A nonholonomic multi-mobile robot fixed-time consensus control method considering external disturbance

Abstract

The application discloses a kind of nonholonomic multi-mobile robot fixed time consensus control method considering external disturbance in the field of automatic control and multi-robot technology, step one: establishing mobile robot model;Step two: task description: introduce virtual leader, give the mathematical model for realizing specified formation;Step three: formation problem conversion: the formation tracking control problem of NMMR is converted into the consensus control problem of new variable;Step four: the design of formation control protocol: first, the error tracking system of leader and follower is defined;Second, the system is decomposed into two subsystems, a fixed-time consensus control protocol is designed by designing double-power approaching law and nonlinear function;Step five: stability proof of formation control protocol, so that the robot formation converges stably in fixed time——double-power approaching law, nonlinear function and fixed-time consensus control protocol therein.

Description

Technical Field

[0001] This invention relates to the fields of automatic control and multi-robot technology, specifically to a fixed-time consistency control method for nonholonomic multi-mobile robots that takes into account external disturbances. Background Technology

[0002] In recent years, cooperative control of nonholonomic wheeled mobile robots has attracted widespread attention, and its application in various fields, including military surveillance and environmental monitoring, has been rapid. Currently, the mainstream formation control methods include: the leader-follower method, the virtual structure method, and the behavior-based method. Among these, the leader-follower method, with its easily understood mathematical model and strong scalability, is widely used.

[0003] The consistency problem, as a core issue in the cooperative control of multi-robot systems, has attracted widespread research interest. Its aim is to design consistent control strategies to drive robots to achieve the same system state. Therefore, the convergence speed undoubtedly becomes a performance indicator for evaluating the quality of the control method. However, most previous studies only considered asymptotic consistency, meaning that convergence is achieved when time approaches infinity. But in practical applications, such as UAV navigation, vehicle tracking, and robot environmental monitoring, completing all planned tasks within a finite time is crucial. To improve convergence speed, researchers have introduced finite-time control protocols, enabling the system to achieve rapid convergence.

[0004] However, under finite-time control protocols, the convergence time set by the system depends on the initial state. If the initial state information is inaccurate or the environment is unknown, the effect of the control protocol will no longer be significant. In response, some researchers have proposed a fixed-time consensus protocol to obtain a convergence time upper limit independent of the initial state. However, most research focuses on multi-agent systems under external disturbances and model uncertainties. For nonholonomic multi-wheeled mobile robots (NMMRs), especially when the system is transformed into two coupled subsystems, the fixed-time consensus problem has not been fully studied.

[0005] This invention studies the fixed-time consistency formation tracking problem of nonholonomic multi-wheeled mobile robots. For nonholonomic multi-wheeled mobile robots (NMMR) under sideslip conditions, a fixed-time active disturbance rejection consistency formation tracking control protocol is proposed. The multi-robot system can overcome disturbances and achieve formation tracking within a fixed time, and the system's fixed convergence time is independent of the robot's initial state; the upper bound of the convergence time can be obtained through pre-set parameters. Simulation experiments show that the control protocol designed in this invention enables the multi-robot system to converge faster and the convergence time is independent of the robot's initial state, thus making it more suitable for practical applications. Based on this, this invention designs a fixed-time consistency control method for nonholonomic multi-wheeled mobile robots that considers external disturbances to solve the aforementioned problem. Summary of the Invention

[0006] The purpose of this invention is to provide a fixed-time consistency control method for nonholonomic multi-mobile robots that takes into account external disturbances, so as to solve the problems mentioned in the background art.

[0007] To achieve the above objectives, the present invention provides the following technical solution: a fixed-time consistency control method for nonholonomic multi-mobile robots considering external disturbances, comprising the following steps:

[0008] Step 1: Mobile Robot Model: Define the navigator robot and the follower robot, and establish a mathematical model of the mobile robot;

[0009] Step 2: Task Description: Introduce a virtual navigator and provide a mathematical model for achieving a specified formation;

[0010] Step 3: Formation Problem Transformation: Considering the influence of external disturbances, a disturbance factor is introduced into the model. A new robot dynamics model is constructed through robot coordinate transformation, thereby transforming the formation tracking control problem of NMMR into a consistency control problem with new variables.

[0011] Step 4: The design of the formation control protocol is carried out in two steps: First, to facilitate the analysis of the consistency problem between the leader and the follower in the robot dynamics system, an error tracking system for the leader and the follower is defined; Second, the system is decomposed into two subsystems, and a fixed-time consistency control protocol is designed for the two subsystems by designing a double power-law approaching law and a nonlinear function, so that the robot formation can converge stably within a fixed time.

[0012] Step 5: Convergence proof of the formation control protocol: This is done in two steps: The stability of the two subsystems in Step 4 is verified by Lyapunov theory. The control protocol designed in this invention enables the NMMR system to form the desired formation within a fixed time T≤T1+T2+T3.

[0013] Preferably, the mobile robot model, for an NMMR system, considers n robots as followers and one robot numbered 0 as the navigator; for the i-th mobile robot, its dynamic model can be established as follows:

[0014]

[0015] Where, i∈Γ∪{0},(x i ,y i )∈R 2 θ represents the Cartesian coordinates of the centroid of the i-th robot. i ∈R represents the deflection angle, v i ∈R, w i ∈R represent linear velocity and angular velocity, respectively.

[0016] Preferably, the task description is to implement a specified formation, defining a formation control matrix P = [P1, P2, ..., P...]. j ,...,P n ] T , 1≤j≤n, where P j =(p jx ,p jy Let be the expected position vector of the following robot j relative to the virtual navigator; if equation (2) holds, then the NMMR system will achieve the specified formation task within a fixed time.

[0017]

[0018] Among them, (x j ,y j ,θ j ), (x0, y0, θ0) represent the poses of the follower and the navigator in the global coordinate system, respectively.

[0019] Preferably, the formation control problem is defined by an NMMR coordinate transformation, as follows:

[0020]

[0021] Where vector P j =(p jx ,p jy From this diagram, the coordinates of the virtual navigator corresponding to the follower in the local coordinate system X'O'Y' can be obtained as follows:

[0022]

[0023] Considering the slight sideslip effect during the robot's actual movement, an interference variable δ=-k0sign(w) is added to equation (3). i )z 3iIf k0 > 0, then the robot coordinate transformation can be defined as:

[0024]

[0025] Where, i∈Γ∪{0}, P0=(p 0x ,p 0y )=(0,0), therefore the dynamic model of the robot can be defined as:

[0026]

[0027] Note 1: For system (6), if a suitable control law u exists j =[u 1j ,u 2j ] T If equation (7) holds, the NMMR system can converge within a fixed time, thus achieving the objective.

[0028]

[0029] Where ρ∈{1,2,3},T is the convergence time function, and the value of T is independent of the robot's initial pose state, but only related to the preset parameters of the control protocol;

[0030] Note 2: Due to the nonlinear characteristics of NMMR systems, traditional linear control theory is difficult to apply to NMMR uniform formation tracking control. This invention uses classical variable transformation to linearize the NMMR system, transforming it into a type of linear chain system. Therefore, the NMMR formation tracking control problem is transformed into a uniform control problem with new variables.

[0031] Preferably, the design of the control protocol is carried out in two steps. First, to facilitate the analysis of the consistency problem between the leader and the follower in the robot dynamics system, an error tracking system for the leader and the follower is defined. Second, the system is decomposed into two subsystems, and then the control protocols are designed for each subsystem.

[0032] Define the tracking error of the following robot as

[0033]

[0034] Differentiating equation (8) and substituting equation (6) into it, we get:

[0035]

[0036] To facilitate the design of the control protocol, equation (9) is decomposed into a first-order subsystem equation (10) and a second-order subsystem equation (11), as shown below;

[0037]

[0038]

[0039] According to the above subsystem design, the follower control law u 1j and u 2j are designed to track the corresponding leader within a fixed time; the design is divided into two parts: Step1: Make z 1j stable; Step2: Make z 2j and z 3j stable;

[0040] Step1: Design the control law u 1j as follows:

[0041]

[0042] where α, β > 0, p > 1, 0 < q < 1 are preset constants, and α ∈ (0, 1);

[0043] Step2: Make z 2j and z 3j stable; It can be seen from Step1 that can converge to 0 at a fixed time T1. When t ≥ T1, then Equation (11) can be written as:

[0044]

[0045] Define the double power reaching law (14), (15), (16) and the nonlinear function (17)

[0046]

[0047]

[0048]

[0049]

[0050] where α η , β η > 0, η ∈ {2, 3, 4}, p > 1, 0 < q < 1;

[0051] Design the control law u 2j as follows

[0052]

[0053] Preferably, for the stability proof of the control protocol, Theorem 1 states that for the first-order subsystem Equation (10), under the control law Equation (12), there is Specifically, for any t > T1, we have z 1j = z 10 , where T1 is a fixed time, and we have:

[0054]

[0055] Proof: Substitute Equation (12) into Equation (10), and we get:

[0056]

[0057] Let Construct the Lyapunov function:

[0058]

[0059] Take the derivative of V1 and combine it with Equation (19) to get

[0060]

[0061] Furthermore, given ξ1, ξ2,..., ξ N ≥ 0, then

[0062]

[0063]

[0064] We can obtain

[0065]

[0066] Since: If there exists a continuous radially unbounded positive definite function V(x), and there exist α, β > 0, p > 1, 0 < q < 1 such that the following holds:

[0067]

[0068] Then the equilibrium point of the system that converges in fixed time is the origin, and the convergence time function satisfies

[0069]

[0070] Therefore

[0071]

[0072] From this, we can know can converge to 0 in fixed time T1. Then, when t ≥ T1, satisfies Equation (7).

[0073] 7. Preferably, under the conditions of satisfying assumptions 1 and 2, Theorem 2, for the second-order subsystems equations (10) and (11), under the control laws equations (12) and (18), has: In particular, for any t ≥ T1 + T2 + T3, we have z 1j =z 10 , z 2j =z 20 , z 3j =z 30 Where T1, T2, and T3 are fixed times, we have:

[0074]

[0075]

[0076]

[0077] Proof: Taking the derivative of equation (17), we get

[0078]

[0079] Substituting equation (13) into equation (26), we get

[0080]

[0081] Then the control law u 2j Substituting into the above equation and combining it with the double power approach law (14), we can obtain

[0082]

[0083] Next, let K = [κ1,κ2,...,κ] j ] T Construct Lyapunov functions

[0084]

[0085] Differentiating V2 and combining it with equation (28) yields

[0086]

[0087] Given ξ1, ξ2, ..., ξ N ≥0, then

[0088]

[0089]

[0090] achievable

[0091]

[0092]

[0093] Therefore, κ j It is stable at a fixed time, and its upper bound on the convergence time is T2, that is, when t≥T2, κ j =0, substituting it into the nonlinear function (17), and combining it with the double power approach law (15), we have

[0094]

[0095] Substituting equation (33) into equation (13), we get

[0096]

[0097] Similar to the proof above, let Constructing Lyapunov functions

[0098]

[0099] Differentiate V3 and combine it with equation (34), then use the given ξ1,ξ2,…,ξ N ≥0, then

[0100]

[0101]

[0102] achievable

[0103]

[0104]

[0105] thereby, It is stable at a fixed time, and its upper bound on the convergence time is T3, that is, when t≥T3,

[0106] When t≥T1 When t≥T2, κ j =0; when t≥T3, Then, when t≥T1+T2+T3, combining with equation (27), we can obtain The designed control protocol enables the NMMR system to form the desired formation within a fixed time, with the upper bound of the convergence time being T≤T1+T2+T3.

[0107] Preferably, the navigator's angular velocity w0 is set to 1 rad / s, and its linear velocity v0 is set to 5 m / s. The control protocol parameters are α1=α2=α3=1, β1=β2=β3=1, p1=p2=p3=1.5, q1=q2=q3=0.5; the formation control matrix P is:

[0108]

[0109] Compared with the prior art, the beneficial effects of this invention are as follows: First, a robot dynamics model is defined through coordinate transformation, and a disturbance factor is introduced into the model. Second, a leader-follower strategy is adopted to transform the formation control problem of the system into a simple consistency tracking problem, and the error tracking system is divided into two subsystems. Third, by designing a double power-law approaching law and a nonlinear function, a fixed-time consistency control protocol is designed for the two subsystems, so that the robot formation converges stably within a fixed time. Finally, the stability of the control protocol is verified by Lyapunov theory.

[0110] By designing a double power-law approaching the system and a nonlinear function, a fixed-time consistency control protocol was designed for the two subsystems, enabling the robot formation to converge stably within a fixed time. The double power-law approaching the system, the nonlinear function, and the fixed-time consistency control protocol are all involved in this design.

[0111] Of course, any product implementing this invention does not necessarily need to achieve all of the advantages described above at the same time. Attached Figure Description

[0112] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0113] The results of the first set of experimental figures are as follows:

[0114] Figure 1 The trajectory of the mobile robot in the X direction is shown in Algorithm A of this invention;

[0115] Figure 2 The B algorithm of this invention represents the trajectory of the mobile robot in the X direction.

[0116] Figure 3 The C algorithm of this invention provides the trajectory of the mobile robot in the X direction.

[0117] Figure 4 The Y-direction trajectory of the mobile robot is represented by Algorithm A of this invention;

[0118] Figure 5 The trajectory of the mobile robot in the Y direction is shown in the B algorithm of this invention;

[0119] Figure 6 The C algorithm of this invention represents the trajectory of the mobile robot in the Y direction.

[0120] Figure 7 The trajectory of the mobile robot's heading angle using Algorithm A of this invention;

[0121] Figure 8 The B algorithm of this invention provides the heading angle trajectory of the mobile robot.

[0122] Figure 9 The C algorithm of this invention provides the heading angle trajectory of the mobile robot.

[0123] The results of the second set of experimental figures are as follows:

[0124] Figure 10 The trajectory of the mobile robot in the X direction is shown in Algorithm A of this invention;

[0125] Figure 11 The B algorithm of this invention represents the trajectory of the mobile robot in the X direction.

[0126] Figure 12 The C algorithm of this invention provides the trajectory of the mobile robot in the X direction.

[0127] Figure 13 The Y-direction trajectory of the mobile robot is represented by Algorithm A of this invention;

[0128] Figure 14 The trajectory of the mobile robot in the Y direction is shown in the B algorithm of this invention;

[0129] Figure 15 The C algorithm of this invention represents the trajectory of the mobile robot in the Y direction.

[0130] Figure 16 The trajectory of the mobile robot's heading angle using Algorithm A of this invention;

[0131] Figure 17 The B algorithm of this invention provides the heading angle trajectory of the mobile robot.

[0132] Figure 18 The C algorithm of this invention represents the heading angle trajectory of a mobile robot. Detailed Implementation

[0133] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0134] Please see Figures 1 to 18This invention provides a technical solution for a fixed-time consistency control method for nonholonomic multi-mobile robots considering external disturbances, comprising the following steps:

[0135] Step 1: Mobile Robot Model: Define the navigator robot and follower robots, and establish a mathematical model for the mobile robots. For the NMMR system, this model considers n robots as followers and one robot numbered 0 as the navigator. For the i-th mobile robot, its dynamic model can be established as follows:

[0136]

[0137] Where, i∈Γ∪{0},(x i ,y i )∈R 2 θ represents the Cartesian coordinates of the centroid of the i-th robot. i ∈R represents the deflection angle, v i ∈R, w i ∈R represent linear velocity and angular velocity, respectively;

[0138] Step Two: Task Description: Introduce a virtual navigator and provide a mathematical model for achieving a specified formation. The task description is to achieve a specified formation, defining the formation control matrix P = [P1, P2, ..., P...]. j ,...,P n ] T , 1≤j≤n, where P j =(p jx ,p jy Let be the expected position vector of the following robot j relative to the virtual navigator; if equation (2) holds, then the NMMR system will achieve the specified formation task within a fixed time.

[0139]

[0140] Among them, (x j ,y j ,θ j ), (x0, y0, θ0) represent the poses of the follower and the navigator in the global coordinate system, respectively.

[0141] Step 3: Formation Problem Transformation: Considering the influence of external disturbances, a disturbance factor is introduced into the model. A new robot dynamics model is constructed through robot coordinate transformation, thereby transforming the NMMR formation tracking control problem into a consistency control problem with new variables. The formation control problem transformation defines the NMMR coordinate transformation as follows:

[0142]

[0143] Where vector P j=(p jx ,p jy From this diagram, the coordinates of the virtual navigator corresponding to the follower in the local coordinate system X'O'Y' can be obtained as follows:

[0144]

[0145] Considering the slight sideslip effect during the robot's actual movement, an interference variable δ=-k0sign(w) is added to equation (9). i )z 3i If k0 > 0, then the robot coordinate transformation can be defined as:

[0146]

[0147] Where, i∈Γ∪{0}, P0=(p 0x ,p 0y )=(0,0), therefore the dynamic model of the robot can be defined as:

[0148]

[0149] Note 1: For system (6), if a suitable control law u exists j =[u 1j ,u 2j ] T If equation (7) holds, the NMMR system can converge within a fixed time, thus achieving the objective.

[0150]

[0151] Where ρ∈{1,2,3},T is the convergence time function, and the value of T is independent of the robot's initial pose state, but only related to the preset parameters of the control protocol;

[0152] Note 2: Due to the nonlinear characteristics of NMMR systems, traditional linear control theory is difficult to apply to NMMR uniform formation tracking control. This invention uses classical variable transformation to linearize the NMMR system, transforming it into a type of linear chain system. Therefore, the formation tracking control problem of NMMR is transformed into a uniform control problem with new variables.

[0153] Step 4: The design of the formation control protocol is carried out in two steps: First, to facilitate the analysis of the consensus problem between the leader and the follower of the robot dynamic system, an error tracking system for the leader and the follower is defined; Second, the system is decomposed into two subsystems, and by designing a double power reaching law and a nonlinear function, a fixed-time consensus control protocol is designed for the two subsystems, so that the robot formation converges stably within a fixed time. The design of the control protocol is carried out in two steps. First, to facilitate the analysis of the consensus problem between the leader and the follower of the robot dynamic system, an error tracking system for the leader and the follower is defined; Second, the system is decomposed into two subsystems, and then the control protocols are designed respectively.

[0154] Define the tracking error of the follower robot as

[0155]

[0156] Taking the derivative of Equation (8) and substituting Equation (6) into it, we can get:

[0157]

[0158] For the convenience of control protocol design, Equation (9) is decomposed into a first-order subsystem Equation (10) and a second-order subsystem Equation (11), as follows:

[0159]

[0160]

[0161] Design the follower control law u 1j and u 2j , so that it can track the corresponding leader within a fixed time; The design is divided into two parts: Step1: Make z 1j stable; Step2: Make z 2j and z 3j stable;

[0162] Step1: Design the control law u 1j as follows:

[0163]

[0164] where, α, β > 0, p > 1, 0 < q < 1 are preset constants, and α ∈ (0, 1);

[0165] Step2: Make z 2j and z 3j stable; It can be seen from Step1 that can converge to 0 within a fixed time T1. When t ≥ T1, Then Equation (11) can be written as:

[0166]

[0167] Define the double power reaching laws (14), (15), (16) and the nonlinear function (17).

[0168]

[0169]

[0170]

[0171]

[0172] Where, α η ,β η >0, η∈{2,3,4}, p>1, 0<q<1;

[0173] Design control law u 2j as follows

[0174]

[0175] Step 5: Convergence proof of the formation control protocol: This is done in two steps: the stability of the two subsystems in Step 4 is verified by Lyapunov theory; the control protocol designed in this invention enables the NMMR system to form the desired formation within a fixed time T≤T1+T2+T3. The stability proof of the control protocol is as follows: Theorem 1, under the conditions of satisfying Assumption 1 and Assumption 2, for the first-order subsystem equation (10), under the control law equation (12), we have In particular, for any t > T1, we have z 1j =z 10 Where T1 is a fixed time, we have:

[0176]

[0177] Proof: Substituting equation (12) into equation (10), we get:

[0178]

[0179] make Constructing Lyapunov functions:

[0180]

[0181] Differentiating with respect to V1 and combining with equation (19) yields

[0182]

[0183] Again, given ξ1, ξ2,..., ξ N ≥0, then

[0184] [[ID=,5]]

[0185]

[0186] it can be obtained that

[0187]

[0188] Since: If there exists a continuously radially unbounded positive definite function V(x), there exist α, β > 0, p > 1, 0 < q < 1 such that the following holds:

[0189]

[0190] then the equilibrium point of the system that converges in fixed time is the origin, and the convergence time function satisfies

[0191]

[0192] So

[0193]

[0194] From this, it can be known that it can converge to 0 in fixed time T1. Then, when t ≥ T1, it satisfies equation (7). Under the conditions of satisfying Assumption 1 and Assumption 2, for the second-order subsystems in equations (10) and (11), under the control laws in equations (12) and (18), there is In particular, for any t ≥ T1 + T2 + T3, there is z 1j = z 10 z 2j = z 20 z 3j = z 30 where T1, T2, and T3 are fixed times, and there is:

[0195]

[0196]

[0197]

[0198] Proof: Differentiating equation (17), it can be obtained that

[0199]

[0200] Substituting equation (13) into equation (26), it can be obtained that

[0201]

[0202] Then the control law u 2j Substituting into the above equation and combining it with the double power approach law (14), we can obtain

[0203]

[0204] Next, let K = [κ1,κ2,...,κ] j ] T Construct Lyapunov functions

[0205]

[0206] Differentiating V2 and combining it with equation (28) yields

[0207]

[0208] Given ξ1, ξ2, ..., ξ N ≥0, then

[0209]

[0210]

[0211] achievable

[0212]

[0213]

[0214] Therefore, κ j It is stable at a fixed time, and its upper bound on the convergence time is T2, that is, when t≥T2, κ j =0, substituting it into the nonlinear function (17), and combining it with the double power approach law (15), we have

[0215]

[0216] Substituting equation (33) into equation (13), we get

[0217]

[0218] Similar to the proof above, let Constructing Lyapunov functions

[0219]

[0220] Differentiate V3 and combine it with equation (34), then use the given ξ1,ξ2,...,ξ N ≥0, then

[0221]

[0222]

[0223] achievable

[0224]

[0225]

[0226] thereby, It is stable at a fixed time, and its upper bound on the convergence time is T3, that is, when t≥T3,

[0227] When t≥T1 When t≥T2, κ j =0; when t≥T3, Then, when t≥T1+T2+T3, combining with equation (27), we can obtain The designed control protocol enables the NMMR system to form the desired formation within a fixed time, with the upper bound of the convergence time being T≤T1+T2+T3.

[0228] Set the navigator's angular velocity w0 to 1 rad / s, linear velocity v0 to 5 m / s, and control protocol parameters α1=α2=α3=1, β1=β2=β3=1, p1=p2=p3=1.5, q1=q2=q3=0.5; formation control matrix P is:

[0229]

[0230] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0231] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims

1. A fixed-time consistency control method for nonholonomic multi-mobile robots considering external disturbances, characterized in that, Includes the following steps: Step 1: Mobile Robot Model: Define the navigator robot and the follower robot, and establish a mathematical model of the mobile robot; Step 2: Task Description: Introduce a virtual navigator and provide a mathematical model for achieving a specified formation; Step 3: Formation Problem Transformation: Considering the influence of external disturbances, a disturbance factor is introduced into the model. A new robot dynamics model is constructed through robot coordinate transformation, thereby transforming the formation tracking control problem of NMMR into a consistency control problem with new variables. Step 4: The design of the formation control protocol is carried out in two steps: First, to facilitate the analysis of the consistency problem between the leader and the follower in the robot dynamics system, an error tracking system for the leader and the follower is defined; Second, the system is decomposed into two subsystems, and a fixed-time consistency control protocol is designed for the two subsystems by designing a double power-law approaching law and a nonlinear function, so that the robot formation can converge stably within a fixed time. Step 5: Convergence Proof of the Formation Control Protocol: This is done in two steps: The stability of the two subsystems from Step 4 is verified using Lyapunov theory; the designed control protocol enables the NMMR system to converge at a fixed time. The desired formation is formed within; The mobile robot model, for an NMMR system, considers n robots as followers and one robot numbered 0 as the navigator; for the i-th mobile robot, its dynamic model can be established as follows: (1) in, , The Cartesian coordinates representing the centroid of the i-th robot. Indicates the deflection angle. , These are linear velocity and angular velocity, respectively. The task description is to implement a specified formation and define a formation control matrix. , ,in To follow the robot j The expected position vector relative to the virtual navigator; if equation (2) holds, the NMMR system achieves the specified formation task within a fixed time. (2) in, , These represent the poses of the follower and navigator in the global coordinate system, respectively. The formation control problem is defined by the NMMR coordinate transformation as follows: (3) Where vector Therefore, it can be concluded that in the local coordinate system The coordinates of the virtual navigator corresponding to the follower are as follows: (4) Considering the slight sideslip effect during the robot's actual movement, an interference variable is added to equation (3). , k 0 If the coordinates are greater than 0, then the robot coordinate transformation can be defined as: (5) in, , Therefore, the dynamic model of the robot can be defined as: (6) For system (6), if a suitable control law exists If equation (7) holds, the NMMR system can converge within a fixed time, thus achieving the objective. (7) in, T is the convergence time function. The value of T is independent of the robot's initial pose state and is only related to the preset parameters of the control protocol. Due to the nonlinear characteristics of NMMR systems, traditional linear control theory is difficult to apply to NMMR uniform formation tracking control. By using classical variable transformation, the NMMR system is linearized into a class of linear chain systems. Therefore, the formation tracking control problem of NMMR is transformed into a uniform control problem with new variables. The design of the control protocol is carried out in two steps. First, to facilitate the analysis of the consistency problem between the leader and the follower in the robot dynamics system, an error tracking system for the leader and the follower is defined. Second, the system is decomposed into two subsystems, and then the control protocols are designed for each subsystem. Define the tracking error of the following robot as (8) Differentiating equation (8) and substituting equation (6) into it, we get: (9) To facilitate the design of the control protocol, equation (9) is decomposed into a first-order subsystem equation (10) and a second-order subsystem equation (11), as shown below; (10) (11) Based on the above subsystem design, a follower control law is implemented. and This allows it to track the corresponding leader within a fixed time period; the design consists of two parts: Step 1: [The text abruptly ends here, likely due to an incomplete sentence or a formatting error.] z 1j Stable; Step 2: Make z 2j and z 3j Stablize; Step 1: Design the control law as follows: (12) in, , p >1, 0< q <1 is a preset constant. ; Step 2: Make z 2j and z 3j Stable; as can be seen from Step 1 Able to do at a fixed time T 1 Converges to 0 when hour, Then equation (11) can be written as: (13) Define the double power reaching laws (14), (15), (16) and the nonlinear function (17). (14) (15) (16) (17) in, , ; Design control law as follows (18)。 2. The fixed-time consistency control method for nonholonomic multi-mobile robots considering external disturbances according to claim 1, characterized in that, Proof of the stability of the control protocol: Theorem 1, under the conditions of Assumptions 1 and 2, for the first-order subsystem equation (10), under the control law equation (12), we have: In particular, for any ,have ,in T 1 It is a fixed time, including: ； Proof: Substituting equation (12) into equation (10), we get: (19) make Construct the Lyapunov function: (20) right Differentiate and combine with equation (19) to obtain (21) Then from the given ,So ； achievable (22) Because: if there exists a continuous radially unbounded positive definite function V(x) ,exist , p >1, 0< q <1, which makes the following equation true: (23) Therefore, the equilibrium point where the system converges in a fixed time is the origin, and the convergence time function satisfies (24) so (25) Therefore, we can know Able to complete in a fixed time T 1 If it converges to 0, then when hour, , satisfying equation (7).

3. The fixed-time consistency control method for nonholonomic multi-mobile robots considering external disturbances according to claim 2, characterized in that, Theorem 2 Under the conditions of satisfying Assumption 1 and Assumption 2, for the second-order subsystems equations (10) and (11), under the control laws equations (12) and (18), we have , , In particular, for any ,have , , ,in T 1 , T 2 , T 3 It is a fixed time, including: ； ； ； Proof: Differentiating equation (17), we get (26) Substituting equation (13) into equation (26), we get (27) Then the control law Substituting into the above equation and combining it with the double power approach law (14), we can obtain (28) Next, let Construct Lyapunov functions (29) right Differentiate and combine with equation (28) to obtain (30) Then from the given ,So ； achievable (31) (32) Therefore, It stabilizes at a fixed time, and its upper bound on the convergence time is: T 2 That is, when hour, Substituting this into the nonlinear function (17) and combining it with the double power approach law (15), we have (33) Substituting equation (33) into equation (13), we get (34) Similar to the proof above, let Construct Lyapunov functions (35) right Differentiate, and combine with equation (34), then use the given... ,So ； achievable (36) (37) thereby, It stabilizes at a fixed time, and its upper bound on the convergence time is: T 3 That is, when hour, ; when hour, ;when hour, ;when hour, ; then, when When combined with equation (27), we can obtain The designed control protocol enables the NMMR system to form the desired formation within a fixed time, with an upper bound on the convergence time. .

4. The fixed-time consistency control method for nonholonomic multi-mobile robots considering external disturbances according to claim 3, characterized in that, Set the navigator's angular velocity for linear velocity Set as Control protocol parameters , , , The formation control matrix P is: 。

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