Event-triggered multi-flexible manipulator system tracking and vibration control method

Abstract

The application discloses a kind of based on event triggering's multi-flexible robot arm system tracking and vibration control method, it is related to flexible robot arm system control technical field, including: based on Hamilton principle establishes multi-flexible robot arm system mathematical model;In undirected communication topological structure, the boundary state of the mathematical model of multi-flexible robot arm system is used to design control signal;Based on multi-flexible robot arm system control signal, design adaptive compensation law and adaptive iterative learning law to reduce the negative influence caused by system parameter uncertainty and disturbance on system performance;Based on multi-flexible robot arm system control signal, adaptive compensation law and adaptive iterative learning law, the consistency tracking and vibration control of multi-flexible robot arm system are carried out.The application can improve the communication efficiency and reliability of system, ensure that the following robot arm accurately tracks leader robot arm while inhibiting the vibration of robot arm.

Description

TECHNICAL FIELD

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

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

[0003] To this end, existing technologies propose to model flexible robot arm systems using fourth-order partial differential equations and propose a series of control strategies to suppress the vibration of flexible robot arms, which can reduce the damage caused by vibration to flexible robot arms to some extent. It should be noted that existing technologies usually only consider single flexible robot arm system vibration control and lack collaborative control methods for multiple flexible robot arm systems. Consistent tracking and vibration control of multi-flexible robot arm systems can improve system performance and stability and quickly complete tasks in complex situations. In practical applications, due to limitations such as system communication network bandwidth, system information loss, delay, and other factors, the control accuracy of the system may decrease and the stability may be compromised. To ensure the safe and reliable operation of the flexible robot arm system, event-triggered control has been used to address the limited communication resources in the system and has made many advances. However, current event-triggered control methods mainly focus on single flexible robot arm systems. Due to the complexity of the structure and communication topology of multi-flexible robot arm systems, existing event-triggered control methods have not been fully extended to this system. Therefore, it is of great value to design an adaptive boundary event-triggered tracking and vibration control method for the established flexible robot arm system. SUMMARY

[0004] The present application relates to the technical field of flexible robot arm control, in particular to a multi-flexible robot arm system tracking and vibration control method based on event triggering.

[0005] To achieve the above-mentioned purpose, the technical solutions adopted by the present application are as follows:

[0006] An event-triggered multi-flexible manipulator system tracking and vibration control method, comprising the following steps:

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

[0008] S2, in a non-directional communication topology, a boundary state of the mathematical model of the multi-flexible manipulator system is used to design a control signal of the multi-flexible manipulator system based on an event-triggered rule;

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

[0010] S4, based on the control signal of the multi-flexible manipulator system, the adaptive compensation law and the adaptive iterative learning law, the multi-flexible manipulator system is subjected to consistent tracking and vibration control.

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

[0012] The mathematical model of the follower manipulator is:

[0013]

[0014] The boundary condition is expressed as:

[0015]

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

[0017] Wherein, 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; x∈[0,L] represents the spatial position of the manipulator, L represents the length of the manipulator, t represents the system running time, m represents the load mass of the manipulator; ρ, E and J represent the unit length mass, bending stiffness and hub inertia of the manipulator, respectively; θ i (t) represents the angular displacement of the i-th follower manipulator; represents the second derivative of θ i (t) with respect to time; p i (x,t) represents the elastic deformation of the i-th follower manipulator at the spatial position x; p i ″″(x,t) represents the fourth derivative of p i (x,t) with respect to space;i (0, t) denotes the elastic deformation of the i-th follower manipulator at spatial position 0; p i (0, t) and p i (0, t) and p i (x, t) denotes the first order derivative of p i (L, t) denotes the elastic deformation of the i-th follower manipulator at spatial position L; p i (L, t) and p i (L, t) and p i (x, t) denotes the second order derivative of p i (x, t) denotes the displacement of the i-th follower manipulator at spatial position x; denotes r i (x, t) denotes the second order derivative of r i (L, t) denotes the displacement of the i-th follower manipulator at spatial position L; denotes r i (L, t) denotes the second order derivative of r 1,i (t) and d 2,i (t) denotes the disturbance of the i-th follower manipulator; w 1,i (t) and w 2,i (t) denotes the control law of the i-th follower manipulator;

[0018] The mathematical model of the leader manipulator is:

[0019]

[0020] The boundary conditions are expressed as:

[0021]

[0022] p0(0, t) = p0'(0, t) = p0"(L, t) = 0

[0023] where r0(x, t) denotes the displacement of the leader manipulator at spatial position x; denotes the second order derivative of r0(x, t) with respect to time; r0(L, t) denotes the displacement of the leader manipulator at spatial position L; Let θ0(L,t) represent the second derivative of r0(L,t) with respect to time; p0(x,t) represent the elastic deformation of the leading robotic arm at spatial position x; p0″″(x,t) represent the fourth derivative of p0(x,t) with respect to space; p0(0,t) represent the elastic deformation of the leading robotic arm at spatial position L; p0″(L,t) and p0″′(L,t) represent the second derivative of p0(x,t) with respect to space and the second derivative of p0(x,t) with respect to space and the third ...t) with respect to space and the third derivative of p0(t) with respect to space and the third derivative of p0(t) with respect to space and the third derivative of p0(t) with respect to space and the third derivative of p< w represents the second derivative of θ0(t) with respect to time. 1,0 (t) and w 2,0 (t) represents the control law of the leading robotic arm.

[0024] Further, in step S2, the undirected communication topology is as follows:

[0025] Let graph G = (P, Γ, A) describe the communication topology among n following robotic arms, where each robotic arm is considered an agent. Let P represent the set of edges, P = {1, ..., n} represent the set of nodes, and (i, j) ∈ Γ indicate that the i-th robotic arm can obtain information from the j-th robotic arm; Ξ i ={P j |(P j ,P i )∈Γ,i≠j} represents the neighborhood set of the i-th robotic arm; A=[a ij ]∈R n×n Let denote the adjacency matrix, where if (i,j)∈Γ, then a ij =1, otherwise a ij =0, i≠j; S=diag{s1,…,s n} denotes the interior degree matrix, where the symbol diag{s1,…,s} n} represents the transition from s1 to s n The diagonal matrix formed Y = SA is the Laplace matrix of graph G; graph G is an undirected and connected graph, i.e., (i,j)∈Γ is equivalent to (j,i)∈Γ; graph This represents the augmented graph of graph G. 0 represents the sequence number of the leader robotic arm. If the leader robotic arm transmits information to the i-th following robotic arm, then a... i0 =1, otherwise a i0 =0; A1=diag{a 10 ,a 20 ,...,aN0 D = Y + A1 a communication matrix.

[0026] Further, in the step S2, the event-triggered rule is:

[0027]

[0028] where h 1,i (t) and h 2,i (t) are the control signals of the i-th follower manipulator to be designed; k is a positive integer; t 1,k and t 2,k represent the k-th triggering time of the control signals, respectively; |w 1,i (t)| and |w 2,i (t)| represent the absolute values of w 1,i (t) and w 2,i (t), respectively; 0 < μ 1,i < 1, 0 < μ 2,i < 1, b 1,i > 0 and b 2,i > 0 are the event-triggered parameters of the i-th follower manipulator; inf represents the infimum of the set.

[0029] Further, the mathematical expressions of the control signals h 1,i (t), h 2,i (t) of the follower manipulator are:

[0030]

[0031] The mathematical expressions of the control laws w 1,0 (t) and w 2,0 (t) of the leader manipulator are:

[0032]

[0033] where c1, c2, ε, and are normal numbers; tanh is the hyperbolic tangent function; and are the maximum values of the disturbances d 1,i (t) and d 2,i (t), respectively; and are auxiliary variables constructed; represents the first-order derivative of θ i (t) with respect to time; is the defined angle error of the i-th follower manipulator; represents the first-order derivative of Y i (t) with respect to time; and are the estimated values of the i-th following robot with respect to P 1,i , P 2,i , J and E; λ1(t)∈[-1,1] and λ2(t)∈[-1,1] are functions of time t; sup denotes the maximum value of elements in a set; υ 1,i (t) and υ 2,i (t) are iterative terms designed to compensate for disturbances d 1,i (t) and d 2,i (t); p i ′(L,t), and denote the first-order derivative of p i (x,t) with respect to space, the first-order derivative with respect to time first and the second-order derivative with respect to space second, and the first-order derivative with respect to time first and the third-order derivative with respect to space second at spatial position L, respectively; and r i ″′(L,t) denote the first-order derivative with respect to time and the third-order derivative with respect to space of r i (x,t) at spatial position L; e1(t) = θ0(t) - θ d ; θ d is a constant, representing the desired angular displacement; denotes the first-order derivative of θ0(t) with respect to time; p0′(L,t) denotes the first-order derivative of p0(x,t) with respect to space at spatial position L; and denote the first-order derivative with respect to time first and the second-order derivative with respect to space second, and the first-order derivative with respect to time first and the third-order derivative with respect to space second of p0(x,t) at spatial position L, respectively; and r0″′(L,t) denote the first-order derivative with respect to time and the third-order derivative with respect to space of r0(x,t) at spatial position L, respectively.

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

[0035]

[0036] where l1, l2, l3 and l4 are normal numbers; and denote and the first-order derivative with respect to time.

[0037] Further, in the step S3, the adaptive iterative learning law is:

[0038]

[0039] where ζ1∈(0,1) and ζ2∈(0,1) are constants; F is the disturbance d 1,i (t) and d 2,i (t) is periodic; and denote the first order derivative of υ 1,i (t) and υ 2,i (t) with respect to time; and denote the first order derivative of β 1,i (t) and β 2,i (t) with respect to time.

[0040] Further, the method further comprises performing stability analysis of the flexible-joint manipulator system by establishing Lyapunov functions V a (t) and V b (t), which are respectively:

[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 are normal numbers; 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 transposition; and denote parameter estimation errors, and satisfy and D -1 denotes the inverse matrix of D; denotes the first order derivative of r i (x,t) with respect to time; p i ′(x,t) and p i ″(x,t) denote the first order derivative and the second order derivative of p i (x,t) with respect to space, respectively; ∑(·) represents summing the elements in the parentheses. This represents the first derivative of e1(t) with respect to time. Let r0(x,t) denote the first derivative of r0(x,t) with respect to time; p0′(x,t) and p0″(x,t) denote the first derivative of p0(x,t) with respect to space and the second derivative with respect to space, respectively.

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

[0046] This invention discloses an adaptive tracking and vibration control method for a multi-flexible robotic arm system under event-triggered communication. Compared with existing mathematical models of a single flexible robotic arm system, this invention establishes separate mathematical models for the leading and following robotic arms, exhibiting better applicability and robustness. Compared with existing tracking and vibration control methods for flexible robotic arm systems, this invention reduces the negative impact of system parameter uncertainties and disturbances by designing adaptive compensation and adaptive iterative learning laws, thereby improving the system's control performance and reliability. Compared with traditional continuous control methods, this invention designs event-triggered control to improve the system's communication efficiency, while ensuring that the following robotic arm accurately tracks the leading robotic arm and suppresses the robotic arm's vibration. Using this method, control of the process object can be achieved, exhibiting excellent robustness and stability, and effectively improving economic benefits. Attached Figure Description

[0047] Figure 1 This is a schematic diagram of the multi-flexible robotic arm system in this invention;

[0048] Figure 2 This is a network communication topology diagram in this invention;

[0049] Figure 3 This is a simulation diagram of the elastic deformation of the robotic arm without control in this invention;

[0050] Figure 4 This is a simulation diagram of the elastic deformation of the following robotic arm without control in this invention;

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

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

[0053] Figure 7 This is a simulation diagram of the elastic deformation of the following robotic arm under the control method of the present invention. Detailed Implementation

[0054] The application will be further described in connection with the accompanying drawings and examples, and the modes of the application include but are not limited to the following examples.

[0055] Embodiment

[0056] As shown in Figure 1 and Figure 2 , the application provides an event-triggered multi-flexible manipulator system tracking and vibration control method, specifically as follows:

[0057] Step 1, establish a mathematical model of the multi-flexible manipulator system based on the Hamilton principle; including the mathematical model of the follower manipulator and the mathematical model of the leader manipulator.

[0058] The mathematical model of the follower manipulator 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] Wherein, 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; x∈[0,L] represents the spatial position of the manipulator, L represents the length of the manipulator, t represents the system running time, m represents the load mass of the manipulator; ρ, E 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; represents the second derivative of θ i (t) with respect to time; p i (x,t) represents the elastic deformation of the i-th follower manipulator at the spatial position x; p i ″″(x,t) represents the fourth derivative of p i (x,t) with respect to space; p i (0,t) represents the elastic deformation of the i-th follower manipulator at the spatial position 0; p i ′(0,t) and p i ″(0,t) respectively represent the first derivative and the second derivative of p i (x,t) with respect to space at the spatial position 0; p i(L, t) represents the elastic deformation of the i-th follower manipulator at spatial position L; p i (L, t) and p i (L, t) and p i (x, t) represents the second order derivative of the displacement of the i-th follower manipulator with respect to space at spatial position x and the third order derivative of the displacement of the i-th follower manipulator with respect to space; r i (x, t) represents the displacement of the i-th follower manipulator at spatial position x; represents r i (x, t) represents the second order derivative of the displacement of the i-th follower manipulator with respect to time; r i (L, t) represents the displacement of the i-th follower manipulator at spatial position L; represents r i (L, t) represents the second order derivative of the displacement of the i-th follower manipulator with respect to time; d 1,i (t) and d 2,i (t) represents the disturbance of the i-th follower manipulator; w 1,i (t) and w 2,i (t) represents the control law of the i-th follower manipulator.

[0065] The mathematical model of the leader manipulator is:

[0066]

[0067] The boundary conditions are represented as:

[0068]

[0069] p0(0, t) = p0'(0, t) = p0"(L, t) = 0 (8)

[0070] wherein r0(x, t) represents the displacement of the leader manipulator at spatial position x; represents the second order derivative of the displacement of the leader manipulator with respect to time; r0(L, t) represents the displacement of the leader manipulator at spatial position L; represents the second order derivative of the displacement of the leader manipulator with respect to time; p0(x, t) represents the elastic deformation of the leader manipulator at 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 leader manipulator at spatial position 0; p0'(0, t) and p0"(0, t) respectively represent the first order derivative and the second order derivative of p0(x, t) with respect to space at spatial position 0; p0(L, t) represents the elastic deformation of the leader manipulator at spatial position L; p0"(L, t) and p0"'(L, t) respectively represent the second order derivative and the third order derivative of p0(x, t) with respect to space at spatial position 0; θ0(t) represents the angular displacement of the leader manipulator; denotes the second derivative of θ0(t) with respect to time; w 1,0 (t) and w 2,0 (t) denotes the control law of the leader manipulator.

[0071] Step 2, design the control signal of the multi-flexible manipulator system using the boundary state of the system under the undirected communication topology.

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

[0073] The communication topology between the n follower manipulators is described by the graph G=(P,Γ,A), where each manipulator is regarded as an agent, denotes the set of edges, P={1,…,n} denotes the set of nodes, (i,j)∈Γ indicates that the ith manipulator can obtain the information of the jth manipulator; Ξ i ={P j |(P j ,P i )∈Γ,i≠j} denotes the neighbor set of the ith manipulator; A=[a ij ]∈R n×n denotes the adjacency matrix, where a ij =1 if (i,j)∈Γ, otherwise a ij =0, i≠j; S=diag{s1,…,s n} denotes the in-degree matrix, where the symbol diag{s1,…,s n} denotes a diagonal matrix composed of s1 to s n , Y=S-A is the Laplacian matrix of graph G; graph G is undirected and connected, i.e. (i,j)∈Γ is equivalent to (j,i)∈Γ; graph denotes the augmented graph of graph G, 0 denotes the leader manipulator number, where a i0 =1 if the leader manipulator can transmit information to the ith follower manipulator, otherwise a i0 =0; A1=diag{a 10 ,a 20 ,...,a N0}; D=Y+A1 is the communication matrix of .

[0074] Step 2.2, design the event-triggered rule, specifically:

[0075]

[0076] where h 1,i (t) and h 2,i (t) are the control signals of the ith follower manipulator that need to be designed; k is a positive integer; t1,k and t 2,k denote the kth triggering moment of the control signal respectively; |w 1,i (t)| and |w 2,i (t)| denote the absolute value of w 1,i (t) and w 2,i (t) respectively; 0 < μ 1,i < 1, 0 < μ 2,i < 1, b 1,i > 0 and b 2,i > 0 are the event-triggered parameters of the ith follower manipulator; inf denotes the infimum of the elements in a set.

[0077] Step 2.3, based on the boundary state information of the system, the undirected communication topology structure constructed in step 2.1 and the event-triggered rule constructed in step 2.2, design the control signal of the multi-flexible manipulator system, specifically:

[0078] The control signal h 1,i (t) of the multi-flexible manipulator system, h 2,i (t) and the control law w 1,0 (t), w 2,0 (t) are respectively:

[0079]

[0080] Wherein, c1, c2, ε, and are normal numbers; tanh is the hyperbolic tangent function; and are the maximum values of the disturbances d 1,i (t) and d 2,i (t) respectively; and are auxiliary variables constructed; denotes the first order derivative of θ i (t) with respect to time; is the defined angle error of the ith follower manipulator; denotes the first order derivative of Y i (t) with respect to time; and are the estimated values of the ith follower manipulator with respect to P 1,i , P 2,i , J and E respectively; λ1(t) ∈ [-1, 1] and λ2(t) ∈ [-1, 1] are functions of time t; sup denotes the maximum value of the elements in a set; υ 1,i (t) and υ 2,i (t) are used to compensate for the disturbances d 1,i (t) and d2,i (t) the designed iteration term; p i ′(L, t), and respectively represent the first-order derivative of p i (x, t) with respect to space at spatial position L, the first-order derivative with respect to time first and the third-order derivative with respect to space second, and the first-order derivative with respect to time first and the fourth-order derivative with respect to space second; and r i ″′(L, t) represent the first-order derivative of r i (x, t) with respect to time and the third-order derivative with respect to space at 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) with respect to space at spatial position L; and respectively represent the first-order derivative with respect to time first and the third-order derivative with respect to space second, and the first-order derivative with respect to time first and the fourth-order derivative with respect to space second; 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 spatial position L.

[0081] Step 3, design an adaptive compensation law to reduce the negative impact of system parameter uncertainty on control performance, specifically:

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

[0083]

[0084]

[0085] where l1, l2, l3 and l4 are normal numbers; and respectively represent and the first-order derivative with respect to time.

[0086] Step 4, design an adaptive iterative learning law to reduce the negative impact of disturbance on control performance.

[0087] The adaptive iterative learning law is:

[0088]

[0089] where ζ1∈(0,1) and ζ2∈(0,1) are constants; F is the disturbance d 1,i (t) and d 2,i (t) are periodic; and denote the first order derivatives of υ 1,i (t) and υ 2,i (t) with respect to time; and denote the first order derivatives of β 1,i (t) and β 2,i (t) with respect to time.

[0090] Step 5, establish Lyapunov function, and analyze the stability of the flexible manipulator system.

[0091] Lyapunov functions V a (t) and V b (t) are established as follows:

[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] In the formula:

[0095]

[0096]

[0097] where dx and dκ represent differential operators; and are normal numbers; 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 transposition; and denote parameter estimation errors, and satisfy and D -1 represents the inverse matrix of D; denotes the first order derivative of r i (x,t) with respect to time; p i ′(x,t) and p i ″(x,t) respectively denote pi (x, t) and the second order derivative with respect to space; ∑(·) denotes the summation of the elements in the parentheses; denotes the first order derivative of e1(t) with respect to time; denotes the first order derivative of r0(x, t) with respect to time; p0'(x, t) and p0"(x, t) denote the first order derivative and the second order derivative of p0(x, t) with respect to space, respectively.

[0098] Further, define the constant where the function max denotes the maximum function and the function min denotes the minimum function; the designed Lyapunov function can be obtained in combination with the constant and satisfies and the designed Lyapunov function V a (t) and V b (t) are positive, i.e., the Lyapunov function V(t) in formula (23) satisfies:

[0099]

[0100] Further, by using the method of partial integration and the Young inequality, the derivative of the Lyapunov function with respect to time can be obtained in combination with the designed Lyapunov functions (23)-(24):

[0101]

[0102] where the constant and satisfy C is a normal number;

[0103]

[0104]

[0105] ζ = min{l1, l2, l3, l4}.

[0106]

[0107] where λ max (D -1 ) denotes the maximum eigenvalue of the matrix D -1 ; λ min (D) denotes the minimum eigenvalue of the matrix D; τ1, τ2, τ3, τ4, τ5, κ1 and κ2 > 1 are normal numbers.

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

[0109]

[0110] where V a (0) is the value of V a (t) at t = 0.

[0111] Further, based on the formula (27), it can be obtained that

[0112]

[0113] where |p i (x,t)| and |Υ i (t)| represent the absolute values of p i (x,t) and Υ i (t), respectively; when time tends to infinity, |p i (x,t)| and |Υi(t)| satisfy and where lim represents the limit symbol.

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

[0115] In addition, it is assumed that time t i * satisfies Definition Then it can be obtained that:

[0116]

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

[0118] According to the formulas (11-12), w z,i (t) is differentiable, and the derivative of w z,i (t) is a smooth function, therefore, it is assumed that a constant q > 0 satisfies At the triggering time, it can be obtained that when t tends to t i k+1 , |e z,i (t)| = X i , X i is a normal number, therefore, it can be obtained that the lower bound of the triggering interval ti * satisfies Therefore, the Zeno phenomenon does not occur.

[0119] To illustrate the control effect of the event-triggered multi-flexible manipulator system tracking and vibration control method, the event-triggered multi-flexible manipulator system tracking and vibration control method is simulated in MATLAB.

[0120] The present application considers five flexible manipulator systems in the simulation process, including one leader manipulator and four follower manipulators. The communication topology structure of the multi-flexible manipulator system is as shown in Figure 2

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

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

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

[0124] p i (x, 0) = 0.3x, x e [0, L], 0 0 (0) = 0.5 rad, 0 1 (0) = 1.8 rad, 0 2 (0) = 1.5 rad, 0 3 (0) = 2.2 rad,

[0125] 0 4 (0) = 2.1 rad.

[0126] Wherein, 0 0 (0), 0 1 (0), 0 2 (0), 0 3 (0) and 0 4 (0) represent the angular displacement of the leader manipulator and the follower manipulators 1-4 at 0 time, respectively.

[0127] The external disturbance in the system is:

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

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

[0130] Wherein, sin and cos represent the sine and cosine functions, respectively.

[0131] It can be obtained​

[0132] The simulation diagram of the elastic deformation of the leading robot arm under the action of no control is as shown in Figure 3 The simulation diagram of the elastic deformation of the following robot arm under the action of no control is as shown in Figure 4 .

[0133] From Figure 4 and Figure 5 It can be found that the vibration amplitude of the elastic deformation of the leading robot arm and the following robot arm is large under the action of no control, and therefore it is necessary to exert appropriate control on the system.

[0134] Further, given the parameters required in the design process of the control method of the application μ 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 action of the designed control signals (11)-(12) and (15)-(16), the adaptive compensation law (17)-(20), and the adaptive iterative learning law (21) and (22), the simulation diagram of the angular displacement of the leading robot arm and the following robot arm is as shown in Figure 5 The simulation diagram of the elastic deformation of the leading robot arm and the following robot arm is as shown in Figure 6 and Figure 7 According to Figure 5 , Figure 6 and Figure 7It can be found that, under the control method designed in the application, the following flexible manipulator can complete the tracking of the angular displacement of the leading manipulator, and the elastic deformation of each flexible manipulator is inhibited (i.e., the vibration of the flexible manipulator is inhibited). Therefore, the event-triggered multi-flexible manipulator system tracking and vibration control method designed in the application is effective, that is, the consistency tracking of the multiple flexible manipulators can be realized, and the vibration of the flexible manipulator is inhibited.

[0136] The above embodiment is only one of the preferred embodiments of the application and should not be used to limit the protection scope of the application, but any modification or polishing without substantial meaning made within the main design idea and spirit of the application, and the technical problems solved are still consistent with the application, should be included in the protection scope of the application.

Claims

1. A tracking and vibration control method for a multi-flexible robotic arm system based on event triggering, characterized in that, Includes the following steps: S1, A mathematical model of a multi-flexible robotic arm system is established based on Hamilton's principle; wherein, the mathematical model of the multi-flexible robotic arm system includes a mathematical model of the following robotic arm and a mathematical model of the leading robotic arm; The mathematical model for the following robotic arm is: Its boundary conditions are expressed as follows: p i (0,t)=p i ′(0,t)=p i ″(L,t)=0 Where i∈{1,2,...,n} represents the i-th following robotic arm, n is an integer greater than 2 and represents the maximum sequence number of the following robotic arm; x∈[0,L] represents the spatial position of the robotic arm, L represents the length of the robotic arm, t represents the system running time, and m represents the load mass of the robotic arm; ρ, E, and J represent the mass per unit length, bending stiffness, and hub inertia of the robotic arm, respectively; θ i (t) represents the angular displacement of the i-th following robotic arm; Represents θ i (t) Second derivative with respect to time; p i (x,t) represents the elastic deformation of the i-th following robotic arm at spatial position x; p i """(x,t) represents p i (x,t) is the fourth derivative with respect to space; p i (0,t) represents the elastic deformation of the i-th following robotic arm at spatial position 0; p i ′(0,t) and p i "(0,t) represent p" i (x,t) represents the first and second derivatives of (x,t) with respect to space at spatial location 0; p i (L,t) represents the elastic deformation of the i-th following robotic arm at spatial position L; p i "(L,t) and p i "′(L,t) represent p i The second and third derivatives of (x,t) with respect to space at spatial location L; r i (x,t) represents the displacement of the i-th following robotic arm at spatial position x; Indicates r i The second derivative of (x,t) with respect to time; r i (L,t) represents the displacement of the i-th following robotic arm at spatial position L; Indicates 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 i-th following robotic arm; w 1,i (t) and w 2,i (t) represents the control law of the i-th following robotic arm; The mathematical model of the robotic arm is as follows: The boundary conditions are expressed as follows: p0(0,t)=p0′(0,t)=p0″(L,t)=0 Where r0(x,t) represents the displacement of the leading robotic arm at spatial position x; Let r0(x,t) represent the second derivative of r0(x,t) with respect to time; r0(L,t) represent the displacement of the robotic arm at spatial position L. Let θ0(L,t) represent the second derivative of r0(L,t) with respect to time; p0(x,t) represent the elastic deformation of the leading robotic arm at spatial position x; p0″″(x,t) represent the fourth derivative of p0(x,t) with respect to space; p0(0,t) represent the elastic deformation of the leading robotic arm at spatial position L; p0″(L,t) and p0″′(L,t) represent the second derivative of p0(x,t) with respect to space and the second derivative of p0(x,t) with respect to space and the third ...t) with respect to space and the third derivative of p0(t) with respect to space and the third derivative of p0(t) with respect to space and the third derivative of p0(t) with respect to space and the third derivative of p< w represents the second derivative of θ0(t) with respect to time. 1,0 (t) and w 2,0 (t) represents the control law of the leading robotic arm; S2, under an undirected communication topology, utilize the boundary states of the mathematical model of the multi-flexible robotic arm system to design control signals for the multi-flexible robotic arm system based on event-triggered rules; wherein, the undirected communication topology is: Let graph G = (P, Γ, A) describe the communication topology among n following robotic arms, where each robotic arm is considered an agent. Let P represent the set of edges, P = {1, ..., n} represent the set of nodes, and (i, j) ∈ Γ indicate that the i-th robotic arm can obtain information from the j-th robotic arm; Ξ i ={P j |(P j ,P i )∈Γ,i≠j} represents the neighborhood set of the i-th robotic arm; A=[a ij ]∈R n×n Let denote the adjacency matrix, where if (i,j)∈Γ, then a ij =1, otherwise a ij =0, i≠j; S=diag{s1,…,s n } denotes the interior degree matrix, where the symbol diag{s1,…,s} n } represents the transition from s1 to s n The diagonal matrix formed Y = SA is the Laplace matrix of graph G; graph G is an undirected and connected graph, i.e., (i,j)∈Γ is equivalent to (j,i)∈Γ; graph This represents the augmented graph of graph G. 0 represents the sequence number of the leader robotic arm. If the leader robotic arm transmits information to the i-th following robotic arm, then a... i0 =1, otherwise a i0 =0; A1=diag{a 10 ,a 20 ,...,a N0 };D=Y+A1 is The communication matrix; The event triggering rules are as follows: Among them, h 1,i (t) and h 2,i (t) is the control signal for the i-th following robotic arm that needs to be designed; k is a positive integer; t 1,k and t 2,k These represent the k-th trigger time of the control signal, respectively; |w 1,i (t)| and |w 2,i (t)| represents w respectively 1,i (t) and w 2,i The 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 trigger parameter for the i-th event following the robotic arm; inf represents the lower bound of the elements in the set; S3, based on the control signal of the multi-flexible robotic arm system, designs 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 signals of the multi-flexible robotic arm system, adaptive compensation law and adaptive iterative learning law, performs consistency tracking and vibration control on the multi-flexible robotic arm system.

2. The event-triggered multi-flexible robotic arm system tracking and vibration control method according to claim 1, characterized in that, The control signal h of the following robotic arm 1,i (t), h 2,i The mathematical expression for (t) is: The control law of the robotic arm w 1,0 (t) and w 2,0 The mathematical expression for (t) is: Among them, c1, c2, ε, and It is a positive constant; tanh is the hyperbolic tangent function; and These are the perturbations d 1,i (t) and d 2,i The maximum value of (t); and These are auxiliary variables constructed; Represents θ i (t) is the first derivative with respect to time; It is the defined angular error of the i-th following robotic arm; express i (t) is the first derivative with respect to time; and These are the i-th following robotic arm with respect to P. 1,i P 2,i Estimates of J and E; 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 υ 2,i (t) is used to compensate for disturbance d. 1,i (t) and d 2,i (t) is the designed iteration term; p i ′(L,t), and They represent p respectively i The first derivative of (x,t) with respect to space at spatial location L, the derivative with respect to first-order time followed by first-order space, and the derivative with respect to first-order time followed by third-order space. and r i "′(L,t) represents r i (x,t) represents the first derivative of (x,t) with respect to time and the third derivative with respect to space at spatial location L; e1(t) = θ0(t) - θ d ;θ d is a constant, representing the desired angular displacement; This represents the first derivative of θ0(t) with respect to time; p0′(L,t) represents the first derivative of p0(x,t) with respect to space at the spatial location L; and Let p0(x,t) be the derivative of p0(x,t) at spatial position L with respect to the first-order time and the first-order space, and the derivative with respect to the first-order time and the third-order space, respectively. r0″′(L,t) and r0″′(L,t) represent the first derivative of r0(x,t) with respect to time and the third derivative with respect to space at the spatial location L, respectively.

3. The event-triggered multi-flexible robotic arm system tracking and vibration control method according to claim 2, characterized in that, In step S3, the adaptive compensation rule is as follows: Among them, l1, l2, l3, and l4 are positive constants; and They represent and The first derivative with respect to time.

4. The event-triggered multi-flexible robotic arm system tracking and vibration control method according to claim 3, characterized in that, In step S3, the adaptive iterative learning law is as follows: Where ζ1∈(0,1) and ζ2∈(0,1) are constants; F is the perturbation d 1,i (t) and d 2,i The period of (t); and They represent υ respectively 1,i (t) and υ 2,i (t) is the first derivative with respect to time; and They represent β respectively 1,i (t) and β 2,i (t) is the first derivative with respect to time.

5. The event-triggered multi-flexible robotic arm system tracking and vibration control method according to claim 4, characterized in that, This also includes establishing the Lyapunov function V. a (t) and V b (t), to perform stability analysis of the multi-flexible robotic arm system, the Lyapunov functions are as follows: V a (t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t), V b (t)=V6(t)+V7(t)+V8(t), Where dx and dκ represent differential operators; and It 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 Let represent the parameter estimation error, and satisfy respectively. and D -1 Describe the inverse matrix of D; Indicates r i (x,t) is the first derivative of (x,t) with respect to time; p i ′(x,t) and p i "(x,t) represent p" i (x,t) represents the first and second derivatives of (x,t) with respect to space; ∑(·) represents the summation of the elements within the parentheses; This represents the first derivative of e1(t) with respect to time. Let r0(x,t) denote the first derivative of r0(x,t) with respect to time; p0′(x,t) and p0″(x,t) denote the first derivative of p0(x,t) with respect to space and the second derivative with respect to space, respectively.

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