Non-singular predetermined time formation control method for incomplete mobile robots

Abstract

The invention discloses a non-singular predetermined time formation control method for an incomplete mobile robot, which is applied to the technical field of mobile robot formation control and aims to solve the problem that convergence time of an existing finite time and fixed time formation control method is generally represented by a complex mathematical expression. An exact time value is difficult to provide to represent the time required by a system to reach an expected formation form from an initial state; the difficulty of system behavior prediction is increased, and the complexity of parameter tuning is also improved; according to the preset time formation control method provided by the invention, the formation error can be converged to the neighborhood of zero within the time defined by the user, so that the expression of the convergence time is simplified, and the predictability of the system is improved; and a smooth differentiable piecewise function is introduced into the design of the virtual controller, and the piecewise function ensures the continuity and existence of a first-order derivative of the virtual controller, so that the singularity and buffeting phenomena are effectively avoided, and the stability and performance of the system are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of formation control of mobile robots, and particularly relates to a predefined-time formation control technology. Background Art

[0002] In recent years, the formation control of mobile robots has been widely applied in civilian and military fields and has attracted the attention of many scholars. The goal of multi-mobile robot formation control is to make all robots move along a predefined trajectory while maintaining a specific geometric shape. According to different control strategies, formation control methods are mainly divided into three categories: behavior-based method, virtual structure method, and leader-follower method. Among them, the leader-follower method has become a popular choice in multi-mobile robot formation control due to its simplicity and good scalability.

[0003] The convergence speed is one of the key indicators to measure the response efficiency and performance of a control system, and it directly affects the stability and dynamic characteristics of the system. Although the existing finite-time and fixed-time formation control methods have significantly improved the error convergence rate of multi-agent systems when forming a specific formation and achieved a faster synchronization effect than traditional asymptotic control, the convergence time of these advanced control strategies is usually described by complex mathematical expressions, making it difficult to provide an exact time value in practical applications to represent the time required for the system to reach the expected formation shape from the initial state. This not only increases the difficulty of predicting the system behavior but also improves the complexity of parameter tuning. Summary of the Invention

[0004] To solve the above technical problems, the present invention proposes a nonsingular predefined-time formation control method for nonholonomic mobile robots.

[0005] The technical solution adopted by the present invention is: a nonsingular predefined-time formation control method for nonholonomic mobile robots, including:

[0006] Modeling the formation control problem of mobile robots by using a leader-follower based method. Specifically: each robot R i has a unique and fixed leader R j , i ∈ Ω = {1, 2,..., N}, where N represents the number of mobile robots, is a set, and when j = 0, R j is a virtual leader; the formation control method includes the following steps:

[0007] S1. Consider that the leader and the followers have the same kinematic model;

[0008] S2. Based on the kinematic model in step S1, establish the desired trajectory of the followers;

[0009] S3. Define the formation position error according to the expected trajectory of the follower, and design a virtual controller and a position controller based on the formation position error;

[0010] S4. Obtain the attitude angle controller according to the virtual controller;

[0011] S5. Control the moving speed of the robot according to the position controller, and control the heading angle of the robot according to the attitude angle controller.

[0012] Advantages of the present invention: First, the leader-follower mobile robot is modeled to obtain the kinematic equation of the robot; then, based on the backstepping control technology, a predefined-time formation control method for nonholonomic mobile robots is developed in two steps; finally, the feasibility of the proposed formation control method is verified through the results of simulation experiments and physical experiments; the method of the present invention has the following advantages:

[0013] 1. A predefined-time formation control method proposed by the present invention can converge the formation error to the neighborhood of zero within the time defined by the user, thus simplifying the expression of the convergence time and improving the predictability of the system;

[0014] 2. To avoid the singularity problem, a smooth and differentiable piecewise function is introduced in the design of the virtual controller; this piecewise function ensures the continuity and existence of the first derivative of the virtual controller, effectively avoiding the singularity and chattering phenomena, and improving the stability and performance of the system. Description of the Drawings

[0015] Figure 1 It is a schematic diagram of the leader-follower formation structure of the present invention;

[0016] Figure 2 It is a flow block diagram of the control scheme of the present invention;

[0017] Figure 3 It is the communication topology structure between the robots of the present invention;

[0018] Figure 4 It is the simulation formation trajectory of the present invention;

[0019] Figure 5 It is the formation error of the simulation robot R1 of the present invention;

[0020] Figure 6 It is the formation error of the simulation robot R2 of the present invention;

[0021] Figure 7 It is the formation error of the simulation robot R3 of the present invention;

[0022] Figure 8 It is the formation error of the robot R1 of the present invention under different Ts;

[0023] Figure 9 This is the formation error of the robot R1 under different initial states when T = 10 in the present invention;

[0024] Figure 10 This is the experimental environment of the present invention;

[0025] Figure 11 This is the communication topology diagram of the mobile robot Qbot3 in the present invention;

[0026] Figure 12 This is the formation trajectory of the physical experiment in the present invention;

[0027] Figure 13 This is the formation error of the mobile robot R1 in the present invention;

[0028] Figure 14 This is the formation error of the mobile robot R2 in the present invention;

[0029] Figure 15 This is the formation error of the mobile robot R3 in the present invention;

[0030] Figure 16 These are the control inputs of the three mobile robots R1, R2, and R3 in the present invention. Detailed implementation manner

[0031] To facilitate the understanding of the technical content of the present invention by those skilled in the art, the content of the present invention will be further explained below with reference to the accompanying drawings.

[0032] A non-singular predefined-time formation control method for nonholonomic mobile robots in the present invention. First, a model of the leader-follower mobile robots is established to obtain the kinematic equations of the robots. Then, based on the backstepping control technique, a predefined-time formation control method for nonholonomic mobile robots is developed in two steps. Finally, the feasibility of the proposed formation control method is verified through the results of simulation experiments and physical experiments.

[0033] As Figure 2 shown, the implementation process of the method of the present invention includes the following steps:

[0034] S1: Establish a model of the mobile robot to obtain its kinematic equations.

[0035] The present invention relates to a multi-robot formation control problem consisting of N mobile robots and a virtual leader. For ease of description, a set Ω = {1, 2,..., N} is defined, where the indices 1 to N correspond to the N mobile robots respectively, and the virtual leader is not included in the set Ω. Therefore, the virtual leader is represented by the index 0, assuming its state is stable, and the mobile robots and the virtual leader are collectively denoted as R i(i = 0, 1, 2, ..., N). The present invention models the formation control problem of mobile robots using a leader-follower based method. Specifically, each robot R i has a unique and fixed leader In practical applications, the leader of the current robot is determined by any designated method. Specifically: Any robot other than the current robot itself is arbitrarily designated as the leader of the current robot, where is a set, and when j = 0, R j is the virtual leader. In addition, assume that the leader R j and the follower R i have the same kinematic model, and the corresponding kinematic model is as follows:

[0036]

[0037] where, i = 0, 1, 2, ..., N, (x i , y i ) represents the position coordinates of R i in the earth-fixed coordinate system {F}, θ i represents the heading angle, v i represents the linear velocity of R i , and ω i represents the angular velocity. In addition, and respectively represent the velocities of R i in the x and y directions, represents the angular velocity of the follower R i .

[0038] As shown by Figure 1 , the desired trajectory of the follower R i is described as:

[0039]

[0040] where, (x di , y di ) are the desired trajectory position coordinates of the follower R i , θ di represents the desired heading angle of the follower R i . The angle and the distance λ i are formation parameters and are both constants. (x j , y j ) and θ j respectively represent the position coordinates and heading angle of the leader R j .

[0041] S2: The formation position error is defined as xei = x di -x i and y ei = y di -y i Substituting these into formula (2), we get:

[0042]

[0043] Differentiating formula (3) and substituting the kinematic model described in formula (1) into it, we obtain the position error dynamic equation as follows:

[0044]

[0045] where i ∈ Ω, and α i is a virtual controller. The role of the virtual controller α i is to adjust the movement direction of the follower R i such that the position errors x ei and y ei gradually converge to the neighborhood of zero, thereby achieving the goal of mobile robot formation control.

[0046] To make the system stable, design the virtual controller α i such that the following equation holds:

[0047]

[0048] where, T > 0, π is the pi. c 1i and c 2i satisfy c 2i > c 1i > 0, and c 1i is an even number, c 2i is an odd number, S xei and S yei are smooth piecewise functions, and the specific forms are as follows:

[0049]

[0050] where, τ xi > 0, τ yi > 0, and the values of r i,1 and r i,2 are obtained by solving the equations r i,1 + r i,2 = 1 and r i,1 + 2r i,2 = β 2i Here, in order to avoid singularities near the zero point, a piecewise function is set on the right side of the zero point to avoid the occurrence of singularities. In this embodiment, our τxi , τ yi takes a value of a very small positive number, such as 0.1, 0.01, etc.

[0051] Define m xi and m yi as follows:

[0052]

[0053]

[0054] According to formulas (5), (6), (7) and (8), we can get:

[0055] v i cosα i = m xi (9)

[0056] v i sinα i = m yi (10)

[0057] According to formulas (9) and (10), the expressions of the position controller v i and the virtual controller α i are:

[0058]

[0059] Among them, the function of the position controller v i is to generate a resultant velocity v i by calculating the motion components m xi and m yi of the follower R i in the x and y directions, so as to control the motion speed of the follower R i and enable it to track the desired trajectory and achieve the formation control goal. arctan(·) is the arctangent function.

[0060] S3: Define the angle error as θ ei = α i - θ i .

[0061] Take the derivative of the angle error and substitute the kinematic model described in formula (1) into it, and the angle error dynamic equation is obtained as:

[0062]

[0063] Among them, represents the derivative of the virtual controller α i .

[0064] To make θ ei gradually converge within the neighborhood of zero, the attitude angle controller ω i is designed as:

[0065]

[0066] S4: Simulation experiments were carried out based on the Matlab / Simulink simulation platform, and physical experiments were carried out using the QBot3 mobile robot produced by Quanser to verify the proposed algorithm.

[0067] Furthermore, the step S4 includes:

[0068] S41: To verify the proposed control method, a formation control scenario involving 3 mobile robots and a virtual leader was analyzed. The communication topology among the robots is as Figure 3 shown.

[0069] The formation parameters are defined as λ1 = 0, λ2 = 0.5, λ3 = 0.5 and the trajectory parameters of the virtual leader are given as x0(0) = -2m, y0(0) = 0m, θ0(0) = 0rad,

[0070] and the initial positions of the robots are x1(0) = -2.10m, y1(0) = 0.10m, θ1(0) = -π / 30rad, x2(0) = -2.45m, y2(0) = 0.45m, θ2(0) = -π / 30rad, x3(0) = -2.45m, y3(0) = -0.25m, θ3(0) = π / 30rad, where the controller parameters are set as c 11 = 12, c 12 = 35, c 21 = 6, c 22 = 19, c 31 = 6, c 32 = 23, T = 10.

[0071] The simulation results are as Figures 4 to 7 shown. As Figure 4 shown, the robot R1 accurately follows the trajectory of the virtual leader R0 and guides the movement of robots R1 and R2. Figure 4 The relative positions of robots R1, R2, and R3 at 0s, 8s, 16s, 24s, 32s, and 40s are marked in Figure 5 、 Figure 6 、Figure 7 They are the formation errors of three robots respectively. It can be seen from the results that the formation errors can converge quickly. However, the formation errors of robots R2 and R3 show overshoot during the turning process, which is mainly caused by ω j resulting in m xi and m yi suddenly increasing.

[0072] Theoretically, the formation errors x ei , y ei and θ ei can converge to a small neighborhood of zero within 2Ts. Figure 8 shows the convergence of the formation error of robot R1 at different T values. When T = 2, the formation error converges to a small neighborhood of zero within 2s, but there will be chattering phenomenon; when T = 5, the formation error converges to a small neighborhood of zero within 4s, and the chattering phenomenon is significantly weakened; when T = 10, the formation error converges to a small neighborhood of zero within 3s, and the chattering phenomenon is eliminated; when T = 15, the formation error converges to a small neighborhood of zero within 4s, and the chattering phenomenon is also eliminated. After experimental tests and performance evaluations, the appropriate T value selected in this experiment is 10, which can avoid the chattering phenomenon while ensuring the convergence speed. The results show that the smaller the T value, the faster the error converges, but too small a T value will lead to the chattering phenomenon. Therefore, choosing an appropriate T is very important for obtaining ideal control performance.

[0073] To evaluate the convergence of the formation error under different initial states, simulation experiments under various initial states were carried out. As shown in Table 1, four different initial states were considered for comparison. Figure 9 shows the formation error response curves of robot R1 under these four different initial states. The simulation results show that the larger the initial error, the slower the convergence speed of the formation error. However, all formation errors can converge within the preset 10 seconds, which proves the effectiveness of the proposed fixed-time formation control method of the present invention.

[0074] Table 1 Simulation experiments

[0075]

[0076] S42: To evaluate the effectiveness of the proposed control method in practical applications, the present invention built a physical experimental device, as Figures 10 to 11 shown, where Figure 10 provides the hardware environment and setting background of the experiment, Figure 11Further illustrates the communication topology among the robots. The two together constitute a complete description of the physical experiment, gradually demonstrating the effectiveness of the implementation process of the present invention from the experimental environment to the communication architecture and then to the experimental results. In the physical experiment, the formation parameters and the trajectory of the virtual leader are consistent with those in the simulation experiment. The controller parameters are set as c 11 = 6, c 12 = 39, c 21 = 8, c 22 = 43, c 31 = 6, c 32 = 55, T = 10.

[0077] The results of the physical experiment are as Figures 12 to 16 shown. As Figure 12 shown, under the proposed formation control method, the three robots successfully completed the formation task. Figure 13 , Figure 14 , Figure 15 respectively show the formation errors of the three robots, indicating that the method of the present invention can effectively control the formation error and make it quickly converge to zero, reflecting high control accuracy and stability. Figure 16 Shows the control input responses of the three mobile robots, indicating that the control input fluctuates within the expected reasonable range, further verifying the robustness and practicality of the method of the present invention. Generally speaking, the effectiveness of the method proposed by the present invention is verified.

[0078] Those of ordinary skill in the art will realize that the embodiments described herein are for helping the reader understand the principles of the present invention and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. For those skilled in the art, various changes and modifications can be made to the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the scope of the claims of the present invention.

Claims

1. A nonsingular predefined-time formation control method for nonholonomic mobile robots, characterized in that Model the formation control problem of mobile robots using a leader-follower based approach. Specifically, each robot R i has a unique and fixed leader R j , j ≠ i, i ∈ Ω = {1, 2,..., N}, where N represents the number of mobile robots, is a set, and when j = 0, T j is a virtual leader; The formation control method includes the following steps: S1. Consider that the leader and the follower have the same kinematic model; S2. Based on the kinematic model in step S1, establish the desired trajectory of the follower; S3. Define the formation position error according to the desired trajectory of the follower, and design a virtual controller and a position controller based on the formation position error; S4. Obtain the attitude angle controller according to the virtual controller; S5. Control the moving speed of the robot according to the position controller, and control the heading angle of the robot according to the attitude angle controller.

2. The non-singular predefined-time formation control method for a non-integral mobile robot according to claim 1, wherein, The desired trajectory of the follower in step S2 is expressed as: Among them, (x di , y di ) is the expected trajectory position coordinate of the i-th follower, and θ di represents the expected heading angle of the i-th follower. represents an angle, and λ i represents a distance. (x j , y j ) and θ j respectively represent the position coordinate and the heading angle of the j-th leader.

3. A non-singular prescribed-time formation control method for a non-holonomic mobile robot according to claim 2, characterized in that, The formation position error in step S3 is defined as: x ei = x di -x i y ei = y di -y i Among them, (x i , y i ) represents the actual position coordinates of the i-th follower.

4. A non-singular predefined-time formation control method for a non-integral mobile robot according to claim 3, characterized in that The virtual controller is defined as: where α i is a virtual controller, m xi and m yi represent the motion components of the i-th follower in the x and y directions T > 0, π is the ratio of a circle's circumference to its diameter, c 1i and c 2i satisfies c 2i > c 1i > 0, and c 1i is even, c 2i is odd, S xei and S yei are smooth piecewise functions.

5. The non-singular predefined-time formation control method for a non-integral mobile robot according to claim 4, characterized in that, T takes the value of 10.

6. The non-singular predefined-time formation control method for a non-holonomic mobile robot according to claim 5, characterized in that, S xei and S yei The expressions are as follows: wherein, τ xi >0, τ yi >0, and r i,1 and r i,2 are obtained by solving the equations r i,1 + r i,2 = 1 and r i,1 + 2r i,2 = β 2i for the solution.

7. A non-singular pre-defined time formation control method for a non-complete mobile robot according to claim 6, characterized in that, The position controller is defined as:

8. A non-singular prescribed-time formation control method for a non-integral mobile robot according to claim 7, characterized in that The attitude angle controller in step S4 is expressed as: Among them, θ ei represents the angular error, and θ ei = α i - θ i .

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