A preset time formation control method of input delay multi-omni-directional mobile robots

Abstract

The application discloses a preset time formation control method for input delay multi-universal mobile robots, and belongs to the field of adaptive control of universal mobile robots. The control method comprises the following steps: establishing a dynamic model of multi-universal mobile robots with input delay, unknown nonlinear terms and external disturbance; introducing auxiliary vectors based on Padé approximation and Laplace transformation to simplify input delay analysis, and reconstructing the dynamic model based on the auxiliary variables; combining dynamic surface control technology to construct a preset time filter; step-by-step constructing Lyapunov functions to obtain an adaptive law and a preset time formation controller, wherein the preset time formation controller is used to output a control vector, the control vector is input into the reconstructed dynamic model, and formation control is realized. The preset time formation control method effectively suppresses the influence of input delay on the dynamic characteristics of the system, ensures the real-time performance of the control signal, and also ensures that each robot completes the formation task within the preset time.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of adaptive control of omni-directional mobile robots (OMRs), and particularly relates to a preset time formation control method for a multi-OMR model with input delay. BACKGROUND

[0002] In recent years, with the rapid evolution of intelligent control fields such as unmanned system control technology and robot formation coordination technology, adaptive control of omni-directional mobile robots has gradually become the focus of academia. However, with the continuous breakthroughs in control technology, a single omni-directional mobile robot is limited by its own hardware resources and processing capacity, and it is difficult to independently complete the target task when dealing with complex task requirements. Under this background, multi-omni-directional mobile robot formation control technology emerged as the times require, through the fusion of intelligent communication protocols and distributed control algorithms, the robots in the formation can dynamically adjust the formation according to the task requirements and accurately execute the instructions, which shows broad application potential in high-precision operation, complex environment tasks and other scenarios.

[0003] The core of omni-directional mobile robot formation control is to build an adaptive fuzzy control framework to realize stable motion of the formation system and adapt to environmental uncertainties. Specifically, the control objective is to design a controller with adaptivity to ensure the semi-global uniform ultimate boundedness of the closed-loop system signals, while making the system output effectively track the reference signal. Current mainstream control strategies include radial basis neural networks, fuzzy logic systems and PID control, etc. However, the existing schemes still face the following technical challenges:

[0004] First, in the existing adaptive formation control methods of omni-directional mobile robots, the influence of nonlinear dynamics and external disturbances on the system is not considered, and the convergence time problem of multi-omni-directional mobile robot formation control is not considered. As a key indicator for evaluating the dynamic performance of a control system, convergence time is of key significance in engineering fields such as industrial automation and robot control. In actual engineering applications, the control system usually needs to converge quickly to a stable state within a limited time to meet the performance requirements of task real-time, operation reliability, etc.

[0005] Second, in the existing adaptive formation control methods of omni-directional mobile robots, the system input is usually assumed to respond in real time, and there is a lack of effective modeling and compensation for the key non-ideal factor of input delay. In actual engineering scenarios, the input delay problem of omni-directional mobile robots is multi-source and complex, and the existence of input delay will further amplify the phase difference of dynamic response, leading to decreased trajectory tracking accuracy, accumulated formation keeping error, and even stability deterioration of the closed-loop system. SUMMARY

[0006] In order to solve the problems in the prior art, the present application provides a preset time formation control method for multi-omni-directional mobile robots with input delay, which effectively suppresses the influence of input delay on the dynamic characteristics of the system, and ensures that each robot completes the formation task within the preset time, thereby improving the precision and robustness of the formation control.

[0007] The preset time formation control method for multi-omni-directional mobile robots with input delay comprises the following steps:

[0008] A dynamic model of multi-omni-directional mobile robots with input delay, unknown nonlinear terms and external disturbances is established;

[0009] An auxiliary vector is introduced based on Padé approximation and Laplace transform to simplify the analysis of input delay, and the dynamic model is reconstructed based on the auxiliary vector;

[0010] A preset time filter is constructed in combination with dynamic surface control technology;

[0011] Lyapunov functions are constructed step by step to obtain an adaptive law and a preset time formation controller, and the preset time formation controller is used to output a control vector, which is input to the reconstructed dynamic model to realize formation control.

[0012] Further, the dynamic model of multi-omni-directional mobile robots with input delay, unknown nonlinear terms and external disturbances is established, the omni-directional mobile robot is a Mecanum wheel robot, and the dynamic model of multi-omni-directional mobile robots is modeled for a follower multi-omni-directional mobile robot, and is expressed as follows:

[0013]

[0014] The posture state of the i-th omni-directional mobile robot is represented as:

[0015] The velocity state of the i-th omni-directional mobile robot is represented as:

[0016] x wi and y wi represent the coordinate values of the i-th omni-directional mobile robot in the X direction and the Y direction of the world coordinate system, respectively;

[0017] The angle between the positive direction of the X axis of the world coordinate system of the i-th omni-directional mobile robot and the positive direction of the X R axis of the robot coordinate system is represented as:

[0018] y i represents the output vector of the i-th omni-directional mobile robot;

[0019] τi (t - i) = [τ i1 (t - i) τ i2 (t - i) τ i3 (t - i) τ i4 (t - i)] T is the input control vector of the system;

[0020] τ i1 (t - i), τ i2 (t - i), τ i3 (t - i) and τ i4 (t - i) represent the input control vector of the four wheels of the omni-directional mobile robot, respectively;

[0021] t represents time; i denotes the input delay term;

[0022] f i,1 (x i ) and f i,2 (x i , v i ) represent unknown nonlinear terms;

[0023] Λ i,1 (t, x i ) and Λ i,2 (t, x i , v i ) represent unknown external disturbances;

[0024] B i (x i , v i ) and F i (x i , v i ) represent the control input gain matrix and the nonlinear terms inherent to the omni-directional mobile robot, respectively.

[0025] Mathematical expressions of B i (x i , v i ) and F i (x i , v i ) are as follows:

[0026]

[0027] is a 3 x 3 matrix representing the transformation matrix between the world coordinate system and the robot coordinate system;

[0028] is a 3 x 4 matrix representing the Jacobian matrix of the forward kinematics model of the omni-directional mobile robot;

[0029] is a 4x3 matrix representing the Jacobian matrix of the inverse kinematics model of the omni-directional mobile robot;

[0030] is a 4x4 matrix representing the gain matrix of the angular acceleration of the Mecanum wheel;

[0031] is a 4x4 matrix representing the inverse matrix of

[0032] represents the static friction force experienced by the Mecanum wheel;

[0033] represents the angular velocity of each wheel of the omni-directional mobile robot;

[0034] represents the angular acceleration of the Mecanum wheel;

[0035] R represents the radius of the Mecanum wheel;

[0036] is a 3x3 matrix representing the first derivative of the inverse matrix of

[0037] D θ represents the viscous friction coefficient of the Mecanum wheel.

[0038] Further, the auxiliary vector is a variable υ i , which is represented as follows:

[0039]

[0040] wherein, τ i represents the actual control input;

[0041] The dynamics model is reconstructed based on the auxiliary vector as follows:

[0042]

[0043] Further, the preset time filter is represented by the following differential equation:

[0044]

[0045] η∈(0,1) is a design parameter;

[0046] ρ and are design parameters;

[0047] z i,2 is a three-dimensional vector representing the filtering error;

[0048] ​​φ is a filter design parameter;

[0049] T p is a predetermined time;

[0050] c i,2 is a positive constant;

[0051] α i,1 is a virtual controller;

[0052] α i,1 (0) represents the initial value of the virtual controller α i,1 .

[0053] Further, the Lyapunov function is constructed step by step, specifically as follows:

[0054] The first construction of the Lyapunov function can be expressed in the following form:

[0055]

[0056] N represents the number of omnidirectional mobile robots;

[0057] s i,1 is a three-dimensional vector representing the tracking error;

[0058] is the estimation error of the ideal adjustment scalar Θ i,1 and its estimated value Θ

[0059] The second construction of the Lyapunov function can be expressed in the following form:

[0060]

[0061] s i,2 is a three-dimensional vector representing the error surface, z i,2 is a three-dimensional vector representing the filtering error;

[0062] is the estimation error of the ideal adjustment scalar Θ i,2 and its estimated value Θ

[0063] Further, the first construction of the adaptive law is expressed in the following form:

[0064]

[0065] wherein Θ i,1 is the estimated value of the ideal adjustment scalar ;

[0066] The second construction of the adaptive law is expressed in the following form:

[0067]

[0068] where, Θ i,2 is the estimated value of ideal adjustment scalar , λ = 2 + η, λ and are design parameters, s i,1 represents tracking error, s i,2 represents error surface;

[0069]

[0070] a ij represents the communication between the i-th omnidirectional mobile robot and the j-th omnidirectional mobile robot, if there is information transmission between the two omnidirectional mobile robots, a ij = 1, otherwise a ij = 0;

[0071] b i represents the communication between the leader and the i-th omnidirectional mobile robot, if there is information transmission, b i = 1, otherwise b i = 0;

[0072] and are the output vectors of the radial basis neural network.

[0073] Further, the preset time formation controller is represented as:

[0074]

[0075] c i,3 is a positive design parameter.

[0076] Further, based on the backstepping recursive framework, the unknown nonlinear term is approximated by using the radial basis function neural network, and the Lyapunov function is obtained.

[0077] Compared with the prior art technology, the present application has the following beneficial effects:

[0078] First, most of the existing omnidirectional mobile robot adaptive formation control methods do not consider the influence of nonlinear dynamics and external disturbances on the system, and also do not pay attention to the convergence time problem in multi-robot formation control. The present application comprehensively considers the influence of nonlinear dynamics and external disturbances, and proposes an adaptive pre-defined time formation control strategy based on neural network. The strategy can effectively deal with nonlinear dynamics and external disturbances, and ensure that the system completes the formation task within the preset time, thereby improving the efficiency of the system.

[0079] Second, most existing adaptive formation control methods for omnidirectional mobile robots assume that the system input can be ideally and instantaneously responsive, but lack effective modeling and compensation for the key non-ideal factor of input delay. To solve the common input delay problem in practical engineering applications, the invention studies the adaptive formation control problem of the input delay multi-omnidirectional mobile robot model. By combining the Padé approximation method and introducing auxiliary vectors, the analysis complexity of the input delay system is effectively simplified. The designed control strategy can effectively overcome the adverse effects of input delay.

[0080] The method effectively suppresses the influence of input delay on the dynamic characteristics of the system, ensures the real-time performance of the control signal, and meets the semi-global uniformly ultimately boundedness by designing a preset time filter and an adaptive control strategy. With the help of neural network processing nonlinear dynamics and external disturbances, the Padé approximation technique is used to process input delay, and a Lyapunov function is constructed to prove energy dissipation. By using Backstepping recursive technology and dynamic surface control, the derivative of the system state error satisfies a certain inequality, ensuring convergence to a bounded residual set within a preset time, and the convergence time is independent of the initial condition. For any bounded initial condition, the state finally enters a bounded region dependent on the disturbance bound, thus meeting the semi-global uniformly ultimately boundedness, and the system state is stable to the equilibrium point within the preset time. BRIEF DESCRIPTION OF DRAWINGS

[0081] The present invention has Figure 9 Zhang, wherein:

[0082] Figure 1 is a simplified diagram of omnidirectional mobile robot motion analysis;

[0083] Figure 2 is a communication topology diagram of three omnidirectional mobile robots;

[0084] Figure 3 is a trajectory diagram of three omnidirectional mobile robots in the X-Y plane;

[0085] Figure 4 is a velocity state diagram of three omnidirectional mobile robots in the X direction;

[0086] Figure 5 is a velocity state diagram of three omnidirectional mobile robots in the Y direction;

[0087] Figure 6 is a tracking error diagram of three omnidirectional mobile robots in the X direction;

[0088] Figure 7 is a tracking error diagram of three omnidirectional mobile robots in the Y direction;

[0089] Figure 8is the input torque diagram of the 1st omni-directional mobile robot with 4 wheels;

[0090] Figure 9 is the input torque diagram of the 1st wheel of the 3rd omni-directional mobile robot. DETAILED DESCRIPTION

[0091] The application combines backstepping recursive technology and dynamic surface control technology, and provides a preset time formation control method of input delay multi-omni-directional mobile robots based on state feedback, realizes preset time stability of the system, offsets the influence of input delay on the system, and ensures that the multiple omni-directional mobile robots in the system complete the formation task within the preset time.

[0092] The preset time formation control method of input delay multi-omni-directional mobile robots based on state feedback provided by the application firstly selects a system composed of multiple omni-directional mobile robots as a control object, establishes a dynamic model with unknown nonlinear terms and unknown external disturbances, and can reconstruct the dynamic model through Laplace transform and Padé approximation method. In the framework of the backstepping recursive method, the unknown nonlinear terms in the dynamic model of the multiple omni-directional mobile robots are approximated by using a radial basis function neural network, a Lyapunov function is established, and a preset time filter is constructed by combining the dynamic surface control technology. The adaptive law of the input delay multi-omni-directional mobile robot model and the preset time formation controller can be obtained on the basis of the above work.

[0093] The omni-directional mobile robot is a Mecanum wheel type robot;

[0094] The preset time formation control method of input delay multi-omni-directional mobile robots comprises the following steps: A, establishing an input delay multi-omni-directional mobile robot control model

[0095] In the input delay multi-omni-directional mobile robot system, in order to distinguish multiple omni-directional mobile robots, each follower robot is numbered, and the dynamic model of the i-th omni-directional mobile robot is given as follows:

[0096]

[0097] represents the pose state of the i-th omni-directional mobile robot;

[0098] represents the speed state of the i-th omni-directional mobile robot;

[0099] x wi and y wi respectively represent the coordinate values of the i-th omni-directional mobile robot in the X direction and the Y direction of the world coordinate system;

[0100] This indicates that the positive direction of the world coordinate system X-axis of the i-th omnidirectional mobile robot is perpendicular to the robot coordinate system X. R The angle between the positive axes; the robot coordinate system is centered at the geometric center of the robot. R Let the robot's direction of motion be X. R The direction, with Y being perpendicular to the direction of the robot's movement. R direction.

[0101] y i This represents the output vector of the i-th omnidirectional mobile robot;

[0102] τ i (t-ι)=[τ i1 (t-ι) τ i2 (t-ι) τ i3 (t-ι) τ i4 (t-ι)] T It is the system's input control vector;

[0103] τ i1 (t-ι), τ i2 (t-ι), τ i3 (t-ι) and τ i4 (t-ι) represent the input control vectors for the four wheels of the omnidirectional mobile robot, respectively;

[0104] t represents time; ι represents the input delay term;

[0105] f i,1 (x i ) and f i,2 (x i ,v i ) represents an unknown nonlinear term;

[0106] Λ i,1 (t,x i ) and Λ i,2 (t,x i ,v i () indicates unknown external interference;

[0107] B i (x i ,v i ) and F i (x i ,v i The numbers and represent the control input gain matrix and the inherent nonlinear term of the omnidirectional mobile robot, respectively, which can be expressed by the following mathematical formula:

[0108]

[0109] It is a 3×3 matrix representing the transformation matrix between the world coordinate system and the robot coordinate system;

[0110] It is a 3×4 matrix, representing the Jacobian matrix of the forward kinematics model of the omnidirectional mobile robot;

[0111] It is a 4×3 matrix, representing the Jacobian matrix of the inverse kinematics model of the omnidirectional mobile robot;

[0112] It is a 4×4 matrix representing the gain matrix of Mecanum wheel angular acceleration;

[0113] It is a 4×4 matrix, representing The inverse matrix;

[0114] This represents the static friction force acting on the Mecanum wheel;

[0115] This represents the angular velocity of each wheel of the omnidirectional mobile robot;

[0116] Represents the angular acceleration of the Mecanum wheel;

[0117] R represents the radius of the Mecanum wheel;

[0118] It is a 3×3 matrix, representing The first derivative of the inverse matrix;

[0119] D θ This represents the coefficient of viscous friction of the Mecanum wheel.

[0120] For the sake of brevity, f will be used in the following description. i,1 (x i ), f i,2 (x i ,v i ), B i (x i ,v i ) and F i (x i ,v i ) are abbreviated as f i,1 f i,2 B i and F i .

[0121] B. Model Reconstruction

[0122] To solve the input delay problem and obtain the actual control input τ iIntroducing the Padé approximation technique helps reduce the analytical complexity of input-delay systems. The Padé approximation can be expressed as follows:

[0123]

[0124] Represents τ i The Laplace transform of;

[0125] δ represents the Laplace variable;

[0126] Define a new variable υ i as follows:

[0127]

[0128] According to the inverse Laplace transform, we can obtain:

[0129]

[0130] Represents the actual control input τ i The first derivative;

[0131] This represents the newly defined variable υ. i The first derivative;

[0132] make It can be obtained

[0133]

[0134] Based on the above transformation, the input delay multi-omnidirectional mobile robot system (1) can be rewritten as:

[0135]

[0136] C. Establishment of Lyapunov functions

[0137] For a second-order input-delay multi-omnidirectional mobile robot model, only two Lyapunov functions need to be constructed to obtain its adaptive law and preset time formation controller.

[0138] In multi-omnidirectional mobile robot systems, radial basis function neural networks (RBF neural networks) are used to approximate unknown nonlinear terms in the dynamic model. The approximation error generated in this process can serve as a key element in constructing the Lyapunov function. Specifically, tracking error, RBF neural network approximation error, and filtering error are constructed as positive definite functions, and their convergence is analyzed using backstepping recursion techniques, thereby proving the system stability. This method utilizes the universal approximation characteristic of RBF neural networks to handle system uncertainties, and combined with the structured design of error terms, provides a rigorous theoretical analysis framework for the preset-time stable control of multi-omnidirectional mobile robots.

[0139] The first construction of the Lyapunov function can be expressed in the following form:

[0140]

[0141] N represents the number of omnidirectional mobile robots. It is an ideal adjustment scalar Its estimated value Θ i,1 The estimation error.

[0142] s i,1 It is a three-dimensional vector representing the tracking error;

[0143]

[0144] a ij Let a represent the communication between the i-th follower robot and the j-th follower robot. If there is information exchange between the two robots, then a ij =1, if there is no communication between the two robots then a ij =0; κ i κ represents the positional bias vector of the i-th follower robot relative to the leader robot; j y represents the positional bias vector of the j-th follower robot relative to the leader robot; r This indicates a reference signal given by the leader; b i This represents communication between the leader robot and the follower robot; if there is direct communication between them, then b. i =1, if there is no direct communication then b i =0.

[0145] The second construction of the Lyapunov function can be expressed in the following form:

[0146]

[0147] It is an ideal adjustment scalar Its estimated value Θ i,2The estimation error;

[0148] s i,2 It is a three-dimensional vector representing the error surface;

[0149]

[0150] It is the output of the preset time filter.

[0151] z i,2 It is a three-dimensional vector representing the filtering error:

[0152]

[0153] D. Establishing a preset time filter

[0154] The preset time filter is represented by the following differential equation:

[0155]

[0156] η∈(0,1) is a design parameter;

[0157] ρ and These are design parameters;

[0158] φ is a filter design parameter;

[0159] T p It is the scheduled time;

[0160] c i,2 It is a positive constant;

[0161] α i,1 It is a virtual controller, and its specific form will be given later.

[0162] E. Establishment of Adaptive Law and Preset Time Formation Controller

[0163] The first construction of the adaptive law can be expressed in the following form:

[0164]

[0165] The second construction of the adaptive law can be expressed in the following form:

[0166]

[0167] λ = 2 + η, λ and It is a design parameter, s i,1 Represents the tracking error, s i,2 Indicates the error surface;

[0168]

[0169] a ij Let a represent the communication between the i-th omnidirectional mobile robot and the j-th omnidirectional mobile robot. If there is information transmission between the two omnidirectional mobile robots, then a ij =1, otherwise a ij =0; N i =1,2,…N,j≠i;

[0170] b i Let b represent the communication between the leader and the i-th omnidirectional mobile robot. If there is information transmission, then b i =1, otherwise b i =0;

[0171] and It is the output vector of the radial basis function neural network.

[0172] Based on previous work, the virtual controller α can be obtained. i,1 and preset time formation controller τ i as follows:

[0173]

[0174] c i,1 These are positive design parameters;

[0175] Θ i,1 It is an ideal adjustment scalar The estimated value;

[0176] and It is the position offset vector κ i and κ j The derivative;

[0177] It is the reference signal y r The derivative of .

[0178]

[0179] c i,3 These are positive design parameters;

[0180] Θ i,2 It is an ideal adjustment scalar The estimated value.

[0181] The simplified diagram of motion analysis for the omnidirectional mobile robot involved in this invention is shown below. Figure 1 As shown in the diagram. A schematic diagram of the communication topology between the three omnidirectional mobile robots involved in this invention is shown in the diagram. Figure 2As shown, "0" represents the navigator robot, and "1", "2", and "3" all represent robots in the multi-omnidirectional mobile robot system of this invention. Simulation results are as follows. Figures 3-9 As shown. Figure 3 The movement trajectories of three omnidirectional mobile robots in formation are displayed; Figure 4 The velocity state curves of the three omnidirectional robots in the X direction are shown; Figure 5 The velocity state curves of three omnidirectional mobile robots in the Y direction are shown; Figure 6 The output state tracking performance of three omnidirectional mobile robots in the X direction is shown, where s 1,1x s 2,1x and s 3,1x These represent the tracking errors of the three omnidirectional mobile robots in the X direction, respectively. Figure 7 The output state tracking performance of three omnidirectional mobile robots in the Y direction is shown, where s 1,1y s 2,1y and s 3,1y These represent the tracking errors of the three omnidirectional mobile robots in the X direction, respectively. Figure 8 The input torque of the four wheels of the first omnidirectional mobile robot is shown; Figure 9 The simulation results show the input torque of the wheels at the same position of the three omnidirectional mobile robots. It can be seen that a large input torque is required at the start, but then stabilizes. The simulation results also demonstrate that the multi-omnidirectional mobile robot system can complete the formation task within a preset time and maintain the formation. Furthermore, the velocity states of each omnidirectional mobile robot in the X and Y directions stabilize within the preset time, and the tracking errors of the output states of each omnidirectional mobile robot in the X and Y directions also stabilize within the preset time. The designed control scheme effectively counteracts the impact of input delay and achieves the expected preset time formation control effect.

[0182] This invention is not limited to this embodiment. Any equivalent concept or modification within the technical scope disclosed in this invention shall be included within the protection scope of this invention.

Claims

1. A preset time formation control method of an input delay multi-omni-directional mobile robot, characterized by, The method comprises the following steps: A multi-omni-directional mobile robot dynamics model with input delay, unknown nonlinear terms and external disturbance is established, the multi-omni-directional mobile robot dynamics model is modeling of a follower multi-omni-directional mobile robot, and is expressed as follows: represents the pose state of the i-th omnidirectional mobile robot; vi represents the velocity state of the i-th omnidirectional mobile robot; x wi and y wi Xi and yi represent the coordinate values of the i-th omnidirectional mobile robot in the X direction and the Y direction of the world coordinate system, respectively; represents the angle between the positive direction of the X-axis of the world coordinate system Xw and the X-axis of the i-th omnidirectional mobile robot R represents the angle between the positive direction of the X-axis of the world coordinate system Xw and the X-axis of the i-th omnidirectional mobile robot y i represents the output vector of the i-th omnidirectional mobile robot; τ i (t-ι) = [τ i1 (t-ι) τ i2 (t-ι) τ i3 (t-ι) τ i4 (t-ι)] T is the input control vector of the system; t i1 (t-ι), τ i2 (t-ι), τ i3 (t-ι) and τ i4 (t-ι) represent the input control vectors for the four wheels of the omni-directional mobile robot, respectively. t represents time; I represents an input delay term; f i,1 (x i ) and f i,2 (x i , v i ) represent unknown nonlinear terms; Λ i,1 (t,x i ) and Λ i,2 (t,x i ,v i ) represent unknown external disturbances; B i (x i ,v i ) and F i (x i ,v i ) represent the control input gain matrix and the non-linear term inherent to the omni-directional mobile robot itself, respectively; An auxiliary vector is introduced based on Padé approximation and Laplace transform to simplify input delay analysis, and the dynamics model is reconstructed based on the auxiliary vector; The auxiliary vector is the variable υ i is represented as follows: wherein τ i denotes the actual control input; The dynamics model is reconstructed based on the auxiliary vector as follows: A preset time filter is constructed in combination with dynamic surface control technology; Lyapunov functions are constructed step by step to obtain an adaptive law and a preset time formation control controller, and the preset time formation control controller is used to output a control vector, the control vector is input to the reconstructed dynamics model, and formation control is realized; The step-by-step construction of the Lyapunov function is as follows: The first construction of the Lyapunov function is expressed in the following form: N represents the number of omni-directional mobile robots; s i,1 is a three-dimensional vector representing the tracking error; is the ideal adjustment scalar with its estimate Θ i,1 the estimation error; The second construction of the Lyapunov function is expressed in the following form: s i,2 is a three-dimensional vector representing the error surface, z i,2 is a three-dimensional vector representing the filtered error; is the ideal adjustment scalar with its estimate Θ i,2 the estimation error; The first construction of the adaptive law is expressed in the following form: where η ∈ (0, 1) is a design parameter, ρ and are design parameters, T p is a predetermined time, Θ i,1 is an estimate of the ideal adjustment scalar . The second construction of the adaptive law is expressed in the following form: where Θ i,2 is an estimate of the ideal adjustment scalar , λ = 2 + η, λ and are design parameters, s i,1 denotes the tracking error, s i,2 denotes the error surface; a ij denotes the communication between the ith follower robot and the jth follower robot, a ij = 1 if there is information exchange between the two robots, and a ij = 0 if there is no communication between the two robots; y i denotes the position bias vector of the ith follower robot with respect to the leader robot; y j denotes the position bias vector of the jth follower robot with respect to the leader robot; y r denotes the reference signal given by the leader; b i denotes the communication between the leader robot and the follower robots, b i = 1 if there is direct communication between the two, and b i = 0 if there is no direct communication; y j denotes the output vector of the jth omnidirectional mobile robot; is the output of the preset time filter; and is an output vector of the neural network.

2. The preset time formation control method of the input delay omnidirectional mobile robot according to claim 1, wherein B i (x i ,v i ) and F i (x i ,v i ) are mathematically represented as follows: is a 3x3 matrix representing the transformation matrix between the world coordinate system and the robot coordinate system; is a 3x4 matrix representing the Jacobian matrix of the forward kinematics model of the omnidirectional mobile robot; is a 4x3 matrix representing the Jacobian matrix of the inverse kinematics model of the omnidirectional mobile robot; is a 4x4 matrix representing the gain matrix of the Mecanum wheel angular acceleration; is a 4x4 matrix representing the inverse matrix of This represents the static friction force acting on the Mecanum wheel; This represents the angular velocity of each wheel of the omnidirectional mobile robot; Represents the angular acceleration of the Mecanum wheel; R represents the radius of the Mecanum wheel; is a 3x3 matrix representing the first derivative of the inverse matrix of D θ denotes the viscous friction coefficient of the gimbals.

3. The pre-set time formation control method of the input-delayed multi-omni-directional mobile robot according to claim 1, wherein, The preset time filter is expressed by the following differential equation: η ∈ (0, 1) is a design parameter; p and are design parameters; z i,2 is a three-dimensional vector representing the filtering error; φ is a filter design parameter; T p is a predetermined time; c i,2 is a positive constant; α i,1 is a virtual controller; α i,1 (0) denotes the initial value of the virtual controller α i,1 .

4. The preset time formation control method of the input delay omnidirectional mobile robot according to claim 3, wherein, The preset time formation control controller is expressed as follows: c i,3 is a positive design parameter.

5. The pre-set time platoon control method of an input-delayed multi-omni-directional mobile robot according to claim 1, wherein, Based on a backstepping recursive framework, unknown nonlinear terms are approximated by using a radial basis function neural network to obtain the Lyapunov function.

6. The pre-set time platoon control method of an input-delayed multi-omni-directional mobile robot according to claim 1, wherein, The omni-directional mobile robot is a Mecanum wheel robot.

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