Fixed-Time Consensus Control Method for Multi-Robot Based on Dynamic Event-Triggering

Abstract

The present invention discloses a multi-robot fixed-time consistency control method based on dynamic event driving; the method is as follows: 1. Set the dynamic event triggering mechanism of the controlled robot. When any controlled robot triggers a dynamic event, the position and speed of the controlled robot at the triggering moment are recorded. 2. Establish a consistency control protocol for the multi-robot system; 3. Each controlled robot adjusts the moving speed direction according to the basic acceleration obtained by the consistency control protocol. The present invention addresses the problem of constructing a magnetic map by developing a dynamic event-triggered fixed-time consistency control method for a second-order nonlinear multi-robot system. First, a new fixed-time stability condition is derived, and a less conservative sedimentation time is given, so that the theoretical sedimentation time can well reflect the actual situation. The theoretical stability time can well reflect the actual convergence time.

Description

Technical Field

[0001] The present invention belongs to the technical field of multi-robot collaborative control, and in particular relates to a multi-robot fixed-time consistency control method based on dynamic event driving. Background Art

[0002] Multi-robot systems enable independent robots to collaborate to solve complex tasks, boasting advantages such as wide monitoring range and extensive information collection. Consensus algorithms are distributed control algorithms whose goal is to achieve consensus on the states of each individual in a group through local communication. Therefore, how to quickly and efficiently achieve state consensus among multiple robots is currently a key research topic in multi-robot collaborative control.

[0003] Currently, multi-robot consistency control can be divided into finite-time control, fixed-time control, static event-driven control, and static event-driven fixed-time control. The finite-time consistency control method is a distributed consistency control method that can achieve rapid consistency among multiple robots mainly due to its following characteristics: finite-time convergence, higher control accuracy, better interference suppression, and robustness to uncertainty. However, for the finite-time consistency method, the estimated consistency time is related to the initial state of the multi-robot system, which affects the estimation of the convergence time. To address this situation, a fixed-time consistency control method was designed, in which the settling time of the system is independent of the initial state. At present, the stabilization time obtained by the fixed-time method is too conservative and cannot well reflect the actual consistency time.

[0004] However, real-time updates of the controller and communication between robots are still required. In order to effectively reduce the use of chip and network resources, a static event triggering mechanism is introduced in the fixed-time consistency control method. It should be noted that the static event triggering mechanism can only work before consistency is achieved. After consistency is achieved. Zeno behavior is inevitable. This behavior refers to the fact that in static event triggered control, the control is triggered an infinite number of times within a finite number of events. Therefore, some researchers often assume that in an ideal working environment, the rules of static event triggering can be canceled after consistency is achieved. However, this assumption is not true in a dynamic environment, and at this stage, event triggering mechanisms are mostly applied to first-order linear multi-robot systems. Summary of the Invention

[0005] The present invention aims to provide a multi-robot collaborative control method based on dynamic event-driven fixed-time consistency. This collaborative control method exhibits high robustness and effectively reduces chip and communication resources in a second-order nonlinear multi-robot system by leveraging a dynamic event-triggered mechanism and fixed-time consistency control, thus avoiding unnecessary information exchange.

[0006] The present invention is based on a multi-robot fixed time consistency control method driven by dynamic events, comprising the following steps:

[0007] Step 1: Set up a virtual leader for multiple controlled robots; all controlled robots follow the virtual leader. Set up a dynamic event triggering mechanism for the controlled robots. When any controlled robot triggers a dynamic event, record the position and velocity of the controlled robot at the time of triggering.

[0008] The dynamic event triggering mechanism is as follows:

[0009]

[0010] Among them, inf{·} is the lower bound operation; is the moment when the dynamic event is triggered; i=1,2,...,n; n is the number of controlled robots.

[0011]

[0012] is the most recent triggering moment of the i-th controlled robot;

[0013] is the position error of the i-th controlled robot at the current moment;

[0014] is the velocity error of the i-th controlled robot at the current moment;

[0015] is the sum of the position errors between the ith controlled robot and the multi-robot system;

[0016] are the position and velocity of the i-th controlled robot at the most recent triggering moment; is the position of the i-th controlled robot at the most recent triggering moment of the j-th controlled robot; x i (t), v i (t) are the position and velocity of the i-th controlled robot at the current moment.

[0017] is the sum of the velocity errors of the multi-robot system; a ij is the association weight between the i-th controlled robot and the j-th controlled robot.

[0018] γ, δ, h1, h2, h3, θ i (0) are all preset parameters, and the range of values is: γ>0, δ>0, h1>0, h2>0, h3>0, θ i (0)>0.

[0019] Step 2: Establish the consistency control protocol of the multi-robot system as follows:

[0020]

[0021] Among them, u i (t) is the basic acceleration output to the i-th controlled robot;

[0022] sig(φ i (t)) α =[sign(φ i1 (t))|φ i1 (t)| α ,...,sign(φ im (t))|φ im (t)| α ] T

[0023] sig(φ i (t)) β =[sign(φ i1 (t))|φ i1 (t)| β ,...,sign(φ im (t))|φ im (t)| β ] T ;

[0024] sign(·) is the sign function;

[0025] φ i (t) is the linear combination of the velocity error and position error between the ith controlled robot and the virtual leader and all controlled robots, and its expression is:

[0026]

[0027] Among them, α, β, p1, and p2 are all preset parameters with a value range of 0<α<1, β>1, p1>0, and p2>0; h[i,j] is the distance vector between the i-th controlled robot and the j-th controlled robot; and m is the dimension of position and velocity.

[0028] Step 3: Each controlled robot is based on the corresponding basic acceleration u i (t) Adjust the moving speed and direction.

[0029] Preferably, the controlled robot in step 3 is controlled by the following dynamic model:

[0030]

[0031] Among them, x i (t) and v i (t) represents the position and velocity of the i-th controlled robot respectively; is the nonlinear term of the i-th controlled robot; i = 1, 2, ..., n.

[0032] Nonlinear terms The following conditions must be met:

[0033]

[0034] in, is the nonlinear term of the virtual leader; l1>0,l2>0.

[0035] As a preferred method, a group of n controlled robots is composed of a group of n controlled robots, and the information interaction between the controlled robots is described as a directed graph G n (D, E, A), where D = {d1, d2, ..., d n} is a set of nodes, and E∈D×D is a set of edges connecting the nodes.

[0036] Preferably, only one of the controlled robots directly exchanges information with the virtual leader.

[0037] Preferably, in steps 2 and 3, the parameters γ, δ, and h1 satisfy the following two relationships:

[0038]

[0039] Among them, b t The upper bound of the maximum time interval for all event triggers; a ij is the association weight of the ith controlled robot relative to the jth controlled robot; if the ith controlled robot can receive the information of the jth controlled robot, the association weight a ij >0; otherwise the associated weight a ij =0. a i0 is the association weight of the ith controlled robot relative to the virtual leader; if the ith controlled robot can receive information from the virtual leader, the association weight a i0 >0; otherwise, the associated weight a i0 =0.

[0040] A is the characteristic matrix, and the expression is M is the definition of the transformation matrix, the expression is is a directed graph Laplacian matrix of diag{a 10 ,...,a n0} is a diagonal matrix; the elements are a 10 ,...,a n0 ; represents the Kronecker product, I m is the identity matrix of order m. min is the matrix Δ+Δ T The minimum eigenvalue of .

[0041] Preferably, the time T0 for each controlled robot to reach consistency satisfies the following relationship:

[0042]

[0043] in,

[0044]

[0045] λ max yes The maximum eigenvalue of .

[0046] The beneficial effects of the present invention are as follows:

[0047] 1. This paper addresses the magnetic map construction problem by developing a dynamic event-triggered fixed-time consistency control method for a second-order nonlinear multi-robot system. First, a new fixed-time stability condition is derived, along with a less conservative settling time, so that the theoretical settling time closely reflects the actual convergence time. The theoretical settling time closely reflects the actual convergence time.

[0048] 2. The present invention designs a dynamic event triggering rule, which saves chip and communication resources by introducing an internal variable. When using finite / fixed time consistency, Zeno behavior can be avoided after reaching consistency.

[0049] 3. Based on the developed dynamic event-triggered rules, this paper proposes a fixed-time consistency control method that introduces new items to coordinate a multi-robot system to achieve consistency within a fixed time interval. Based on Lyapunov theory and the designed fixed-time stability conditions, the corresponding stability of the multi-robot system using the proposed control method and dynamic event-triggered rules is analyzed. Finally, simulation and experimental results demonstrate the effectiveness of the proposed dynamic event-triggered fixed-time consistency control method. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 is a flow chart of the present invention;

[0051] Figure 2 A communication topology diagram of the present invention;

[0052] Figure 3 Graph showing the displacement changes of a virtual leader and four controlled robots in the present invention. DETAILED DESCRIPTION

[0053] The present invention will be further described below.

[0054] like Figure 1 As shown, a multi-robot fixed-time consistency control method based on dynamic event driving includes the following steps:

[0055] Step 1: Determine the position set x = {x1,...,x n} and the velocity set v={v1,...,v n}, where n represents the number of controlled robots. i is the state of the i-th controlled robot, specifically the position coordinates; v i is the velocity of the i-th controlled robot; i = 1, 2, ..., n.

[0056] For a group of n controlled robots, the information interaction between each controlled robot is described as a directed graph G n (D, E, A), where D = {d1, d2, ..., d n} is a set of nodes, E∈D×D is a set of edges connecting nodes, A=[a ij ] is an adjacency matrix, i=1,2,...,n;j=1,2,...,n. a ij is the association weight of the ith controlled robot relative to the jth controlled robot; if the ith controlled robot can receive the information of the jth controlled robot, the association weight a ij >0; otherwise the associated weight a ij =0(i,j=1,2,...,n). G n+1 Represents an extended graph with the virtual leader x0 as the root node. Similarly, if the i-th controlled robot can receive information from the virtual leader, the associated weight a i0 >0; otherwise, the associated weight a i0 =0.

[0057] Define the transformation matrix and the feature matrix in, Represents a directed graph G n The Laplacian matrix of (D, E, A); diag{a 10 ,...,a n0} is a diagonal matrix with elements a 10 ,...,a n0 ,symbol represents the Kronecker product, I m is the identity matrix of order m. min is Δ+Δ T The minimum eigenvalue of .

[0058] Step 2: Determine the dynamic model of the second-order nonlinear multi-robot system:

[0059]

[0060] Among them, x i (t)∈R m and v i (t)∈R m They represent the position and velocity of the i-th controlled robot at time t; u i (t)∈R m represents the control input of the i-th controlled robot at time t; is the nonlinear term of the i-th controlled robot; i = 1, 2, ..., n.

[0061] Constructing a dynamic model of virtual leaders:

[0062]

[0063] Where x0(t)∈R m and v0(t)∈R m represent the position and velocity of the virtual leader at time t respectively. is the nonlinear term of the virtual leader.

[0064] For the convenience of calculation, the nonlinear term of the present invention must meet the following conditions:

[0065] Among them, l1>0,l2>0.

[0066] Step 3: Based on the observability of the multi-robot system state, introduce a new internal dynamic variable θ i (t), the given dynamic event triggering mechanism is as follows:

[0067]

[0068] Among them, inf{·} is the lower bound operation;

[0069]

[0070] is the time index of the sth event triggered by the i-th controlled robot; s is the sequence number of the most recent event triggered;

[0071] is the position error of the i-th controlled robot at time t, a scalar with the unit of m;

[0072] is the velocity error of the i-th controlled robot at time t, a scalar in m / s;

[0073] is the sum of the position errors of the multi-robot system;

[0074] is the sum of the velocity errors of the multi-robot system;

[0075] γ, δ, h1, h2, h3, θ i (0) are all preset parameters, and the range of values is: γ>0, δ>0, h1>0, h2>0, h3>0, θ i (0)>0. The following non-essential limitations are provided in this embodiment: the specific values of the above parameters are preferably: γ=8, δ=8.6, h1=0.01, h2=1, h3=0.1, θ i (0)=18.

[0076] Step 4: Based on the dynamic event-driven approach in step 3, determine the consistency control protocol for the multi-robot system:

[0077]

[0078] Among them, u i (t) is the next-moment base acceleration output to the i-th controlled robot. The next-moment actual acceleration of the i-th controlled robot is determined by the dynamic model provided in step 2. 2 is the square operation of the two-norm.

[0079] φ i (t) is the linear combination of the velocity error and position error between the ith controlled robot and the virtual leader and all controlled robots, and its expression is:

[0080]

[0081] sig(φ i (t)) α =[sign(φ i1 (t))|φ i1 (t)| α ,…,sign(φ im (t))|φ im (t)| α ] T ;

[0082] sig(φ i (t))β =[sign(φ i1 (t))|φ i1 (t)| β ,...,sign(φ im (t))|φ im (t)| β ] T ;

[0083] Where h[i,j] is the distance vector between the i-th controlled robot and the j-th controlled robot. The distance vector is a two-dimensional vector consisting of the preset distance in the x-axis direction and the preset distance in the y-axis direction between the i-th controlled robot and the j-th controlled robot. The distance vector h[i,j] is used to avoid collisions between the controlled robots during consistency control. sign(·) is the sign function. is the most recent triggering moment of the i-th controlled robot; is the moment when the jth controlled robot was last triggered; α, β, p1, and p2 are all preset parameters with a value range of 0 < α < 1, β > 1, p1 > 0, and p2 > 0; m is the dimension of position and velocity; the following non-essential limitations are provided in this embodiment: the specific values of the above parameters are preferably: α = 0.5, β = 1.9, p1 = 20, and p2 = 0.1. In addition, this embodiment sets φ i (t)=0, then

[0084] Step 5: Determine the stability conditions of the above multi-robot system through Lyapunov stability theorem.

[0085] When satisfied and

[0086]

[0087] When , the fixed consistency time converges to

[0088] in,

[0089]

[0090] b t It is the upper bound of the maximum time interval of all event triggering. In this embodiment, the following non-essential limitations are provided: b t The specific value of is preferably 0.4.

[0091] λ max yes The maximum eigenvalue of .

[0092] Step 6: Write the multi-controlled robot consistency control method based on dynamic event-driven fixed-time consistency into each controlled robot through code, and realize distributed information interaction between controlled robots through directed communication topology. According to the multi-robot system consistency control protocol determined in step 4, the n controlled robots are collaboratively controlled so that the n controlled robots meet the stability conditions determined in step 5 and the multi-controlled robot consistency requirements of the control performance requirements. Simulation and experiments have proved that Zeno behavior does not exist. The communication topology diagram of the multi-robot system is shown in the figure below. Figure 2 As shown, the virtual leader only communicates with the first controlled robot.

[0093] In this embodiment, there are four controlled robots; the virtual leader is not a physical robot but a fictitious setting with a certain controllable position used to guide the controlled robots to quickly reach a consistent position.

[0094] To verify the reliability of the present invention, the initial positions of the four controlled robots are randomly generated in the range of [-17,17]. The displacement of the virtual leader and the four controlled robots is shown in Figure 3 As shown, Figure 3 is the exclusion distance matrix (i.e. in the linear combination φ i (t) The simulation result diagram after h[i,j] is set to 0 to facilitate observation of the motion consistency of the controlled robot). Figure 3 In the figure, the vertical axis represents the distance from the robot to the preset origin (m), and the horizontal axis represents the running time (s); Figure 3 This figure illustrates the convergence process of the displacements of a virtual leader and four robots. It clearly demonstrates that the multi-robot system quickly reaches consensus. After 50 calculations, the average time to reach consensus was 7.3933 seconds, demonstrating the high efficiency of the proposed multi-robot fixed-time consensus control method.

[0095] The comparison of the stabilization time of the method provided in this embodiment and the conventional method is shown in Table 1 below:

[0096] Table 1 Estimated stabilization time

[0097]

[0098] It can be seen from Table 1 that the method provided in this embodiment can greatly improve the speed at which multiple controlled robots reach consistency requirements.

[0099] The comparison of the event-driven rate between the method provided in this embodiment and the existing static event-driven fixed-time method is shown in Table 2 below:

[0100] Table 2 The average value of event-driven rate after 50 calculations

[0101] method Controlled robot 1 Controlled Robot 2 Controlled Robot 3 Controlled Robot 4 The present invention 4.06 2.68 2.61 2.62 Static event-driven fixed-time method 13.63 16.17 14.4 20.04

[0102] It can be seen from Table 2 that the method provided in this embodiment can greatly reduce the amount of calculation required for the collaboration of multiple controlled robots.

Claims

1. A multi-robot fixed-time consistency control method based on dynamic event-driven, characterized by: The following steps are involved: Step 1: Set a virtual leader for multiple controlled robots; all controlled robots follow the virtual leader; set a dynamic event triggering mechanism for the controlled robots; when any controlled robot triggers a dynamic event, record the position and speed of the controlled robot at the time of triggering; The dynamic event triggering mechanism is as follows: Among them, inf{·} is the lower bound operation; is the time when the dynamic event is triggered; i=1,2,...,n; n is the number of controlled robots; is the most recent triggering moment of the i-th controlled robot, is the position error of the i-th controlled robot at the current moment; is the velocity error of the i-th controlled robot at the current moment; is the sum of the position errors between the ith controlled robot and the multi-robot system, are the position and velocity of the i-th controlled robot at the most recent triggering moment; is the position of the i-th controlled robot at the most recent triggering moment of the j-th controlled robot; x i (t), v i (t) are the position and velocity of the i-th controlled robot at the current moment; is the sum of the velocity errors of the multi-robot system; a ij is the association weight between the i-th controlled robot and the j-th controlled robot; γ, δ, h1, h2, h3, θ i (0) are all preset parameters, and the range of values is: γ>0, δ>0, h1>0, h2>0, h3>0, θ i (0)>0; Step 2: Establish the consistency control protocol of the multi-robot system as follows: Among them, u i (t) is the basic acceleration output to the i-th controlled robot; sig(φ i (t)) α =[sign(φ i1 (t))|φ i1 (t)| α ,…,sign(φ im (t))|φ im (t)| α ] T sig(φ i (t)) β =[sign(φ i1 (t))|φ i1 (t)| β ,...,sign(φ im (t))|φ im (t)| β ] T ; sign(·) is the sign function; φ i (t) is the linear combination of the velocity error and position error between the ith controlled robot and the virtual leader and all controlled robots, and its expression is: Among them, α, β, p1, and p2 are all preset parameters with a value range of 0 < α < 1, β > 1, p1 > 0, and p2 > 0; h[i, j] is the distance vector between the i-th controlled robot and the j-th controlled robot; m is the dimension of position and velocity; Step 3: Each controlled robot is based on the corresponding basic acceleration u i (t) Adjust the moving speed and direction.

2. The multi-robot fixed-time consistency control method based on dynamic event driving according to claim 1 is characterized in that: The controlled robot in step 3 is controlled by the following dynamic model: Among them, x i (t) and v i (t) represents the position and velocity of the i-th controlled robot respectively; is the nonlinear term of the i-th controlled robot; i=1,2,...,n; Nonlinear terms The following conditions must be met: in, is the nonlinear term of the virtual leader; l1>0,l2>0.

3. The multi-robot fixed-time consistency control method based on dynamic event driving according to claim 1 is characterized in that: A group of n controlled robots, the information interaction between the controlled robots is described as a directed graph G n (D, E, A), where D = {d1, d2, ..., d n } is a set of nodes, and E∈D×D is a set of edges connecting the nodes.

4. The method for controlling multiple robots with fixed time consistency based on dynamic event driving according to claim 3 is characterized in that: Only one of the controlled robots directly interacts with the virtual leader.

5. The multi-robot fixed-time consistency control method based on dynamic event driving according to claim 3 is characterized in that: In steps 2 and 3, the parameters γ, δ, and h1 satisfy the following two relationships: Among them, b t The upper bound of the maximum time interval for all event triggers; a ij is the association weight of the ith controlled robot relative to the jth controlled robot; if the ith controlled robot can receive the information of the jth controlled robot, the association weight a ij >0; otherwise the associated weight a ij =0;a i0 is the association weight of the ith controlled robot relative to the virtual leader; if the ith controlled robot can receive information from the virtual leader, the association weight a i0 >0; otherwise, the associated weight a i0 =0; Δ is the characteristic matrix, and the expression is M is the definition of the transformation matrix, the expression is For a directed graph G n Laplacian matrix of (v,E,A); diag{a 10 ,...,a n0 } is a diagonal matrix; the elements are a 10 ,...,a n0 ; represents the Kronecker product, I m is the identity matrix of order m; μ min is the matrix Δ+Δ T The minimum eigenvalue of .

6. The multi-robot fixed-time consistency control method based on dynamic event driving according to claim 1 is characterized in that: The time T0 for each controlled robot to reach consistency satisfies the following relationship: in, λ max yes The maximum eigenvalue of .

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