Preset time formation control method for multiple omnidirectional mobile robots with input delay

Abstract

The invention discloses a preset time formation control method for multiple omnidirectional mobile robots with input delay, and belongs to the field of omnidirectional mobile robot adaptive control, and the control method comprises the following steps: building a multiple omnidirectional mobile robot kinetic model with input delay, unknown nonlinear terms and external interference; introducing an auxiliary vector based on Pade approximation and Laplacian transformation to simplify input delay analysis, and reconstructing the kinetic model based on the auxiliary vector; constructing a preset time filter in combination with a dynamic surface control technology; a Lyapunov function is constructed step by step, an adaptive law and a preset time formation controller are obtained, the preset time formation controller is used for outputting a control vector, and the control vector is input into the reconstructed dynamic model to achieve formation control. The influence of input delay on the dynamic characteristics of the system is effectively suppressed, the real-time performance of the control signal is ensured, and each robot can be ensured to complete the formation task within the preset time.

Description

Technical Field

[0001] The present invention belongs to the field of adaptive control of omnidirectional mobile robots (OMRs), and in particular to a preset time formation control method for a multi-omnidirectional mobile robot model with input delay. Background Art

[0002] In recent years, with the rapid advancement of intelligent control technologies such as unmanned system control and robot formation collaboration, research on the adaptive control of omnidirectional mobile robots has gradually become a focus of academic research. However, with the continuous breakthroughs in control technology, a single omnidirectional mobile robot, limited by its own hardware resources and processing power, has become unable to independently complete the target task when dealing with complex tasks. In this context, multi-omnidirectional mobile robot formation control technology has emerged. By integrating intelligent communication protocols with distributed control algorithms, robots in the formation can dynamically adjust their formation according to task requirements and accurately execute commands, showing broad application potential in scenarios such as high-precision operations and tasks in complex environments.

[0003] The core of omnidirectional mobile robot formation control lies in building an adaptive fuzzy control framework to achieve stable motion of the formation system and adapt to environmental uncertainty. Specifically, the control goal is to design an adaptive controller to ensure the semi-global consistency and ultimate boundedness of the closed-loop system signal, while ensuring that the system output effectively tracks the reference signal. Current mainstream control strategies include radial basis function neural networks, fuzzy logic systems, and PID control. However, existing solutions still face the following technical challenges:

[0004] First, most existing adaptive formation control methods for omnidirectional mobile robots fail to consider the impact of nonlinear dynamics and external disturbances on the system, and they also fail to address the convergence time of multi-omnidirectional mobile robot formation control. As a core metric for evaluating the dynamic performance of control systems, convergence time is crucial in engineering fields such as industrial automation and robotic control. In practical engineering applications, control systems often need to converge quickly to a stable state within a limited time to meet performance requirements such as real-time mission performance and operational reliability.

[0005] Second, existing adaptive formation control methods for omnidirectional mobile robots mostly assume an ideal real-time response to system inputs, lacking effective modeling and compensation for input delay, a key non-ideal factor. In real-world engineering scenarios, input delay in omnidirectional mobile robots is multi-source and complex. The presence of input delay can further amplify phase differences in the dynamic response, leading to decreased trajectory tracking accuracy, accumulated formation maintenance errors, and even deteriorating the stability of the closed-loop system. Summary of the Invention

[0006] To solve the problems existing in the prior art, the present invention proposes a preset time formation control method for multiple omnidirectional mobile robots with input delay, which effectively suppresses the impact of input delay on the dynamic characteristics of the system. The method can also ensure that each robot completes the formation task within the preset time, thereby improving the accuracy and robustness of formation control.

[0007] A preset time formation control method for input-delayed multi-omnidirectional mobile robots of the present invention comprises the following steps:

[0008] Establish a dynamic model of multiple omnidirectional mobile robots with input delay, unknown nonlinear terms and external disturbances;

[0009] introducing auxiliary vectors based on Padé approximation and Laplace transform to simplify input delay analysis, and reconstructing the dynamic model based on the auxiliary variables;

[0010] Combining dynamic surface control technology to construct a preset time filter;

[0011] The Lyapunov function is constructed step by step to obtain an adaptive law and a preset time formation controller. The preset time formation controller is used to output a control vector, which is input into a reconstructed dynamic model to achieve formation control.

[0012] Furthermore, the multi-omnidirectional mobile robot dynamics model with input delay, unknown nonlinear terms and external interference is established. The omnidirectional mobile robot is a Mecanum wheeled robot. The multi-omnidirectional mobile robot dynamics model is a model of the follower multi-omnidirectional mobile robot, which is expressed as follows:

[0013]

[0014] represents the posture state of the i-th omnidirectional mobile robot;

[0015] represents the speed state of the i-th omnidirectional mobile robot;

[0016] x wi and y wi Respectively represent the coordinate values ​​of the i-th omnidirectional mobile robot in the X direction and Y direction of the world coordinate system;

[0017] Indicates the positive direction of the world coordinate system X axis of the i-th omnidirectional mobile robot and the positive direction of the robot coordinate system X R The angle between the positive directions of the axes;

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

[0019] τi (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;

[0020] τ i1 (t-ι), τ i2 (t-ι), τ i3 (t-ι) and τ i4 (t-ι) represent the input control vectors of the four wheels of the omnidirectional mobile robot;

[0021] t represents time; ι represents the input delay term;

[0022] f i,1 (x i ) and f i,2 (x i ,v i ) represents the unknown nonlinear term;

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

[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 in the omnidirectional mobile robot itself.

[0025] B i (x i ,v i ) and F i (x i ,v i ) is expressed as follows:

[0026]

[0027] It is a 3×3 matrix, representing the transformation matrix between the world coordinate system and the robot coordinate system; It is a 3×4 matrix, representing the Jacobian matrix of the forward kinematic model of the omnidirectional mobile robot;

[0028] It is a 4×3 matrix, representing the Jacobian matrix of the inverse kinematics model of the omnidirectional mobile robot;

[0029] It is a 4×4 matrix, representing the gain matrix of the Mecanum wheel angular acceleration;

[0030] is a 4×4 matrix, indicating The inverse matrix of

[0031] represents the static friction force on the Mecanum wheel;

[0032] represents the angular velocity of each wheel of the omnidirectional mobile robot;

[0033] represents the angular acceleration of the Mecanum wheel;

[0034] R represents the radius of the Mecanum wheel;

[0035] It is a 3×3 matrix, representing The first derivative of the inverse matrix of ;

[0036] D θ represents the viscous friction coefficient of the Mecanum wheel.

[0037] Furthermore, the auxiliary vector is the variable υ i , which is expressed as follows:

[0038]

[0039] in, τ i represents the actual control input;

[0040] The kinetic model is reconstructed based on the auxiliary variables as follows:

[0041]

[0042] Furthermore, the preset time filter is expressed by the following differential equation:

[0043]

[0044] η∈(0,1) is a design parameter;

[0045] ρ and is a design parameter;

[0046] z i,2 is a three-dimensional vector representing the filtering error;

[0047] φ is the filter design parameter;

[0048] T p It is the scheduled time;

[0049] c i,2 is a positive constant;

[0050] α i,1 It is a virtual controller;

[0051] α i,1 (0) indicates virtual controller α i,1 The initial value of .

[0052] Furthermore, the Lyapunov function is constructed step by step, specifically:

[0053] The first construction of the Lyapunov function can be expressed in the following form:

[0054]

[0055] N represents the number of omnidirectional mobile robots;

[0056] s i,1 is a three-dimensional vector representing the tracking error;

[0057] is the ideal adjustment scalar Its estimated value Θ i,1 The estimation error of

[0058] The second construction of the Lyapunov function can be expressed as follows:

[0059]

[0060] s i,2 is a three-dimensional vector representing the error surface, z i,2 is a three-dimensional vector representing the filtering error;

[0061] is the ideal adjustment scalar Its estimated value Θ i,2 The estimation error.

[0062] Furthermore, the adaptive law is constructed for the first time and expressed in the following form:

[0063]

[0064] Among them, Θ i,1 Ideally adjusted scalar estimated value of;

[0065] The second adaptive law is constructed and expressed in the following form:

[0066]

[0067] Among them, Θ i,2 is the ideal adjustment scalar The estimated value of ,λ=2+η, λ and is the design parameter, s i,1 represents the tracking error, s i,2 represents the error surface;

[0068]

[0069] 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, then a ij =1, otherwise a ij =0;

[0070] b i Represents 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;

[0071] and is the output vector of the radial basis neural network.

[0072] Furthermore, the preset time formation controller is expressed as:

[0073]

[0074] c i,3 is a positive design parameter.

[0075] Furthermore, based on the backstepping recursive framework, the radial basis function neural network is used to approximate the unknown nonlinear terms to obtain the Lyapunov function.

[0076] Compared with the existing working technology, the present invention has the following beneficial effects:

[0077] First, most existing adaptive formation control methods for omnidirectional mobile robots fail to consider the impact of nonlinear dynamics and external interference on the system, nor do they address the convergence time issue in multi-robot formation control. This paper comprehensively considers the impact of nonlinear dynamics and external interference, proposing a neural network-based adaptive predefined-time formation control strategy. This strategy effectively addresses nonlinear dynamics and external interference, ensuring that the system completes the formation task within the preset time, thereby improving system efficiency.

[0078] Second, existing adaptive formation control methods for omnidirectional mobile robots mostly assume that system inputs respond perfectly in real time. However, they lack effective modeling and compensation for input delay, a key non-ideal factor. To address the common input delay problem in practical engineering applications, this paper investigates the adaptive formation control problem for a multi-omnidirectional mobile robot model with input delay. By combining the Padé approximation method and introducing auxiliary vectors, the analytical complexity of the input delay system is effectively simplified. The designed control strategy can effectively overcome the adverse effects of input delay.

[0079] The method of the present invention effectively suppresses the influence of input delay on the dynamic characteristics of the system and ensures the real-time performance of the control signal. The method satisfies the semi-global consistent final boundedness by designing a preset time filter and an adaptive control strategy. Nonlinear dynamics and external interference are processed with the help of neural networks, and input delay is processed in combination with the Padé approximation technique. A Lyapunov function is constructed to prove energy dissipation. Backstepping recursion technology and dynamic surface control are used to make the system state error derivative satisfy specific inequalities, ensuring that it converges to a bounded residual set within a preset time, and the convergence time is independent of the initial conditions. For any bounded initial conditions, the state eventually enters a bounded region that depends on the interference bound, thereby satisfying the semi-global consistent final boundedness and stabilizing the system state to an equilibrium point within a preset time. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] The present invention has the following Figure 9 Zhang, among which:

[0081] Figure 1 This is a simplified diagram of the motion analysis of an omnidirectional mobile robot;

[0082] Figure 2 It is a schematic diagram of the communication topology between three omnidirectional mobile robots;

[0083] Figure 3 It is the motion trajectory diagram of three omnidirectional mobile robots in the XY plane;

[0084] Figure 4 is the velocity state diagram of the three omnidirectional mobile robots in the X direction;

[0085] Figure 5 is the velocity state diagram of the three omnidirectional mobile robots in the Y direction;

[0086] Figure 6 is the tracking error diagram of the three omnidirectional mobile robots in the X direction;

[0087] Figure 7 is the tracking error diagram of three omnidirectional mobile robots in the Y direction;

[0088] Figure 8This is the input torque diagram of the four wheels of the first omnidirectional mobile robot;

[0089] Figure 9 This is the input torque diagram of the first wheel of the three omnidirectional mobile robots. DETAILED DESCRIPTION

[0090] The present invention combines backstepping technology and dynamic surface control technology to propose a preset time formation control method for multiple omnidirectional mobile robots with input delay based on state feedback, which achieves the preset time stability of the system and offsets the impact of input delay on the system, while ensuring that multiple omnidirectional mobile robots in the system complete the formation task within the preset time.

[0091] The present invention proposes a state-feedback-based method for controlling the preset time formation of multiple omnidirectional mobile robots with input delay. First, a system consisting of multiple omnidirectional mobile robots is selected as the control object. A dynamic model with unknown nonlinear terms and unknown external disturbances is established. This dynamic model can be reconstructed using the Laplace transform and Padé approximation method. Within the framework of the backstepping recursive method, a radial basis function neural network is used to approximate the unknown nonlinear terms in the dynamic model of the multiple omnidirectional mobile robots. This Lyapunov function is then established, and combined with dynamic surface control technology, a preset time filter is constructed. Based on the above work, an adaptive law for the input-delayed multiple omnidirectional mobile robot model and a preset time formation controller are derived.

[0092] The omnidirectional mobile robot is a Mecanum wheeled robot;

[0093] The preset time formation control method of the multi-omnidirectional mobile robot with input delay comprises the following steps: A. establishing a control model of the multi-omnidirectional mobile robot with input delay

[0094] In the input-delay multi-omnidirectional mobile robot system, each follower robot is numbered to distinguish multiple omnidirectional mobile robots. The dynamic model of the i-th omnidirectional mobile robot is given as follows:

[0095]

[0096] represents the posture state of the i-th omnidirectional mobile robot;

[0097] represents the speed state of the i-th omnidirectional mobile robot;

[0098] x wi and y wi Respectively represent the coordinate values ​​of the i-th omnidirectional mobile robot in the X direction and Y direction of the world coordinate system;

[0099] Indicates the positive direction of the world coordinate system X axis of the i-th omnidirectional mobile robot and the positive direction of the robot coordinate system X R The robot coordinate system is based on the geometric center of the robot as the circle center O R , with the robot's movement direction as X R Direction, with the direction perpendicular to the robot's movement as Y R direction.

[0100] y i represents the output vector of the i-th omnidirectional mobile robot;

[0101] 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;

[0102] τ i1 (t-ι), τ i2 (t-ι), τ i3 (t-ι) and τ i4 (t-ι) represent the input control vectors of the four wheels of the omnidirectional mobile robot;

[0103] t represents time; ι represents the input delay term;

[0104] f i,1 (x i ) and f i,2 (x i ,v i ) represents the unknown nonlinear term;

[0105] Λ i,1 (t,x i ) and Λ i,2 (t,x i ,v i ) indicates unknown external interference;

[0106] B i (x i ,v i ) and F i (x i ,v i ) represent the control input gain matrix and the nonlinear term inherent in the omnidirectional mobile robot itself, which can be expressed by the following mathematical formula:

[0107]

[0108] It is a 3×3 matrix representing the transformation matrix between the world coordinate system and the robot coordinate system;

[0109] is a 3×4 matrix representing the Jacobian matrix of the forward kinematics model of the omnidirectional mobile robot;

[0110] Is a 4×3 matrix, representing the Jacobian matrix of the inverse kinematics model of the omnidirectional mobile robot;

[0111] Is a 4×4 matrix, representing the gain matrix of the Mecanum wheel angular acceleration;

[0112] Is a 4×4 matrix, representing The inverse matrix of

[0113] represents the static friction force on the Mecanum wheel;

[0114] represents the angular velocity of each wheel of the omnidirectional mobile robot;

[0115] represents the angular acceleration of the Mecanum wheel;

[0116] R represents the radius of the Mecanum wheel;

[0117] Is a 3×3 matrix, representing The first derivative of the inverse matrix of ;

[0118] D θ represents the viscous friction coefficient of the Mecanum wheel.

[0119] For the sake of simplicity, f 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 .

[0120] B. Model Reconstruction

[0121] In order to solve the input delay problem and obtain the actual control input τ i, introducing the Padé approximation technique helps reduce the analytical complexity of the input delay system. The Padé approximation can be expressed as follows:

[0122]

[0123] Represents τ i Laplace transform of

[0124] δ represents the Laplace variable;

[0125] Define a new variable υ i as follows:

[0126]

[0127] According to the inverse Laplace transform, we can get:

[0128]

[0129] represents the actual control input τ i The first derivative of ;

[0130] Indicates the definition of a new variable υ i The first derivative of ;

[0131] make Can get

[0132]

[0133] Based on the above transformation, the input delay multi-omnidirectional mobile robot system (1) can be rewritten as:

[0134]

[0135] C. Establishment of Lyapunov function

[0136] For the second-order input-delay multi-omnidirectional mobile robot model, it is only necessary to construct the Lyapunov function twice to obtain its adaptive law and preset time formation controller.

[0137] In a multi-omnidirectional mobile robot system, a radial basis function neural network is used to approximate the unknown nonlinear terms in the dynamic model. The approximation error generated by this process serves as a key element in constructing the Lyapunov function. Specifically, the tracking error, radial basis function neural network approximation error, and filtering error are constructed as positive definite functions. Their convergence is analyzed using the backstepping recursive technique, and the system stability is proven. This method utilizes the universal approximation properties of radial basis function neural networks to handle system uncertainty. Combined with the structured design of the error terms, it provides a rigorous theoretical analysis framework for the preset time stability control of multi-omnidirectional mobile robots.

[0138] The first construction of the Lyapunov function can be expressed in the following form:

[0139]

[0140] N represents the number of omnidirectional mobile robots, is the ideal adjustment scalar Its estimated value Θ i,1 The estimation error.

[0141] s i,1 is a three-dimensional vector representing the tracking error;

[0142]

[0143] a ij Represents 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 interaction between the two robots, then a ij =0;κ i represents the position bias vector of the i-th follower robot relative to the leader robot; κ j represents the position bias vector of the jth follower robot relative to the leader robot; y r Indicates the reference signal given by the leader; b i Indicates the communication between the leader robot and the follower robot. If there is direct communication between the two, then b i =1, if there is no direct communication then b i =0.

[0144] The second construction of the Lyapunov function can be expressed as follows:

[0145]

[0146] is the ideal adjustment scalar Its estimated value Θ i,2The estimation error of

[0147] s i,2 is a three-dimensional vector representing the error surface;

[0148]

[0149] is the output of the preset time filter.

[0150] z i,2 is a three-dimensional vector representing the filtering error:

[0151]

[0152] D. Preset time filter establishment

[0153] The preset time filter is expressed by the following differential equation:

[0154]

[0155] η∈(0,1) is a design parameter;

[0156] ρ and is a design parameter;

[0157] φ is the filter design parameter;

[0158] T p It is the scheduled time;

[0159] c i,2 is a positive constant;

[0160] α i,1 It is a virtual controller, and its specific form will be given later.

[0161] E. Establishment of adaptive law and preset time formation controller

[0162] The first construction of the adaptive law can be expressed in the following form:

[0163]

[0164] The second construction of the adaptive law can be expressed in the following form:

[0165]

[0166] λ=2+η, λ and is the design parameter, s i,1 represents the tracking error, s i,2 represents the error surface;

[0167]

[0168] 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, then a ij =1, otherwise a ij =0;N i =1,2,...N,j≠i;

[0169] b i Represents 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;

[0170] and is the output vector of the radial basis function neural network.

[0171] Based on the previous work, the virtual controller α can be obtained i,1 and preset time formation controller τ i as follows:

[0172]

[0173] c i,1 is a positive design parameter;

[0174] Θ i,1 is the ideal adjustment scalar estimated value of;

[0175] and is the position bias vector κ i and κ j The derivative of

[0176] is the reference signal y r The derivative of .

[0177]

[0178] c i,3 is a positive design parameter;

[0179] Θ i,2 is the ideal adjustment scalar estimated value.

[0180] The motion analysis diagram of the omnidirectional mobile robot involved in the present invention is as follows Figure 1 The communication topology diagram between the three omnidirectional mobile robots involved in the present invention is shown in FIG. Figure 2As shown in the figure, "0" represents the pilot robot, and "1", "2" and "3" all represent the robots in the multi-directional mobile robot system of the present invention. The simulation results are shown in the figure. Figure 3-9 shown. Figure 3 The motion trajectories of three omnidirectional mobile robots in formation are shown; 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 effect of three omnidirectional mobile robots in the X direction is shown, where s 1,1x 、s 2,1x and s 3,1x Represent the tracking errors of the three omnidirectional mobile robots in the X direction; Figure 7 The output state tracking effect of three omnidirectional mobile robots in the Y direction is shown, where s 1,1y 、s 2,1y and s 3,1y Represent the tracking errors of the three omnidirectional mobile robots in the X direction; Figure 8 The input torques of the four wheels of the first omnidirectional mobile robot are shown; Figure 9 The input torques of the wheels at the same position on the three omnidirectional mobile robots are shown. It can be seen that a very high input torque is required during the initial phase, but then levels off. The simulation results above demonstrate that the multi-omnidirectional mobile robot system can complete the formation task within the preset time and maintain formation. Furthermore, the velocity states of each omnidirectional mobile robot in the system in the X and Y directions remain stable within the preset time. Furthermore, the tracking errors of the output states of each omnidirectional mobile robot in the system in the X and Y directions also stabilize within the preset time. The designed control scheme effectively offsets the effects of input delay, achieving the desired formation control effect within the preset time.

[0181] The present invention is not limited to this embodiment, and any equivalent concepts or modifications within the technical scope disclosed by the present invention are included in the protection scope of the present invention.

Claims

1. A preset time formation control method for multiple omnidirectional mobile robots with input delay, characterized in that: The following steps are involved: Establish a dynamic model of multiple omnidirectional mobile robots with input delay, unknown nonlinear terms and external disturbances; introducing auxiliary vectors based on Padé approximation and Laplace transform 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; The Lyapunov function is constructed step by step to obtain an adaptive law and a preset time formation controller. The preset time formation controller is used to output a control vector, which is input into a reconstructed dynamic model to achieve formation control.

2. The preset time formation control method of a multi-omnidirectional mobile robot with input delay according to claim 1, characterized in that: A multi-omnidirectional mobile robot dynamics model with input delay, unknown nonlinear terms and external interference is established. The multi-omnidirectional mobile robot dynamics model is used to model the follower multi-omnidirectional mobile robot, which is expressed as follows: represents the posture state of the i-th omnidirectional mobile robot; represents the speed state of the i-th omnidirectional mobile robot; x wi and y wi Respectively represent the coordinate values ​​of the i-th omnidirectional mobile robot in the X direction and Y direction of the world coordinate system; Indicates the positive direction of the world coordinate system X axis of the i-th omnidirectional mobile robot and the positive direction of the robot coordinate system X R The angle between the positive directions of the axes; 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; τ i1 (t-ι), τ i2 (t-ι), τ i3 (t-ι) and τ i4 (t-ι) represent the input control vectors of the four wheels of the omnidirectional mobile robot; t represents time; ι represents the input delay term; f i,1 (x i ) and f i,2 (x i ,v i ) represents the unknown nonlinear term; Λ i,1 (t,x i ) and Λ i,2 (t,x i ,v i ) indicates unknown external interference; B i (x i ,v i ) and F i (x i ,v i ) represent the control input gain matrix and the nonlinear terms inherent in the omnidirectional mobile robot itself.

3. The preset time formation control method of a multi-omnidirectional mobile robot with input delay according to claim 2, characterized in that: B i (x i ,v i ) and F i (x i ,v i ) is expressed as follows: It is a 3×3 matrix, representing the transformation matrix between the world coordinate system and the robot coordinate system; It is a 3×4 matrix, representing the Jacobian matrix of the forward kinematic model of the omnidirectional mobile robot; It is a 4×3 matrix, representing the Jacobian matrix of the inverse kinematics model of the omnidirectional mobile robot; It is a 4×4 matrix, representing the gain matrix of the Mecanum wheel angular acceleration; is a 4×4 matrix, indicating The inverse matrix of represents the static friction force on the Mecanum wheel; 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; It is a 3×3 matrix, representing The first derivative of the inverse matrix of ; D θ represents the viscous friction coefficient of the Mecanum wheel.

4. The preset time formation control method of a multi-omnidirectional mobile robot with input delay according to claim 3, characterized in that: The auxiliary vector is the variable υ i , which is expressed as follows: in, τ i represents the actual control input; The kinetic model is reconstructed based on the auxiliary variables as follows:

5. The preset time formation control method of a multi-omnidirectional mobile robot with input delay according to claim 3, characterized in that: The preset time filter is expressed by the following differential equation: η∈(0,1) is a design parameter; ρ and is a design parameter; z i,2 is a three-dimensional vector representing the filtering error; φ is the filter design parameter; T p It is the scheduled time; c i,2 is a positive constant; α i,1 It is a virtual controller; α i,1 (0) indicates virtual controller α i,1 The initial value of .

6. The preset time formation control method of a multi-omnidirectional mobile robot with input delay according to claim 4, characterized in that: Construct the Lyapunov function step by step, specifically: The first construction of the Lyapunov function can be expressed in the following form: N represents the number of omnidirectional mobile robots; s i,1 is a three-dimensional vector representing the tracking error; is the ideal adjustment scalar Its estimated value Θ i,1 The estimation error of The second construction of the Lyapunov function can be expressed as follows: s i,2 is a three-dimensional vector representing the error surface, z i,2 is a three-dimensional vector representing the filtering error; is the ideal adjustment scalar Its estimated value Θ i,2 The estimation error.

7. The preset time formation control method of a multi-omnidirectional mobile robot with input delay according to claim 4, characterized in that: The first time we construct the adaptive law, we can express it in the following form: Among them, Θ i,1 Ideally adjusted scalar estimated value of; The second adaptive law is constructed and expressed in the following form: Among them, Θ i,2 is the ideal adjustment scalar The estimated value of ,λ=2+η, λ and is the design parameter, s i,1 represents the tracking error, s i,2 represents the error surface; 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, then a ij =1, otherwise a ij =0; b i Represents 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; and is the output vector of the neural network.

8. The preset time formation control method of the input delayed multi-omnidirectional mobile robot according to claim 7, characterized in that: The preset time formation controller is expressed as: c i,3 is a positive design parameter.

9. The preset time formation control method of a multi-omnidirectional mobile robot with input delay according to claim 1, characterized in that: Based on the backstepping recursive framework, the radial basis function neural network is used to approximate the unknown nonlinear terms to obtain the Lyapunov function.

10. The preset time formation control method of a multi-omnidirectional mobile robot with input delay according to claim 1, characterized in that: The omnidirectional mobile robot is a Mecanum wheeled robot.

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