A Non-Complete Multi-Mobile Robot Prescribed-Time Consensus Control Method

Abstract

The present invention discloses a nonholonomic multi-mobile robot pre-determined time consensus control method in the technical field of automation systems. Step 1: Establish a mathematical model of mobile robots by using a directed graph in graph theory. Step 2: Task description: Introduce a virtual leader and give a mathematical model for achieving a specified formation. Step 3: Formation problem transformation: Use classical variable substitution to linearize the nonholonomic multi-mobile robot system and transform it into a class of linear chained systems, that is, transform the formation tracking control problem of nonholonomic multi-mobile robots into a consensus control problem of new variables. Step 4: Design a distributed pre-determined time observer so that all followers can obtain the leader state and input information. Step 6: Design of the formation pre-determined time control protocol: Define an error tracking system for nonholonomic multi-mobile robots, decompose this system into two subsystems, and design a control protocol by referring to the double power reaching law, nonlinear manifold, and switching protocol, so that the robot formation converges stably within a pre-determined time, that is, enable the followers to track the estimated leader state within a pre-determined time independent of the initial conditions. Step 7: Prove the stability of the formation control protocol.

Description

Technical Field

[0001] The present invention relates to the technical field of automation systems, and particularly to a predetermined time consistency control method for nonholonomic multi-mobile robots. Background Art

[0002] In recent years, the cooperative control problem of nonholonomic multi-mobile robots has received extensive attention. The formation control of multi-robot systems can complete complex tasks that cannot be completed by single robots, such as exploration, surveillance and security, search and rescue operations, cooperative transportation, etc. The current mainstream formation control methods include: leader-following method, virtual structure method and behavior-based method. Among them, the leader-following method has an easy-to-understand mathematical model and strong scalability, so it is widely used.

[0003] The speed of convergence is an important performance criterion for evaluating the quality of the designed controller. At present, most of the designed consensus controllers can only achieve asymptotic consensus, so they are not suitable for some applications with high control accuracy requirements and strict convergence time requirements. For this reason, scholars have proposed a finite-time consensus controller with high control accuracy, fast convergence speed, and strong robustness to disturbances and uncertainties. However, the time required to reach consensus is determined by the initial state difference. Usually, the initial state is unknown, so it is difficult to estimate the convergence time. In addition, if the initial state error is large, then the time required to reach consensus is greater, so the controller cannot be applied to the case where the initial state error is very large.

[0004] To solve this problem, researchers have proposed a fixed-time control method, in which the convergence time is bounded by a positive constant independent of the system initial value. However, there are still two main problems in the existing research: First, the convergence time bound determined by Lyapunov stability analysis is very conservative, and the estimated bound is about several times the bound determined by simulation. Second, the convergence time limit of fixed-time consensus is a complex function of system parameters, and the direct relationship between the tuning parameters and the timing consensus is not clear.

[0005] For this reason, scholars have introduced the concept of predetermined time control, but most of them solve the predetermined time tracking consensus problem of multi-agent systems. For nonholonomic multi-mobile robots, especially when the system is transformed into two coupled subsystems, the problem of predetermined time formation consensus has not been fully studied. Based on this, the present invention designs a predetermined time consistency control method for nonholonomic multi-mobile robots to solve the above problems. Summary of the Invention

[0006] The purpose of the present invention is to provide a predetermined time consistency control method for nonholonomic multi-mobile robots to solve the problems raised in the above background art.

[0007] To achieve the above object, the present invention provides the following technical solutions: A nonholonomic multi-mobile robot pre-determined time consensus control method, comprising the following steps:

[0008] Step 1: Establish a mathematical model of the mobile robot using a directed graph in graph theory;

[0009] Step 2: Task description: Introduce a virtual leader and give a mathematical model for achieving a specified formation;

[0010] Step 3: Formation problem transformation: In this paper, the nonholonomic multi-mobile robot system is linearized using classical variable substitution and transformed into a class of linear chained systems; Therefore, the formation tracking control problem of nonholonomic multi-mobile robots is transformed into a consensus control problem of new variables;

[0011] Step 4: Design of a distributed pre-determined time observer: Divided into two steps: (1) Estimation of the leader input by the follower; (2) Estimation of the leader state by the follower;

[0012] Step 5: Stability proof of the distributed pre-determined time observer: Divided into two steps: (1) Proof for the observer estimating the leader input; (2) Proof for the observer estimating the leader state;

[0013] Step 6: The design of the formation pre-determined time control protocol is carried out in two steps. First, to facilitate the analysis of the consensus problem between the leader and the followers of the robot dynamics system, an error tracking system for nonholonomic multi-mobile robots is defined; Second, the system is decomposed into two subsystems, and the double power reaching law, nonlinear manifold, and switching protocol are used to design the control protocol, so that the robot formation converges stably within a pre-determined time;

[0014] Step 7: Stability proof of the formation control protocol: Carried out in two steps: The stability of the two subsystems in Step 4 is verified respectively through the Lyapunov theory; The control protocol designed by the present invention can make the nonholonomic multi-mobile robot system form the desired formation within a fixed time T≤T 1 +T 2 +T 3 inside.

[0015] Preferably, the nonholonomic multi-mobile robot system in establishing the mathematical model of the mobile robot is described by a directed graph G=(V,ε,A), where V={v 1 ,v 2 ,…,v m} is the node set, represents the edge set, A=[a ij m×m is a weighted adjacency matrix; each node v i represents a robot i, and each edge set (v i ​,v j ) ∈ ε indicates that robot i is a neighbor of robot j, and robot j can obtain the state information of robot i; for the weighted adjacency matrix A = [a ij m×m , when (v j ,v i ) ∈ ε, a ij = 1, otherwise a ij = 0; assume there is no self-loop, that is then a jj = 0; the Laplacian matrix L of the directed graph G is defined as L = D - A, where D = diag{d 1 ,d 2 ,…,d m},

[0016] Let B = diag{b 1 ,b 2 ,…,b m}, b j represents the connection weight between robot j (1 ≤ j ≤ m) and the leader 0; if robot j can obtain the state information of the leader 0, then b j > 0, otherwise b j = 0;

[0017] Mathematical Model

[0018] Mobile Robot Model

[0019] For the nonholonomic multi-mobile robot system, consider n robots as followers and a robot numbered 0 as the leader; for the i-th robot, its dynamic model is

[0020]

[0021] where, i ∈ Γ ∪ {0}, (x i ,y i ) ∈ R 2 represents the Cartesian coordinates of the centroid of the i-th robot, θ i ∈ R represents the yaw angle, v i ∈ R, w i ∈ R are the linear velocity and angular velocity respectively.

[0022] Preferably, the task description is to achieve a specified formation, and the formation control matrix P = [P 1 ,P 2 ,...,P j ,...,P n T, 1 ≤ j ≤ n, where P j = (p jx ​, p jy ) is the expected position vector of follower j relative to the virtual leader; if equation (2) holds, the nonholonomic multi-mobile robot system can achieve the specified formation task within a predetermined time;

[0023]

[0024] where, (x j , y j , θ j ), (x 0 , y 0 , θ 0 ) represent the poses of the follower and the leader in the global coordinate system respectively.

[0025] Preferably, the formation control problem is transformed to define the coordinate transformation of the nonholonomic multi-mobile robot as follows

[0026]

[0027] Figure 1 is the schematic diagram of the coordinate transformation of the nonholonomic multi-mobile robot, where the vector P j = (p jx , p jy ). From this figure, the coordinates of the virtual leader corresponding to the follower in the local coordinate system X'O'Y' can be obtained as follows

[0028]

[0029] Define the robot coordinate transformation as

[0030]

[0031] where, i ∈ Γ ∪ {0}, P 0 = (p 0x , p 0y ) = (0, 0), so the dynamic model of the robot can be defined as

[0032]

[0033] Note 1: For system (6), if there exists a suitable control law u j = [u 1j , u 2j T such that equation (7) holds, the nonholonomic multi-mobile robot system can converge within a predetermined time and thus complete the goal;

[0034]

[0035] where, ρ ∈ {1, 2, 3}, T is the convergence time constant;

[0036] Note 2: Due to the non - linear characteristics of the non - holonomic multi - mobile robot system, it is difficult to apply traditional linear control theory to the consensus formation tracking control of non - holonomic multi - mobile robots. In this paper, by using the classical variable substitution, the non - holonomic multi - mobile robot system is linearized and transformed into a class of linear chained systems. Therefore, the formation tracking control problem of non - holonomic multi - mobile robots is transformed into the consensus control problem of new variables.

[0037] To achieve our control objective, the following assumptions are given;

[0038] Assumption 1 The angular velocity ω i and the linear velocity v i of robot i are bounded, i ∈ Γ ∪ {0};

[0039] Assumption 2 The directed graph G of n followers is detail - balanced (i.e., there exist some scalars pi > 0, i ∈ Γ, such that for all i, j ∈ Γ, p j a ij = p i a ji ), and at least one follower can obtain the state information of the leader;

[0040] Assumption 3 The directed graph of n + 1 robots contains a spanning tree rooted at the leader (that is, there exists a vertex that has a directed path to any other vertex).

[0041] Preferably, the distributed pre - determined time observer designs an observer for each follower to estimate the leader state and input information. The observer is as follows

[0042]

[0043]

[0044] where, and are the estimates of the leader state z 0 = [z 1,0 , z 2,0 , z 3,0 T and the input u 0 = [u 1,0 , u 2,0 T respectively, 0 < α 0 < 1, α, γ l , l = 1, 2 satisfy

[0045]

[0046] where, α​​0 is a preset positive constant, p max and p min are the maximum diagonal element and the minimum diagonal element of the diagonal matrix P respectively, λ min (·) is the minimum eigenvalue of the matrix, c l satisfies the following formula

[0047]

[0048] Define

[0049] Preferably, under the conditions of Assumption 1 and Assumption 2 in the stability proof theorem 1 of the distributed pre-determined time observer, each follower can accurately estimate the leader input through the observers (8) - (10) within the pre-determined time, that is In particular, for any t > T 1 , there is where T 1 is the pre-determined time;

[0050] Proof: Combining the in formula (9) gives is

[0051]

[0052] Define Then formula (12) can be written as

[0053]

[0054] Construct the Lyapunov function Differentiating V 1 yields

[0055]

[0056] where

[0057] Also, considering a dynamic system

[0058]

[0059] If there exists a continuous radially unbounded positive definite function V(x), and there exists 0 < α < 1 such that the following formula holds

[0060]

[0061] It can be obtained that V 1 is stable at the pre-determined time T 1 , and then it can be obtained that Similarly, the above similar proof can obtain that at the pre-determined time T1 Inside

[0062] In summary, the follower can accurately estimate the leader's input information within T 1 ;

[0063] Theorem 2 Under the conditions of Assumptions 1 and 2, each follower can accurately estimate the leader's state through the observers (8)-(10) within a predetermined time, that is Specifically, for any t > T 2 = 3T 1 , there is

[0064] Proof: Combining the in Equation (8) gives as

[0065]

[0066] According to Theorem 1, when t ≥ T 1 , Define Then Equation (17) can be written as

[0067]

[0068] Construct the Lyapunov function Differentiate V 2 with respect to t to get

[0069]

[0070] Also, considering a dynamic system

[0071]

[0072] If there exists a continuous radially unbounded positive definite function V(x), and there exists 0 < α < 1 such that the following equation holds

[0073]

[0074] It can be obtained that V 2 is stable within the predetermined time 2T 1 , and then it can be obtained that Then, combining the in Equation (8) gives as

[0075]

[0076] When t ≥ 2T 1 , there is Therefore Thus, it can be obtained that For

[0077]

[0078] Define By constructing a Lyapunov function Similarly, we can also obtain V 3 At the predetermined time 3T 1 Stable; similarly, through the above similar proof of equations (17)-(19), it can be concluded that at the predetermined time 2T 1 Within

[0079] In summary, it can be obtained that the follower can accurately estimate the leader's state within T 2 = 3T 1 To achieve an accurate estimate of the leader's state within

[0080] Preferably, the design of the formation predetermined-time control protocol is carried out in two steps. First, to facilitate the analysis of the consistency problem between the leader and the follower of the robot dynamic system, an error tracking system for non-holonomic multi-mobile robots is defined; second, the system is decomposed into two subsystems, and the double power-law reaching law, nonlinear manifold, and switching protocol are used to design the control protocol, so that the robot formation can converge stably within the predetermined time;

[0081] Define the tracking error of the follower robot as

[0082]

[0083] Differentiate the tracking error equation (24) and substitute equation (8) into it to obtain

[0084]

[0085] According to Theorem 2, when t > T 2 When k = 1, 2, 3, at this time, equation (25) can be decomposed into two subsystems after simplification

[0086]

[0087]

[0088] According to the above subsystems, design the follower control laws u 1j And u 2j , so that it can track the corresponding leader within the predetermined time; the design is divided into two parts: Step1: Make e 1,j Stable; Step2: Make e 2,j And e 3,j Stable;

[0089] Step1: Design the control law u 1j as follows

[0090]

[0091] where 0 < α 0 < 1 is a preset constant;

[0092] Step2: Stabilize e 2,j and e 3,j ; It can be seen from Step1 that e 1,j can converge to 0 within a predetermined time T 1 Then when t ≥ T 2 + T 3 , Equation (27) can be written as

[0093]

[0094] Define the double power reaching law (30), (31), (32) and the nonlinear manifold (33)

[0095]

[0096]

[0097]

[0098]

[0099] where

[0100] Design the control law u 2,j as follows

[0101]

[0102] Note 3: The control law equation (34) contains Therefore, when the system solution makes e j = 0 before κ 3,j = 0, a singular solution will occur at this time; Therefore, the singularity problem can be solved by designing a switching protocol;

[0103] Define the linear manifold

[0104] ρ j = 0, ρ j = e 2,j (35)

[0105] Design the control law u 2,j as follows

[0106]

[0107] Preferably, for the stability proof of the formation control protocol, Theorem 3 states that for the first-order subsystem in Equation (26), under the predefined-time control law in Equation (28), there is Specifically, for any t > T 3 , there is where T 3 is the predefined time;

[0108] Proof: Substitute Equation (28) into Equation (26) to obtain

[0109]

[0110] Construct the Lyapunov function

[0111] V 11 = |e 1,j | (38)

[0112] Take the derivative of V 11 and combine it with Equation (37) to get

[0113]

[0114] From this, it can be obtained that e 1,j can converge to 0 at the predefined time T 3 . Then, when t ≥ T 3 ,

[0115] Theorem 4 states that for the two subsystems in Equations (26) and (27), under the predefined-time control laws in Equations (28) (36) and the distributed observer in Equations (8) (9), there is where k = 1, 2, 3; specifically, for any t ≥ T = T 2 + T 3 + T 4 + T 5 , there is z 1j = z 10 , z 2j = z 20 , z 3j = z 30 ;

[0116] Proof: Take the derivative of Equation (33) to obtain

[0117]

[0118] Substitute Equation (27) into Equation (40) to obtain

[0119]

[0120] Substituting Equation (34) into the above equation and combining with the double power reaching law Equation (30), we can obtain

[0121]

[0122] Next, construct the Lyapunov function

[0123] V 12 = |κ j | (43)

[0124] Differentiate V 12 and combine with Equation (42) to get

[0125]

[0126] It can be seen from this that κ j converges within the predetermined time T 2 + T 3 + T 4 , that is, κ j = 0. Substitute it into the nonlinear flow form (33) and combine with the double power reaching law Equation (31), we have

[0127]

[0128] Substitute Equation (45) into Equation (29) to obtain

[0129]

[0130] Similar to the above proof, construct the Lyapunov function

[0131] V 13 = |e 3,j | (47)

[0132] Differentiate V 13 and combine with Equation (46) to get

[0133]

[0134] Thus is stable at the predetermined time, and the upper bound of its convergence time is T 5 , that is, when t ≥ T 2 + T 3 + T 4 + T 5 ,

[0135] In summary, when t ≥ T 2 + T 3 + T 4 + T 5When e 1,j = 0, κ j = 0, e 3,j = 0, then combining equations (31) and (33) gives e 2,j = 0;

[0136] Theorem 5 Under the conditions of satisfying Assumptions 1, 2, and 3, for the two subsystems in equations (26) and (27), under the control laws in equations (28), (36) and the observers in equations (8), (9), there is where k = 1, 2, 3; In particular, for any t ≥ T = T 2 + T 3 + T 4 + T 5 + T 6 , there is

[0137] Proof: As known from Theorem 3, when t ≥ T 2 + T 3 e 1,j = 0; Substituting the above equation of equation (36) into equation (29)'s gives

[0138]

[0139] Therefore, e 2,j is stable in a predetermined time, and the upper bound of its convergence time is T 6 , then when t ≥ T 2 + T 3 + T 6 At this time, the control law switches to the following equation of equation (36), and there is ρj = 0, ρ j belongs to the non-singular set of the system, avoiding the problem of singular solutions. Combining Theorem 4 again, it can be known that when t ≥ T 2 + T 3 + T 4 + T 5 + T 6 e 1,j = 0, e 2,j = 0, e 3,j = 0.

[0140] Compared with the prior art, the beneficial effects of the present invention are:

[0141] In order to enable all followers to obtain the leader state and input information, a distributed observer is proposed - the distributed observer therein;

[0142] ​To enable the followers to track the estimated leader state within a predetermined time independent of the initial conditions, a double-power reaching law, a nonlinear manifold, and a switching protocol are introduced, and a predetermined-time controller is designed, including the double-power reaching law, the nonlinear manifold, the switching protocol, and the predetermined-time consensus control protocol;

[0143] A new predetermined-time control protocol is proposed, which enables all followers to obtain the leader state and input information within a predetermined time, enables the followers to track the estimated leader state within a predetermined time independent of the initial conditions, and stabilizes the system state within a predetermined time. Compared with most existing asymptotic convergence control protocols, finite-time convergence control protocols, and fixed-time convergence control protocol technical solutions, the technical solution of the present invention has better convergence performance, simpler convergence time control and is independent of the initial conditions, is more applicable, and has more value for actual mobile robot applications.

[0144] Of course, it is not necessary for any product implementing the present invention to achieve all the above-mentioned advantages simultaneously. BRIEF DESCRIPTION OF THE DRAWINGS

[0145] To more clearly illustrate the technical solutions of the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0146] Figure 1 Schematic diagram of coordinate transformation of nonholonomic multi-mobile robots of the present invention;

[0147] Figure 2 Directed communication graph of the present invention;

[0148] Figures 3 - 5 Error curve graph of the estimated value and the actual value of the leader state under the observer of the present invention;

[0149] Figures 6 - 8 Error curve between the state of the virtual leader robot and the estimated state of the leader under the distributed predetermined-time controller of the present invention;

[0150] Figures 9 - 11 Curve graph showing that the follower robot can still achieve tracking of the leader within 11 s after changing the initial state of the robot of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0151] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts belong to the scope of protection of the present invention.

[0152] Please refer to Figures 1 to 11 , the present invention provides a technical solution for a non - complete multi - mobile robot pre - determined time consistency control method: including the following steps:

[0153] Step 1: Use a directed graph in graph theory to establish a mathematical model of mobile robots. The non - complete multi - mobile robot system in the established mathematical model of mobile robots is described by a directed graph G=(V, ε, A), where V = {v 1 , v 2 , …, v m} is the node set, represents the edge set, A = [a ij m×m is a weighted adjacency matrix; each node v i represents a robot i, and each edge set (v i , v j ) ∈ ε means that robot i is a neighbor of robot j, and robot j can obtain the state information of robot i; for the weighted adjacency matrix A = [a ij m×m , when (v j , v i ) ∈ ε, a ij = 1, otherwise a ij = 0; assume that there is no self - loop, that is then a jj = 0; the Laplacian matrix L of the directed graph G is defined as L = D - A, where D = diag{d 1 , d 2 , …, d m},

[0154] Let B = diag{b 1 , b 2 , …, b m}, b j represents the connection weight between robot j (1 ≤ j ≤ m) and the leader 0; if robot j can obtain the state information of the leader 0, then b j > 0, otherwise b j = 0;

[0155] Mathematical model

[0156] ​​Mobile robot model

[0157] For a nonholonomic multi-mobile robot system, consider that there are n robots as followers and a robot numbered 0 as the leader; for the i-th robot, its dynamic model is

[0158]

[0159] where i ∈ Γ ∪ {0}, (x i , y i ) ∈ R 2 represents the Cartesian coordinates of the centroid of the i-th robot, θ i ∈ R represents the yaw angle, v i ∈ R, w i ∈ R are the linear velocity and angular velocity respectively;

[0160] Step 2: Task description: Introduce a virtual leader and give a mathematical model for achieving a specified formation. The task description is to achieve a specified formation. Define the formation control matrix P = [P 1 , P 2 ,..., P j ,..., P n T , 1 ≤ j ≤ n, where P j = (p jx , p jy ) is the desired position vector of follower j relative to the virtual leader; if equation (2) holds, the nonholonomic multi-mobile robot system achieves the specified formation task within a predetermined time;

[0161]

[0162] where, (x j , y j , θ j ), (x 0 , y 0 , θ 0 ) represent the poses of the follower and the leader in the global coordinate system respectively;

[0163] Step 3: Formation problem transformation: In this paper, the classical variable substitution is used to linearize the nonholonomic multi-mobile robot system and transform it into a class of linear chained systems; therefore, the formation tracking control problem of nonholonomic multi-mobile robots is transformed into a consensus control problem of new variables. The formation control problem transformation defines the coordinate transformation of nonholonomic multi-mobile robots as follows

[0164]

[0165] Figure 1 is the schematic diagram of the coordinate transformation of nonholonomic multi-mobile robots, where the vector P​j = (p jx , p jy ), from which the coordinates of the virtual leader corresponding to the follower in the local coordinate system X'O'Y' can be obtained as follows

[0166]

[0167] Define the robot coordinate transformation as

[0168]

[0169] where i ∈ Γ ∪ {0}, P 0 = (p 0x , p 0y ) = (0, 0), thus the dynamic model of the robot can be defined as

[0170]

[0171] Note 1: For system (6), if there exists a suitable control law u j = [u 1j , u 2j T such that equation (7) holds, then the nonholonomic multi-mobile robot system can converge within a predetermined time and thus achieve the goal;

[0172]

[0173] where ρ ∈ {1, 2, 3}, T is the convergence time constant;

[0174] Note 2: Due to the non-linear characteristics of the nonholonomic multi-mobile robot system, it is difficult to apply traditional linear control theory to the consensus formation tracking control of nonholonomic multi-mobile robots; in this paper, classical variable substitution is used to linearize the nonholonomic multi-mobile robot system and convert it into a class of linear chained systems; therefore, the formation tracking control problem of nonholonomic multi-mobile robots is converted into a consensus control problem of new variables;

[0175] To achieve our control goal, the following assumptions are given;

[0176] Assumption 1 The angular velocity w i and linear velocity v i of robot i are bounded, i ∈ Γ ∪ {0};

[0177] Assumption 2 The directed graph G of n followers is detailed balanced, that is, there exist some scalars pi > 0, i ∈ Γ, such that for all i, j ∈ Γ, p j a ij = p i a ji ​, at least one follower can obtain the status information of the leader;

[0178] Assume a directed graph of 3n + 1 robots contains a spanning tree rooted at the leader;

[0179] Step 4: Design of distributed pre-determined time observer: It is divided into two steps: (1) Estimation of the leader input by the follower; (2) Estimation of the leader status by the follower. In the design of the distributed pre-determined time observer, each follower proposes an observer to estimate the leader status and input information. The observer is as follows

[0180]

[0181]

[0182] where and are the estimates of the leader status z 0 = [z 1,0 , z 2,0 , z 3,0 T and the input u 0 = [u 1,0 , u 2,0 T by the j-th follower, 0 < α 0 < 1, α, γ l , l = 1, 2 satisfy

[0183]

[0184] where α 0 is a preset positive constant, p max and p min are the maximum diagonal element and the minimum diagonal element of the diagonal matrix P respectively, λ min (·) is the minimum eigenvalue of the matrix, and c l satisfies the following equation

[0185]

[0186] Define

[0187] Step 5: Stability proof of the distributed pre-determined time observer: It is divided into two steps: (1) Proof of the observer for estimating the leader input; (2) Proof of the observer for estimating the leader status. The stability proof of the distributed pre-determined time observer Theorem 1 Under the conditions of Assumption 1 and Assumption 2, each follower can accurately estimate the leader input within a pre-determined time through the observers (8) - (10), that is In particular, for any t > T​​1 , there is where T 1 is a predetermined time;

[0188] Proof: Combining the in Equation (9) gives as

[0189]

[0190] Define Then Equation (12) can be written as

[0191]

[0192] Construct the Lyapunov function Differentiate V 1 to obtain

[0193]

[0194] where,

[0195] Also, from: Consider a dynamical system

[0196]

[0197] If there exists a continuous radially unbounded positive definite function V(x), and there exists 0 < α < 1 such that the following holds

[0198]

[0199] It can be obtained that V 1 is stable at the predetermined time T 1 , and then it can be obtained that Similarly, through a proof similar to the above, it can be concluded that at the predetermined time T 1 inside

[0200] In summary, it can be obtained that the follower can accurately estimate the leader's input information within T 1 ;

[0201] Theorem 2 Under the conditions of Assumption 1 and Assumption 2, each follower can accurately estimate the leader's state through the observers (8) - (10) within the predetermined time, that is In particular, for any t > T 2 = 3T 1 , there is

[0202] Proof: Combining the in Equation (8) gives as

[0203]

[0204] As can be seen from Theorem 1, when t ≥ T 1 , Define Then equation (17) can be written as

[0205]

[0206] Construct the Lyapunov function For V 2 Taking the derivative gives

[0207]

[0208] Also from: Consider a dynamic system

[0209]

[0210] If there exists a continuous radially unbounded positive definite function V(x), and there exists 0 < α < 1 such that the following equation holds

[0211]

[0212] We can obtain V 2 Stable at the predetermined time 2T 1 Subsequently, we can obtain Then, combining with that in equation (8) We can obtain As

[0213]

[0214] When t ≥ 2T 1 At this time, there is Therefore Thus, we can obtain As

[0215]

[0216] Define By constructing the Lyapunov function Similarly, we can obtain V 3 Stable at the predetermined time 3T 1 Stable; Similarly, through the above similar proof for equations (17) - (19), it can be concluded that within the predetermined time 2T 1 Inside

[0217] In summary, the follower can achieve accurate estimation of the leader's state within T 2 = 3T 1 ;

[0218] Step 6: The design of the formation scheduled-time control protocol is carried out in two steps. First, to facilitate the analysis of the consensus problem between the leader and followers in the robot dynamics system, an error tracking system for nonholonomic multi-mobile robots is defined. Second, the system is decomposed into two subsystems, and the double power reaching law, nonlinear manifold, and switching protocol are used to design the control protocol, so that the robot formation converges stably within the scheduled time. The design of the formation scheduled-time control protocol is carried out in two steps. First, to facilitate the analysis of the consensus problem between the leader and followers in the robot dynamics system, an error tracking system for nonholonomic multi-mobile robots is defined. Second, the system is decomposed into two subsystems, and the double power reaching law, nonlinear manifold, and switching protocol are used to design the control protocol, so that the robot formation converges stably within the scheduled time;

[0219] Define the tracking error of the follower robot as

[0220]

[0221] Take the derivative of the tracking error equation (24) and substitute equation (8) into it to obtain

[0222]

[0223] According to Theorem 2, when t > T 2 there is k = 1, 2, 3. At this time, equation (25) can be decomposed into two subsystems after simplification

[0224]

[0225]

[0226] According to the above subsystems, design the follower control laws u 1j and u 2j , so that it can track the corresponding leader within the scheduled time. The design is divided into two parts: Step1: Make e 1,j stable; Step2: Make e 2,j and e 3,j stable;

[0227] Step1: Design the control law u 1j as follows

[0228]

[0229] where 0 < α 0 < 1 is a preset constant;

[0230] Step2: Make e 2,j and e 3,j stable; It can be seen from Step1 that e1,j converges to 0 at a predetermined time T 1 then when t ≥ T 2 + T 3 when Equation (27) can be written as

[0231]

[0232] Define the double power reaching law (30), (31), (32) and the nonlinear manifold (33)

[0233]

[0234]

[0235]

[0236]

[0237] where

[0238] Design the control law u 2,j as follows

[0239]

[0240] Note 3: The control law in Equation (34) contains Therefore, when the system solution makes e j = 0 before κ 3,j = 0, a singular solution will occur at this time; therefore, the singularity problem can be solved by designing a switching protocol;

[0241] Define the linear manifold

[0242] ρ j = 0, ρ j = e 2,j (35)

[0243] Design the control law u 2,j as follows

[0244]

[0245] Step 7: Stability proof of the formation control protocol: It is carried out in two steps: The stability of the two subsystems in Step 4 is verified respectively through the Lyapunov theory; The control protocol designed in the present invention can make the nonholonomic multi-mobile robot system reach stability at a fixed time T ≤ T 1 + T 2 + T 3Form the desired formation. The stability proof theorem 3 of the formation control protocol states that, under the conditions of Assumptions 1, 2, and 3, for the first-order subsystem in Equation (26), under the predefined-time control law in Equation (28), there is Specifically, for any t > T 3 , there is where T 3 is the predefined time;

[0246] Proof: Substitute Equation (28) into Equation (26) to obtain

[0247]

[0248] Construct the Lyapunov function

[0249] V 11 = |e 1,j | (38)

[0250] Take the derivative of V 11 and combine it with Equation (37) to obtain

[0251]

[0252] From this, it can be obtained that e 1,j can converge to 0 at the predefined time T 3 , then, when t ≥ T 3 ,

[0253] Theorem 4 states that, under the conditions of Assumptions 1, 2, and 3, for the two subsystems in Equations (26) and (27), under the predefined-time control laws in Equations (28)(36) and the distributed observer in Equations (8)(9), there is where k = 1, 2, 3; specifically, for any t ≥ T = T 2 + T 3 + T 4 + T 5 , there is z 1j = z 10 , z 2j = z 20 , z 3j = z 30 ;

[0254] Proof: Take the derivative of Equation (33) to obtain

[0255]

[0256] Substitute Equation (27) into Equation (40) to obtain

[0257]

[0258] Substituting Equation (34) into the above equation and combining with the double power reaching law Equation (30), we can obtain

[0259]

[0260] Next, construct the Lyapunov function

[0261] V 12 = |κ j | (43)

[0262] Take the derivative of V 12 and combine with Equation (42) to get

[0263]

[0264] It can be seen from this that κ j converges within the predetermined time T 2 +T 3 +T 4 i.e., κ j = 0. Substitute it into the non-linear flow form (33) and combine with the double power reaching law Equation (31), we have

[0265]

[0266] Substitute Equation (45) into Equation (29), we can obtain

[0267]

[0268] Similar to the above proof, construct the Lyapunov function

[0269] V 13 = |e 3,j | (47)

[0270] Take the derivative of V 13 and combine with Equation (46) to get

[0271]

[0272] Thus,[[]] is stable at the predetermined time, and the upper bound of its convergence time is T 5 , i.e., when t ≥ T 2 +T 3 +T 4 +T 5

[0273] In summary, when t ≥ T 2 +T 3 +T 4 +T 5 1,j = 0, κ​​j = 0, e 3,j = 0, then combining equations (31) and (33) gives e 2,j = 0;

[0274] Theorem 5 Under the conditions of satisfying Assumptions 1, 2, and 3, for the two subsystems in equations (26) and (27), under the control laws in equations (28), (36) and the observers in equations (8), (9), there is where k = 1, 2, 3; in particular, for any t ≥ T = T 2 + T 3 + T 4 + T 5 + T 6 , there is

[0275] Proof: As known from Theorem 3, when t ≥ T 2 + T 3 , e 1,j = 0; substituting the above equation of equation (36) into equation (29)'s gives

[0276]

[0277] Therefore, e 2,j is stable in a predetermined time, and its upper bound of the convergence time is T 6 , then when t ≥ T 2 + T 3 + T 6 , At this time, the control law switches to the following equation of equation (36), and there is ρ j = 0, ρ j belongs to the non-singular set of the system, avoiding the problem of singular solutions. Combining Theorem 4 again, it can be known that when t ≥ T 2 + T 3 + T 4 + T 5 + T 6 , e 1,j = 0, e 2,j = 0, e 3,j = 0.

[0278] In the description of this specification, the descriptions referring to the terms "an embodiment", "example", "specific example", etc. mean that the specific features, structures, materials or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic expressions of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in a suitable manner in any one or more embodiments or examples.

[0279] The preferred embodiments of the present invention disclosed above are only used to help illustrate the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the invention to the specific embodiments described. Obviously, many modifications and variations can be made according to the content of this specification. These embodiments are selected and specifically described in this specification to better explain the principles and practical applications of the present invention, so that those skilled in the art can well understand and utilize the present invention. The present invention is only limited by the claims and their full scope and equivalents.

Claims

1. A method for pre-specified-time consensus control of nonholonomic multi-mobile robots, characterized in that, it includes the following steps: Step 1: Establish a mathematical model of mobile robots using a directed graph in graph theory; Step 2: Task description: Introduce a virtual leader and give a mathematical model for achieving a specified formation; Step 3: Formation problem transformation: Using classical variable substitution, linearize the nonholonomic multi-mobile robot system and transform it into a class of linear chained systems; thus, the formation tracking control problem of nonholonomic multi-mobile robots is transformed into a consensus control problem of new variables; Step 4: Design of a distributed pre-specified-time observer: Divided into two steps: (1) Estimation of the leader's input by the follower; (2) Estimation of the leader's state by the follower; Step 5: Stability proof of the distributed pre-specified-time observer: Divided into two steps: (1) Proof for the observer estimating the leader's input; (2) Proof for the observer estimating the leader's state; Step 6: The design of the formation pre-specified-time control protocol is carried out in two steps. First, to facilitate the analysis of the consensus problem between the leader and the followers of the robot dynamics system, an error tracking system for nonholonomic multi-mobile robots is defined. Second, decompose this system into two subsystems, and design the control protocol by introducing the double power reaching law, nonlinear manifold, and switching protocol, so that the robot formation converges stably within a pre-specified time; Step 7: Stability proof of the formation control protocol: It is carried out in two steps: the stability of the two subsystems in Step 4 is verified respectively through the Lyapunov theory; the designed control protocol can make the nonholonomic multi-mobile robot system form the desired formation within a fixed time \(T\leq T\) 1 +T 2 +T 3 and \(T\leq T + T\).

2. A method for pre-specified-time consensus control of nonholonomic multi-mobile robots according to claim 1, characterized in that: The non - holonomic multi - mobile robot system in the mathematical model of mobile robots is described by a directed graph \(G=(V,\varepsilon,A)\), where \(V = \{v 1 ,v 2 ,\cdots,v m \}\) is the node set, denotes the edge set, \(A = [a ij m×m is a weighted adjacency matrix; each node \(v i \) represents a robot \(i\), and each edge set \((v i ,v j )\in\varepsilon\) means that robot \(i\) is a neighbor of robot \(j\), and robot \(j\) can obtain the state information of robot \(i\); for the weighted adjacency matrix \(A = [a ij m×m \), when \((v j ,v i )\in\varepsilon\), \(a ij = 1\), otherwise \(a ij = 0\); assuming there is no self - loop, that is then \(a jj = 0\); the Laplacian matrix \(L\) of the directed graph \(G\) is defined as \(L = D - A\), where \(D=\text{diag}\{d 1 ,d 2 ,\cdots,d m \}\), j\in\{1,2,\cdots,m\}\);​​ Let \(B = diag\{b 1 , b 2 , \cdots, b m \}\), where \(b j \) represents the connection weight between robot \(j\) and leader \(0\), where \(1\leq j\leq m\); if robot \(j\) can obtain the state information of leader \(0\), then \(b j > 0\), otherwise \(b j = 0\). Mathematical model Mobile robot model For the nonholonomic multi-mobile robot system, consider that there are n robots as followers and a robot numbered 0 as the leader; for the i-th robot, its dynamic model is where \(i\in\Gamma\cup\{0\},(x i ,y i )\in\mathbb{R} 2 represents the Cartesian coordinates of the centroid of the \(i\)-th robot, \(\theta i \in\mathbb{R}\) represents the deviation angle, \(v i \in\mathbb{R}, w i \in\mathbb{R}\) are the linear velocity and the angular velocity respectively.

3. A method for pre-specified-time consensus control of nonholonomic multi-mobile robots according to claim 1, characterized in that: The task is described as achieving a specified formation, and the formation control matrix P = [P 1 , P 2 , …, P j , …, P n T , 1 ≤ j ≤ n, where P j = (p jx , p jy ) is the desired position vector of the follower j relative to the virtual leader; if Equation (2) holds, the nonholonomic multi-mobile robot system achieves the specified formation task within a predetermined time;​ where (x j , y j , θ j ) and (x 0 , y 0 , θ 0 ) represent the poses of the follower and the leader in the global coordinate system, respectively.

4. A method for pre-specified-time consensus control of nonholonomic multi-mobile robots according to claim 1, characterized in that: The transformation of the formation control problem defines the coordinate transformation of nonholonomic multi-mobile robots as follows Among them, vector P j =(p jx , p jy ), which is the coordinate of the virtual leader corresponding to the follower under the local coordinate system X'O'Y', as follows Define the robot coordinate transformation as where \(i\in\Gamma\cup\{0\}\), \(P\) 0 =(p 0x , p 0y ) = (0, 0), so the dynamic model of the robot can be defined as For the system (6), if there exists an appropriate control law u j = [u 1j , u 2j T such that equation (7) holds, then the nonholonomic multi-mobile robot system can converge within a predetermined time, thus achieving the goal;​ where ρ ∈ {1, 2, 3}, and T is the convergence time constant; Due to the non-linear characteristics of the nonholonomic multi-mobile robot system, it is difficult to apply traditional linear control theory to the consensus formation tracking control of nonholonomic multi-mobile robots; using classical variable substitution, linearize the nonholonomic multi-mobile robot system and transform it into a class of linear chained systems; thus, the formation tracking control problem of nonholonomic multi-mobile robots is transformed into a consensus control problem of new variables; To achieve the control objective, the following assumptions are given; Hypothesis 1: The angular velocity w of robot i i and the linear velocity v i are bounded, i ∈ Γ ∪ {0}; Suppose the directed graph G of 2n followers is detailed balanced, that is, there exist some scalars pi > 0, i ∈ Γ, such that for all i, j ∈ Γ, p j a ij = p i a ji , and at least one follower can obtain the state information of the leader; Directed graph of 3n + 1 robots Contains a spanning tree rooted at the leader.

5. A method for pre-specified-time consensus control of nonholonomic multi-mobile robots according to claim 1, characterized in that: In the design of the distributed pre-specified-time observer, each follower proposes an observer for estimating the leader's state and input information, and the observer is as follows wherein, and are the estimates of the state z 0 = [z 1,0 , z 2,0 , z 3,0 T and the input u 0 = [u 1,0 , u 2,0 T of the j-th follower, 0 < α 0 < 1, α, γ l , l = 1, 2 satisfy​​ where α 0 is a preset positive constant, p max and p min are the maximum diagonal element and the minimum diagonal element of the diagonal matrix P respectively, λ min (·) is the minimum eigenvalue of the matrix, and c l satisfies the following equation Definition 6. A method for pre-specified-time consensus control of nonholonomic multi-mobile robots according to claim 1, characterized in that: The stability proof of the distributed pre - defined - time observer Theorem 1 Under the conditions of Assumption 1 and Assumption 2, each follower can accurately estimate the leader input through the observers (8)-(10) within the pre - defined time, that is Specifically, for any t > T 1 , there is where T 1 is the pre - defined time; Proof: Combining the one in Equation (9), we can obtain which is Definition Then Equation (12) can be written as Construct the Lyapunov function For V 1 Taking the derivative gives Among them, Also, consider a dynamic system If there exists a continuously radially unbounded positive definite function V(x), and there exists 0 < α < 1 such that the following holds V can be obtained 1 at a predetermined time T 1 stable, and then it can be obtained that Similarly, it can be concluded by a similar proof as above that at a predetermined time T 1 within In summary, the follower can accurately estimate the leader's input information within T 1 ; Theorem 2 Under the conditions of Assumption 1 and Assumption 2, each follower can accurately estimate the leader's state within a predetermined time through the observers (8)-(10), that is Specifically, for any t > T 2 = 3T 1 , there is Proof: Combining the gives as As can be seen from Theorem 1, when t≥T 1 , Define Then equation (17) can be written as Construct the Lyapunov function For V 2 Taking the derivative gives Also, considering a dynamic system If there exists a continuously radially unbounded positive definite function V(x), and there exists 0 < α < 1 such that the following holds V can be obtained 2 at a predetermined time 2T 1 stable, and then Then, combining with that in Equation (8) it can be obtained is When t ≥ 2T 1 there is Therefore Thus, it can be obtained that is Definition By constructing a Lyapunov function Similarly, V can also be obtained 3 At the predetermined time 3T 1 Stable; Similarly, by the similar proof of the above equations (17)-(19), it can be obtained that at the predetermined time 2T 1 Within In summary, the follower can accurately estimate the leader's state within T 2 = 3T 1 ​ 7. A nonholonomic multi-mobile robot preset-time consensus control method according to claim 1, characterized in that: The design of the preset-time control protocol for formation is carried out in two steps. First, to facilitate the analysis of the consensus problem between the leader and followers of the robot dynamic system, an error tracking system for nonholonomic multi-mobile robots is defined; second, the system is decomposed into two subsystems, and the double power-law reaching law, nonlinear manifold and switching protocol are used to design the control protocol, so that the robot formation converges stably within the preset time; Define the tracking error of the follower robot as Differentiate the tracking error formula (24) and substitute formula (8) into it to obtain As can be seen from Theorem 2, when t > T 2 , there is k = 1, 2, 3. At this time, after simplification, Equation (25) can be decomposed into two subsystems According to the above subsystems, the follower control laws u 1j and u 2j are designed respectively to make them track the corresponding leader within a predetermined time. The design is divided into two parts: Step1: Make e 1,j stable; Step2: Make e 2,j and e 3,j stable; Step1: Design the control law u 1j As follows where, 0 < α 0 < 1 is a preset constant; Step2: Make e 2,j and e 3,j stable; As known from Step1, e 1,j can converge to 0 within the predetermined time T 1 ; then when t ≥ T 2 + T 3 , Equation (27) can be written as Define the double power-law reaching laws (30), (31), (32) and the nonlinear manifold (33) Among them, Design the control law u 2,j as follows The control law in Equation (34) contains Therefore, when the system solution makes e j = 0 before κ 3,j = 0, a singular solution will occur at this time. Therefore, the singularity problem can be solved by designing a switching protocol. Define the linear manifold ρ j = 0, ρ j = e 2,j (35) Design the control law u 2,j as follows 8. A nonholonomic multi-mobile robot preset-time consensus control method according to claim 1, characterized in that: Stability proof of the formation control protocol Theorem 3 Under the conditions of Assumption 1, Assumption 2, and Assumption 3, for the first-order subsystem in Equation (26), under the predefined-time control law in Equation (28), there is Specifically, for any t > T 3 , there is where T 3 is the predefined time; Proof: Substitute formula (28) into formula (26) to get Construct the Lyapunov function V 11 = |e 1,j | (38) Derive with respect to V 11 Take the derivative and combine with Equation (37) to obtain It can be obtained that e 1,j converges to 0 at a predetermined time T 3 then, when t ≥ T 3 at this time Theorem 4 Under the conditions of Assumptions 1, 2, and 3, for the two subsystems in Eqs. (26) and (27), under the predefined-time control laws in Eqs. (28) and (36) and the distributed observers in Eqs. (8) and (9), there is where k = 1, 2, 3; in particular, for any t ≥ T = T 2 + T 3 + T 4 + T 5 , there is z 1j = z 10 , z 2j = z 20 , z 3j = z 30 ; Proof: Differentiate formula (33) to obtain Substitute formula (27) into formula (40) to get Substitute formula (34) into the above formula and combine with the double power-law reaching law formula (30) to get Next, construct the Lyapunov function V 12 = |κ j | (43) Derive with respect to V 12 Take the derivative and combine with Equation (42) to obtain It can be seen from this that κ j converges within a predetermined time T 2 +T 3 +T 4 That is, κ j = 0. Substituting it into the non-linear flow form (33) and combining with the double power approach law formula (31), we have Substitute formula (45) into formula (29) to get Similar to the above proof, construct the Lyapunov function V 13 = |e 3,j | (47) Derive with respect to V 13 Take the derivative and combine with Equation (46) to obtain Thus, it is stable at a predetermined time, and the upper bound of its convergence time is T 5 , that is, when t ≥ T 2 +T 3 +T 4 +T 5 then In summary, when t≥T 2 +T 3 +T 4 +T 5 , e 1,j = 0, κ j = 0, e 3,j = 0. Then, combining equations (31) and (33), we can obtain e 2,j = 0; Theorem 5 Under the conditions of satisfying Assumption 1, Assumption 2, and Assumption 3, for the two subsystems in Eqs. (26) and (27), under the control laws in Eqs. (28) and (36) and the observers in Eqs. (8) and (9), there is where k = 1, 2, 3; in particular, for any t ≥ T = T 2 + T 3 + T 4 + T 5 + T 6 , there is Proof: As known from Theorem 3, when t≥T 2 +T 3 e 1,j = 0; Substituting the above equation of Equation (36) into in Equation (29), we can obtain Therefore, e 2,j is stable at a predetermined time, and the upper bound of its convergence time is T 6 , then when t ≥ T 2 + T 3 + T 6 When At this time, the control law switches to the following formula of Equation (36), and there is ρ j = 0, ρ j belongs to the non-singular set of the system, avoiding the problem of singular solutions. Combining Theorem 4, it can be known that when t ≥ T 2 + T 3 + T 4 + T 5 + T 6 When, e 1,j = 0, e 2,j = 0, e 3,j = 0.

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