Multi-flexible mechanical arm system tracking and vibration control method based on event triggering

Abstract

The invention discloses a multi-flexible mechanical arm system tracking and vibration control method based on event triggering, and relates to the technical field of flexible mechanical arm system control, and the method comprises the following steps: establishing a multi-flexible mechanical arm system mathematical model based on the Hamiltonian principle; a control signal is designed by utilizing the boundary state of the mathematical model of the multi-flexible mechanical arm system under the undirected communication topological structure; based on a multi-flexible mechanical arm system control signal, an adaptive compensation rule and an adaptive iterative learning rule are designed to reduce negative effects of system parameter uncertainty and disturbance on system performance; and performing consistency tracking and vibration control on the multi-flexible mechanical arm system based on the control signal of the multi-flexible mechanical arm system, the self-adaptive compensation rule and the self-adaptive iterative learning rule. The communication efficiency and reliability of the system can be improved, and it is guaranteed that the following mechanical arm accurately tracks the leading mechanical arm and meanwhile vibration of the mechanical arm is restrained.

Description

Technical Field

[0001] The present invention relates to the technical field of flexible robotic arm control, and in particular to a multi-flexible robotic arm system tracking and vibration control method based on event triggering. Background Art

[0002] In application scenarios such as precision manufacturing and medical operations that require high flexibility and precision, traditional rigid robotic arms are difficult to adapt to complex and dynamically changing working environments due to their high structural rigidity and limited motion paths. For example, in narrow or irregular spaces, rigid robotic arms are often difficult to operate flexibly, and are prone to mission failure or damage to equipment due to collisions or confined spaces. In this context, flexible robotic arms, as an emerging solution, can better adapt to mission requirements in complex environments with their flexible and deformable characteristics. Flexible robotic arms can not only move flexibly in narrow spaces, but also avoid damage caused by collisions through their own flexible characteristics. However, the motion and vibration control of flexible robotic arms is more complex than that of rigid robotic arms, and traditional rigid control models cannot accurately simulate the dynamic behavior of flexible robotic arms.

[0003] To this end, the prior art proposes to model the flexible manipulator system using a fourth-order partial differential equation, and proposes a series of control strategies to suppress the vibration of the flexible manipulator, which can reduce the loss caused by vibration to the flexible manipulator to a certain extent. It should be pointed out that the prior art usually only considers the vibration control of a single flexible manipulator system, and lacks a collaborative control method for multiple flexible manipulator systems. The consistency tracking and vibration control of multiple flexible manipulator systems can improve the performance and stability of the system, and can quickly complete tasks in the face of complex situations. In practical applications, due to the limitations of factors such as the bandwidth of the system communication network, system information loss and delay may occur, resulting in a decrease in system control accuracy and impaired stability. In order to ensure the safe and reliable operation of the flexible manipulator system, event-triggered control has been used to solve the problem of limited communication resources in the system, and has made many advances. However, the current event-triggered control method mainly focuses on a single flexible manipulator system. Due to the complexity of the structure and communication topology of the multi-flexible manipulator system, the existing event-triggered control method has not been fully extended to the system. Therefore, for the established flexible manipulator system, how to design an adaptive boundary event-triggered tracking and vibration control method is of great value. Summary of the invention

[0004] The purpose of the present invention is to provide a multi-flexible robotic arm system tracking and vibration control method based on event triggering, mainly to solve the problem that the prior art lacks a collaborative control method for multiple flexible robotic arm systems.

[0005] To achieve the above purpose, the technical solution adopted by the present invention is as follows:

[0006] A method for tracking and vibration control of a multi-flexible manipulator system based on event triggering comprises the following steps:

[0007] S1, mathematical model of multi-flexible manipulator system is established based on Hamilton principle;

[0008] S2, using the boundary state of the mathematical model of the multi-flexible manipulator system under the undirected communication topology, the control signal of the multi-flexible manipulator system is designed based on the event triggering rules;

[0009] S3, based on the control signal of the multi-flexible manipulator system, design adaptive compensation law and adaptive iterative learning law to reduce the negative impact of system parameter uncertainty and disturbance on system performance;

[0010] S4, based on the control signal, adaptive compensation law and adaptive iterative learning law of the multi-flexible robot arm system, the consistency tracking and vibration control of the multi-flexible robot arm system are carried out.

[0011] Further, in the step S1, the mathematical model of the multi-flexible robotic arm system includes a mathematical model of a follower robotic arm and a mathematical model of a leader robotic arm;

[0012] The mathematical model of the following robot arm is:

[0013]

[0014] Its boundary conditions are expressed as:

[0015]

[0016] p i (0,t)=p i ′(0,t)=p i ″(L,t)=0

[0017] Where i∈{1,2,...,n} represents the i-th follower robot, n is an integer greater than 2 and represents the maximum serial number of the follower robot; x∈[0,L] represents the spatial position of the robot, L represents the length of the robot, t represents the system running time, and m represents the load mass of the robot; ρ, E, 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; Represents θ i (t) the second-order derivative with respect to time; p i (x, t) represents the elastic deformation of the ith follower robot at the spatial position x; p i ″″(x,t) represents p i The fourth-order derivative of (x, t) with respect to space; pi (0,t) represents the elastic deformation of the i-th follower robot at the spatial position 0; p i ′(0,t) and p i ″(0,t) respectively represents p i (x, t) is the first-order derivative and the second-order derivative of (x, t) with respect to space at the spatial position 0; p i (L, t) represents the elastic deformation of the ith follower robot at the spatial position L; p i ″(L,t) and p i ″′(L,t) respectively represents p 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; r i (x, t) represents the displacement of the ith follower robot at spatial position x; Represents r i The second derivative of (x,t) with respect to time; r i (L, t) represents the displacement of the i-th follower robot at the spatial position L; Represents r i The second derivative of (L,t) with respect to time; d 1,i (t) and d 2,i (t) represents the disturbance of the ith follower robot; w 1,i (t) and w 2,i (t) represents the control law of the i-th follower robot;

[0018] The mathematical model of the leadership robot arm is:

[0019]

[0020] The boundary conditions are expressed as:

[0021]

[0022] p0(0,t)=p0′(0,t)=p0″(L,t)=0

[0023] Among them, r0(x,t) represents the displacement of the leader robot at the spatial position x; represents the second-order derivative of r0(x, t) with respect to time; r0(L, t) represents the displacement of the leading manipulator at the spatial position L; represents the second-order derivative of r0(L,t) with respect to time; p0(x,t) represents the elastic deformation of the leading manipulator at the spatial position x; p0″″(x,t) represents the fourth-order derivative of p0(x,t) with respect to space; p0(0,t) represents the elastic deformation of the leading manipulator at the spatial position 0; p0′(0,t) and p0″(0,t) represent the first-order derivative and the second-order derivative of p0(x,t) at the spatial position 0 with respect to space, respectively; p0(L,t) represents the elastic deformation of the leading manipulator at the spatial position L; p0″(L,t) and p0″′(L,t) represent the second-order derivative and the third-order derivative of p0(x,t) at the spatial position 0 with respect to space, respectively; θ0(t) represents the angular displacement of the leading manipulator; represents the second-order derivative of θ0(t) with respect to time; w 1,0 (t) and w 2,0 (t) represents the control law of the leading robot arm.

[0024] Further, in step S2, the unidirectional communication topology is:

[0025] The communication topology between n follower robots is described by graph G = (P, Γ, A), where each robot is regarded as an agent. represents the set of edges, P = {1, ..., n} represents the set of nodes, (i, j) ∈ Γ represents that the i-th robot can obtain the information of the j-th robot; i = {P j |(P j ,P i )∈Γ,i≠j} represents the neighbor set of the i-th robot; A=[a ij ]∈R n×n represents the adjacency matrix, where if (i,j)∈Γ, then a ij =1, otherwise a ij =0, i≠j; S=diag{s1,…,s n} represents the inner degree matrix, where the symbol diag{s1,…,s n} means from s1 to s n The diagonal matrix composed of Y=SA is the Laplacian matrix of the graph G; the graph G is an undirected and connected graph, that is, (i,j)∈Γ is equivalent to (j,i)∈Γ; represents the enhanced graph of graph G, 0 represents the serial number of the leading robot. If the leading robot passes the information to the i-th follower robot, then a i0 =1, otherwise a i0 =0; A1 = diag{a 10 ,a 20 ,...,aN0}; D = Y + A1 is The communication matrix.

[0026] Furthermore, in step S2, the event triggering rule is:

[0027]

[0028] Among them, h 1,i (t) and h 2,i (t) is the control signal of the i-th follower robot to be designed; k is a positive integer; t 1,k and t 2,k represent the kth triggering moment of the control signal respectively; |w 1,i (t)| and|w 2,i (t)|represents w 1,i (t) and w 2,i Absolute value of (t); 0<μ 1,i <1、0<μ 2,i <1, b 1,i >0 and b 2,i >0 is the event trigger parameter of the i-th follower robot; inf represents the infimum of the elements in the set.

[0029] Furthermore, the control signal h of the following robot arm 1,i (t), h 2,i The mathematical expression of (t) is:

[0030]

[0031] The control law of the leading robot arm w 1,0 (t) and w 2,0 The mathematical expression of (t) is:

[0032]

[0033] Among them, c1, c2, ε, and is a positive constant; tanh is the hyperbolic tangent function; and are the perturbations d 1,i (t) and d 2,i The maximum value of (t); and is an auxiliary variable of the construction; Represents θ i (t) the first derivative with respect to time; is the angle error of the defined i-th follower robot; Indicates Υ i (t) the first derivative with respect to time; and are the i-th follower robot arm about P 1,i , P 2,i , J, and E estimates; λ1(t)∈[-1,1] and λ2(t)∈[-1,1] are functions of time t; sup represents the maximum value of the elements in the set; υ 1,i (t) and v 2,i (t) is used to compensate for the disturbance d 1,i (t) and d 2,i (t) the designed iteration item; p i ′(L,t), and Respectively represent p i (x, t) is the first-order derivative of space, the derivative of first-order time and then first-order space, and the derivative of first-order time and then third-order space at the spatial position L; and r i ″′(L,t) represents r i The first derivative of (x, t) with respect to time and the third derivative with respect to space at the spatial position L; e1(t) = θ0(t) - θ d θ d is a constant, representing the desired angular displacement; represents the first-order derivative of θ0(t) with respect to time; p0′(L,t) represents the first-order derivative of p0(x,t) at the spatial position L with respect to space; and They represent the derivative of p0(x,t) at the spatial position L with respect to the first-order time and then the first-order space and the derivative with respect to the first-order time and then the third-order space respectively; and r0″′(L,t) represent the first-order derivative of r0(x,t) with respect to time and the third-order derivative with respect to space at the spatial position L, respectively.

[0034] Further, in step S3, the adaptive compensation rule is:

[0035]

[0036] Among them, l1, l2, l3 and l4 are positive constants; and Respectively and First derivative with respect to time.

[0037] Furthermore, in step S3, the adaptive iterative learning rule is:

[0038]

[0039] Among them, ζ1∈(0,1) and ζ2∈(0,1) are constants; F is the disturbance d 1,i (t) and d 2,i (t) period; and Respectively represent υ 1,i (t) and v 2,i (t) the first derivative with respect to time; and Respectively represent β 1,i (t) and β 2,i (t) The first derivative with respect to time.

[0040] Furthermore, the method further comprises establishing a Lyapunov function V a (t) and V b (t), and conduct stability analysis of the multi-flexible manipulator system. The Lyapunov functions are:

[0041] V a (t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t),

[0042] V b (t)=V6(t)+V7(t)+V8(t),

[0043]

[0044] Where dx and dκ represent differential operators; and is a positive constant; δ1(t)=[δ 1,i (t),...,δ 1,n (t)] T ,δ2(t)=[δ 2,i (t),...,δ 2,n (t)] T , the superscript T represents transpose; and represents the parameter estimation error, and satisfies and D -1 represents the inverse matrix of D; Represents r i The first derivative of (x,t) with respect to time; p i ′(x,t) and p i ″(x,t) represents p i (x, t) is the first-order derivative and the second-order derivative with respect to space; ∑(·) represents the sum of the elements in the brackets; represents the first-order derivative of e1(t) with respect to time; represents the first-order derivative of r0(x, t) with respect to time; p0′(x, t) and p0″(x, t) represent the first-order derivative and the second-order derivative of p0(x, t) with respect to space, respectively.

[0045] Compared with the prior art, the present invention has the following beneficial effects:

[0046] The present invention discloses an adaptive tracking and vibration control method for a multi-flexible robotic arm system under event-triggered communication; compared with the existing mathematical model of a single flexible robotic arm system, the present invention respectively establishes mathematical models of a leading robotic arm and a following robotic arm, which have good applicability and robustness; compared with the existing flexible robotic arm system tracking and vibration control method, the present invention reduces the negative impact caused by system parameter uncertainty and disturbance by designing an adaptive compensation law and an adaptive iterative learning law, thereby improving the control performance and reliability of the system; compared with the traditional continuous control method, the present invention designs event-triggered control to improve the communication efficiency of the system, while ensuring that the following robotic arm accurately tracks the leading robotic arm while suppressing the vibration of the robotic arm; using the method of the present invention, the control of the process object can be achieved, with good robustness and stability, and can effectively improve the economic benefits. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 is a schematic diagram of a multi-flexible mechanical arm system in the present invention;

[0048] Figure 2 It is the network communication topology diagram in the present invention;

[0049] Figure 3 It is a simulation diagram of elastic deformation of the leading mechanical arm without control in the present invention;

[0050] Figure 4 It is a simulation diagram of elastic deformation of the follower mechanical arm without control in the present invention;

[0051] Figure 5 It is a simulation diagram of the angular displacement of the leading mechanical arm and the following mechanical arm under the control method of the present invention;

[0052] Figure 6 It is a simulation diagram of the elastic deformation of the leading mechanical arm under the control method of the present invention.

[0053] Figure 7 It is a simulation diagram of the elastic deformation of the follower robot arm under the control method of the present invention. DETAILED DESCRIPTION

[0054] The present invention is further described below in conjunction with the accompanying drawings and embodiments. The embodiments of the present invention include but are not limited to the following embodiments.

[0055] Example

[0056] like Figure 1 and Figure 2 As shown, the present invention provides a multi-flexible mechanical arm system tracking and vibration control method based on event triggering, which is as follows:

[0057] Step 1: Establish a mathematical model of the multi-flexible robotic arm system based on the Hamiltonian principle, including the mathematical model of the follower robotic arm and the mathematical model of the leading robotic arm.

[0058] Among them, the mathematical model of the following robot arm is:

[0059]

[0060] The boundary conditions are expressed as:

[0061]

[0062]

[0063] p i (0,t)=p i ′(0,t)=p i ″(L,t)=0 (4)

[0064] Where i∈{1,2,...,n} represents the i-th follower robot, n is an integer greater than 2 and represents the maximum serial number of the follower robot; x∈[0,L] represents the spatial position of the robot, L represents the length of the robot, t represents the system running time, and m represents the load mass of the robot; ρ, E, 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; Represents θ i (t) the second-order derivative with respect to time; p i (x, t) represents the elastic deformation of the ith follower robot at the spatial position x; p i ″″(x,t) represents p i The fourth-order derivative of (x, t) with respect to space; p i (0,t) represents the elastic deformation of the i-th follower robot at the spatial position 0; p i ′(0,t) and p i ″(0,t) respectively represents p i (x, t) is the first-order derivative and the second-order derivative of (x, t) with respect to space at the spatial position 0; p i(L, t) represents the elastic deformation of the ith follower robot at the spatial position L; p i ″(L,t) and p i ″′(L,t) respectively represents p 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; r i (x, t) represents the displacement of the ith follower robot at spatial position x; Represents r i The second derivative of (x,t) with respect to time; r i (L, t) represents the displacement of the i-th follower robot at the spatial position L; Represents r i The second derivative of (L,t) with respect to time; d 1,i (t) and d 2,i (t) represents the disturbance of the ith follower robot; w 1,i (t) and w 2,i (t) represents the control law of the i-th following robot.

[0065] The mathematical model of the leadership robot arm is:

[0066]

[0067] The boundary conditions are expressed as:

[0068]

[0069] p0(0,t)=p0′(0,t)=p0″(L,t)=0 (8)

[0070] Among them, r0(x,t) represents the displacement of the leader robot at the spatial position x; represents the second-order derivative of r0(x, t) with respect to time; r0(L, t) represents the displacement of the leading manipulator at the spatial position L; represents the second-order derivative of r0(L,t) with respect to time; p0(x,t) represents the elastic deformation of the leading manipulator at the spatial position x; p0″″(x,t) represents the fourth-order derivative of p0(x,t) with respect to space; p0(0,t) represents the elastic deformation of the leading manipulator at the spatial position 0; p0′(0,t) and p0″(0,t) represent the first-order derivative and the second-order derivative of p0(x,t) at the spatial position 0 with respect to space, respectively; p0(L,t) represents the elastic deformation of the leading manipulator at the spatial position L; p0″(L,t) and p0″′(L,t) represent the second-order derivative and the third-order derivative of p0(x,t) at the spatial position 0 with respect to space, respectively; θ0(t) represents the angular displacement of the leading manipulator; represents the second-order derivative of θ0(t) with respect to time; w 1,0 (t) and w 2,0 (t) represents the control law of the leading robot arm.

[0071] Step 2: Design the control signal of the multi-flexible robotic arm system using the boundary state of the system under the undirected communication topology.

[0072] Step 2.1: Establish an undirected communication topology structure, specifically:

[0073] The communication topology between n follower robots is described by graph G = (P, Γ, A), where each robot is regarded as an agent. represents the set of edges, P = {1, ..., n} represents the set of nodes, (i, j) ∈ Γ represents that the i-th robot can obtain the information of the j-th robot; i = {P j |(P j ,P i )∈Γ,i≠j} represents the neighbor set of the i-th robot; A=[a ij ]∈R n×n represents the adjacency matrix, where if (i,j)∈Γ, then a ij =1, otherwise a ij =0, i≠j; S=diag{s1,…,s n} represents the inner degree matrix, where the symbol diag{s1,…,s n} means from s1 to s n The diagonal matrix composed of Y=SA is the Laplacian matrix of the graph G; the graph G is an undirected and connected graph, that is, (i,j)∈Γ is equivalent to (j,i)∈Γ; represents the enhanced graph of graph G, 0 represents the serial number of the leader robot. If the leader robot can pass information to the i-th follower robot, then a i0 =1, otherwise a i0 =0; A1 = diag{a 10 ,a 20 ,...,a N0}; D = Y + A1 is The communication matrix.

[0074] Step 2.2: Design event triggering rules, specifically:

[0075]

[0076] Among them, h 1,i (t) and h 2,i (t) is the control signal of the i-th follower robot to be designed; k is a positive integer; t1,k and t 2,k represent the kth triggering moment of the control signal respectively; |w 1,i (t)| and|w 2,i (t)|represents w 1,i (t) and w 2,i Absolute value of (t); 0<μ 1,i <1、0<μ 2,i <1, b 1,i >0 and b 2,i >0 is the event trigger parameter of the i-th follower robot; inf represents the infimum of the elements in the set.

[0077] Step 2.3: Design the control signal of the multi-flexible manipulator system based on the boundary state information of the system, the undirected communication topology structure constructed in step 2.1, and the event triggering rules constructed in step 2.2. Specifically:

[0078] Multi-flexible robotic arm system control signal h 1,i (t), h 2,i (t) and control law w 1,0 (t), w 2,0 (t) are:

[0079]

[0080] Among them, c1, c2, ε, and is a positive constant; tanh is the hyperbolic tangent function; and are the perturbations d 1,i (t) and d 2,i The maximum value of (t); and is an auxiliary variable of the construction; Represents θ i (t) the first derivative with respect to time; is the angle error of the defined i-th follower robot; Indicates Υ i (t) the first derivative with respect to time; and are the i-th follower robot arm about P 1,i , P 2,i , J, and E estimates; λ1(t)∈[-1,1] and λ2(t)∈[-1,1] are functions of time t; sup represents the maximum value of the elements in the set; υ 1,i (t) and v 2,i (t) is used to compensate for the disturbance d 1,i (t) and d2,i (t) the designed iteration item; p i ′(L,t), and Respectively represent p i (x, t) is the first-order derivative of space, the derivative of first-order time and then first-order space, and the derivative of first-order time and then third-order space at the spatial position L; and r i ″′(L,t) represents r i The first derivative of (x, t) with respect to time and the third derivative with respect to space at the spatial position L; e1(t) = θ0(t) - θ d θ d is a constant, representing the desired angular displacement; represents the first-order derivative of θ0(t) with respect to time; p0′(L,t) represents the first-order derivative of p0(x,t) at the spatial position L with respect to space; and They represent the derivative of p0(x,t) at the spatial position L with respect to the first-order time and then the first-order space and the derivative with respect to the first-order time and then the third-order space respectively; and r0″′(L,t) represent the first-order derivative of r0(x,t) with respect to time and the third-order derivative with respect to space at the spatial position L, respectively.

[0081] Step 3: Design an adaptive compensation rule to reduce the negative impact of system parameter uncertainty on control performance. Specifically:

[0082] Based on the designed control signals (11) and (12), the adaptive compensation law is designed:

[0083]

[0084]

[0085] Among them, l1, l2, l3 and l4 are positive constants; and Respectively and First derivative with respect to time.

[0086] Step 4: Design an adaptive iterative learning rule to reduce the negative impact of disturbances on control performance.

[0087] The adaptive iterative learning rule is:

[0088]

[0089] Among them, ζ1∈(0,1) and ζ2∈(0,1) are constants; F is the disturbance d 1,i (t) and d 2,i (t) period; and Respectively represent υ 1,i (t) and v 2,i (t) the first derivative with respect to time; and Respectively represent β 1,i (t) and β 2,i (t) The first derivative with respect to time.

[0090] Step 5: Establish the Lyapunov function and conduct stability analysis on the multi-flexible robotic arm system.

[0091] Establish the Lyapunov function V a (t) and V b (t):

[0092] V a (t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t) (23)

[0093] V b (t)=V6(t)+V7(t)+V8(t) (24)

[0094] Where:

[0095]

[0096]

[0097] Where dx and dκ represent differential operators; and is a positive constant; δ1(t)=[δ 1,i (t),...,δ 1,n (t)] T ,δ2(t)=[δ 2,i (t),...,δ 2,n (t)] T , the superscript T represents transpose; and represents the parameter estimation error, and satisfies and D -1 represents the inverse matrix of D; Represents r i The first derivative of (x,t) with respect to time; p i ′(x,t) and p i ″(x,t) represents pi (x, t) is the first-order derivative and the second-order derivative with respect to space; ∑(·) represents the sum of the elements in the brackets; represents the first-order derivative of e1(t) with respect to time; represents the first-order derivative of r0(x, t) with respect to time; p0′(x, t) and p0″(x, t) represent the first-order derivative and the second-order derivative of p0(x, t) with respect to space, respectively.

[0098] 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 a (t) and V b (t) are all positive definite, that is, the Lyapunov function V(t) in formula (23) satisfies:

[0099]

[0100] Furthermore, by using the method of integration by parts and Young’s inequality, combined with the designed Lyapunov function (23)-(24), the derivative of the Lyapunov function with respect to time can be obtained:

[0101]

[0102] Among them, the constant and Satisfy C is a positive constant;

[0103]

[0104]

[0105] ζ=min{l1,l2,l3,l4}.

[0106]

[0107] Among them, λ max (D -1 ) represents the matrix D -1 The maximum eigenvalue of min (D) represents the smallest eigenvalue of the matrix D; τ1, τ2, τ3, τ4, τ5, κ1 and κ2>1 are positive constants.

[0108] Furthermore, according to formulas (25), (26) and (29), by using the integration technique, the Lyapunov function satisfies:

[0109]

[0110] Among them, V a (0) is V a (t) The value at time t=0.

[0111] Furthermore, based on formula (27), we can get

[0112]

[0113] Among them, |p i (x,t)| and |Υ i (t)|represents p i (x,t) and Υ i (t) is the absolute value; when time tends to infinity, |p i (x,t)| and |Υi(t| satisfy and Where lim represents the limit symbol.

[0114] Moreover, according to formulas (24) and (30), the elastic deformation p0(x, t) and angular displacement error e1(t) of the leading manipulator are asymptotically stable. Therefore, the multi-flexible manipulator system is stable.

[0115] In addition, suppose time t i * satisfy definition Then we can get:

[0116]

[0117] Wherein, sign represents the sign function; z=1,2.

[0118] According to formula (11-12), we can get w z,i (t) is differentiable, and w z,i The derivative of (t) is a smooth function, so let the constant q>0 satisfy At the triggering moment, it can be obtained that when t approaches t i k+1 Time|e z,i (t)|=X i , X i is a positive number, so the trigger interval can be obtained ti * The lower bound satisfies Therefore, Zeno's phenomenon will not occur.

[0119] In order to illustrate in detail the control effect of the event-triggered multi-flexible manipulator system tracking and vibration control method of the present invention, the event-triggered multi-flexible manipulator system tracking and vibration control method is simulated in MATLAB.

[0120] The present invention considers five flexible manipulator systems in the simulation process, including one leading manipulator and four following manipulators. The communication topology of the multi-flexible manipulator system is as follows: Figure 2 shown.

[0121] The specific parameters of the multi-flexible robotic arm system are:

[0122] L = 1m, J = 0.6kg·m 2 , E=8N·m 2 , ρ=0.5kg / m, m=0.2kg, θ d =1rad.

[0123] The initial conditions of the multi-flexible manipulator system are:

[0124] p i (x,0)=0.3x,x∈[0,L], θ0(0)=0.5rad, θ1(0)=1.8rad, θ2(0)=1.5rad, θ3(0)=2.2rad,

[0125] θ4(0)=2.1rad.

[0126] Among them, θ0(0), θ1(0), θ2(0), θ3(0) and θ4(0) represent the angular displacements of the leading robot and the following robot 1-4 at time 0, respectively.

[0127] The external disturbance in the system is:

[0128] d 1,1 (t) = 0.2cosπt, d 1,2 (t) = 0.4cosπt, d 1,3 (t) = 0.3cosπt, d 1,4 (t) = 0.1cosπt,

[0129] d 2,1 (t) = 0.3sinπt, d 2,2 (t) = 0.2sinπt, d 2,3 (t) = 0.2sinπt, d 2,4 (t) = 0.1sinπt,

[0130] Among them, sin and cos represent sine and cosine functions respectively.

[0131] Can get

[0132] The simulation diagram of the elastic deformation of the leading robot arm without control is shown in the figure Figure 3 As shown in the figure, the elastic deformation simulation diagram of the following robot arm without control is as follows Figure 4 shown.

[0133] from Figure 4 and Figure 5 It can be found that without control, the elastic deformation vibration amplitude of the leading robot arm and the following robot arm is large, so it is necessary to apply appropriate control to the system.

[0134] Furthermore, given the parameters μ required in the design process of the control method of the present invention, 1,1 =0.11, μ 1,2 =0.12, μ 1,3 =0.12, μ 1,4 =0.12, μ 2,1 =0.015, μ 2,2 =0.018, μ 2,3 =0.02, μ 2,4 =0.015, b 1,1 =0.43, b 1,2 =0.47, b 1,3 =0.5, b 1,4 =0.45, b 2,1 =0.515, b 2,2 =0.5, b 2,3 =0.65, b 2,4 =0.45, c1=c2=1, ε=0.01, l1=5, l2=4, l3=l4=1, ζ1=0.001, ζ2=0.001.

[0135] Then, under the effects of the designed control signals (11)-(12) and (15)-(16), the adaptive compensation rules (17)-(20), and the adaptive iterative learning rules (21) and (22), the angular displacement simulation diagram of the leading robot arm and the following robot arm is shown in FIG. Figure 5 As shown in the figure, the elastic deformation simulation diagrams of the leading robot arm and the following robot arm are shown in Figure 6 and Figure 7 As shown. Figure 5 , Figure 6 and Figure 7It can be found that under the control method designed by the present invention, the follower robot arm can complete the tracking of the angular displacement of the leading robot arm, and the elastic deformation of each flexible robot arm is suppressed (that is, the vibration of the flexible robot arm is suppressed). Therefore, the event-triggered multi-flexible robot arm system tracking and vibration control method designed by the present invention is effective, that is, the consistency tracking of multiple flexible robot arms can be achieved, and the vibration of the flexible robot arms is suppressed.

[0136] The above embodiment is only one of the preferred implementation modes of the present invention and should not be used to limit the protection scope of the present invention. Any changes or modifications that are made to the main design concept and spirit of the present invention and have no substantive significance, and the technical problems they solve are still consistent with the present invention, should be included in the protection scope of the present invention.

Claims

1. A tracking and vibration control method for a multi-flexible manipulator system based on event triggering, characterized in that: The following steps are involved: S1, mathematical model of multi-flexible manipulator system is established based on Hamilton principle; S2, using the boundary state of the mathematical model of the multi-flexible manipulator system under the undirected communication topology, the control signal of the multi-flexible manipulator system is designed based on the event triggering rules; S3, based on the control signal of the multi-flexible manipulator system, design adaptive compensation law and adaptive iterative learning law to reduce the negative impact of system parameter uncertainty and disturbance on system performance; S4, based on the control signal, adaptive compensation law and adaptive iterative learning law of the multi-flexible robot arm system, the consistency tracking and vibration control of the multi-flexible robot arm system are carried out.

2. The event-triggered multi-flexible manipulator system tracking and vibration control method according to claim 1 is characterized in that: In the step S1, the mathematical model of the multi-flexible robotic arm system includes a mathematical model of a follower robotic arm and a mathematical model of a leader robotic arm; The mathematical model of the following robot arm is: Its boundary conditions are expressed as: p i (0,t)=p i ′(0,t)=p i ″(L,t)=0 Where i∈{1,2,...,n} represents the i-th follower robot, n is an integer greater than 2 and represents the maximum serial number of the follower robot; x∈[0,L] represents the spatial position of the robot, L represents the length of the robot, t represents the system running time, and m represents the load mass of the robot; ρ, E, 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; Represents θ i (t) the second derivative with respect to time; p i (x, t) represents the elastic deformation of the ith follower robot at the spatial position x; p i ″″(x,t) represents p i The fourth-order derivative of (x, t) with respect to space; p i (0,t) represents the elastic deformation of the i-th follower robot at the spatial position 0; p i ′(0,t) and p i ″(0,t) respectively represent p i (x, t) is the first-order derivative and the second-order derivative of (x, t) with respect to space at the spatial position 0; p i (L, t) represents the elastic deformation of the ith follower robot at the spatial position L; p i ″(L,t) and p i ″′(L,t) respectively represents p 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; r i (x, t) represents the displacement of the ith follower robot at spatial position x; Represents r i The second derivative of (x,t) with respect to time; r i (L, t) represents the displacement of the i-th follower robot at the spatial position L; Represents r i The second derivative of (L,t) with respect to time; d 1,i (t) and d 2,i (t) represents the disturbance of the ith follower robot; w 1,i (t) and w 2,i (t) represents the control law of the i-th follower robot; The mathematical model of the leadership robot arm is: The boundary conditions are expressed as: p0(0,t)=p0′(0,t)=p0″(L,t)=0 Among them, r0(x,t) represents the displacement of the leader robot at the spatial position x; represents the second-order derivative of r0(x, t) with respect to time; r0(L, t) represents the displacement of the leading manipulator at the spatial position L; represents the second-order derivative of r0(L,t) with respect to time; p0(x,t) represents the elastic deformation of the leading manipulator at the spatial position x; p0″″(x,t) represents the fourth-order derivative of p0(x,t) with respect to space; p0(0,t) represents the elastic deformation of the leading manipulator at the spatial position 0; p0′(0,t) and p0″(0,t) represent the first-order derivative and the second-order derivative of p0(x,t) at the spatial position 0 with respect to space, respectively; p0(L,t) represents the elastic deformation of the leading manipulator at the spatial position L; p0″(L,t) and p0″′(L,t) represent the second-order derivative and the third-order derivative of p0(x,t) at the spatial position 0 with respect to space, respectively; θ0(t) represents the angular displacement of the leading manipulator; represents the second-order derivative of θ0(t) with respect to time; w 1,0 (t) and w 2,0 (t) represents the control law of the leading robot arm.

3. The event-triggered multi-flexible manipulator system tracking and vibration control method according to claim 2 is characterized in that: In step S2, the unidirectional communication topology is: The communication topology between n follower robots is described by graph G = (P, Γ, A), where each robot is regarded as an agent. represents the set of edges, P = {1, ..., n} represents the set of nodes, (i, j) ∈ Γ represents that the i-th robot can obtain the information of the j-th robot; i = {P j |(P j ,P i )∈Γ,i≠j} represents the neighbor set of the i-th robot; A=[a ij ]∈R n×n represents the adjacency matrix, where if (i,j)∈Γ, then a ij =1, otherwise a ij =0, i≠j; S=diag{s1,…,s n } represents the inner degree matrix, where the symbol diag{s1,…,s n } means from s1 to s n The diagonal matrix composed of Y=SA is the Laplacian matrix of the graph G; the graph G is an undirected and connected graph, that is, (i,j)∈Γ is equivalent to (j,i)∈Γ; represents the enhanced graph of graph G, 0 represents the serial number of the leading robot. If the leading robot passes the information to the i-th follower robot, then a i0 =1, otherwise a i0 =0; A1 = diag{a 10 ,a 20 ,...,a N0 }; D = Y + A1 is The communication matrix.

4. The event-triggered multi-flexible manipulator system tracking and vibration control method according to claim 3 is characterized in that: In step S2, the event triggering rule is: Among them, h 1,i (t) and h 2,i (t) is the control signal of the i-th follower robot to be designed; k is a positive integer; t 1,k and t 2,k Respectively represent the kth triggering moment of the control signal; |w 1,i (t)| and|w 2,i (t)|represents w 1,i (t) and w 2,i Absolute value of (t); 0<μ 1,i <1、0<μ 2,i <1, b 1,i >0 and b 2,i >0 is the event trigger parameter of the i-th follower robot; inf represents the infimum of the elements in the set.

5. The event-triggered multi-flexible manipulator system tracking and vibration control method according to claim 4 is characterized in that: The control signal h of the following robot arm 1,i (t), h 2,i The mathematical expression of (t) is: The control law of the leading robot arm w 1,0 (t) and w 2,0 The mathematical expression of (t) is: Among them, c1, c2, ε, and is a positive constant; tanh is the hyperbolic tangent function; and are the perturbations d 1,i (t) and d 2,i The maximum value of (t); and is the auxiliary variable of the construction; Represents θ i (t) the first derivative with respect to time; is the angle error of the defined i-th follower robot; Indicates Υ i (t) the first derivative with respect to time; and are the i-th follower robot arm about P 1,i , P 2,i , J, and E estimates; λ1(t)∈[-1,1] and λ2(t)∈[-1,1] are functions of time t; sup represents the maximum value of the elements in the set; υ 1,i (t) and v 2,i (t) is used to compensate for the disturbance d 1,i (t) and d 2,i (t) the designed iteration item; p i ′(L,t), and Respectively represent p i (x, t) is the first-order derivative of space, the derivative of first-order time and then first-order space, and the derivative of first-order time and then third-order space at the spatial position L; and r i ″′(L,t) represents r i The first derivative of (x, t) with respect to time and the third derivative with respect to space at the spatial position L; e1(t) = θ0(t) - θ d θ d is a constant, representing the desired angular displacement; represents the first-order derivative of θ0(t) with respect to time; p0′(L,t) represents the first-order derivative of p0(x,t) at the spatial position L with respect to space; and They represent the derivative of p0(x,t) at the spatial position L with respect to the first-order time and then the first-order space and the derivative with respect to the first-order time and then the third-order space respectively; and r0″′(L,t) represent the first-order derivative of r0(x,t) with respect to time and the third-order derivative with respect to space at the spatial position L, respectively.

6. The event-triggered multi-flexible manipulator system tracking and vibration control method according to claim 5, characterized in that: In step S3, the adaptive compensation rule is: Among them, l1, l2, l3 and l4 are positive constants; and Respectively and First derivative with respect to time.

7. The event-triggered multi-flexible manipulator system tracking and vibration control method according to claim 5, characterized in that: In step S3, the adaptive iterative learning rule is: Among them, ζ1∈(0,1) and ζ2∈(0,1) are constants; F is the disturbance d 1,i (t) and d 2,i (t) period; and Respectively represent υ 1,i (t) and v 2,i (t) the first derivative with respect to time; and Respectively represent β 1,i (t) and β 2,i (t) The first derivative with respect to time.

8. The event-triggered multi-flexible manipulator system tracking and vibration control method according to claim 7, characterized in that: It also includes the establishment of the Lyapunov function V a (t) and V b (t), and conduct stability analysis of the multi-flexible manipulator system. The Lyapunov functions are: V a (t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t), V b (t)=V6(t)+V7(t)+V8(t), Among them, dx and dκ represent differential operators; and is a positive constant; δ1(t)=[δ 1,i (t),...,δ 1,n (t)] T ,δ2(t)=[δ 2,i (t),...,δ 2,n (t)] T , the superscript T represents transpose; and represents the parameter estimation error, and satisfies and D -1 represents the inverse matrix of D; Represents r i The first derivative of (x,t) with respect to time; p i ′(x,t) and p i ″(x,t) represents p i (x, t) is the first-order derivative and the second-order derivative with respect to space; ∑(·) represents the sum of the elements in the brackets; represents the first-order derivative of e1(t) with respect to time; represents the first-order derivative of r0(x, t) with respect to time; p0′(x, t) and p0″(x, t) represent the first-order derivative and the second-order derivative of p0(x, t) with respect to space, respectively.

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