Fixed-time robust encircling control method for multi-robot system under switching topology

Abstract

The invention discloses a fixed-time robust encircling control method for a multi-robot system under a switching topology, and the method comprises the steps: building a dynamic model of the multi-robot system under a multi-leader mode, defining the related concept and control target of encircling control, building a multi-robot system encircling control model under the switching topology, and carrying out the fixed-time robust encircling control of the multi-robot system under the switching topology; according to the method, switching topology attributes are defined, conditions which need to be met by switching topology are provided, setting time of fixed time convergence constraint under the switching topology conditions is defined, then an estimator based on an RBF neural network is designed, compensation is carried out through an unknown interference estimation item obtained by substituting the estimator, and a robust encircling controller under the switching topology is designed. And the controller is substituted into the multi-robot system dynamic model to obtain a closed-loop error system, and fixed-time encircling control under the switching topology is realized. According to the method, adverse effects caused by topology switching and external disturbance in the multi-robot system can be effectively suppressed, so that the robustness and effectiveness of a multi-robot encircling control scheme are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of enclosure control of a multi-robot system, and in particular relates to a fixed-time robust enclosure control method for a multi-robot system under a switching topology. Background Art

[0002] With the rapid development of computer, electronic information and other fields in recent decades, the application of cooperative control of multi-robot systems in various industries has become more and more extensive. Encirclement, as one of the basic cooperative control behaviors, aims to make multiple follower robots enter the convex hull formed by multiple leaders by designing appropriate encirclement control laws. However, most of the current research on encirclement control only considers the situation where the topological structure between each robot remains unchanged. However, in actual scenarios, the topological structure may switch over time due to the disconnection or connection of links between adjacent robots. For example, when multiple submarines are operating, the interaction between submarines is likely to be limited due to the complex underwater environment, and the topological structure changes; in the biological field, different pathways will form between different neurons over time; in traffic management, the topological structure used for communication perception between different vehicles may change. Moreover, in most actual systems, people have limited time requirements for multi-robot encirclement control, and hope that the convergence time is not affected by its initial state. Therefore, from an application perspective, it is obviously more meaningful to explore fixed-time encirclement control methods under switching topology conditions.

[0003] However, in actual engineering applications, in addition to receiving the signal sent by the leader, the follower robot itself is often affected by adverse factors such as unknown interference, which is usually unpredictable. Therefore, it is necessary to consider the above issues comprehensively and design a fixed-time enclosure controller with robust performance.

[0004] In response to the above problems, the problem of robust encirclement control under switching topology was studied. However, in existing research, on the one hand, it did not consider how to ensure that the robot can achieve encirclement control within a fixed time under the condition of topology switching; on the other hand, it did not comprehensively consider adverse factors such as the follower robot being subject to unknown interference. Summary of the invention

[0005] In order to solve the above technical problems, the present invention provides a fixed-time robust encirclement control method for a multi-robot system under switching topology, which can effectively suppress the adverse effects of topology switching and external disturbances in the multi-robot system, thereby improving the robustness and effectiveness of the multi-robot encirclement control scheme.

[0006] The technical solution adopted by the present invention is: a fixed-time robust encirclement control method for a multi-robot system under a switching topology, and the specific steps are as follows:

[0007] S1. Establish the dynamic model of the multi-robot system under the multi-leader mode, that is, construct the time-varying multi-leader dynamic model and the follower dynamic model with nonlinear uncertainty, and define the relevant concepts and control objectives of encirclement control;

[0008] The individuals in the multi-robot system include: a leader robot and a follower robot.

[0009] S2. Based on step S1, a multi-robot system enclosure control model under the switching topology is constructed, that is, the switching topology attributes are defined and the conditions that the switching topology needs to meet are proposed, the switching topology cycle is designed, and the setting time of the fixed time convergence constraint under the switching topology condition is clarified;

[0010] S3, based on the dynamic model of the multi-robot system in step S1, design a nonlinear estimator based on RBF neural network and an adaptive update law of the neural network weight matrix to achieve effective approximation of the nonlinear part of the follower;

[0011] S4. By defining the expression of auxiliary variable error, the tracking error expression of follower position error and velocity in encirclement control is proposed, so that the encirclement control error converges to a bounded value;

[0012] S5, based on the switching topology condition and the fixed time convergence constraint proposed in step S2, by using the unknown interference estimation term obtained by the nonlinear estimator in step S3 as the compensation term, designing a fixed time adaptive robust enclosure controller under the switching topology;

[0013] S6, substituting the robust encirclement controller described in step S5 into the dynamic model of the multi-robot system constructed in step S1, constructing a closed-loop error system and analyzing the encirclement reachability conditions;

[0014] S7. Apply the robust enclosure tracking controller described in step S5 to the multi-robot system, so that the followers in the system can effectively track the leader and be driven into the convex hull formed by the leader, thereby realizing fixed-time enclosure control under the switching topology.

[0015] Furthermore, the step S1 is specifically as follows:

[0016] S11. Construct a multi-leader dynamics model and a follower dynamics model with nonlinear uncertainty, and establish an estimation model for uncertain terms.

[0017] The multi-robot system in the multi-leader mode has N+M individuals, which are connected through an undirected network. to connect.

[0018] Among them, the number of follower robots is N, and the number of leader robots is M. Then the dynamic model of the follower robot, that is, the follower dynamic model with nonlinear uncertainty, is expressed as formula (1), and the dynamic model of the leader robot, that is, the dynamic model of the instant variable multi-leader, is expressed as formula (2).

[0019]

[0020] Where t represents time, Respectively represent followers position and velocity. Respectively represent leaders The position and velocity of [] T represents the transposition operation, n is a non-negative integer, representing the dimension of the state space of the multi-robot system, represents an n-dimensional real vector space; represents the control input of the leader, represents the control input of the follower. g i (p i (t),v i (t)): is a continuous nonlinear function, representing the model uncertainty of the follower. That is, p 0i (t), v 0i (t), p i (t), v i The first derivative of (t).

[0021] S12, based on step S11, after the dynamics are modeled, the relevant concepts in the enclosure control are defined;

[0022] set up Represents a real vector space A set in which, if for any vector p in the set i =[p x ,p y ] T exist In, p i =(1-α)p x +αp y And α∈[0,1], then the set is called a convex set. The point set P in q} is the smallest convex set containing all the points in P. i ,i=1,…,q} represents the convex hull of P. Then when Co{pi ,i=1,…,q}={p|p∈[min i p i ,max i p i ]}.

[0023] S13, based on the “convex hull” in the encirclement control proposed in step S12, defining the control target of the encirclement control problem;

[0024] Assume that for a group of robots, the leader cannot be informed by other robots, and there is always at least one leader with a directed path to the followers.

[0025] Then set ‖v i (t)‖ t→∞ , ‖g i (p i (t),v i (t))‖ t→∞ , And i=N+1,…,N+M are all bounded, let Then the control objective of the encirclement control problem is to design u i (t) makes all followers satisfy equation (3), which is expressed as follows:

[0026]

[0027] in, i.e. g i (p i (t),v i (t)) is the first-order derivative.

[0028] Furthermore, the step S2 is specifically as follows:

[0029] S21, define switching topology attributes;

[0030] First, we set up a robot interaction network under the condition of topological switching, assuming that the information exchange between robots is composed of a switchable undirected weighted graph simulation.

[0031] in, Respectively represent the graph The robot set, edge set, and weighted adjacency matrix of , 0<i,j<N+M, σ(t):[τ0,+∞)→{1,2,...,e} represents the topology switching signal, and the initial time τ0>0, the number of topology switching e>1, and the use of τ s (s=0,1,...,e) represents the topology switching time.

[0032] in, Right now elements of

[0033] S22, after defining the switching topology attributes in step S21, designing the switching topology cycle;

[0034] Set the dwell time to 0<T in a topology switching cycle. c ≤τ s+1 -τ s , then in the time period The number of topology switching times

[0035] Among them, t a ,t b represents two randomly selected moments, and satisfies t a <t b .

[0036] S23. Based on step S22, clarify the setting time of the fixed time convergence constraint under the switching topology condition;

[0037] Based on step S22, the topology residence time T under the fixed time convergence constraint is obtained. c Conditions to be met: 0<T c ≤τ s+1 -τ s , according to the fixed time theory, set a non-negative function The differential inequalities satisfied are as follows:

[0038]

[0039] in, Right now The first derivative of s ,s=0,1,…,e represents the switching time sequence, there is Hengyou Time adjustment parameters 0<ι<1,j>1, d1, d2, d3, d4 are all positive numbers.

[0040] Then the dwell time of the switching signal σ(t) involved satisfies the following conditional expression:

[0041]

[0042] set up At a fixed time T f is bounded, so that the multi-robot system can be f Stability is achieved within , and fixed-time convergence is achieved under the switching topology. Then the residual set converged to The expression is as follows:

[0043]

[0044] Among them, if it is a positive constant χ∈(0,1), then the fixed time convergence constraint condition expression is as follows:

[0045]

[0046] Furthermore, the step S3 is specifically as follows:

[0047] The uncertainty in the dynamic model of the follower robot is modeled, and the approximation theorem of radial basis neural network is used to calculate the g i (p i (t),v i (t)) is linearly parameterized so that for any (p i (t),v i (t)) are both in a tight set In the equation, the nonlinear estimator expression is as follows:

[0048]

[0049] in, represents the ideal weight matrix of the Gaussian radial basis function. The number of neurons in the network is m>1. Let Then there exists a positive constant θ M ,satisfy ∈ represents a positive constant. Represents the basis function vector composed of known radial basis functions, and its specific expression is as follows:

[0050]

[0051] Among them, x i =[p i ,v i ] T , Represents the function δ ik The center of Represents the function δ ik The width of , and ‖∈‖≤∈ M .

[0052] Re-order and represents the ideal weight matrix and the estimated value of the nonlinear function, and the expression is as follows:

[0053]

[0054] Then in the estimator, the weight matrix The adaptive update law expression is as follows:

[0055]

[0056] in, Right now The first-order derivative of , σ and ρ represent positive constants. Represents a defined helper function.

[0057] Furthermore, the step S4 is specifically as follows:

[0058] S41. Propose an expression for the auxiliary variable error in encirclement control;

[0059] The auxiliary variable error expression is defined as follows:

[0060]

[0061] in, represents the communication status between the ith robot and the jth robot at time t. If communication is possible, then On the contrary,

[0062] S42. Based on the definition of convex hull, the position and velocity errors of the follower in encirclement control are proposed;

[0063] First, the Laplace matrix of the system is expressed as

[0064] Among them, when i=j, When i≠j,

[0065] Re-order in is a symmetric matrix, represents the zero matrix, then we know that the matrix The sum of each column is 1. Then the position error of the follower can be defined as and speed The error expression is as follows:

[0066]

[0067] in, represents the Kronecker product of matrices, represents the unit matrix, position error Speed ​​error Follower Position Follower speed Leader Position Leader Speed

[0068] Finally, the auxiliary variable error w i (t), si (t) is simplified to the following expression:

[0069]

[0070] in,

[0071] Furthermore, the step S5 is specifically as follows:

[0072] The fixed-time adaptive robust enclosure controller under the switching topology is constructed, that is, the controller of the adaptive neural network, and the expression is as follows:

[0073]

[0074] Among them, the function ε represents a sufficiently small constant to prevent the control term from Zero denominator term appears, for any Satisfy λ≥‖u 0i ‖, κ1, κ2, κ3, κ4, κ5 and κ6 are all positive constants, and 0<a<1, b>1.

[0075] Furthermore, the step S6 is specifically as follows:

[0076] S61. According to the error estimation method for the position and velocity of the follower robot in step S4, an estimate of the influence of the leader on the follower state is obtained, and the expression is as follows:

[0077]

[0078] in,

[0079] S62. Based on step S61, the state error of a single follower robot is analyzed, and the expression is as follows:

[0080]

[0081] in,

[0082] S63. Based on the fixed time theory, a neural network estimator and a robust enclosure controller are combined under the switching topology condition to construct a closed-loop error system and analyze the enclosure reachability condition.

[0083] Let matrix Then equation (20) is described as a closed-loop error system, expressed as follows:

[0084]

[0085] in, and δ={δ1 T (p1(t),v1(t)),δ2 T (p2(t),v2(t)),…,δ N T (p N (t),v N (t))} T ,

[0086] Finally, analyzing the closed-loop error system described by equation (21), we can see that if the derivative of the disturbance is at a fixed time T f After that, the corresponding quantity is bounded asymptotically, and its final bound is consistent with the artificially designed time adjustment parameter related.

[0087] The beneficial effects of the present invention are as follows: the method of the present invention first establishes a dynamic model of a multi-robot system under a multi-leadership mode, defines relevant concepts and control objectives of enclosure control, then constructs an enclosure control model of the multi-robot system under a switching topology, defines the switching topology attributes and proposes conditions that the switching topology needs to meet, clarifies the setting time of the fixed-time convergence constraint under the switching topology condition, and then designs an estimator based on an RBF neural network, compensates by substituting the unknown interference estimation term obtained by the estimator, designs a robust enclosure controller under the switching topology, substitutes the controller into the dynamic model of the multi-robot system, obtains a closed-loop error system, and realizes fixed-time enclosure control under the switching topology. The method of the present invention solves the instability problem of a multi-robot system caused by unknown interference under switching topology conditions. For a multi-robot system with unknown nonlinear dynamics, an adaptive distributed estimation scheme based on RBFNN is designed, which effectively completes the estimation of the unknown nonlinear dynamics in the system, thereby suppressing the adverse effects of nonlinear terms on the system and improving the robustness of the enclosure control of the multi-robot system. By proposing a robust enclosure control scheme based on fixed time and neural network, the enclosure control problem of a multi-robot system with unknown nonlinear dynamics under switching topology is solved, the robustness and convergence rate of the system are effectively improved, and the method is suitable for the field of robust enclosure control of multi-robots. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] Figure 1 The present invention is a flow chart of a fixed-time robust encirclement control method for a multi-robot system under a switching topology.

[0089] Figure 2 This is a communication topology diagram used by the robot before performing the first topology switch in an embodiment of the present invention.

[0090] Figure 3 This is a communication topology diagram used by the robot after performing the first topology switch in an embodiment of the present invention.

[0091] Figure 4 This is a multi-robot encirclement control trajectory diagram under switching topology in an embodiment of the present invention. DETAILED DESCRIPTION

[0092] The method of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0093] like Figure 1 As shown, a flow chart of a fixed-time robust encirclement control method for a multi-robot system under a switching topology of the present invention is shown, and the specific steps are as follows:

[0094] S1. Establish the dynamic model of the multi-robot system under the multi-leader mode, that is, construct the time-varying multi-leader dynamic model and the follower dynamic model with nonlinear uncertainty, and define the relevant concepts and control objectives of encirclement control;

[0095] The individuals in the multi-robot system include: a leader robot and a follower robot.

[0096] S2. Based on step S1, a multi-robot system enclosure control model under the switching topology is constructed, that is, the switching topology attributes are defined and the conditions that the switching topology needs to meet are proposed, the switching topology cycle is designed, and the setting time of the fixed time convergence constraint under the switching topology condition is clarified;

[0097] S3, based on the dynamic model of the multi-robot system in step S1, design a nonlinear estimator based on RBF neural network and an adaptive update law of the neural network weight matrix to achieve effective approximation of the nonlinear part of the follower;

[0098] S4. By defining the expression of auxiliary variable error, the tracking error expression of follower position error and velocity in encirclement control is proposed, so that the encirclement control error converges to a bounded value;

[0099] S5. Based on the switching topology condition and the fixed time convergence constraint proposed in step S2, a fixed time adaptive robust enclosing controller under the switching topology is designed by using the unknown interference estimation term (the estimation value obtained by the nonlinear estimator) obtained by the nonlinear estimator in step S3 as a compensation term;

[0100] S6, substituting the robust enclosure controller described in step S5 into the dynamic model of the multi-robot system constructed in step S1, constructing a closed-loop error system and analyzing the enclosure reachability condition;

[0101] S7. Apply the robust enclosure tracking controller described in step S5 to the multi-robot system, so that the followers in the system can effectively track the leader and be driven into the convex hull formed by the leader, thereby realizing fixed-time enclosure control under the switching topology.

[0102] In this embodiment, the step S1 is specifically as follows:

[0103] S11. Construct a multi-leader dynamics model and a follower dynamics model with nonlinear uncertainty, and establish an estimation model for uncertain terms.

[0104] The multi-robot system in the multi-leader mode has N+M individuals, which are connected through an undirected network. to connect.

[0105] Among them, the number of follower robots is N, and the number of leader robots is M. Then the dynamic model of the follower robot, that is, the follower dynamic model with nonlinear uncertainty, is expressed as formula (1), and the dynamic model of the leader robot, that is, the dynamic model of the instant variable multi-leader, is expressed as formula (2).

[0106]

[0107] Where t represents time, Respectively represent followers position and velocity. Respectively represent leaders The position and velocity of [] T represents the transposition operation, n is a non-negative integer, representing the dimension of the state space of the multi-robot system, represents an n-dimensional real vector space; represents the control input of the leader, represents the control input of the follower. g i (p i (t),v i (t)): is a continuous nonlinear function, representing the model uncertainty of the follower. That is, p 0i (t), v 0i (t), p i (t), v i The first derivative of (t).

[0108] S12, based on step S11, after the dynamics are modeled, the relevant concepts in the enclosure control are defined;

[0109] set up Represents a real vector space A set in which, if for any vector p in the set i =[p x ,p y ] T exist In, p i =(1-α)p x +αp y And α∈[0,1], then the set is called a convex set. The point set P in q} is the smallest convex set containing all the points in P. i ,i=1,…,q} represents the convex hull of P. Then when Co{p i ,i=1,…,q}={p|p∈[min i p i ,max i p i ]}.

[0110] S13, based on the “convex hull” in the encirclement control proposed in step S12, defining the control target of the encirclement control problem;

[0111] Assume that for a group of robots, the leader cannot be informed by other robots, and there is always at least one leader with a directed path to the followers.

[0112] Then set ‖v i (t)‖ t→∞ , ‖g i (p i (t),v i (t))‖ t→∞ , And i=N+1,…,N+M are all bounded, let Then the control objective of the encirclement control problem is to design u i (t) makes all followers satisfy equation (3), which is expressed as follows:

[0113]

[0114] in, i.e. g i (p i (t),v i (t)) is the first-order derivative.

[0115] In this embodiment, step S2 is specifically as follows:

[0116] S21, define switching topology attributes;

[0117] First, we set up a robot interaction network under the condition of topological switching, assuming that the information exchange between robots is composed of a switchable undirected weighted graph simulation.

[0118] in, Respectively represent the graph The robot set, edge set, and weighted adjacency matrix of , 0<i,j<N+M, σ(t):[τ0,+∞)→{1,2,...,e} represents the topology switching signal, and the initial time τ0>0, the number of topology switching e>1, and the use of τ s (s=0,1,...,e) represents the topology switching time.

[0119] in, Right now elements of

[0120] S22, after defining the switching topology attributes in step S21, designing the switching topology cycle;

[0121] Set the dwell time to 0<T in a topology switching cycle. c ≤τ s+1 -τ s , then in the time period The number of topology switching times

[0122] Among them, t a ,t b represents two randomly selected moments, and satisfies t a <t b .

[0123] S23. Based on step S22, clarify the setting time of the fixed time convergence constraint under the switching topology condition;

[0124] Based on step S22, the topology residence time T under the fixed time convergence constraint is obtained. c Conditions to be met: 0<T c ≤τ s+1 -τ s , according to the fixed time theory, set a non-negative function The differential inequalities satisfied are as follows:

[0125]

[0126] in, Right now The first derivative of s ,s=0,1,...,e represents the switching time sequence, there is Hengyou Time adjustment parameters d1, d2, d3, d4 are all positive numbers.

[0127] Then the dwell time of the switching signal σ(t) involved satisfies the following conditional expression:

[0128]

[0129] set up At a fixed time T f is bounded, so that the multi-robot system can be f Stability is achieved within , and fixed-time convergence is achieved under the switching topology. Then the residual set converged to The expression is as follows:

[0130]

[0131] Among them, if it is a positive constant χ∈(0,1), then the fixed time convergence constraint condition expression is as follows:

[0132]

[0133] In this embodiment, step S3 is specifically as follows:

[0134] The uncertainty in the dynamic model of the follower robot is modeled, and the approximation theorem of radial basis neural network is used to calculate the g i (p i (t),v i (t)) is linearly parameterized so that for any (p i (t),v i (t)) are both in a tight set In the equation, the nonlinear estimator expression is as follows:

[0135]

[0136] in, represents the ideal weight matrix of the Gaussian radial basis function. The number of neurons in the network is m>1. Let Then there exists a positive constant θ M ,satisfy ∈ represents a positive constant. Represents the basis function vector composed of known radial basis functions, and its specific expression is as follows:

[0137]

[0138] Among them, x i =[p i ,v i ]T , Represents the function δ ik The center of Represents the function δ ik The width of M , theoretically, when m is infinite, that is, when the number of radial basis functions is large enough, the positive constant ∈ M Approaching 0.

[0139] Re-order and represents the ideal weight matrix and the estimated value of the nonlinear function, and the expression is as follows:

[0140]

[0141] Then in the estimator, the weight matrix The adaptive update law expression is as follows:

[0142]

[0143] in, Right now The first-order derivative of , σ and ρ represent positive constants (set according to actual conditions). Represents a defined helper function.

[0144] In this embodiment, step S4 is specifically as follows:

[0145] S41. Propose an expression for the auxiliary variable error in encirclement control;

[0146] The control goal is to design the encirclement control law so that the encirclement control error converges to a bounded value; the auxiliary variable error expression is defined as follows:

[0147]

[0148] in, represents the communication status between the ith robot and the jth robot at time t. If communication is possible, then On the contrary,

[0149] S42. Based on the definition of convex hull, the position and velocity errors of the follower in encirclement control are proposed;

[0150] First, the Laplace matrix of the system is expressed as

[0151] Among them, when i=j, When i≠j,

[0152] Re-order in is a symmetric matrix, represents the zero matrix, then we know that the matrix The sum of each column is 1. Then the position error of the follower can be defined as and speed The error expression is as follows:

[0153]

[0154] in, represents the Kronecker product of matrices, represents the unit matrix, position error Speed ​​error Follower Position Follower speed Leader Position Leader Speed

[0155] Finally, the auxiliary variable error w i (t), s i (t) is simplified to the following expression:

[0156]

[0157] in,

[0158] In this embodiment, step S5 is specifically as follows:

[0159] The fixed-time adaptive robust enclosure controller under the switching topology is constructed, that is, the controller of the adaptive neural network, and the expression is as follows:

[0160]

[0161] Among them, the function ε represents a sufficiently small constant to prevent the control term from Zero denominator term appears, for any Satisfy λ≥‖u 0i ‖, κ1, κ2, κ3, κ4, κ5 and κ6 are all positive constants, and 0<a<1, b>1.

[0162] In this embodiment, step S6 is specifically as follows:

[0163] S61. According to the error estimation method for the position and velocity of the follower robot in step S4, an estimate of the influence of the leader on the follower state is obtained, and the expression is as follows:

[0164]

[0165] in,

[0166] S62. Based on step S61, the state error of a single follower robot is analyzed, and the expression is as follows:

[0167]

[0168] in,

[0169] S63. Based on the fixed time theory, a neural network estimator and a robust enclosure controller are combined under the switching topology condition to construct a closed-loop error system and analyze the enclosure reachability condition.

[0170] Let matrix Then equation (20) is described as a closed-loop error system, expressed as follows:

[0171]

[0172] in, and

[0173] δ={δ1 T (p1(t),v1(t)),δ2 T (p2(t),v2(t)),...,δ N T (p N (t),v N (t))} T ,

[0174]

[0175] Analyzing the closed-loop error system described by equation (21), we can see that if the derivative of the disturbance is at a fixed time T f After that, the corresponding quantity is bounded asymptotically, and its final bound is consistent with the artificially designed time adjustment parameters i,j, It can be said that the control input shown in formula (18) can achieve robust enclosure control.

[0176] In this embodiment, step S7 is specifically as follows:

[0177] The control protocol of step S5 (18) is applied to the individual dynamic equation. When the structure of the switching topology is connected, any follower can eventually enter the convex hull surrounded by the leader, and at the same time, the nonlinear disturbance part can be estimated under the constraint of fixed time.

[0178] The communication topology used by the multi-robot system in this embodiment before the first topology switching is as follows: Figure 2 As shown, the communication topology used after the first topology switching is as follows Figure 3 shown.

[0179] As can be seen from the figure, there are 9 robot individuals, of which 4 robots are leaders and 5 robots are followers. The dynamic model is given by equations (1) and (2). In this embodiment, it is set to p i (t) = [p i1 (t),p i2 (t)] T , v i (t) = [v i1 (t),v i2 (t)] T , p 0i (t) = [p 0i1 (t),p 0i2 (t)] T , v 0i (t) = [v 0i1 (t),v 0i2 (t)] T Then the unknown disturbance expression in the follower individual dynamics model is as follows:

[0180]

[0181] In this embodiment, the initial values ​​of the position states of the followers are p1(t)=[-3,-4] T , p2(t)=[-2,-2] T p3(t)=[-2,-3] T , p4(t)=[0,2] T , p5(t)=[1,4] T , the initial values ​​of the velocity states of the followers are v1(t)=[0.6,2] T ,v2(t)=[0.9,-1.1] T , v3(t)=[1.8,1.5] T ,v4(t)=[-1,-6] T ,v5(t)=[-2.6,-4] T ; The initial values ​​of the position states of the leader individuals are p 01 (t) = [-2.2, 0] T , p 02 (t) = [-1.2, 3.4] T , p 0i (t) = [0.6, 0] T , p 0i(t) = [-0.8, -4.4] T , the initial value of the leader's speed state is v 0i (t) = [0.2, -1.4] T , the input to the leader individual is

[0182] In order to approximate the unknown function g i (p i (t),v i (t)), each neural network selects 9 neurons, and the center and width of the Gaussian function in formula (9) are respectively taken as and For the weights of the adaptive neural network in formula (11) In this embodiment, σ=10, ρ=0.5 is set. The initial weight matrix is selected as the zero matrix. In addition, in the controller of the adaptive neural network shown in equation (18), the control parameters selected in this embodiment are as follows: κ1=10, κ2=8, κ3=9, κ4=11, κ5=0.775, κ6=3.2457×10 -6 ,a=0.5,b=1.5.

[0183] Figure 4 The figure is a multi-robot encirclement control trajectory diagram under the switching topology in this embodiment. From the simulation results, it can be seen that the followers in the system can effectively track the leader and be driven into the convex hull formed by the leader. Finally, the fixed-time robust encirclement control of the multi-robot system under the switching topology is realized in this embodiment.

[0184] In summary, the method of the present invention focuses on the influence of switching topology conditions on fixed-time convergence and the adverse effects of unknown interference on individual robots. First, the dynamic models of the leader robot and the follower robot are constructed to clarify the research objectives of the encirclement control problem. Then, the nonlinear part of the follower model is estimated based on the RBF neural network, and the expressions of the robot speed and position errors in the encirclement control are proposed. Then, based on the fixed-time stability theory, a robust encirclement controller is designed under the switching topology condition. The method of the present invention can effectively suppress the adverse effects of topology switching and external disturbances in a multi-robot system, thereby improving the robustness and effectiveness of the multi-robot encirclement control scheme.

[0185] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A fixed-time robust encirclement control method for a multi-robot system under switching topology, the specific steps are as follows: S1. Establish the dynamic model of the multi-robot system under the multi-leader mode, that is, construct the time-varying multi-leader dynamic model and the follower dynamic model with nonlinear uncertainty, and define the relevant concepts and control objectives of encirclement control; in, The individuals of the multi-robot system include: a leader robot and a follower robot; S2. Based on step S1, a multi-robot system enclosure control model under the switching topology is constructed, that is, the switching topology attributes are defined and the conditions that the switching topology needs to meet are proposed, the switching topology cycle is designed, and the setting time of the fixed time convergence constraint under the switching topology condition is clarified; S3, based on the dynamic model of the multi-robot system in step S1, design a nonlinear estimator based on RBF neural network and an adaptive update law of the neural network weight matrix to achieve effective approximation of the nonlinear part of the follower; S4. By defining the expression of auxiliary variable error, the tracking error expression of follower position error and velocity in encirclement control is proposed, so that the encirclement control error converges to a bounded value; S5, based on the switching topology condition and the fixed time convergence constraint proposed in step S2, by using the unknown interference estimation term obtained by the nonlinear estimator in step S3 as the compensation term, designing a fixed time adaptive robust enclosure controller under the switching topology; S6, substituting the robust encirclement controller described in step S5 into the dynamic model of the multi-robot system constructed in step S1, constructing a closed-loop error system and analyzing the encirclement reachability conditions; S7. Apply the robust enclosure tracking controller described in step S5 to the multi-robot system, so that the followers in the system can effectively track the leader and be driven into the convex hull formed by the leader, thereby realizing fixed-time enclosure control under the switching topology.

2. According to claim 1, a fixed-time robust encirclement control method for a multi-robot system under switching topology, characterized in that: The step S1 is specifically as follows: S11. Construct a multi-leader dynamics model and a follower dynamics model with nonlinear uncertainty, and establish an estimation model for uncertain terms. The multi-robot system in the multi-leader mode has N+M individuals, which are connected through an undirected network. Make a connection; Among them, the number of follower robots is N, and the number of leader robots is M; then the dynamic model of the follower robot individual, that is, the follower dynamic model with nonlinear uncertainty, is expressed as formula (1), and the dynamic model of the leader robot individual, that is, the dynamic model of the instant variable leader, is expressed as formula (2); Where t represents time, Respectively represent followers position and velocity; Respectively represent leaders The position and velocity of [] T represents the transposition operation, n is a non-negative integer, representing the dimension of the state space of the multi-robot system, represents an n-dimensional real vector space; represents the control input of the leader, represents the control input of the follower; is a continuous nonlinear function, representing the model uncertainty of the follower; That is, p 0i (t), v 0i (t), p i (t), v i The first derivative of (t); S12, based on step S11, after the dynamics are modeled, the relevant concepts in the enclosure control are defined; set up Represents a real vector space A set in which, if for any vector p in the set i =[p x ,p y ] T exist In, p i =(1-α)p x +αp y And α∈[0,1], then the set is called a convex set; for The point set P in q } is the smallest convex set containing all the points in P; Co{p i , i=1,…,q} represents the convex hull of P; then when Co{p i ,i=1,…,q}={p|p∈[min i p i ,max i p i ]}; S13, based on the "convex hull" in the encirclement control proposed in step S12, defining the control target of the encirclement control problem; Assume that for a group of robots, the leader cannot be obtained by other robots, and there is always at least one leader with a directed path to the followers; Then set ‖v i (t)‖ t→∞ , ‖g i (p i (t),v i (t))‖ t→∞ , And i=N+1,…,N+M are all bounded, let Then the control objective of the encirclement control problem is to design u i (t) makes all followers satisfy equation (3), which is expressed as follows: in, i.e. g i (p i (t),v i (t)) is the first-order derivative.

3. The fixed-time robust encirclement control method for a multi-robot system under switching topology according to claim 1 is characterized in that: The step S2 is specifically as follows: S21, define switching topology attributes; First, we set up a robot interaction network under the condition of topological switching, assuming that the information exchange between robots is composed of a switchable undirected weighted graph simulation; in, N+1,...,N+M, Respectively represent the graph The robot set, edge set, and weighted adjacency matrix of , 0<i,j<N+M, σ(t):[τ0,+∞)→{1,2,...,e} represents the topology switching signal, and the initial time τ0>0, the number of topology switching e>1, and the use of τ s (s=0,1,...,e) represents the topology switching time; in, Right now elements of S22, after defining the switching topology attributes in step S21, designing the switching topology cycle; Set the dwell time to 0<T in a topology switching cycle. c ≤τ s+1 -τ s , then in the time period The number of topology switching times Among them, t a ,t b represents two randomly selected moments, and satisfies t a <t b ; S23. Based on step S22, clarify the setting time of the fixed time convergence constraint under the switching topology condition; Based on step S22, the topology residence time T under the fixed time convergence constraint is obtained. c Conditions to be met: 0<T c ≤τ s+1 -τ s , according to the fixed time theory, set a non-negative function The differential inequalities satisfied are as follows: in, Right now The first derivative of s ,s=0,1,...,e represents the switching time sequence, there is Hengyou Time adjustment parameter 0<ι<1, d1, d2, d3, d4 are all positive numbers. Then the dwell time of the switching signal σ(t) involved satisfies the following conditional expression: set up At a fixed time T f is bounded, so that the multi-robot system can be f Stability is achieved within , and fixed-time convergence is achieved under the switching topology. Then the residual set converged to The expression is as follows: Among them, if it is a positive constant χ∈(0,1), then the fixed time convergence constraint condition expression is as follows:

4. The fixed-time robust encirclement control method for a multi-robot system under switching topology according to claim 1 is characterized in that: The step S3 is specifically as follows: The uncertainty in the dynamic model of the follower robot is modeled, and the approximation theorem of radial basis neural network is used to calculate the g i (p i (t),v i (t)) is linearly parameterized so that for any (p i (t),v i (t)) are both in a tight set In the equation, the nonlinear estimator expression is as follows: in, represents the ideal weight matrix of the Gaussian radial basis function. The number of neurons in the network is m>

1. Let Then there exists a positive constant θ M ,satisfy ∈ represents a positive constant; Represents the basis function vector composed of known radial basis functions, and its specific expression is as follows: Among them, x i =[p i ,v i ] T , Represents the function δ ik The center of Represents the function δ ik The width of , and ‖∈‖≤∈ M ; Re-order and represents the ideal weight matrix and the estimated value of the nonlinear function, and the expression is as follows: Then in the estimator, the weight matrix The adaptive update law expression is as follows: in, Right now The first derivative of , σ and ρ represent positive constants; Represents a defined helper function.

5. The fixed-time robust encirclement control method for a multi-robot system under switching topology according to claim 1 is characterized in that: The step S4 is specifically as follows: S41. Propose an expression for the auxiliary variable error in encirclement control; The auxiliary variable error expression is defined as follows: in, represents the communication status between the ith robot and the jth robot at time t. If communication is possible, then On the contrary, S42. Based on the definition of convex hull, the position and velocity errors of the follower in encirclement control are proposed; First, the Laplace matrix of the system is expressed as Among them, when i=j, When i≠j, Re-order in is a symmetric matrix, represents the zero matrix, then we know that the matrix The sum of each column is 1; then the position error of the follower can be defined and speed The error expression is as follows: in, represents the Kronecker product of matrices, represents the unit matrix, position error Speed ​​error Follower Position Follower speed Leader Position Leader Speed Finally, the auxiliary variable error w i (t), s i (t) is simplified to the following expression: in, 6. The fixed-time robust encirclement control method for a multi-robot system under switching topology according to claim 1 is characterized in that: The step S5 is specifically as follows: The fixed-time adaptive robust enclosure controller under the switching topology is constructed, that is, the controller of the adaptive neural network, and the expression is as follows: Among them, the function ε represents a sufficiently small constant to prevent the control term from Zero denominator term appears, for any Satisfy λ≥‖u 0i ‖, κ1, κ2, κ3, κ4, κ5 and κ6 are all positive constants, and 0<a<1, b>1.

7. The fixed-time robust encirclement control method for a multi-robot system under switching topology according to claim 1 is characterized in that: The step S6 is specifically as follows: S61. According to the error estimation method for the position and velocity of the follower robot in step S4, an estimate of the influence of the leader on the follower state is obtained, and the expression is as follows: in, S62. Based on step S61, the state error of a single follower robot is analyzed, and the expression is as follows: in, S63. Based on the fixed time theory, a neural network estimator and a robust enclosure controller are combined under the switching topology condition to construct a closed-loop error system and analyze the enclosure reachability condition. Let matrix Then equation (20) is described as a closed-loop error system, expressed as follows: in, and Finally, analyzing the closed-loop error system described by equation (21), we can see that if the derivative of the disturbance is at a fixed time T f After that, the corresponding quantity is bounded asymptotically, and its final bound is consistent with the artificially designed time adjustment parameter ι, related.

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