Event triggering cooperation index practical tracking control method of multi-mechanical-arm system

Abstract

The invention discloses an event triggering cooperation index practical tracking control method for a multi-mechanical-arm system, and the method comprises the following steps: constructing a multi-agent system which is composed of a follower subsystem described by an Euler-Lagrange equation and a leader system described by a linear system, the Euler-Lagrange equation represents a kinetic model of the mechanical arm system, and the linear system generates reference trajectory information of the multi-mechanical-arm system; designing a distributed event trigger observer to observe the state information of the leader system for the follower subsystem; based on state information, a distributed event triggering feedback control law and an event triggering mechanism are designed, the event triggering cooperation index practical tracking problem of the multi-Euler-Lagrange system is solved, and it is ensured that trajectory tracking errors can be converged to any small neighborhood of an original point in an index mode. It is guaranteed that the distributed event triggering observer only needs to depend on sampling information of a neighbor subsystem, and trajectory tracking errors can be converged to any designated neighborhood of an original point in an exponential mode.

Description

Technical Field

[0001] The present invention relates to the field of automatic control technology, and in particular to a practical tracking control method for event-triggered synergy index of a multi-manipulator system. Background Art

[0002] The robotic arm system is a complex system with high nonlinearity and strong coupling, which can be described by the Euler-Lagrangian system. Due to its operational flexibility, the robotic arm has been used in many engineering fields such as industrial assembly and aerospace, and the control problem of the robotic arm system has also received widespread attention in the control community. Therefore, how to design a control strategy to ensure that the robotic arm system can achieve high-precision and effective control is an important and challenging research topic. In recent years, the use of adaptive control, H ∞ Great progress has been made in the research on trajectory tracking control problems of robotic arm systems by combining methods such as adaptive control and sliding mode control, and combining new sampling control methods such as event triggering and time triggering. Combined with the sampling control method based on event triggering, it is not only possible to design control strategies that are directly executed on the digital platform, but also possible to utilize the event triggering mechanism to reduce the number of sampling times and the use of communication resources. With the development of robotics technology, higher requirements are also placed on robotic arms in terms of the speed and accuracy of trajectory tracking. Based on distributed observers, adaptive control technology and event-triggered control strategy, the present invention proposes a practical tracking control method of event-triggered collaborative index for robotic arm systems, and performs simulation verification on it.

[0003] Based on the event-triggered control method, neural network control method and backstepping design method, the fixed-time consistent tracking problem of multi-manipulator system (CN202210536923, CN202311111574), finite-time consistent tracking problem (CN202210991150), given performance control problem (CN202310496159) and preset time control problem (CN202211690101) were studied, but the design of its event triggering mechanism still needs to rely on the continuous information of neighboring subsystems. Based on the event-triggered control method, the consistency control problem of multi-Eulerian-Lagrangian system under Dos attack was studied (CN202210854102), but the design of its controller is based on continuous time signals. Summary of the invention

[0004] In view of the problems existing in the prior art, the purpose of the present invention is to provide an event-triggered collaborative index practical tracking control method for a multi-manipulator system. The present invention uses distributed observers, adaptive control technology and event-triggered control strategy to study the problem of collaborative index practical tracking of a multi-manipulator system. Under the event-triggered mechanism, a distributed event-triggered observer and a distributed event-triggered state feedback control law are designed to realize trajectory tracking control of the multi-manipulator system, ensuring that the trajectory tracking error can converge to any small neighborhood of the origin in an exponential form.

[0005] The present invention is achieved by at least one of the following technical solutions.

[0006] A practical tracking control method for event-triggered synergy index of a multi-manipulator system, comprising the following steps:

[0007] Step 1: Construct a multi-agent system consisting of N follower subsystems described by Euler-Lagrange equations and a leader system described by a linear system, where the Euler-Lagrange equations represent the dynamic model of the robotic arm system and the linear system generates reference trajectory information of the multi-robotic arm system;

[0008] Step 2: Design a distributed event-triggered observer to observe the status information of the leader system for each follower subsystem;

[0009] Step 3: Design distributed event-triggered feedback control law and event-triggered mechanism to solve the practical tracking problem of event-triggered cooperative index of multi-Euler-Lagrangian system;

[0010] Step 4: Use the Lyapunov method to prove the stability of the closed-loop system and ensure that the trajectory tracking error can converge exponentially to any small neighborhood of the origin;

[0011] Step 5: Verify the effectiveness of the proposed control strategy through MATLAB simulation.

[0012] Furthermore, the multi-manipulator system can be described by the following N Euler-Lagrange equations:

[0013]

[0014] in represents the generalized position vector, represents the generalized velocity vector, represents the generalized inertia matrix, denotes the Coriolis force and the centripetal force vector, represents the gravity vector, represents the generalized force vector. The Euler-Lagrangian system satisfies the following three standard properties:

[0015] Property 1: Matrix Mi (q i ) is a symmetric positive definite matrix, the matrix is an antisymmetric matrix.

[0016] Property 2: There exist some positive constants μ 1 ,μ 2 ,μ 3 ,μ 4 and the n-dimensional identity matrix I, so that for any q i , There is μ 1 I≤M i (q i )≤μ 2 I, and ‖G i (q i )‖≤μ 4 .

[0017] Property 3: For any in is a known regression matrix, is an uncertain parameter vector.

[0018] Assuming reference trajectory Generated by a leader system as follows:

[0019]

[0020] in is the state of the leader system, and are the system matrix and output matrix respectively, and both are composed of constants.

[0021] A multi-agent system with N+1 agents can be composed of a follower subsystem and a leader system, where the follower subsystem and the leader system are represented by formula (1) and formula (2) respectively. We can use To define a multi-agent system, where the node set Edge Set The i-th follower subsystem in the follower subsystem can be represented by node i=1,…,N, and the leader system can be represented by node 0. The neighbor set of the i-th subsystem can be defined as The weighted adjacency matrix can be defined as in And for i≠j, when hour otherwise

[0022] For each follower system i=1,…,N and the leader system, the position tracking error of the trajectory is defined as e i =q i -q 0 and the speed tracking error is

[0023] In order to solve the event-triggered collaborative tracking control problem of a multi-manipulator system, we need the following two assumptions:

[0024] Assumption 1: Matrix S 0 All eigenvalues ​​of are semisimple and have zero real part.

[0025] Assumption 2: In In the example, every node i=1,…,N is reachable from node 0.

[0026] Defining symbols λ min (X) and λ max (X) represent the minimum and maximum eigenvalues ​​of matrix X, respectively, ‖x 0 ‖ or ‖X 0 ‖ represents the vector x 0 or matrix X 0 The Euclidean norm of .

[0027] Furthermore, we introduce a distributed event-triggered observer, which can be viewed as a dynamic compensator and can estimate the state of the leader system for each follower system i=1,…,N. The distributed event-triggered observer can be described as follows:

[0028]

[0029] in is the state of the distributed event-triggered observer, μ 0 is any positive number, yes The estimated state and only the discrete state Right now Decide, represents the set of triggering times of the ith subsystem, represents the set of non-negative integers, represents the event trigger sampling time of the i-th subsystem and It is determined by the event triggering mechanism.

[0030] For i = 0, 1, ..., N, define the observation error and estimation error So, we get

[0031]

[0032] set up and We get

[0033]

[0034] in For i≠j, and I N and N-dimensional and n-dimensional respectively 0 -dimensional unit vector.

[0035] Due to N 1 is the Hurwitz matrix. For any given symmetric positive definite matrix Q 1 , there always exists another symmetric positive definite matrix P 1 , so that

[0036] Furthermore, for i=1,…,N, let the position estimation error between the follower subsystem and the event-triggered observer be and velocity estimation error set up

[0037]

[0038] where α 0 is a positive real number. We further get

[0039]

[0040] According to Property 3, there exists a known matrix and an uncertain parameter vector Θ i , so that

[0041]

[0042] set up

[0043]

[0044] From (3), (6) and (9), we have

[0045]

[0046] in

[0047] Then, for i=1,…,N, we consider the distributed event-triggered control law as follows:

[0048]

[0049] in

[0050]

[0051] is a symmetric positive definite matrix, γ i , k i1 and k i2 are some positive real numbers.

[0052] Furthermore, define

[0053]

[0054] Then, substituting the control law (11) into formula (1) yields

[0055]

[0056] Combining (8) and (10), we can further obtain

[0057]

[0058] in

[0059] Furthermore, the Lyapunov function is selected as

[0060]

[0061] Based on property 2, we have

[0062]

[0063] set up

[0064]

[0065] in k 1 >1 and 0<λ 1 <1.

[0066] For i=1,…,N, we consider the event triggering mechanism as follows:

[0067]

[0068] where β 1 , β 2 and β 3 is positive and satisfies β 1 >0, and β 4 >0.

[0069] Furthermore,

[0070]

[0071] c 7 =c 2 +α 0 c 6 +c 5 ‖C 0 ‖,b 8 =b 6 +b 1 ‖C 0 ‖,c 8 =c 6 +c 1 ‖C 0 ‖#

[0072] b 9 =b 7 +b 4 ‖C 0 ‖,c 9 =c 7 +c 4 ‖C 0 ‖#(20)

[0073] in is a compact set, α 0 >a 0 .

[0074] According to (15), (17), (18) and Property 1, we obtain that the Lyapunov function (15) satisfies

[0075]

[0076]

[0077] Combining (21), (22) and (23), we obtain

[0078]

[0079] Furthermore, based on the event triggering mechanism (19), we obtain

[0080]

[0081] According to (25), we have

[0082]

[0083] According to V 0 (t), we get satisfy

[0084]

[0085] and i (t) Satisfaction

[0086]

[0087] According to the event trigger mechanism (19), we get

[0088]

[0089] According to (29), we can get

[0090]

[0091] because Therefore According to (5), we can get

[0092]

[0093] According to (4), and χ i The definition of Therefore

[0094]

[0095] According to (10), Therefore, we get

[0096]

[0097] From (28), (32) and (33), we can get

[0098]

[0099] Therefore, according to the definition of trajectory tracking error, we get the position tracking error e i (t) Satisfaction

[0100]

[0101] and velocity tracking error satisfy

[0102]

[0103] Furthermore, from (35) and (36), it can be seen that the trajectory tracking error of the multi-manipulator system can converge to any small neighborhood of the origin in an exponential form. The steady-state tracking error is adjusted by γ, and the convergence speed is adjusted by a 0 adjust.

[0104] Furthermore, from (11), it can be seen that the distributed event-triggered control law is discrete and can be directly executed on the digital platform. In addition, from (3), (11) and (19), it can be seen that the designed distributed event-triggered observer, distributed event-triggered control law and event triggering mechanism only need to rely on the discrete information of neighbor subsystems, thus greatly reducing the use of communication resources.

[0105] Compared with the existing technology, the beneficial effects of the present invention are as follows: based on the event-triggered control strategy, the distributed event-triggered observer and the adaptive control technology, the present invention designs a distributed event-triggered control law that can be directly executed on a digital platform, solves the problem of practical tracking of event-triggered collaborative exponentials of multi-manipulator systems, and ensures that the trajectory tracking error can converge to any small neighborhood of the origin in an exponential form, and both the steady-state error and the convergence speed can be adjusted. In addition, the designed distributed event-triggered observer, distributed event-triggered control law and event-triggered mechanism only need to rely on the discrete information of the neighbor subsystem, thus greatly reducing the use of communication resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0106] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0107] Figure 1 is a communication network topology diagram in this embodiment;

[0108] Figure 2 is a schematic diagram of a reference trajectory generated by the leader system in this embodiment;

[0109] FIG3(a) and FIG3(b) are the parameters γ in this embodiment. i =400 and γ i = 800 is a schematic diagram of the event trigger function curve;

[0110] FIG4(a) and FIG4(b) are the parameters γ in this embodiment. i = 400 under the condition of two position state components tracking error curve schematic diagram;

[0111] FIG5(a) and FIG5(b) are the parameters γ in this embodiment. i = 800 under the condition of two position state components tracking error curve schematic diagram;

[0112] FIG6(a) and FIG6(b) are the parameters γ in this embodiment. iSchematic diagram of the tracking error curves of the two speed state components under the condition of =400;

[0113] FIG. 7( a ) and FIG. 7( b ) are the parameters γ in this embodiment. i Schematic diagram of the tracking error curves of the two speed state components under the condition of =800;

[0114] Figure 8 It is a flow chart of a practical tracking control method of an event-triggered collaborative index of a multi-robotic arm system of this embodiment. DETAILED DESCRIPTION

[0115] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0116] The technical solution of the present invention is described in detail below in conjunction with the accompanying drawings.

[0117] This embodiment provides an event-triggered collaborative index practical tracking control method for a multi-manipulator system, which solves the collaborative trajectory tracking problem of the multi-manipulator system, and includes the following steps:

[0118] Step 1: Construct a multi-agent system consisting of N follower subsystems described by Euler-Lagrange equations and a leader system described by a linear system, where the Euler-Lagrange equations represent the dynamic model of the robotic arm system and the linear system generates reference trajectory information of the multi-robotic arm system;

[0119] Step 2: Design a distributed event-triggered observer to observe the status information of the leader system for each follower subsystem;

[0120] Step 3: Design distributed event-triggered feedback control law and event-triggered mechanism to solve the practical tracking problem of event-triggered cooperative index of multi-Euler-Lagrangian system;

[0121] Step 4: Use the Lyapunov method to prove the stability of the closed-loop system and ensure that the trajectory tracking error can converge exponentially to any small neighborhood of the origin;

[0122] Step 5: Verify the effectiveness of the proposed control strategy through MATLAB simulation.

[0123] As an embodiment, step 1, further, the multi-manipulator system can be described by the following N Euler-Lagrange equations:

[0124]

[0125] in represents the generalized position vector, represents the generalized velocity vector, represents the generalized inertia matrix, denotes the Coriolis force and the centripetal force vector, represents the gravity vector, represents the generalized force vector. The Euler-Lagrangian system satisfies the following three standard properties:

[0126] Property 1: Matrix M i (q i ) is a symmetric positive definite matrix, the matrix is an antisymmetric matrix.

[0127] Property 2: There exist some positive constants μ 1 ,μ 2 ,μ 3 ,μ 4 and the n-dimensional identity matrix I, so that for any q i , There is μ 1 I≤M i (q i )≤μ 2 I, and ‖G i (q u )‖≤μ 4 .

[0128] Property 3: For any in is a known regression matrix, is an uncertain parameter vector.

[0129] Assuming reference trajectory Generated by a leader system as follows:

[0130]

[0131] in is the state of the leader system, and are the system matrix and output matrix respectively, and both are composed of constants.

[0132] A multi-agent system with N+1 agents can be composed of a follower subsystem and a leader system, where the follower subsystem and the leader system are represented by formula (1) and formula (2) respectively. We can use To define a multi-agent system, where the node set Edge Set The i-th follower subsystem in the follower subsystem can be represented by node i=1,…,N, and the leader system can be represented by node 0. The neighbor set of the i-th subsystem can be defined as The weighted adjacency matrix can be defined as in And for i≠j, when hour otherwise

[0133] For each follower system i=1,…,N and the leader system, the position tracking error of the trajectory is defined as e i =q i -q 0 and the speed tracking error is

[0134] In order to solve the event-triggered collaborative tracking control problem of a multi-manipulator system, we need the following two assumptions:

[0135] Assumption 1: Matrix S 0 All eigenvalues ​​of are semisimple and have zero real part.

[0136] Assumption 2: In In the example, every node i=1,…,N is reachable from node 0.

[0137] Defining symbols λ min (X) and λ max (X) represent the minimum and maximum eigenvalues ​​of matrix X, respectively, ‖x 0 ‖ or ‖X 0 ‖ represents the vector x 0 or matrix X 0 The Euclidean norm of .

[0138] Step 2: Further, we introduce a distributed event-triggered observer, which can be regarded as a dynamic compensator and can estimate the state of the leader system for each follower system i=1,…,N. The distributed event-triggered observer can be described as follows:

[0139]

[0140] in is the state of the distributed event-triggered observer, μ 0 is any positive number, yes The estimated state and only the discrete state Right now Decide, represents the set of triggering times of the ith subsystem, represents the set of non-negative integers, represents the event trigger sampling time of the i-th subsystem and It is determined by the event triggering mechanism.

[0141] For i = 0, 1, ..., N, define the observation error and estimation error So, we get

[0142]

[0143] set up and We get

[0144]

[0145] in For i≠j, and I N and N-dimensional and n-dimensional respectively 0 -dimensional unit vector.

[0146] Due to N 1 is the Hurwitz matrix. For any given symmetric positive definite matrix Q 1 , there always exists another symmetric positive definite matrix P 1 , so that

[0147] Step 3: Further, for i=1,…,N, let the position estimation error between the follower subsystem and the event-triggered observer be and velocity estimation error set up

[0148]

[0149] where α 0 is a positive real number. We further get

[0150]

[0151] According to Property 3, there exists a known matrix and an uncertain parameter vector Θ i , so that

[0152]

[0153] set up

[0154]

[0155] From (3), (6) and (9), we have

[0156]

[0157] in

[0158] Then, for i=1,…,N, we consider the distributed event-triggered control law as follows:

[0159]

[0160] in

[0161]

[0162] is a symmetric positive definite matrix, γ i , k i1 and k i2 are some positive real numbers.

[0163] Furthermore, define

[0164]

[0165] Then, substituting the control law (11) into formula (1) yields

[0166]

[0167] Combining (8) and (10), we can further obtain

[0168]

[0169] in

[0170] Furthermore, the Lyapunov function is selected as

[0171]

[0172] Based on property 2, we have

[0173]

[0174] set up

[0175]

[0176] in k 1 >1 and 0<λ 1 <1.

[0177] Furthermore, for i=1,…,N, we consider the event triggering mechanism as follows:

[0178]

[0179] where β 1 , β 2 and β 3 is positive and satisfies β 1 >0, and β 4 >0.

[0180] Step 4: Further, set

[0181]

[0182] c 7 =c 2 +α 0 c 6 +c 5 ‖C 0 ‖,b 8 =b 6 +b 1 ‖C 0 ‖,c 8 =c 6 +c 1 ‖C 0 ‖#

[0183] b 9 =b 7 +b 4 ‖C 0 ‖,c 9 =c 7 +c 4 ‖C 0 ‖#(20)

[0184] in is a compact set, α 0 >a 0 .

[0185] According to (15), (17), (18) and Property 1, we obtain that the Lyapunov function (15) satisfies

[0186]

[0187]

[0188] Combining (21), (22) and (23), we obtain

[0189]

[0190] Furthermore, based on the event triggering mechanism (19), we obtain

[0191]

[0192] According to (25), we have

[0193]

[0194] According to V 0 By definition of (t), we get satisfy

[0195]

[0196] and i (t) Satisfaction

[0197]

[0198] According to the event trigger mechanism (19), we get

[0199]

[0200] According to (29), we can get

[0201]

[0202] because Therefore According to (5), we can get

[0203]

[0204] According to (4), and χ i The definition of Therefore

[0205]

[0206] According to (10), Therefore, we get

[0207]

[0208] From (28), (32) and (33), we can get

[0209]

[0210] Therefore, according to the definition of trajectory tracking error, we get the position tracking error e i (t) Satisfaction

[0211]

[0212] and velocity tracking error satisfy

[0213]

[0214] Furthermore, from (35) and (36), it can be seen that the trajectory tracking error of the multi-manipulator system can converge to any small neighborhood of the origin in an exponential form. The steady-state tracking error is adjusted by γ, and the convergence speed is adjusted by a 0 adjust.

[0215] Furthermore, from (11), it can be seen that the distributed event-triggered control law is discrete and can be directly executed on the digital platform. In addition, from (3), (11) and (19), it can be seen that the designed distributed event-triggered observer, distributed event-triggered control law and event triggering mechanism only need to rely on the discrete information of neighbor subsystems, thus greatly reducing the use of communication resources.

[0216] Step 5. In this example, we consider a multi-Eulerian-Lagrangian system consisting of three two-DOF manipulators, whose dynamic equations are as follows:

[0217]

[0218] where q i =col(q i1 ,q i2 ), τ i =col(τ i1 ,τ i2 ),

[0219]

[0220] Where g is the gravity vector, Θ i =col(Θ i1 ,Θ i2 ,Θ i3 ,Θ i4 ,Θ i5 ). Let the reference trajectory q 0 =col(q 01 ,q 02 ), where q 01 (t) = sin 0.5t and q 02 (t) = cos 0.5t, which can be generated by (2),

[0221]

[0222] And v 0 (0) = col(0,1). Obviously, for i = 1, 2, 3, we have ei =col(e i1 ,e i2 )and It is represented as a communication network topology diagram, such as Figure 1 shown. From (38) and Figure 1 It is easy to see that assumptions 1 and 2 are satisfied. For i=1,2,3, the controller parameters are selected as follows:

[0223] g=9.8,μ 0 =1,α 0 =1,γ i =400,γ i =800,k i1 =96,k i2 =92#

[0224] Λ i =100I,λ=0.2,Q 1 =2I,β 1 =50,β 2 =2,β 4 =200#(40)

[0225] Where I is an identity matrix with appropriate dimensions. The initial values ​​are chosen as follows:

[0226] Θ 1 =col(0.6,1.1,0.1,0.5,0.3),Θ 2 =col(0.8,1.2,0.1,0.8,0.4)#

[0227]

[0228] According to the above parameters, we perform simulation based on the distributed event-triggered control law (11) under the event-triggered mechanism (19) and obtain the simulation results as follows: Figure 2 -As shown in Figure 7. Figure 2 represents the reference trajectory generated by the leader system. Figure 3(a) and Figure 3(b) respectively represent γ i =400 and γ i =800. Figure 4(a) and Figure 4(b) show the event triggering function under two different parameter conditions. i =400, the tracking error curves of the two position state components, Figure 5(a) and Figure 5(b) show the parameter γ i = 800, the tracking error curves of the two position state components, Figure 6 (a) and Figure 6 (b) show the parameter γ i =400, the tracking error curves of the two velocity state components, Figure 7(a) and Figure 7(b) show the parameter γ i= 800 under the condition of tracking error curves of the two speed state components. As shown in Figures 4 to 7, under γ i =400 and γ i =800, the position and velocity tracking errors of the system can converge in an exponential form, and by selecting different parameters, the trajectory tracking error can be guaranteed to converge to an arbitrarily small neighborhood of the origin.

[0229] The above is the content of the present invention. Any changes made according to the technical solution of the present invention, as long as the functional effects produced do not exceed the scope of the technical solution of the present invention, belong to the protection scope of the present invention.

Claims

1. A practical tracking control method for event-triggered synergy index of a multi-manipulator system, characterized in that: The following steps are involved: Step 1: Construct a multi-agent system consisting of N follower subsystems described by Euler-Lagrange equations and a leader system described by a linear system, where the Euler-Lagrange equations represent the dynamic model of the robotic arm system and the linear system generates reference trajectory information of the multi-robotic arm system; Step 2: Design a distributed event-triggered observer to observe the status information of the leader system for each follower subsystem; Step 3: Based on the state information, a distributed event-triggered feedback control law and an event-triggered mechanism are designed to solve the event-triggered cooperative exponential practical tracking problem of the multi-Euler-Lagrangian system, ensuring that the trajectory tracking error can converge to any small neighborhood of the origin in an exponential form.

2. The event-triggered synergy index practical tracking control method of a multi-manipulator system according to claim 1, characterized in that: The multi-manipulator system can be described by the following N Euler-Lagrange equations: in represents the generalized position vector, represents the generalized velocity vector, represents the generalized inertia matrix, denotes the Coriolis force and the centripetal force vector, represents the gravity vector, represents the generalized force vector; the Euler-Lagrange system satisfies the following three standard properties: Property 1: Matrix M i (q i ) is a symmetric positive definite matrix, the matrix is an antisymmetric matrix; Property 2: There exist some positive constants μ1, μ2, μ3, μ4 and the n-dimensional identity matrix I such that for any q i , μ1I≤M i (q i )≤μ2I, and ‖G i (q i )‖≤μ4; Property 3: For any vector in is a known regression matrix, is an uncertain parameter vector; Reference trajectory Generated by a leader system as follows: q0=C0v0;#(2) in is the state of the leader system, and are the system matrix and output matrix respectively, and both are composed of constants.

3. The event-triggered synergy index practical tracking control method of a multi-manipulator system according to claim 2, characterized in that: A multi-agent system with N+1 agents consists of a follower subsystem and a leader system, where the follower subsystem and the leader system are represented by formula (1) and formula (2) respectively; To define a multi-agent system, where the node set Edge Set The i-th follower subsystem in the follower subsystem is represented by node i=1,…,N, and the leader system is represented by node 0; The neighbor set of the i-th subsystem is defined as The weighted adjacency matrix is ​​defined as in And for i≠j, when hour otherwise For each follower system i=1,…,N and the leader system, the position tracking error of the trajectory is defined as e i =q i -q0 and velocity tracking error are In order to solve the event-triggered collaborative tracking control problem of multi-manipulator systems, the following two assumptions are proposed: Assumption 1: All eigenvalues ​​of the matrix S0 are semisimple and have zero real part; Assumption 2: In In the example, from node 0 to every node i=1,…,N is reachable; Defining symbols λ min (X) and λ max (X) represents the smallest eigenvalue and the largest eigenvalue of the matrix X, respectively, and ‖x0‖ or ‖X0‖ represents the Euclidean norm of the vector x0 or the matrix X0.

4. The event-triggered synergy index practical tracking control method of a multi-manipulator system according to claim 3, characterized in that: A distributed event-triggered observer is introduced as a dynamic compensator and can estimate the state of the leader system for each follower system i=1,…,N; the distributed event-triggered observer is described as follows: in is the state of the distributed event-triggered observer, μ0 is an arbitrary positive number, θ p (t) Yes The estimated state and only the discrete state Right now Decide, represents the set of triggering times of the ith subsystem, represents the set of non-negative integers, represents the event trigger sampling time of the i-th subsystem and It is determined by the event triggering mechanism; For i = 0, 1, ..., N, define the observation error and estimation error get set up and get in For i≠j, and I N and are N-dimensional and n0-dimensional unit vectors respectively; Since N1 is a Hurwitz matrix, for any given symmetric positive definite matrix Q1, there always exists another symmetric positive definite matrix P1 such that 5. The event-triggered synergy index practical tracking control method of a multi-manipulator system according to claim 4, characterized in that: For i = 1, ..., N, let the position estimation error between the follower subsystem and the event-triggered observer be and velocity estimation error set up Where α0 is a positive real number, we further get According to Property 3, there exists a known matrix and an uncertain parameter vector Θ i , so that set up According to formula (3), formula (6) and formula (9), we have in 6. The event-triggered synergy index practical tracking control method of a multi-manipulator system according to claim 5, characterized in that: For i=1,…,N, the distributed event-triggered control law is as follows: in φ i (t)=Y i T s i (t);#(12) is a symmetric positive definite matrix, γ i , k i1 and k i2 are some positive real numbers; definition Substituting the control law (11) into formula (1), we can obtain Combining formula (8) and formula (10), we can further obtain in 7. The event-triggered synergy index practical tracking control method of a multi-manipulator system according to claim 6, characterized in that: Select the Lyapunov function as V0=V1+V2+V3;#(16) Based on property 2, we have set up α1=‖P1N2‖, # in k1>1 and 0<λ1<1; For i=1,…,N, the event triggering mechanism is as follows: Among them, β1, β2 and β3 are positive numbers and satisfy β1>0, and β4>

0.

8. The event-triggered synergy index practical tracking control method of a multi-manipulator system according to claim 7, characterized in that: definition b5=b1‖S0‖+b4,c5=c1‖S0‖+c4; b7=b2+α0b6+b5‖C0‖; c7=c2+α0c6+c5‖C0‖, b8=b6+b1‖C0‖, c8=c6+c1‖C0‖; b9=b7+b4‖C0‖,c9=c7+c4‖C0‖;#(20) in It is a compact set, α0>a0; According to formula (15), formula (17), formula (18) and property 1, we can get that the Lyapunov function (15) satisfies Combining formula (21), formula (22) and formula (23), we can get Furthermore, based on formula (19), we get According to formula (25), we can get 9. The event-triggered synergy index practical tracking control method of a multi-manipulator system according to claim 8, characterized in that: According to the definition of V0(t), we get satisfy and i (t) Satisfaction According to formula (19), we can get According to formula (29), we can get because Therefore According to (5), we can get According to formula (4), and χ i The definition of Therefore According to formula (10), So we get From formula (28), formula (32) and formula (33), we can get Therefore, according to the definition of trajectory tracking error, the position tracking error e is obtained i (t) Satisfaction and velocity tracking error satisfy 10. The event-triggered synergy index practical tracking control method of a multi-manipulator system according to claim 9, characterized in that: It can be seen from formula (35) and formula (36) that the trajectory tracking error of the multi-manipulator system converges to any small neighborhood of the origin in an exponential form. The steady-state tracking error is adjusted by γ, and the convergence speed is adjusted by a0.

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