Iterative learning fault-tolerant consistency control method for multi-flexible mechanical arm system

Abstract

An iterative learning fault-tolerant consistency control method for a multi-flexible-mechanical-arm system comprises the following steps that S1, a directed communication topological structure is constructed, flexible mechanical arms serve as followers, and a virtual leader is established; a mathematical model of the multi-flexible mechanical arm system is established based on the Hamiltonian principle; s2, constructing a finite time observer for estimating the angular displacement and the angular velocity of the virtual leader for each follower of the multi-flexible mechanical arm system; s3, designing an iterative learning rule, a parameter adaptive rule and a control signal of the multi-flexible mechanical arm system based on the virtual leader state estimated by the finite time observer in combination with an actuator fault model and an event triggering rule based on sampling data; according to the control method, fault-tolerant consistency control design based on a sampling data event triggering mechanism is carried out in combination with an estimated value of a finite time observer, so that communication resources are saved, and meanwhile, consistency control of the system is realized.

Description

Technical Field

[0001] The present invention relates to the technical field of flexible robotic arms, and in particular to an iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system. Background Art

[0002] As an important part of modern automation technology, robotic arms have been widely used in many fields such as industry, medical care, and services, driven by scientific and technological progress and the development of advanced productivity. Traditional rigid robotic arms have high structural rigidity and limited motion paths, making them difficult to adapt to complex and changing working environments and unable to meet people's increasingly stringent requirements for lightweight, precision, and intelligence. In this context, flexible robotic arms have been widely studied due to their advantages of light weight, high flexibility, strong operability, and the ability to complete tasks in complex environments. Flexible robotic arms can not only move flexibly in narrow spaces, but also avoid damage caused by collisions through their own flexible characteristics. However, in actual production activities, due to the complexity of the external environment and the flexibility of the flexible robotic arm structure, the flexible robotic arm is prone to vibration and deformation during operation, which in turn affects the control accuracy of the system and even destroys the structural integrity of the system.

[0003] To this end, the prior art proposes to use fourth-order partial differential equations to model the flexible manipulator, 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 extend its service life. It should be pointed out that the existing control methods usually only consider the trajectory tracking and vibration control of a single flexible manipulator system, and there is little research on the consistency 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 factors such as system communication bandwidth, system communication delays and network congestion may occur, increasing the burden on network sensors; (2) During the control process of multiple flexible manipulators, the control accuracy of the entire system will decrease due to the accumulation of errors of a single flexible manipulator. Summary of the invention

[0004] The purpose of the present invention is to propose an iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system, and to design a fault-tolerant consistency control based on a sampled data event trigger mechanism in combination with the estimated value of a finite-time observer, so as to achieve consistency control of the system while saving communication resources.

[0005] The technical solution adopted by the present invention is: a multi-flexible mechanical arm system iterative learning fault-tolerant consistency control method, comprising the following steps:

[0006] S1. Construct a directed communication topology, use the flexible robotic arm as a follower, and establish a virtual leader for the followers;

[0007] Establish a mathematical model of a multi-flexible robotic arm system based on the Hamiltonian principle;

[0008] 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;

[0009] S3. Based on the virtual leader state estimated by the finite time observer, the iterative learning law, parameter adaptation law and control signal of the multi-flexible manipulator system are designed in combination with the actuator fault model and the event triggering rule based on the sampling data;

[0010] S4. Based on the designed iterative learning law, parameter adaptive law and control signal, the follower in the multi-flexible robotic arm system is controlled so that the angular position tracking error between the follower and the virtual leader and the elastic deformation of the follower itself are bounded.

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

[0012]

[0013] The boundary conditions are:

[0014]

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

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

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

[0018] The communication topology of the multi-flexible robotic arm system is described by a graph G = (P, Z). The virtual leader and each robotic arm are regarded as an agent, and P = {0, 1, ..., N} represents the set of nodes. Represents the set of edges. Assume 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 means that the jth flexible manipulator can obtain information from the ith flexible manipulator; the adjacency matrix A of the graph G = [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 followers and cannot obtain information from followers, that is, a 0j =0, j∈P. The Laplace matrix of graph G is where L1∈R N×N represents the Laplacian matrix between the follower manipulators, L2∈R N represents the Laplacian matrix between the leader and follower manipulators; L1 = [l ij ], and when i≠j, l ij =-aij ,otherwise

[0019] As a preferred solution, the constructed finite time observer is:

[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 denote the estimated values ​​of the angular displacement and angular velocity of the virtual leader by the finite-time observer of the i-th follower flexible manipulator, and Respectively The first derivative with respect to time and The first-order derivative with respect to time; h1, h2 and ι are all constants, and ι>k, k is the upper bound of the angular acceleration of the flexible manipulator; θ0(t) represents the angular displacement of the virtual leader at time t, Represents the first-order derivative of θ0(t) with respect to time.

[0022] As a preferred solution, the actuator fault model is:

[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 solution, the event triggering rule based on the sampling data is:

[0026]

[0027] in, is the control signal of the i-th follower robot arm to be designed, a=1,2; Indicates the control signal of the i-th follower robot The kth triggering moment, k is a positive integer; express The absolute value of a,i <1 and μa,i >0 is the event trigger parameter of the i-th follower robot; 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.

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

[0029]

[0030] Among them, 0<γ1<1, 0<γ2<1; p a,i (t) is used to compensate r a,i (t) is the iterative term designed for is the disturbance d a,i (t) is an upper bound, and and Respectively represent p a,i (t) The first-order derivative with respect to time and r a,i (t) is the first-order derivative with respect to time, a=1,2, i=1,2,...,N; F is the disturbance d a,i (t) period; Indicates e 1,i (t) the first derivative with respect to time;

[0031] As a preferred solution, the parameter adaptation rule is:

[0032]

[0033] in, and Respectively represent o 1,i , o 2,i , 1,i and λ 2,i The estimated value of

[0034] q a,i =sup t≥0 v a,i (t), a=1,2, i=1,2,...,N; and Respectively and The first derivative with respect to time; and is a part of the control signal to be designed; l1, l2, l3, l4, η1, η2, η3 and η4 are positive constants.

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

[0036]

[0037] Wherein, tanh represents the hyperbolic tangent function; c1, c2, τ4, τ5 and ε are positive constants; express The second derivative with respect to time; y′ i (L,t) represents y i The first-order derivative of (x, t) with respect to space L.

[0038] As a preferred solution, the Lyapunov function V(t) is established to analyze the stability of the multi-flexible manipulator system. The Lyapunov functions are:

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

[0040] Where:

[0041]

[0042] Where dx and dκ represent differential operators; τ1, τ2 and τ3 are positive constants; and Respectively and The estimation error of Indicates z i The first derivative of (x,t) with respect to time; y′ i (x,t) and y″ i (x,t) represent y respectively i The first-order derivative of (x, t) with respect to space and the second-order derivative with respect to space, a=1,2, i=1,2,...,N.

[0043] Compared with the prior art, the present invention has the following beneficial effects:

[0044] 1. The multi-flexible robotic arm system iterative learning fault-tolerant consistency control method established by the present invention can realize the control of the multi-flexible robotic arm system with good robustness and stability. Even when the robotic arm part fails or is disturbed, the stability and function of the overall system can still be maintained;

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

[0046] 3. Different from other methods, the present invention designs a finite-time observer for each follower based on the directed communication topology of the system, transforms the consistency control problem into a trajectory tracking problem, and improves the control accuracy of the system; compared with the continuous control method, the present invention uses an event trigger mechanism to design the controller, thereby improving the communication efficiency of the system.

[0047] 4. Compared with the existing methods of trajectory tracking and vibration suppression of a single flexible robotic arm system, the present invention establishes an iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system based on a sampling data event trigger mechanism. In the event triggered control, the robotic arm will only send status information when a specific event occurs. Compared with the traditional time trigger mechanism, this trigger mechanism effectively reduces the frequency of event triggering, saves computer system communication resources, reduces network bandwidth pressure, and solves the problem of network congestion to a certain extent. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

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

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

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

[0052] Figure 4 It is a simulation diagram of the angular displacement of the follower mechanical arm under the control method of the present invention;

[0053] Figure 5 It is a simulation diagram of the elastic deformation of the follower mechanical arm under the control method of the present invention;

[0054] Figure 6 is the control signal u of the present invention 1,i (t) simulation diagram;

[0055] Figure 7 is the control signal u of the present invention 2,i (t) simulation diagram. DETAILED DESCRIPTION

[0056] The present invention is described in detail below by way of exemplary embodiments. However, it should be understood that, without further description, elements, structures and features in one embodiment may also be beneficially combined in other embodiments.

[0057] It should be noted that: unless otherwise defined, the technical terms or scientific terms used in this document shall have the usual meanings understood by people with ordinary skills in the field to which the invention belongs. The words "one", "an" or "the" and the like used in the patent application specification and claims of the present invention do not express quantity restrictions, but indicate the existence of at least one; the words "first", "second" and "third" used in this document shall not be regarded as restrictions on the order of components, but are only used to distinguish different components; words such as "include" or "comprise" and the like indicate that the elements or objects appearing before "include" or "comprise" include the elements or objects listed after "include" or "comprise" and their equivalents, but do not exclude other elements or objects with the same function.

[0058] In order to more clearly describe the iterative learning fault-tolerant consistency control method of the multi-flexible robot system, combined with the attached Figure 1-6 Describe this embodiment:

[0059] See also Figure 1 and Figure 2 , an iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system, comprising the following steps:

[0060] S1. Construct a directed communication topology, use the flexible robotic arm as a follower, and establish a virtual leader for the followers;

[0061] Establish a mathematical model of a multi-flexible robotic arm system based on the Hamiltonian principle;

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

[0063] The communication topology of the multi-flexible robotic arm system is described by a graph G = (P, Z). The virtual leader and each robotic arm are regarded as an agent, and P = {0, 1, ..., N} represents the set of nodes. Represents the set of edges. Assume 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 means that the jth flexible manipulator can obtain information from the ith flexible manipulator; the adjacency matrix A of the graph G = [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 followers and cannot obtain information from followers, that is, a 0j =0, j∈P. The Laplace matrix of graph G is where L1∈R N×N represents the Laplacian matrix between the follower manipulators, L2∈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 established multi-flexible robotic arm system is:

[0065]

[0066] The boundary conditions are:

[0067]

[0068] y i (0,t)=y′ i (0,t)=y″ i (L,t)=0 (4)

[0069] Where x∈[0,L] represents the spatial position of the robot, L represents the length of the robot, i∈{1,2,...,N} represents the i-th follower robot, N is an integer greater than 2 and represents the maximum number of follower robots; t represents the system running time, m represents the load mass of the robot; ρ, EI and J represent the unit length mass, bending stiffness and hub inertia of the robot, respectively; θ i (t) represents the angular displacement of the i-th follower robot at time t; Represents θ i (t) the second derivative with respect to time; z i (x, t) represents the displacement of the ith follower robot at spatial position x, Indicates z i The second derivative of (x,t) with respect to time; z i (L,t) represents the displacement of the ith follower robot at the spatial position L, Indicates z i The second derivative of (x, t) with respect to time at the spatial position L; y i (x, t) represents the elastic deformation of the ith follower robot at the spatial position x; y″″ i (x,t) represents y i The fourth-order derivative of (x, t) with respect to space; y i(0,t) represents the elastic deformation of the ith follower robot at spatial position 0; y′ i (0,t) and y″ i (0,t) represent y i (x, t) is the first-order derivative and the second-order derivative of (x, t) with respect to space at spatial position 0; y i (L, t) represents the elastic deformation of the ith follower robot at the spatial position L; y″ i (L,t) and y″′ i (L,t) respectively represent y i (x, t) is the second-order derivative and the third-order derivative of (x, t) with respect to space at the spatial position L; d 1,i (t) and d 2,i (t) represents the disturbance of the ith follower robot; u 1,i (t) and u 2,i (t) represents the control law of the i-th following robot.

[0070] 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;

[0071] In a multi-flexible robotic arm system under a directed communication topology, the error may be accumulated and the control accuracy of the system may be reduced in the process of the leader's state information being transmitted step by step to the end of the communication topology. Therefore, the present invention designs a finite-time observer for each follower in combination with the directed communication topology to estimate the state of the leader. Then, the fault-tolerant consistency control design based on the sampled data event trigger mechanism is performed in combination with the estimated value of the finite-time observer, so as to realize the consistency control of the system while saving communication resources;

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

[0073]

[0074] 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 denote the estimated values ​​of the angular displacement and angular velocity of the virtual leader by the finite-time observer of the i-th follower flexible manipulator, and Respectively 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 manipulator; θ0(t) represents the angular displacement of the virtual leader at time t, Represents the first-order derivative of θ0(t) with respect to time.

[0076] S3. Based on the virtual leader state estimated by the finite time observer, the iterative learning law, parameter adaptation law and control signal of the multi-flexible manipulator system are designed in combination with the actuator fault model and the event triggering rule based on the sampling data;

[0077] Specifically, the actuator fault model used is:

[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 sampling data are:

[0081]

[0082] in, is the control signal of the i-th follower robot arm to be designed, a=1,2; Indicates the control signal of the i-th follower robot The kth triggering moment, k is a positive integer; express The absolute value of a,i <1 and μ a,i >0 is the event trigger parameter of the i-th follower robot; 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 rule of the design is:

[0084]

[0085] Among them, 0<γ1<1, 0<γ2<1; p a,i (t) is used to compensate r a,i (t) is the iterative term designed for is the disturbance d a,i(t) is an upper bound, and and Respectively represent p a,i (t) The first-order derivative with respect to time and r a,i (t) is the first-order derivative with respect to time, a=1,2, i=1,2,...,N; F is the disturbance d a,i (t) period; Indicates e 1,i (t) the first derivative with respect to time;

[0086] Specifically, the designed parameter adaptation rule is:

[0087]

[0088] in, and Respectively represent o 1,i , o 2,i , 1,i and λ 2,i The estimated value of q a,i =sup t≥0 v a,i (t), a=1,2, i=1,2,...,N; and Respectively and The first derivative with respect to time; and is a part of the control signal to be designed; l1, l2, l3, l4, η1, η2, η3 and η4 are positive constants.

[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 positive constants; express The second derivative with respect to time; y′ i (L,t) represents y i The first-order derivative of (x, t) with respect to space L.

[0092] S4, controlling 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 of the followers themselves are bounded;

[0093] The Lyapunov function V(t) is established. In the control process of the designed control signal, the stability analysis of the multi-flexible manipulator system is carried out. In the process of analysis, the parameters of the control signal, parameter adaptation law, iterative learning law and event triggering rule are specifically designed. The Lyapunov functions are:

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

[0095]

[0096] Where dx and dκ represent differential operators; τ1, τ2 and τ3 are positive constants; and Respectively and The estimation error of Indicates z i The first derivative of (x,t) with respect to time; y′ i (x,t) and y″ i (x,t) represent y respectively 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;

[0097] Furthermore, we define the constant The function max represents the maximum value function, and the function min represents the minimum value function; combined with the designed Lyapunov function, we can get the constant and satisfy and In the case of , the designed Lyapunov function V(t) is positive definite, that is, the Lyapunov function V(t) in formula (18) satisfies:

[0098]

[0099] Furthermore, by using the method of integration by parts and Young’s inequality, combined with the designed Lyapunov function (18), we can obtain the derivative of the Lyapunov function with respect to time:

[0100]

[0101] in, ξ3=min{l1n1,l2n2,l3n3,l4n4,} / 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 normal number.

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

[0103]

[0104] Wherein, V(0) is the value of V(t) at time t=0.

[0105] Furthermore, based on formula (22), we can obtain:

[0106]

[0107] Among them, |y i (x,t)| and|e 1,i (t)|represents y i (x,t) and e 1,i (t) is the absolute value; when time tends to infinity, |y i (x,t)|satisfies |e 1,i (t)|Satisfaction Where lim represents the limit symbol. Therefore, it can be obtained that the angular displacement tracking error of the i-th flexible manipulator is uniformly bounded and stable, and the vibration of the flexible manipulator is suppressed, that is, the multi-flexible manipulator system is stable. In addition, since the event trigger mechanism adopted by the present invention is an event trigger mechanism based on sampled data, the Zeno phenomenon will not occur.

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

[0109] Create a Lyapunov function:

[0110]

[0111] in, represents the first-order derivative of θ0(t) with respect to time; T represents the transpose of vectors and matrices.

[0112] Furthermore, substituting (6) into formula (25) and performing an equivalent transformation with the Laplace matrix of the directed communication topology yields:

[0113]

[0114] in, λ max (L1) represents the matrix λ max (L1) Maximum eigenvalue.

[0115] According to (26), (25) can be stabilized within a finite time T1. That is, within the finite time T1, the estimated error of the angular velocity of the virtual leader can be stabilized to 0. When t ≥ T1, we can get

[0116] In addition, establish the Lyapunov function:

[0117]

[0118] Substituting (5) into (27) and performing an equivalent transformation with the Laplace matrix of the directed communication topology, we can obtain:

[0119]

[0120] in,

[0121] According to (28), (27) can be stabilized within a finite time T2. That is, within the finite time T2, the estimated error of the angular displacement of the virtual leader can be stabilized to 0. When t ≥ T2, we can get In summary, the effectiveness of the designed finite-time observer has been demonstrated.

[0122] In order to illustrate in detail the control effect of the iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system of the present invention, the iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system based on a sampling data event trigger mechanism is simulated in MATLAB.

[0123] The present invention considers a virtual leader and four follower flexible manipulators in the simulation process. The directed communication topology of the multi-flexible manipulator system is as follows: Figure 2 shown.

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

[0125] Δx=0.1m,Δt=1×10 -4 s, h = 5 × 10 -4 s, L = 1m, J = 0.6kg·m2 , EI=9N·m 2 , ρ=0.6kg / m, m=0.25kg, θ d =0.1sin(0.12πt)rad;

[0126] The initial condition of the multi-flexible manipulator system is: p i (x,0)=0.3x,x∈[0,L], θ1(0)=1.8rad, θ2(0)=1.5rad, θ3(0)=2.2rad, θ4(0)=2.1rad;

[0127] Among them, θ1(0), θ2(0), θ3(0) and θ4(0) respectively represent the angular displacements of follower manipulators 1-4 at time 0.

[0128] The external disturbance in the system is:

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

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

[0131] Among them, sin and cos represent sine and cosine functions respectively.

[0132] Can get

[0133] The simulation diagram of elastic deformation of the following robot arm without control is shown in the figure Figure 3 shown.

[0134] from Figure 3 It can be found that without control, the elastic deformation of the follower robot arm is large, so it is necessary to apply appropriate control to the system.

[0135] Furthermore, given the parameters ψ required in the design process of the control method of the present invention 1,1 =ψ 1,2 =ψ 1,3 =ψ 1,4 =0.5,ψ 2,1 =ψ 2,2 =0.02,ψ2,3 =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, eta1=eta2=10, eta3=1, eta4=0.005, γ1=γ2=0.001, ε=0.01.

[0136] Then, under the action of the designed iterative learning rules (9) and (10), parameter adaptive rules (11)-(14) and control signals (15)-(17), the angular displacement simulation diagram of the follower robot is shown in Figure 4 As shown in the figure, the elastic deformation simulation diagram of the following robot arm is as follows Figure 5 As shown. Control signal u 1,i The simulation diagram of (t) is shown in Figure 6 As shown, the control signal u 2,i The simulation diagram of (t) is shown in Figure 7 Observation Figure 4 and Figure 5 It can be found that under the control method designed by the present invention, 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 (that is, the vibration of the flexible manipulator is suppressed). Therefore, the multi-flexible manipulator system iterative learning fault-tolerant consistency control method based on the sampling data event trigger mechanism designed by the present invention is effective, that is, the fault-tolerant control of multiple flexible manipulators can be realized, and the vibration of the flexible manipulators is suppressed.

[0137] The parts not described in detail in the above embodiments are prior art.

[0138] It should be noted that although the present invention is described by the above embodiments, the present invention may also have other various embodiments. Without departing from the spirit and scope of the present invention, it is obvious that those skilled in the art may make various corresponding changes and deformations to the present invention, but these changes and deformations should all fall within the scope of protection of the appended claims of the present invention and their equivalents.

Claims

1. A multi-flexible robotic arm system iterative learning fault-tolerant consistency control method, characterized in that: The following steps are involved: S1. Construct a directed communication topology, use the flexible robotic arm as a follower, and establish a virtual leader for the followers; Establish a mathematical model of a multi-flexible robotic arm system based on the Hamiltonian 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; S3. Based on the virtual leader state estimated by the finite time observer, the iterative learning law, parameter adaptation law and control signal of the multi-flexible manipulator system are designed in combination with the actuator fault model and the event triggering rule based on the sampling data; S4. Based on the designed iterative learning law, parameter adaptation law and control signal, the follower in the multi-flexible robotic arm system is controlled so that the angular position tracking error between the follower and the virtual leader and the elastic deformation of the follower itself are bounded.

2. The iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system according to claim 1 is characterized by: The mathematical model of the multi-flexible robotic arm system is: The boundary conditions are: y i (0,t)=y i ′(0,t)=y i ″(L,t)=0 Where x∈[0,L] represents the spatial position of the robot, L represents the length of the robot, i∈{1,2,...,N} represents the i-th follower robot, N is an integer greater than 2 and represents the maximum number of follower robots; t represents the system running time, m represents the load mass of the robot; ρ, EI and J represent the unit length mass, bending stiffness and hub inertia of the robot, respectively; θ i (t) represents the angular displacement of the i-th follower robot at time t; Represents θ i (t) the second derivative with respect to time; z i (x, t) represents the displacement of the ith follower robot at spatial position x, Indicates z i The second derivative of (x,t) with respect to time; z i (L,t) represents the displacement of the ith follower robot at the spatial position L, Indicates z i The second derivative of (x, t) with respect to time at the spatial position L; y i (x, t) represents the elastic deformation of the ith follower robot at the spatial position x; y i ″″(x,t) represents y i The fourth-order derivative of (x, t) with respect to space; y i (0, t) represents the elastic deformation of the i-th follower robot at the spatial position 0; y i ′(0,t) and y i ″(0,t) represents y i (x, t) is the first-order derivative and the second-order derivative of (x, t) with respect to space at spatial position 0; y i (L, t) represents the elastic deformation of the ith follower robot at the spatial position L; y i ″(L,t) and y i ″′(L,t) represents y i (x, t) is the second-order derivative and the third-order derivative of (x, t) with respect to space at the spatial position L; d 1,i (t) and d 2,i (t) represents the disturbance of the ith follower robot; u 1,i (t) and u 2,i (t) represents the control law of the i-th following robot.

3. The iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system according to claim 1 is characterized by: The directed communication topology is: The communication topology of the multi-flexible robotic arm system is described by a graph G = (P, Z). The virtual leader and each robotic arm are regarded as an agent, and P = {0, 1, ..., N} represents the set of nodes. represents the set of edges; assuming 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 means that the jth flexible manipulator can obtain information from the ith flexible manipulator; the adjacency matrix A of the graph G = [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 followers and cannot obtain information from followers, that is, a 0j =0, j∈P; the Laplace matrix of graph G is where L1∈R N×N represents the Laplacian matrix between the follower manipulators, L2∈R N represents the Laplacian matrix between the leader and follower manipulators; L1 = [l ij ], and when i≠j, l ij =-a ij ,otherwise 4. The iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system according to claim 1, characterized in that: The constructed finite-time observer is: in, 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 denote the estimated values ​​of the angular displacement and angular velocity of the virtual leader by the finite-time observer of the i-th follower flexible manipulator, and Respectively The first derivative with respect to time and The first-order derivative with respect to time; h1, h2 and ι are all constants, and ι>k, k is the upper bound of the angular acceleration of the flexible manipulator; θ0(t) represents the angular displacement of the virtual leader at time t, Represents the first-order derivative of θ0(t) with respect to time.

5. The iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system according to claim 1, characterized in that: The actuator fault model is: 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.

6. The iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system according to claim 1, characterized in that: The event triggering rules based on the sampled data are: in, is the control signal of the i-th follower robot arm to be designed, a=1,2; Indicates the control signal of the i-th follower robot The kth triggering moment, k is a positive integer; express The absolute value of a,i <1 and μ a,i >0 is the event trigger parameter of the i-th follower robot; 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.

7. The iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system according to claim 1, characterized in that: The designed iterative learning law is: Among them, 0<γ1<1, 0<γ2<1; p a,i (t) is used to compensate r a,i (t) is the iterative term designed for is the disturbance d a,i (t) is an upper bound, and and Respectively represent p a,i (t) The first-order derivative with respect to time and r a,i (t) is the first-order derivative with respect to time, a=1,2, i=1,2,...,N; F is the disturbance d a,i (t) period; Indicates e 1,i (t) the first derivative with respect to time; 8. The iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system according to claim 1, characterized in that: The parameter adaptation rule is: in, and Respectively represent o 1,i , o 2,i , 1,i and λ 2,i The estimated value of q a,i =sup t≥0 v a,i (t),a=1,2,i=1,2,...,N; and Respectively and The first derivative with respect to time; and is a part of the control signal to be designed; l1, l2, l3, l4, η1, η2, η3 and η4 are positive constants.

9. The iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system according to claim 1, characterized in that: The control signal is: Wherein, tanh represents the hyperbolic tangent function; c1, c2, τ4, τ5 and ε are positive constants; express The second derivative with respect to time; y i ′(L,t) represents y i The first-order derivative of (x, t) with respect to space L.

10. The iterative learning fault-tolerant consistency control method for a multi-flexible robotic arm system according to claim 1, characterized in that: The Lyapunov function V(t) is established to analyze the stability of the multi-flexible manipulator system. The Lyapunov functions are: V(t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t) Where: Where dx and dκ represent differential operators; τ1, τ2 and τ3 are positive constants; and Respectively and The estimation error of Indicates z i The first derivative of (x,t) with respect to time; y i ′(x,t) and y i ″(x,t) represents y i The first-order derivative of (x, t) with respect to space and the second-order derivative with respect to space, a=1,2, i=1,2,...,N.

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