manipulator system iterative learning fault-tolerant consistency control method established by the present application can realize control of the multi-flexible manipulator system, has good robustness and stability, and can still maintain the stability and function of the overall system even when the manipulator is partially failed or disturbed.

[0045] 2. Compared with the existing flexible manipulator system tracking and vibration control method, the present application estimates unknown parameters and disturbances in the system by designing parameter adaptive law and iterative learning law to improve the control accuracy and reliability of the system.

[0046] 3. Unlike other methods, this invention designs a finite-time observer for each follower based on the directed communication topology of the system, transforming the consistency control problem into a trajectory tracking problem and improving the control accuracy of the system. Compared with continuous control methods, this invention uses an event-triggered mechanism for controller design, thereby improving the communication efficiency of the system.

[0047] 4. Compared with existing methods for trajectory tracking and vibration suppression in single flexible robotic arm systems, this invention establishes an iterative learning fault-tolerant consistency control method for multi-flexible robotic arm systems based on a sampled data event triggering mechanism. In event triggering control, the robotic arm only sends status information when a specific event occurs. Compared with the traditional time triggering mechanism, this triggering mechanism effectively reduces the frequency of event triggering, saves computer system communication resources, reduces network bandwidth pressure, and solves the network congestion problem to a certain extent. Attached Figure Description

[0048] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0049] Figure 1 This is a simplified structural diagram of the multi-flexible robotic arm system of the present invention;

[0050] Figure 2 This is the directed communication topology diagram of the present invention;

[0051] Figure 3 This is a simulation diagram of the elastic deformation of a follower robotic arm without control.

[0052] Figure 4 This is a simulation diagram of the angular displacement of the follower robotic arm under the control method of the present invention.

[0053] Figure 5 This is a simulation diagram of the elastic deformation of the follower robotic arm under the control method of the present invention.

[0054] Figure 6 The control signal u of this invention 1,i Simulation diagram of (t);

[0055] Figure 7 The control signal u of this invention 2,i Simulation diagram of (t). Detailed Implementation

[0056] The application will be described in greater detail by way of example with reference to the accompanying drawings. However, it should be understood that elements, structures and features in one embodiment can be beneficially combined with elements, structures and features of another embodiment without further description.

[0057] It should be noted that, unless otherwise defined, technical terms or scientific terms used herein should be understood as having the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terms "one", "an" or "the" and similar referents in the application patent application and claims should not be construed as indicating a quantity of one, but rather indicating the presence of at least one; the terms "first", "second" and "third" should not be construed as indicating a sequential order, but merely as a distinction between different components; the terms "include", "comprise" and similar terms mean that the elements or objects before the "include" or "comprise" encompass the elements or objects listed after the "include" or "comprise" and equivalents thereof, but do not exclude other elements or objects with the same function.

[0058] In order to more clearly describe the multi-flexible manipulator system iterative learning fault-tolerant consensus control method, the accompanying drawings are combined Figures 1-6 This embodiment is described:

[0059] Referring to Figure 1 and Figure 2 , a multi-flexible manipulator system iterative learning fault-tolerant consensus control method includes the following steps:

[0060] S1, a directed communication topology is constructed, and the flexible manipulator is taken as a follower, and a virtual leader is established for the follower;

[0061] A mathematical model of the multi-flexible manipulator system is established based on the Hamilton principle;

[0062] Specifically, the constructed directed communication topology is described as:

[0063] The communication topology of the multi-flexible manipulator system is described by a graph G=(P,Z), the virtual leader and each manipulator are regarded as an agent, P={0,1,…,N} represents the set of nodes, represents the set of edges. It is assumed that the directed graph G has a directed spanning tree with the leader as the root node. 0 represents the serial number of the virtual leader, and 1 to N represent the serial numbers of the follower flexible manipulators. The edge (i,j)∈Z represents that the jth flexible manipulator can obtain information from the ith flexible manipulator; the adjacency matrix A of the graph G is ij ]∈R (N+1)×(N+1) , where i,j∈P and a ii =0. If (j,i)∈Z, then a ij =1, otherwise a ij= 0. The leader can only send information to the followers, and cannot obtain information from the followers, i.e., a 0j = 0, j e P. The Laplacian matrix of the graph G is where L1 e R N×N represents the Laplacian matrix between the follower manipulators, L2 e R N represents the Laplacian matrix between the leader and follower manipulators; L1 = [l ij ], and when i≠j, l ij = -a ij , otherwise

[0064] Specifically, the mathematical model of the multi-flexible manipulator system established is as follows:

[0065]

[0066] The boundary conditions are as follows:

[0067]

[0068] y i (0, t) = y' i (0, t) = y" i (L, t) = 0 (4)

[0069] where x e [0, L] represents the spatial position of the manipulator, L represents the length of the manipulator, i e {1, 2,..., N} represents the i-th follower manipulator, N is an integer greater than 2 and represents the maximum serial number of the follower manipulators; t represents the system running time, m represents the load mass of the manipulator; p, EI and J respectively represent the unit length mass, bending stiffness and hub inertia of the manipulator; q i (t) represents the angular displacement of the i-th follower manipulator at time t; represents the second derivative of q i (t) with respect to time; z i (x, t) represents the displacement of the i-th follower manipulator at the spatial position x, represents the second derivative of z i (x, t) with respect to time; z i (L, t) represents the displacement of the i-th follower manipulator at the spatial position L, represents the second derivative of z i (x, t) with respect to time at the spatial position L; y i (x, t) represents the elastic deformation of the i-th follower manipulator at the spatial position x; y" i (x, t) represents the fourth derivative of y i (x, t) with respect to space; y iy (0, t) represents the elastic deformation of the i-th follower manipulator at the spatial position 0; y' i y (0, t) and y'' i y (0, t) respectively represent the first-order derivative of y i (x, t) with respect to the space at the spatial position 0 and the second-order derivative with respect to the space; y i y (L, t) represents the elastic deformation of the i-th follower manipulator at the spatial position L; y'' i y (L, t) and y''' i y (L, t) respectively represent the second-order derivative of y i (x, t) with respect to the space at the spatial position L and the third-order derivative with respect to the space; d 1,i d (0, t) and d 2,i d (0, t) represents the disturbance of the i-th follower manipulator; u 1,i u (0, t) and u 2,i u (0, t) represents the control law of the i-th follower manipulator.

[0070] S2, a finite-time observer for estimating the virtual leader angular displacement and angular velocity is constructed for each follower of the multi-flexible manipulator system;

[0071] In the multi-flexible manipulator system under the directed communication topology, the state information of the leader is transmitted to the end of the communication topology, and in the process, the error may be accumulated and the control accuracy of the system may be reduced. Therefore, the present application combines the directed communication topology to design a finite-time observer for each follower to estimate the state of the leader. Then, the fault-tolerant consensus control based on the sampling data event trigger mechanism is designed in combination with the estimated value of the finite-time observer, so that the system realizes the consensus control while saving the communication resources;

[0072] Specifically, the constructed finite-time observer is:

[0073]

[0074] wherein, 0 < γ < 1 is a constant; Γ1(t) = [Γ 1,1 (t),...,Γ 1,N (t)] T , Γ2(t) = [Γ 2,1 (t),...,Γ 2,N (t)] T ; and respectively represent the estimated values of the finite-time observer of the i-th follower flexible manipulator for the virtual leader angular displacement and angular velocity, and respectively represent The first derivative with respect to time and The first derivative with respect to time;

[0075] h1, h2, and ι are all constants, and ι > k, where k is the upper bound of the angular acceleration of the flexible robotic arm; θ0(t) represents the angular displacement of the virtual leader at time t. It represents the first derivative of θ0(t) with respect to time.

[0076] S3. Based on the virtual leader state estimated by the finite-time observer, and combined with the actuator failure model and the event triggering rules based on the sampled data, design the iterative learning law, parameter adaptation law and control signal of the multi-flexible manipulator system;

[0077] Specifically, the actuator failure model adopted is as follows:

[0078]

[0079] Among them, w a,i (t) represents the control signal; Represented as an unknown bounded continuous signal; 0 < v a,i (t)<1, a=1,2, i=1,2,...,N.

[0080] Specifically, the event triggering rules based on the sampled data are as follows:

[0081]

[0082] in, This is the control signal for the i-th following robotic arm that needs to be designed, where a = 1, 2; Indicates the i-th control signal of the following robotic arm. The kth trigger time, where k is a positive integer; express The absolute value of ψ; 0 < ψ a,i <1 and μ a,i >0 is the event trigger parameter for the i-th event following the robotic arm; inf represents the infimum of the elements in the set; Υ1 is a positive integer, and h is the sampling interval of the sampled data.

[0083] Specifically, the iterative learning pattern of the design is as follows:

[0084]

[0085] Where 0 < γ1 < 1, 0 < γ2 < 1; p a,i (t) is used to compensate r a,i (t) and the designed iteration term, For the disturbance d a,ithe upper bound of (t) and and respectively represent p a,i (t) the first order derivative of (t) with respect to time and r a,i (t) the first order derivative of (t) with respect to time, a = 1, 2, i = 1, 2, …, N; F represents the disturbance d a,i (t) the period of (t); represent e 1,i (t) the first order derivative of (t) with respect to time;

[0086] Specifically, the designed parameter adaptive law is:

[0087]

[0088] wherein, and respectively represent o 1,i , o 2,i , λ 1,i and λ 2,i are the estimated values of o 1,i , o 2,i , λ 1,i and λ 2,i , q a,i = sup t≥0 v a,i (t), a = 1, 2, i = 1, 2, …, N; and respectively represent and the first order derivative of (t) with respect to time; and are part of the control signal to be designed; l1, l2, l3, l4, η1, η2, η3 and η4 are normal numbers.

[0089] As a preferred solution, the control signal is:

[0090]

[0091] wherein, tanh represents the hyperbolic tangent function; c1, c2, τ4, τ5 and ε are normal numbers; represent the second order derivative of (t) with respect to time; y′ i (L, t) represents the first order derivative of y i (x, t) with respect to space at space L.

[0092] S4, based on the designed iterative learning law, parameter adaptive law and control signal, the follower in the multi-flexible manipulator system is controlled, so that the angular position tracking error between the follower and the virtual leader and the elastic deformation generated by the follower itself are bounded;

[0093] A Lyapunov function V(t) is established, and the stability of the flexible manipulator system is analyzed in the process of the designed control signal control. In the analysis process, the parameters of the control signal, the parameter adaptive law, the iterative learning law and the event-triggered rule are specifically designed, and the Lyapunov function is respectively:

[0094] V(t) = V1(t) + V2(t) + V3(t) + V4(t) + V5(t) (18)

[0095]

[0096] Wherein, dx and dκ represent differential operators; τ1, τ2 and τ3 are normal numbers; And Respectively represent And The estimation error, and z i (x,t) is the first order derivative of z i (x,t) with respect to time; y′ i (x,t) and y″ i (x,t) represent the first order derivative and the second order derivative of y 1,i (x,t) with respect to space, respectively, a = 1, 2, i = 1, 2,..., N;

[0097] Further, the constant Wherein, the function max represents the maximum value function, and the min function represents the minimum value function; in combination with the designed Lyapunov function, it can be obtained that when the constants And Satisfy And The designed Lyapunov function V(t) is positive, that is, the Lyapunov function V(t) in formula (18) satisfies:

[0098]

[0099] Further, by using the method of partial integration and Young inequality, in combination with the designed Lyapunov function (18), the derivative of the Lyapunov function with respect to time can be obtained:

[0100]

[0101] Wherein, ξ3 = min{l1η1, l2η2, l3η3, l4η4,} / 2, Λ1 = (l1o 1,i 2 +l2o 2,i2 +q 1,i l3λ 1,i 2 +q 2,i l4λ 2,i 2 ) / 2, Θ1=c1-τ1EI / 2, 0<ι1<1, and is a constant.

[0102] Further, according to formula (21), by using integral technique, the Lyapunov function satisfies:

[0103]

[0104] where V(0) is the value of V(t) at t=0.

[0105] Further, based on formula (22), it can be obtained that:

[0106]

[0107] where |y i (x,t)| and |e 1,i (t)| represent the absolute values of y i (x,t) and e 1,i (t), respectively; when time tends to infinity, |y i (x,t)| satisfies |e 1,i (t)| satisfies where lim represents the limit symbol. Therefore, it can be obtained that the angle displacement tracking error of the i-th flexible manipulator is uniformly bounded stable, and the vibration of the flexible manipulator is suppressed, that is, the multi-flexible manipulator system is stable. In addition, since the event-triggered mechanism adopted in the present application is an event-triggered mechanism based on sampling data, Zeno phenomenon will not occur.

[0108] In addition, the effectiveness of the designed finite-time observer is proved:

[0109] The Lyapunov function is established:

[0110]

[0111] where, represents the first order derivative of θ0(t) with respect to time; T represents the transpose of a vector and a matrix.

[0112] Further, substituting (6) into (25) and combining the Laplacian matrix of the directed communication topology to make equivalent transformation can obtain:

[0113]

[0114] wherein, λ max (L1) represents the maximum value of the eigenvalue of the matrix λ max (L1).

[0115] According to (26), (25) can be stable in a finite time T1, that is, in a finite time T1, the estimation error of the virtual leader angular velocity can be stabilized to 0. When t >= T1, it can be obtained that

[0116] In addition, the Lyapunov function is established:

[0117]

[0118] Substituting (5) into (27) and combining the Laplacian matrix of the directed communication topology to make equivalent transformation can obtain:

[0119]

[0120] wherein,

[0121] According to (28), (27) can be stable in a finite time T2, that is, in a finite time T2, the estimation error of the virtual leader angular displacement can be stabilized to 0. When t >= T2, it can be obtained that In summary, the effectiveness of the designed finite-time observer is proved.

[0122] In order to explain the control effect of the multi-flexible manipulator system iterative learning fault-tolerant consensus control method, the multi-flexible manipulator system iterative learning fault-tolerant consensus control method based on the sampling data event trigger mechanism is simulated in MATLAB.

[0123] In the simulation process, a virtual leader and four follower flexible manipulators are considered, and the directed communication topology structure of the multi-flexible manipulator system is as shown in Figure 2 .

[0124] The specific parameters of the multi-flexible manipulator system are:

[0125] Delta x = 0.1m, Delta t = 1x10 -4 s, h = 5x10 -4 s, L = 1m, J = 0.6kg.m2 EI = 9 N.m 2 p = 0.6 kg / m, m = 0.25 kg, 0 d = 0.1 sin(0.12pi*t) rad;

[0126] The initial conditions of the flexible multi-arm system are: p i (x,0) = 0.3x, x e [0,L], 0 1(0) = 1.8 rad, 0 2(0) = 1.5 rad, 0 3(0) = 2.2 rad, 0 4(0) = 2.1 rad;

[0127] Wherein, 0 1(0), 0 2(0), 0 3(0) and 0 4(0) represent the angular displacement of the follower arm 1-4 at 0 time respectively.

[0128] The external disturbance in the system is:

[0129] d 1,1 (t) = 0.03 cospi*t, d 1,2 (t) = 0.04 cospi*t, d 1,3 (t) = 0.05 cospi*t, d 1,4 (t) = 0.03 cospi*t

[0130] d 2,1 (t) = 0.04 sinpi*t, d 2,2 (t) = 0.02 sinpi*t, d 2,3 (t) = 0.03 sinpi*t, d 2,4 (t) = 0.03 sinpi*t

[0131] Wherein, sin and cos represent the sine and cosine functions respectively.

[0132] It can be obtained that

[0133] The elastic deformation simulation diagram of the follower arm under the action of no control is shown in Figure 3 .

[0134] It can be found from Figure 3 that the elastic deformation amplitude of the follower arm is large under the action of no control, and therefore it is necessary to exert appropriate control on the system.

[0135] Further, the parameters required in the control method design process of the application are given as follows: 1,1 = 0.5, 0 1,2 = 0.5, 0 1,3 = 0.5, 0 1,4 = 0.5, 0 2,1 = 0.02, 0 2,2 = 0.02, 02,3 = ψ 2,4 = 0.01, μ 1,1 = μ 1,2 = μ 1,3 = μ 1,4 = 0.05, μ 2,1 = μ 2,2 = 0.015, μ 2,3 = μ 2,4 = 0.02, h1 = -1, h2 = -2, τ4 = τ5 = 10, c1 = 0.01, c2 = 0.018, l1 = l2 = 5, l3 = 0.0014, l4 = 0.0001, η1 = η2 = 10, η3 = 1, η4 = 0.005, γ1 = γ2 = 0.001, ε = 0.01.

[0136] Then, under the action of the designed iterative learning law (9) and (10), the parameter adaptive law (11)-(14) and the control signal (15)-(17), the angular displacement simulation diagram of the follower manipulator is as shown in Figure 4 , the elastic deformation simulation diagram of the follower manipulator is as shown in Figure 5 , the simulation diagram of the control signal u 1,i (t) is as shown in Figure 6 , the simulation diagram of the control signal u 2,i (t) is as shown in Figure 7 . Observing Figure 4 and Figure 5 It can be found that under the control method designed in the application, the follower manipulator can complete the tracking of the angular displacement of the virtual leader, and the elastic deformation of each flexible manipulator is suppressed (i.e. the vibration of the flexible manipulator is suppressed). Therefore, the multi-flexible manipulator system iterative learning fault-tolerant consensus control method based on the sampling data event trigger mechanism designed in the application is effective, that is, the fault-tolerant control of multiple flexible manipulators can be realized, and the vibration of the flexible manipulator is suppressed.

[0137] The part not described in detail in the above embodiment is prior art.

[0138] It should be noted that although the application has been described through the above embodiments, the application can also have other various embodiments. Without departing from the spirit and scope of the application, those skilled in the art can obviously make various corresponding changes and modifications to the application, but these changes and modifications should all belong to the scope protected by the appended claims and their equivalents of the application.

Claims

1. A method for iterative learning fault-tolerant consensus control of a multi-flexible manipulator system, characterized in that, The method comprises the following steps: S1, constructing a directed communication topology, taking the flexible manipulator as a follower, and establishing a virtual leader for the follower; A mathematical model of the multi-flexible manipulator system is established based on Hamilton's principle; S2, constructing a finite-time observer for estimating the angular displacement and angular velocity of the virtual leader for each follower of the multi-flexible manipulator system; The constructed finite-time observer is: wherein , , is a constant; , , , ; and denote the estimated values of the virtual leader angular displacement and angular velocity, respectively, by the finite-time observer of the nth follower flexible manipulator, and denote the first order derivative with respect to time and the first order derivative with respect to time, respectively; and are constants and , , , is an upper bound of the angular acceleration of the flexible manipulator; denotes the angular displacement of the virtual leader at the time instant , denotes the first order derivative with respect to time; S3, based on the virtual leader state estimated by the finite-time observer, combining the actuator fault model and the event-triggered rule based on sampling data to design the iterative learning law, the parameter adaptive law and the control signal of the multi-flexible manipulator system; The actuator fault model is: wherein denotes a control signal; denotes an unknown bounded continuous signal; , , ; The event-triggered rule based on sampling data is: wherein, is a control signal of the th following robot arm to be designed, ; represents the th trigger time of the th following robot arm control signal ; is a positive integer; represents the absolute value of ; and are event trigger parameters of the th following robot arm; represents the infimum of elements in the set; is a positive integer, is a sampling interval of the sampled data; The designed iterative learning law is: wherein , ; is an iteration term designed to compensate for , , is an upper bound for the perturbation , and and denote the first order derivative with respect to time and the first order derivative with respect to time, , ; denotes the period of the perturbation ; , , denotes the first order derivative with respect to time; ; The parameter adaptive law is: wherein , , and denote the estimated values of , , and , , , , , ; , , and denote the first order derivatives with respect to time of , , and ; and are parts of the control signal to be designed; , , , , , , and are normal numbers; The control signal is: wherein denotes the hyperbolic tangent function; , , , and are normal numbers; denotes the second derivative with respect to time; denotes the first derivative with respect to space at the space ; S4, based on the designed iterative learning law, parameter adaptive law and control signal, the follower in the multi-flexible manipulator system is controlled, so that the angular position tracking error between the follower and the virtual leader and the elastic deformation generated by the follower itself are bounded. 2.The method according to claim 1, wherein: The mathematical model of the multi-flexible manipulator system is: The boundary condition is: wherein, represents the spatial position of the robot arm, represents the length of the robot arm, represents the i-th following robot arm, is an integer greater than 2 and represents the maximum index of the following robot arms; represents the system running time, represents the load mass of the robot arm; , and represent the unit length mass, the bending stiffness and the hub inertia of the robot arm, respectively; represents the angular displacement of the i-th following robot arm at the time instant ; represents the second derivative with respect to time; represents the displacement of the i-th following robot arm at the spatial position , represents the second derivative with respect to time; represents the displacement of the i-th following robot arm at the spatial position , represents the second derivative with respect to space; represents the displacement of the i-th following robot arm at the spatial position , represents the second derivative with respect to space; represents the displacement of the i-th following robot arm at the spatial position , represents the second derivative with respect to space; represents the displacement of the i-th following robot arm at the spatial position , represents the second derivative with respect to time; represents the elastic deformation of the i-th following robot arm at the spatial position , represents the fourth derivative with respect to space; represents the elastic deformation of the i-th following robot arm at the spatial position 0; and represent the first and second derivatives with respect to space at the spatial position 0, respectively; represents the elastic deformation of the i-th following robot arm at the spatial position , represents the second and third derivatives with respect to space at the spatial position , represents the elastic deformation of the i-th following robot arm at the spatial position , represents the second and third derivatives with respect to space at the spatial position , represents the disturbance of the i-th following robot arm; and represents the disturbance of the i-th following robot arm; and represents the disturbance of the i-th following robot arm; and represents the disturbance of the i-th following robot arm; A control law for a following robot arm.

3. The method of claim 1, wherein the method further comprises: The directed communication topology is: with a figure The communication topology of a multi-flexible manipulator system is described, where the virtual leader and each manipulator are regarded as an agent, denotes a set of nodes, denotes a set of edges; assume a directed graph has a directed spanning tree rooted at the leader, 0 denotes the index of the virtual leader, 1 to denotes the index of a follower flexible manipulator; edge denotes the th flexible manipulator can obtain information from the th flexible manipulator; the adjacency matrix of graph where and ; if then , otherwise ; the leader can only send information to the followers, but cannot obtain information from the followers, i.e. , ; the Laplacian matrix of graph where denotes the Laplacian matrix among the follower manipulators, denotes the Laplacian matrix between the leader and the follower manipulators; and when , , otherwise .​​ 4.The method of claim 1, wherein: Establishing Lyapunov function For the stability analysis of the flexible multi-joint robot system, the Lyapunov functions are respectively: In the formula: in, and Represents the differential operator; , and It is a positive number; , , and They represent , , and The estimation error, and , , , ; express The first derivative with respect to time; and They represent The first derivative with respect to space and the second derivative with respect to space, , .

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