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

Abstract

The application discloses a kind of fixed time robust encirclement control method of multi-robot system under switching topology, first establish the dynamics model of multi-robot system under multi-leader mode, define the related concept and control target of encirclement control, then construct the encirclement control model of multi-robot system under switching topology, define switching topology attribute and propose the condition that switching topology needs to satisfy, clear under the setting time of fixed time convergence constraint in switching topology condition, then design the estimator based on RBF neural network, compensate by the unknown disturbance estimation term obtained by substituting estimator, design out the robust encirclement controller under switching topology, the controller is substituted into the dynamics model of multi-robot system, obtain closed-loop error system, realize fixed time encirclement control under switching topology.The method of the application can effectively inhibit the adverse effects caused by topology switching and external disturbance in multi-robot system, so as to improve the robustness and effectiveness of multi-robot encirclement control scheme.

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 computing, electronic information, and other fields in recent decades, the application of cooperative control in multi-robot systems has become increasingly widespread across various industries. Encirclement, a fundamental cooperative control behavior, aims to enable multiple follower robots to enter the convex hull formed by multiple leaders by designing appropriate encirclement control laws. However, current research on encirclement control has mostly considered the case where the topology of each robot remains unchanged. In real-world scenarios, however, topologies can change over time due to disconnections and connections between adjacent robots. For example, when multiple submarines operate, interactions between them are likely to be limited due to the complex underwater environment, leading to changes in topology. In biology, different pathways between neurons form different pathways over time. In traffic management, the topology used for communication and perception between different vehicles may also change. Furthermore, in most practical systems, the time required to achieve encirclement control for multiple robots is limited, and it is desirable that the convergence time is not affected by their initial state. Therefore, from an application perspective, exploring fixed-time encirclement control methods under the condition of switching topologies is more meaningful.

[0003] However, in practical engineering applications, follower robots, in addition to receiving signals from their leaders, are often themselves subject to unpredictable influences such as unknown interference. Therefore, it is necessary to comprehensively consider these issues and design a robust fixed-time enclosure controller.

[0004] To address the above issues, the robust encirclement control problem under switching topology was studied. However, in existing studies, on the one hand, no consideration was given to how to ensure that the robot can achieve encirclement control within a fixed time under topology switching conditions; on the other hand, no comprehensive consideration was given to adverse factors such as unknown interference to the follower robot. Summary of the Invention

[0005] To solve the above technical problems, the present invention provides a fixed-time robust enclosure 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 enclosure 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 switching topology, the specific steps are as follows:

[0007] S1. Establish a dynamic model of a multi-robot system under a multi-leadership mode, that is, construct a time-varying multi-leader dynamic model and a 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, a nonlinear estimator based on an RBF neural network and an adaptive update law for the neural network weight matrix are designed to achieve an 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 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 disturbance estimation term obtained by the nonlinear estimator in step S3 as a compensation term.

[0013] S6. Substitute the robust encirclement controller described in step S5 into the dynamic model of the multi-robot system constructed in step S1, construct a closed-loop error system and analyze the encirclement reachability condition;

[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 the uncertainties.

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

[0018] Wherein, the number of individual follower robots is N, and the number of individual leader robots is M. The dynamic model of individual follower robots, i.e. the follower dynamic model expression with nonlinear uncertainty, is shown as formula (1), and the dynamic model of individual leader robots, i.e. the time-varying multi-leader dynamic model expression, is shown as formula (2).

[0019]

[0020] Wherein, t represents time, respectively represent the position and velocity of the follower . respectively represent the position and velocity of the leader . T represents a transpose operation, n is a non-negative integer, and represents 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, and represents the model uncertainty of the follower. i.e. the first-order derivative of p 0i (t), v 0i (t), p i (t), v i (t).

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

[0022] Set represents a set in a real vector space , if for any vector p i ∈P, there exists a vector p x ∈P such that y T In , p i =(1-α)p x +αp y and α∈[0,1], then the set is called a convex set. For a point set P={p1,…,p q} in , the convex hull of P is the smallest convex set containing all points in P. Use Co{p i ,i=1,…,q} to represent the convex hull of P. When Co{p i ,i=1,…,q}={p|p∈[min p i ,max p​i ]}.

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

[0024] Assume that for a group of robots, the leader cannot be learned 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 formula (3), which is expressed as follows:

[0026]

[0027] in, That is g i (p i (t),v i (t))'s first-order derivative.

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

[0029] S21, defining switching topology attributes;

[0030] First, a robot interaction network under topology switching conditions is set up, and the information exchange between robots is assumed to be a switchable undirected weighted graph simulation.

[0031] in, Respectively represent the graph The robot set, edge set, weighted adjacency matrix, 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; and

[0033] S22. After defining the switching topology attributes in step S21, design a switching topology cycle;

[0034] Set in a topology switching cycle, the residence time 0<T c ≤τ s+1 -τ s , then in the time period Number of topology switching times

[0035] Among them, t a ,t b Denotes 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 satisfied differential inequalities are as follows:

[0038]

[0039] in, Right now The first derivative of τ s ,s=0,1,…,e represents the switching time sequence, there is Hengyou Time adjustment parameters 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. The residual set that converges to The expression is as follows:

[0043]

[0044] Where, if χ∈(0,1) is a positive constant, the fixed time convergence constraint 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 all in a tight set , then 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 a 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, Represents the function δ ik The width of , and ‖∈‖≤∈ M .

[0052] Again 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 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, Indicates the communication status between the i-th robot and the j-th 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] Again 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), s i (t) is simplified and the expression is as follows:

[0069]

[0070] in,

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

[0072] A fixed-time adaptive robust enclosure controller under switching topology is constructed, that is, a controller of adaptive neural network, which is expressed as follows:

[0073]

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

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

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

[0077]

[0078] in,

[0079] S62: Based on step S61, analyze the state error of a single follower robot. 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 switching topology conditions to construct a closed-loop error system and analyze the enclosure reachability conditions.

[0083] Let matrix Then Equation (20) is described as a closed-loop error system, which is 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 Then it tends to be bounded, then its corresponding quantity is asymptotically bounded, 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 multi-robot systems caused by unknown interference under switching topology conditions. For multi-robot systems with unknown nonlinear dynamics, an adaptive distributed estimation scheme based on RBFNN is designed, which effectively completes the estimation of unknown nonlinear dynamics in the system, thereby suppressing the adverse effects of nonlinear terms on the system and improving the robustness of the multi-robot system's enclosure control. By proposing a robust enclosure control scheme based on fixed time and neural network, the enclosure control problem of multi-robot systems with unknown nonlinear dynamics under switching topology is solved, effectively improving the robustness and convergence rate of the system, and is suitable for the field of multi-robot robust enclosure control. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0089] Figure 2 This is the 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 the communication topology diagram used by the robot after the first topology switch in the 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 with reference to the accompanying drawings and embodiments.

[0093] like Figure 1 As shown in FIG, 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 a dynamic model of a multi-robot system under a multi-leadership mode, that is, construct a time-varying multi-leader dynamic model and a 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, a nonlinear estimator based on an RBF neural network and an adaptive update law for the neural network weight matrix are designed to achieve an 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 disturbance estimate term (the estimated value obtained by the nonlinear estimator) obtained by the nonlinear estimator in step S3 as a compensation term;

[0100] S6. Substitute the robust encirclement controller described in step S5 into the dynamic model of the multi-robot system constructed in step S1, construct a closed-loop error system and analyze the encirclement 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, 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 the uncertainties.

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

[0105] The number of follower robots is N, and the number of leader robots is M. The dynamic model of the follower robot, i.e., the follower dynamic model with nonlinear uncertainty, is expressed as Equation (1), and the dynamic model of the leader robot, i.e., the dynamic model of the leader with multiple instantaneous variables, is expressed as Equation (2).

[0106]

[0107] Where t represents time, Respectively represent followers position and velocity. Respectively represent leaders The position and velocity of [] T represents the transpose 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 that represents 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 modeling the dynamics, define relevant concepts in encirclement control;

[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 yAnd α∈[0,1], then the set is called a convex set. The point set P in the q The convex hull of P 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 p i ,max p i ]}.

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

[0111] Assume that for a group of robots, the leader cannot be learned 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 formula (3), which is expressed as follows:

[0113]

[0114] in, That is g i (p i (t),v i (t))'s first-order derivative.

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

[0116] S21, defining switching topology attributes;

[0117] First, a robot interaction network under topology switching conditions is set up, and the information exchange between robots is assumed to be a switchable undirected weighted graph simulation.

[0118] in, Respectively represent the graph The robot set, edge set, weighted adjacency matrix, 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; and

[0120] S22. After defining the switching topology attributes in step S21, design a switching topology cycle;

[0121] Set in a topology switching cycle, the residence time 0<T c ≤τ s+1 -τ s , then in the time period Number of topology switching times

[0122] Among them, t a ,t b Denotes 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 satisfied differential inequalities are as follows:

[0125]

[0126] in, Right now The first derivative of τ 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. The residual set that converges to The expression is as follows:

[0130]

[0131] Where, if χ∈(0,1) is a positive constant, the fixed time convergence constraint 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 all in a tight set , then 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 a 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, 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] Again 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, Indicates the communication status between the i-th robot and the j-th 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] Again 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]

[0155] where, denotes the Kronecker product of matrices, denotes the identity matrix, position error velocity error follower position follower velocity leader position leader velocity

[0156] Finally, the auxiliary variable error w i (t), s i (t) are simplified, and the expression is as follows:

[0157]

[0158] where,

[0159] In this embodiment, the step S5 is specifically as follows:

[0160] A fixed-time adaptive robust containment controller under switching topology, that is, an adaptive neural network controller, is constructed, and the expression is as follows:

[0161]

[0162] where, the function ε represents a small enough constant, and the purpose is to prevent the zero denominator item of the control term from appearing in the control process, and for any satisfies λ≥‖u 0i ‖, κ1, κ2, κ3, κ4, κ5 and κ6 are all normal numbers, and 0

[0163] In this embodiment, the step S6 is specifically as follows:

[0164] S61, according to the error estimation method of the follower robot position and velocity in step S4, the estimation of the influence of the leader on the follower state is obtained, and the expression is as follows:

[0165]

[0166] where,

[0167] S62: Based on step S61, analyze the state error of a single follower robot. The expression is as follows:

[0168]

[0169] in,

[0170] S63. Based on the fixed time theory, a neural network estimator and a robust enclosure controller are combined under switching topology conditions to construct a closed-loop error system and analyze the enclosure reachability conditions.

[0171] Let matrix Then Equation (20) is described as a closed-loop error system, which is expressed as follows:

[0172]

[0173] 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 ,

[0174] Then, we analyze the closed-loop error system described by Equation (21) and find that if the derivative of the disturbance is at a fixed time T f Then it tends to be bounded, then its corresponding quantity is asymptotically bounded, and its final bound is consistent with the artificially designed time adjustment parameter It can be said that the control input shown in formula (18) can achieve robust encirclement control.

[0175] In this embodiment, step S7 is specifically as follows:

[0176] The control protocol of step S5 (18) is applied to the dynamic equation of the individual. Under the condition of structural connectivity of the switching topology, 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.

[0177] The communication topology used by the multi-robot system before the first topology switching in this embodiment is as follows: Figure 2 As shown, the communication topology used after the first topology switching is as follows Figure 3 shown.

[0178] As can be seen from the figure, there are 9 robot individuals, of which 4 robots serve as leaders and 5 robots serve as 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 The unknown interference expression in the follower individual dynamics model is as follows:

[0179]

[0180] 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 follower individuals 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 leader's position state 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 speed state of the leader individual is v 0i (t)=[0.2,-1.4] T , the input to the leader individual is

[0181] In order to approximate the unknown function g i (p i (t),v i (t)), 9 neurons are selected for each neural network, 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 formula (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.

[0182] Figure 4 Figure 2 shows the trajectory of multi-robot encirclement control under topology switching in this embodiment. The simulation results show that the followers in the system can effectively track the leader and are driven into the convex hull formed by the leader. Ultimately, this embodiment achieves fixed-time robust encirclement control of the multi-robot system under topology switching.

[0183] In summary, the method of the present invention focuses on the impact of switching topology conditions on fixed-time convergence and the adverse effects of unknown disturbances on individual robots. First, a dynamic model of the leader and follower robots is constructed to clarify the research objectives of the encirclement control problem. Then, the nonlinear part of the follower model is estimated based on an RBF neural network, and expressions for the robot velocity and position errors in encirclement control are proposed. Then, based on the fixed-time stability theory, a robust encirclement controller is designed under switching topology conditions. The method of the present invention can effectively suppress the adverse effects of topology switching and external disturbances in multi-robot systems, thereby improving the robustness and effectiveness of the multi-robot encirclement control scheme.

[0184] 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 it should be understood that the scope of protection of the present invention is not limited to such specific descriptions and embodiments. Those skilled in the art can make various other specific variations and combinations based on the technical teachings disclosed in the present invention without departing from the essence of the present invention, and such variations and combinations are still within the scope of protection 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 a dynamic model of a multi-robot system under a multi-leader mode, that is, construct a time-varying multi-leader dynamic model and a 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, 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, a nonlinear estimator based on an RBF neural network and an adaptive update law for the neural network weight matrix are designed to achieve an 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 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 disturbance estimation term obtained by the nonlinear estimator in step S3 as a compensation term. S6. Substitute the robust encirclement controller described in step S5 into the dynamic model of the multi-robot system constructed in step S1, construct a closed-loop error system and analyze the encirclement reachability condition; 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. 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 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 the uncertainties. 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 instantaneous multiple leader dynamic model, 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 transpose 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 that represents 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 modeling the dynamics, define relevant concepts in encirclement control; 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 the 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 p i ,max p i ]}; S13, based on the "convex hull" in the encirclement control proposed in step S12, defining the control objective of the encirclement control problem; Assume that for a group of robots, the leader cannot be learned 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 formula (3), which is expressed as follows: in, That is g i (p i (t),v i (t))'s 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, defining switching topology attributes; First, a robot interaction network under topology switching conditions is set up, and the information exchange between robots is assumed to be a switchable undirected weighted graph simulation; in, Respectively represent the graph The robot set, edge set, weighted adjacency matrix, 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; and S22. After defining the switching topology attributes in step S21, design a switching topology cycle; Set in a topology switching cycle, the residence time 0<T c ≤τ s+1 -τ s , then in the time period 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 satisfied differential inequalities are as follows: in, Right now The first derivative of τ 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. The residual set that converges to The expression is as follows: Where, if χ∈(0,1) is a positive constant, the fixed time convergence constraint expression is as follows:

4. The fixed-time robust encirclement control method for a multi-robot system under switching topology according to claim 1, 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 all in a tight set , then 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 a 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, Represents the function δ ik The width of , and ‖∈‖≤∈ M ; Again 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, 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, Indicates the communication status between the i-th robot and the j-th 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, Again 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 and the expression is as follows: in, 6. The fixed-time robust encirclement control method for a multi-robot system under switching topology according to claim 1, characterized in that: The step S5 is specifically as follows: A fixed-time adaptive robust enclosure controller under switching topology is constructed, that is, a controller of adaptive neural network, which is expressed as follows: Among them, the function ε represents a sufficiently small constant to prevent the control term from There is a zero denominator term, for any Satisfy λ≥‖u 0i ‖, κ1, κ2, κ3, κ4, κ5 and κ6 are all positive numbers, 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, characterized in that: The step S6 is specifically as follows: S61. Based on the error estimation method for the follower robot position and velocity in step S4, an estimate of the influence of the leader on the follower state is obtained, which is expressed as follows: in, S62: Based on step S61, analyze the state error of a single follower robot. 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 switching topology conditions to construct a closed-loop error system and analyze the enclosure reachability conditions. Let matrix Then Equation (20) is described as a closed-loop error system, which is expressed as follows: Among them, and δ = {δ1 T (p1(t), v1(t)), δ2 T (p2(t), v2(t)),..., δ N T (p N (t), v N (t))} T , 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 Then it tends to be bounded, then its corresponding quantity is asymptotically bounded, and its final bound is consistent with the artificially designed time adjustment parameter ι, related.

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