A Fast Finite-Time Binary Formation Obstacle Avoidance Control Method for a Cluster of Time-Delay Mobile Robots

Abstract

The present invention relates to a fast finite-time bipartite formation obstacle avoidance control method for a time-delay mobile robot swarm, belonging to the field of control engineering technology. In order to improve the convergence speed of the formation shape, the present invention proposes a fast finite-time formation control method, which can enable the mobile robot swarm to achieve the desired bipartite formation shape in fast finite time; considering that the output of the actuators in the actual system is usually limited and vulnerable to faults, an anti-saturation technique and a fault-tolerant control strategy are integrated in the design of the control method; due to the lack of prior map knowledge of the operating environment, an obstacle avoidance mechanism is embedded in the control method by using the artificial potential field technique to ensure the safe operation of the robot formation; an improved Lyapunov-Krasovskii functional is constructed to eliminate the influence of time delay on the system. The present invention effectively utilizes the artificial potential field technique to construct a virtual potential field, eliminates the influence of time delay on the system on the premise of ensuring that the saturation constraint conditions are not violated and the actuators have deviation faults, ensures that the mobile robot swarm can achieve the desired bipartite formation in fast finite time and avoid collisions with obstacles, and improves the response speed and safety of the system.

Description

Technical Field

[0001] The present invention relates to a fast finite-time bipartite formation obstacle avoidance control method for a time-delay mobile robot swarm, belonging to the technical field of control engineering. Background Technique

[0002] The industrial modernization has promoted the rapid development of mobile robot research. The research trends at home and abroad have also shifted from single robots to multi-robot swarm cooperative control. Through the mutual cooperation among robots, many complex and dangerous tasks can be completed. Formation control is a fundamental issue in the field of multi-robot cooperative control, and its goal is to enable the entire robot swarm to achieve the desired formation through distributed information perception among robots.

[0003] When mobile robots execute formation tasks, they usually have no prior knowledge of the map, which means that they may be affected by obstacles during operation. To ensure the safe operation of the formation, it is necessary to consider avoiding collisions with obstacles. The artificial potential field method has been widely studied due to its convenience, high efficiency, and strong real-time performance. By simulating potential fields such as gravitational fields and electromagnetic fields, each obstacle is set as a high electric potential point, and then a virtual potential field function is constructed. When a robot enters the potential field, a repulsive force will be generated to make it bypass the obstacle.

[0004] Currently, most of the research on formation control assumes that there is only one type of cooperation relationship among robots, that is, all communication links among robots are non-negative. However, in reality, in addition to the cooperation relationship, there are often adversarial relationships among robots, similar to the interaction between opponents and teammates in multi-player games. Therefore, according to the positive and negative weights in the signed topology graph, the robots are divided into two groups, and bipartite formation control is introduced to describe the formation control of a robot swarm operating in a cooperative adversarial interaction network.

[0005] Due to the limitations of physical devices, actuators are usually affected by faults and saturations, which will have an adverse impact on system performance and even hinder the completion of control tasks. Therefore, it is very necessary to consider actuator faults and saturation constraints in mobile robot formation control. Due to the existence of network communication, time delay is inevitable. It will reduce system performance and increase the complexity of controller design. The Lyapunov-Krasovskii functional method has been used by scholars to solve the time-delay problem due to its wide application ability.

[0006] The convergence speed of formation error is one of the key indicators for evaluating its performance. In the obstacle avoidance problem and actual formation construction, the convergence speed of infinite time cannot guarantee good transient performance. To improve the convergence speed of formation error, scholars have developed finite-time control methods. In addition, a control method called fast finite time has been proposed, which can improve the robustness and convergence speed of traditional finite-time control algorithms.

[0007] In summary, it is necessary to develop a bipartite formation control and obstacle avoidance strategy with fast finite-time stability for the actual mobile robot swarm system under the influence of time delay. Summary of the Invention

[0008] The main purpose of the present invention is: aiming at the deficiencies and blanks of the prior art, the present invention provides a fast finite-time bipartite formation obstacle avoidance control method for mobile robot swarms, considering the communication topology where cooperative and adversarial interactions coexist, and using the advantages of the artificial potential field method to construct a virtual potential field function to achieve the desired obstacle avoidance behavior. In the case of actuator bias faults, saturation constraints, and the system being affected by time delay, the construction of a fast finite-time bipartite formation for mobile robot swarms is realized, improving the system response speed and robustness.

[0009] In order to achieve the above objectives, a fast finite-time bipartite formation obstacle avoidance control method for time-delay mobile robot swarms studied by the present invention includes the following steps:

[0010] S1: Based on the kinematic and dynamic models of mobile robots, combined with the variable transformation method, construct the state space expression of mobile robots with time delay and unknown non-linear terms, and give the desired leader trajectory signal;

[0011] S2: Establish the communication relationship between follower robots and leader signals with the help of graph theory, and define the desired bipartite formation control objective based on the constructed state space expression of mobile robots;

[0012] S3: Model the actuators in the actual scenario, consider the actuator model under saturation constraints and fault effects, and construct a virtual potential field obstacle avoidance function and obstacle avoidance repulsive force based on the artificial potential field method;

[0013] S4: Design the first Lyapunov function candidate, and use the fuzzy logic system to approximate the non-linear terms in the dynamics; design the Lyapunov-Krasovskii functional to eliminate the influence of time delay on the controller design;

[0014] S5: Give the designed control strategy and adaptive update law to achieve the fast finite-time convergence of the error system and ensure the obstacle avoidance ability of the designed formation.

[0015] For the fast finite-time bipartite formation obstacle avoidance control method for time-delay mobile robot swarms as described above, it is characterized in that the operation of S1 is specifically:

[0016] Establish the kinematic and dynamic equations of the wheeled mobile robot, perform model conversion and coordinate transformation, and at the same time consider the influence of time delay on the system to obtain the state space expression of the mobile robot system:

[0017]

[0018] where p i (t), q i (t), u i (t) represent the position, velocity, and control input of the i-th robot, respectively; f i (p i , q i , t) represents a heterogeneous non-linear vector-valued function with uncertainties; g i (p i (t - τ i ), q i (t - τ i )) represents a smooth vector-valued function with time delay; τ i is an unknown bounded time delay, satisfying 0 < τ i ≤ τ max . The reference trajectory or path of the leader is given as follows:

[0019]

[0020] where p0(t) and q0(t) are the position and velocity vectors of the leader, respectively, and f0(t) represents a bounded function set as the reference signal.

[0021] For the fast finite-time bipartite formation and obstacle avoidance control method of a time-delay mobile robot swarm as described above, it is characterized in that the operation of S2 is specifically as follows:

[0022] Express the communication relationship between the follower robots and the leader signal as a signed topological graph which consists of a vertex set an edge set and an adjacency matrix . The degree matrix related to the leader reference signal is defined as The Laplacian matrix is defined as where l ij = -a ij , i ≠ j and The information interaction matrix is expressed as where Graph has a spanning tree with the leader as the root node. If the vertex set can be decomposed into two vertex subsets and satisfying and then the topological graph is structurally balanced. If υ i and υ jBelong to the same vertex subset or then a ij > 0, otherwise a ij < 0. In addition, define a diagonal matrix If the i-th robot has a cooperative relationship with the leader, then s i = 1, otherwise s i = -1. Let If for any bounded initial value r i (t0) there exists a distributed controller u i (t) such that each follower robot can maintain an offset vector with the leader after a bounded settling time T0, then it is said that the mobile robot swarm can achieve the desired fast finite-time bipartite formation control, that is

[0023]

[0024] where represents the offset vector between the i-th robot and the leader, and σ > 0 is the allowable formation error bound.

[0025] For the fast finite-time bipartite formation obstacle avoidance control method of the time-delay mobile robot swarm as described above, it is characterized in that the operation of S3 is specifically as follows:

[0026] Express the actual actuator output as:

[0027]

[0028] where μ i (t) is the original control input signal; ε i (t) is the approximation error, is the actuator bias fault. Applying the artificial potential field method, the position of the s-th obstacle is expressed as Express the relative displacement vector between the i-th robot and the s-th obstacle as:

[0029]

[0030] where s = 1, 2,..., M. Define the repulsive potential function generated by the s-th obstacle on the i-th robot approaching it as a non-negative differentiable function Define the repulsive force function of the obstacle on the robot as the negative gradient along the potential function , that is The repulsive potential function satisfies: i) When , the repulsive force gradually increases, that is where is the minimum obstacle avoidance distance for the s-th obstacle; ii) when the repulsive force gradually weakens to the minimum value, where is the maximum action distance threshold of the repulsive force.

[0031] The fast finite-time bipartite formation obstacle avoidance control method for the time-delay mobile robot swarm as described above is characterized in that the operation of S4 is specifically as follows:

[0032] Define the bipartite formation error as e p,i (t) = p i - s i p0 - h p，i , e q，i (t) = q i - s i q0 - h q，i , e i (t) = e p，i (t) + e q，i (t).

[0033] The distributed bipartite consensus error is defined as:

[0034]

[0035] According to the defined bipartite formation error, design the first Lyapunov function candidate as shown below:

[0036]

[0037] where and satisfies e = e p + e q . Use the fuzzy logic system to approximate the nonlinear function, and there is For the considered time-delay function g i (p i (t - τ i ), q i (t - τ i ), assume it satisfies:

[0038] ||g i (p i (t - τ i ), q i (t - τ i ))|| 2 ≤ ||ψ i (t - τ i )|| 2 ||δ i (p i (t - τ i ), q i (t - τ i))||+δ0(t-τ i ) (8)

[0039] where δ i (·) is a known function, and δ0(·) is a bounded function satisfying 0 < δ0(·) ≤ δ m . The Lyapunov-Krasovskii functional is designed as follows:

[0040]

[0041] For the fast finite-time binary formation obstacle avoidance control method of the time-delay mobile robot swarm as described above, it is characterized in that the operation of S5 is specifically as follows:

[0042] To achieve the desired fast finite-time binary formation control and obstacle avoidance ability, the fuzzy adaptive control strategy is designed as follows:

[0043]

[0044] The specific expressions of the elements in the formula are as follows:

[0045]

[0046] where λ max (·) represents the maximum eigenvalue of the matrix; for a vector x = [x1, x2,..., x n T , <x> a Defined as <x> a = [|x1| a sgn(x1), |x2| a sgn(x2),..., |x n | a sgn(x n )] T ; κ1 > λ max (H) + σ e / 2 + 2 and κ2 > 0 are control gains; is the repulsive force intensity parameter; 0 < β < 1 is a design parameter; is the actuator fault compensation term, which satisfies for estimating the unknown constant for updating The adaptive law for

[0047]

[0048] where γ2 > 0 is a design parameter.

[0049] Based on the control strategy designed by the Lyapunov function, the fast finite-time stability is verified, and the obstacle avoidance ability of the designed formation is guaranteed.

[0050] Compared with the prior art, the beneficial effects of the present invention are:

[0051] 1) The present invention aims at a mobile robot system affected by time delay, considering the situation of simultaneous cooperative and adversarial interactions between robots, and constructs a fast finite-time bipartite formation control strategy with obstacle avoidance ability, effectively improving the response speed of the system;

[0052] 2) The present invention makes full use of the convenient and efficient advantages of the artificial potential field method, embeds the obstacle avoidance mechanism into the bipartite formation control strategy, and can ensure the safe operation of all robots in an environment without prior map knowledge;

[0053] 3) The present invention avoids the influence of actuator faults and saturation constraints on the controller design and system operation. At the same time, based on the improved Lyapunov-Krasovskii functional, the influence of time delay on the system performance is eliminated, the singularity problem can be avoided, and the robustness of the system is improved.

[0054] To make the above objects, features, and advantages of the present invention more obvious and understandable, the following specifically gives embodiments of the present invention and, in conjunction with the accompanying drawings, makes a detailed description as follows. Description of the Drawings

[0055] To more clearly illustrate the technical solutions of the embodiments of the present invention, the following will briefly introduce the drawings required for the embodiments. It should be understood that the following drawings only show some embodiments of the present invention, and therefore should not be regarded as a limitation of the scope. For those of ordinary skill in the art, without creative efforts, other related drawings can also be obtained based on these drawings.

[0056] Figure 1 It is a flowchart of the control method in the embodiment.

[0057] Figure 2 It is a schematic structural diagram of a wheeled mobile robot in the embodiment.

[0058] Figure 3 It is a communication topology diagram inside a mobile robot cluster in the embodiment.

[0059] Figure 4 It is the motion trajectory and position snapshot of the robot in the embodiment.

[0060] Figure 5 It is the running trajectory of the robot when approaching an obstacle in the embodiment.

[0061] Figure 6 It is the evolution law of the position vector of the robot in the embodiment.

[0062] Figure 7 It is the evolution law of the speed error of the robot in the embodiment. Specific embodiments

[0063] The following will further elaborate on the present invention in conjunction with the drawings.

[0064] Taking a wheeled mobile robot as an example, as Figure 2 shown, the robot includes four mobile wheels, two front wheels are driven wheels, and two rear wheels are driving wheels. (x i , y i ) is the center point position coordinate between the two driving wheels, R i is the wheel radius, L i is half of the distance between the two driving wheels, (p x，i , p y，i ) represents the center point position coordinate of the robot, is the yaw angle.

[0065] The embodiments of the present application provide a fast finite-time bipartite formation and obstacle avoidance control method for a time-delay mobile robot cluster. Refer to Figure 1 , including the following steps:

[0066] S1: Based on the kinematic and dynamic models of a mobile robot, combined with the variable transformation method, construct the state - space expression of the mobile robot with time - delay and unknown non - linear terms, and give the desired leader trajectory signal;

[0067] S2: Establish the communication relationship between the follower robot and the leader signal by means of the topological graph theory, and define the desired bipartite formation control objective based on the constructed state - space expression of the mobile robot;

[0068] S3: Model the actuators in the actual scenario, consider the actuator model under saturation constraints and fault effects, and construct the virtual potential field obstacle - avoidance function and the obstacle - avoidance repulsive force based on the artificial potential field method;

[0069] S4: Design the first Lyapunov function candidate, and use the fuzzy logic system to approximate the non - linear terms in the dynamics; Design the Lyapunov - Krasovskii functional to eliminate the influence of time - delay on the controller design;

[0070] S5: Give the designed control strategy and the adaptive update law, realize the fast finite - time convergence of the error system, and ensure the obstacle - avoidance ability of the designed formation.

[0071] S1: Based on the kinematic and dynamic models of a mobile robot, combined with the variable transformation method, construct the state - space expression of the mobile robot with time - delay and unknown non - linear terms, and give the desired leader trajectory signal. The specific steps are as follows.

[0072] Step 1 - 1: Give the structural schematic diagram of the wheeled mobile robot as Figure 2 shown, and establish the kinematic and dynamic equations of the wheeled mobile robot as follows:

[0073]

[0074] where (x i , y i ) is the coordinate of the center point between the two driving wheels, represents the yaw angle of the robot in the ground coordinate system; and are the angular velocities of the left and right wheels respectively; ι i = [ι l，i , ι r，i T , ι l，i and ι r，ι are the control torques of the left and right wheels respectively; J i (η i ) is the rotation matrix; M i is the symmetric positive - definite inertia matrix; ​is the Coriolis centripetal acceleration matrix; D i is the damping matrix. The specific expressions of the matrices mentioned in the robot model (12) are as follows:

[0075]

[0076]

[0077] where R i is the wheel radius; L i is half of the distance between the two wheels; m 1，i and m 2，i are the inertial masses; c i is the Coriolis centripetal coefficient; d 1，i and d 2，i are the damping coefficients.

[0078] Step 1-2: Note that before performing error conversion, the wheeled mobile robot satisfies the following non-slip kinematic constraints:

[0079]

[0080] This means that the robot moves at a speed perpendicular to the axis of the drive wheels. The angular velocity of the left wheel and the angular velocity of the right wheel can be converted into the linear velocity v i and the angular velocity w i of the robot according to the following relationship:

[0081]

[0082] Based on the above relationship, the robot model (1) can be converted to:

[0083]

[0084] where k 1，i = R i / 2(m 1，i + m 2，i ), ι 1，i = ι l，i + ι r，i , g 2，i = -[2c i v i w i + v i (d 1，i + d 2，i ) + L i w i (d 1，i + d 2，i )] / R i ,k 2，i =R i / 2L i (m 1，i -m 2，i ),i 2，i =ι l，i -ι r，i 。

[0085] Step 1-3: Introduce the following coordinate transformation mechanism:

[0086]

[0087] where (p x，i ,p y，i ) represents the position coordinates of the center point of the robot, and d i represents the distance between the point (x i ,y i ) and the point (p x，i ,p y，i ). Let p i =[p x，i ,p y，i T , then the mobile robot model (15) can be re-expressed as:

[0088]

[0089] where

[0090] Step 1-4: Considering the influence of time delay on the mobile robot system, it can be expressed as the following state-space expression:

[0091]

[0092] where p i (t), q i (t), u i (t) represent the position, velocity, and control input of the i-th robot respectively; f i (p i ,q i ,t) represents a heterogeneous non-linear vector-valued function with uncertainty; g i (p i (t-τ i ),q i (t-τ i )) represents a smooth vector-valued function with time delay; τ i is an unknown bounded time delay, satisfying 0<τ i ≤τ​ max The reference trajectory or path of the leader is given as follows:

[0093]

[0094] where p0(t) and q0(t) are the position and velocity vectors of the leader respectively, and f0(t) represents a bounded function that is artificially set as the reference signal.

[0095] S2: Establish the communication relationship between the follower robots and the leader signal by means of the topological graph theory, and define the desired bipartite formation control objective based on the constructed state space expression of the mobile robot. The specific steps are as follows.

[0096] Step 2-1: The information interaction between the robot swarms can be represented by a signed topological graph where represents the set of vertices; represents the set of edges; represents the adjacency matrix. If the i-th robot can receive the information of the j-th robot, then a ij ≠0, otherwise a ij =0. In addition, if a ij >0, it means that the relationship between the i-th robot and the j-th robot is cooperative. If a ij <0, it means that the relationship between the i-th robot and the j-th robot is adversarial, and there is no self-information transmission, that is, a ii =0. The degree matrix related to the leader reference signal is defined as β = diag{b1, b2,..., b N}}, if the i-th robot can directly receive the leader reference signal, then b i ≠0, otherwise b i =0. If b i >0, it represents that the i-th robot and the leader are in a cooperative relationship. If b i <0, it represents that the i-th robot and the leader are in an adversarial relationship. The Laplacian matrix is defined as where The information interaction matrix is expressed as where Graph has a spanning tree with the leader as the root node.

[0097] Step 2-2: If the vertex set can be decomposed into two vertex subsets and that satisfy and then the topological graph is structurally balanced. If υ i and υ j belong to the same vertex subset or then a ij > 0, otherwise a ij < 0. In addition, define a diagonal matrix If the i-th robot has a cooperative relationship with the leader, then s i = 1, otherwise s i = -1.

[0098] Step 2-3: The desired control objective is to design a distributed control strategy such that: i) all follower robots can achieve the desired binary formation pattern based on the leader's reference trajectory within a fast finite time; ii) the effects of actuator faults, saturation constraints, and time delays on the system can be eliminated; iii) and collisions between each robot and any obstacle in the operating environment can be prevented. Let If for any bounded initial value r i (t0) there exists a distributed controller u i (t) such that each follower robot can maintain a set offset vector from the leader after a bounded settling time T0, then it is said that the mobile robot swarm can achieve the desired fast finite-time binary formation control, that is

[0099]

[0100] where represents the offset vector between the i-th robot and the leader, and σ > 0 is the allowable formation error bound. It should be noted that h i (t) and r i (t) have the same dimension, which means that each item in r i (t) has its corresponding offset vector for the offset vector.

[0101] S3: Model the actuators in the actual scenario, consider the actuator model under saturation constraints and fault effects, and construct a virtual potential field obstacle avoidance function and obstacle avoidance repulsive force based on the artificial potential field method. The specific steps are as follows.

[0102] Step 3-1: In engineering practice, due to power and technological limitations, the output of actuators is usually restricted to a certain extent, and the actuators may malfunction due to factors such as wear and aging during operation. Therefore, model the output of the actuator affected by saturation constraints and faults as follows:

[0103]

[0104] where μ i (t) is the original control input signal; is the actuator bias fault, and both it and its derivative are unknown but bounded; sat(μ i (t)) represents the saturation constraint, which can be described as:

[0105]

[0106] where μ i，max > 0 and μ i，min < 0 represent the maximum and minimum output torques related to the physical limits, respectively. To handle actuator saturation, the saturation constraint is reconstructed as:

[0107]

[0108] where or ε i (t) is the approximation error. Since using the mean value theorem, we can obtain:

[0109]

[0110] where Let then the actual actuator output can be re-expressed as:

[0111]

[0112] and the following results hold:

[0113]

[0114] Therefore, it can be known that the maximum eigenvalue of Γ i is less than 0.712.

[0115] Step 3-2: Using the artificial potential field method, all obstacles in the operating environment are regarded as high potential points. If the robot operates near an obstacle, it will be affected by the repulsive force and move away from the obstacle. The position of the s-th obstacle is expressed as The relative displacement vector between the i-th robot and the s-th obstacle is expressed as:

[0116]

[0117] where s = 1, 2,..., M. The sensor device installed on the robot can obtain the position information of the obstacle, but only knows the specific position of the obstacle when the robot is close enough to the obstacle.

[0118] Step 3-3: Define the repulsive force potential function generated by the s-th obstacle on the i-th robot approaching it as a non-negative differentiable function The repulsive force function of the obstacle on the robot is defined as the negative gradient along the potential function , that is The repulsive potential function satisfies: i) when , the repulsive force gradually increases, that is where is the minimum obstacle avoidance distance of the s-th obstacle; ii) when , the repulsive force gradually weakens to the minimum value, where is the maximum action distance threshold of the repulsive force.

[0119] S4: Design the first Lyapunov function candidate, and use the fuzzy logic system to approximate the nonlinear terms in the dynamics; design the Lyapunov-Krasovskii functional to eliminate the influence of time delay on the controller design;

[0120] Step 4-1: Define the bipartite formation error as e p，i (t) = p i - s i p0 - h p，i , e q，l (t) = q i - s i q0 - h q，i , and e i (t) = e p，i (t) + e q，i (t). The distributed bipartite consensus error is defined as:

[0121]

[0122] According to the defined bipartite formation error, design the first Lyapunov function candidate as shown below:

[0123]

[0124] where and satisfies e = e p + e q .

[0125] Step 4-2: Give the following lemma to approximate the nonlinear function f i (p i , q i , t) in the follower dynamics.

[0126] Lemma 1: For any continuous function r(x) defined on the compact set Ω r , there exists a fuzzy logic system Θ *T Φ(x), which can approximate the function r(x) with arbitrary precision and satisfies

[0127]

[0128] where Θ * is the ideal weight matrix, Φ(x) is the fuzzy basis function, and ε(x) is the approximation error.

[0129] With the aid of Lemma 1, we can obtain Then, by calculating the derivative of V1(t), we can get:

[0130]

[0131] By using Young's inequality, we can get:

[0132]

[0133] where ζ i > 0 is the design parameter, is the square of the Frobenius norm of the matrix , and σ e = exp(στ max ), with σ > 0.

[0134] Step 4 - 3: For the considered time - delay function g i (p i (t - τ i ), q i (t - τ i ), assume it satisfies:

[0135] ||g i (p i (t - τ i ), q i (t - τ i ))|| 2 ≤ ||ψ i (t - τ i )|| 2 ||δ i (p i (t - τ i ), q i (t - τ i ))|| + δ0(t - τ i ) (34)

[0136] where δ i (·) is a known function, and δ0(·) is a bounded function satisfying 0 < δ0(·) ≤ δ m . Design the Lyapunov - Krasovskii functional as follows:

[0137]

[0138] where Z i (z) = exp(σz)||ψ i (z)|| 2 ||δ i (p i (z), q i (z))||, σ > 0. Taking the derivative of V2(t) gives:

[0139]

[0140] S5: Give the designed control strategy and adaptive update law to achieve the fast finite-time convergence of the error system and ensure the obstacle avoidance ability of the designed formation.

[0141] Step 5-1: To achieve the desired fast finite-time bipartite formation control and obstacle avoidance ability, the fuzzy adaptive control strategy is designed as follows:

[0142]

[0143] The specific expressions of the elements in the formula are:

[0144]

[0145] where λ max (·) represents the maximum eigenvalue of the matrix; for a vector x = [x1, x2,..., x n T , <x> a Defined as (x > a = [|x1| a sgn(x1), |x2| a sgn(x2),..., |x n | a sgn(x n )] T ; κ1 > λ max (H) + σ e / 2 + 2 and κ2 > 0 are control gains; is the repulsive force intensity parameter; 0 < β < 1 is a design parameter; is the actuator fault compensation term, which satisfies for estimating the unknown constant for updating The adaptive law of

[0146]

[0147] where γ2 > 0 is a design parameter.

[0148] Step 5 - 2: The total Lyapunov function is designed as follows:

[0149]

[0150] It can be seen through calculation that the derivative of V(t) satisfies:

[0151]

[0152] To handle the saturation constraint suffered by the actuator, the following relationship can be established:

[0153]

[0154] In addition, there are and which hold. With the help of Young's inequality, the following can be obtained:

[0155]

[0156] Substituting the above inequalities into (40), the following can be obtained:

[0157]

[0158] where

[0159] Step 5 - 3: The following lemma is given to handle the power terms:

[0160] Lemma 2: For any two real variables a1 and a2, the following inequality holds:

[0161]

[0162] wherein are all positive design constants.

[0163] Lemma 3: For real number c i , i = 1, 2,..., N and constant 0 < χ ≤ 1, the following inequality can be established

[0164]

[0165] Applying Lemma 2, let then the following inequality holds:

[0166]

[0167] wherein Applying Lemma 3, we can obtain In addition, the following results can also be obtained:

[0168]

[0169] where F = F1 + 4F2, χ1 = min{γ2 / 2, σ / 2, (2κ1 - σ e - 4) / λ max (H)-1, (γ1 - 1) / 2, 1},

[0170] Step 5 - 4: The following lemma is given to illustrate that the designed controller can achieve fast finite - time stability:

[0171] Lemma 4: For the system If there exists a scalar function V(x) > 0 satisfying:

[0172]

[0173] wherein then the system is actually fast finite - time stable. In addition, the stability time T0 of the system can be calculated by the following formula:

[0174]

[0175] where t0 is the initial time of the system, and the constant 0 < c0 < 1.

[0176] According to Lemma 4, it can be known that the designed control strategy can make the error system achieve fast finite - time stability, which means that fast finite - time bipartite formation control of the mobile robot swarm can be realized. In addition, according to Lemma 4, the stability time of the system is:

[0177]

[0178] where the constant \(0 < c < 1\).

[0179] Step 5 - 5: When the robot approaches an obstacle, that is, when it reaches the range of the repulsive force, according to the defined relative displacement vector, the following dynamics can be obtained:

[0180]

[0181] Design the energy function between the \(i\)-th robot and the \(\theta\)-th obstacle as Taking the derivative of it, we can get:

[0182]

[0183] Let It can be seen that all signals in \(\Delta\) i,θ are bounded. In addition, it can be concluded that is also bounded. When the \(i\)-th robot approaches the \(\theta\)-th obstacle along the gradient direction of the repulsive potential, that is, when we can get:

[0184]

[0185] Therefore, when the \(i\)-th robot is close enough to the \(\theta\)-th obstacle, the following results hold:

[0186]

[0187] That is to say, Multiply both sides of by \(\exp(-\omega\) i，θ t), we can get:

[0188]

[0189] Integrating the above inequality from \(t\) to \(t_0\) gives:

[0190] E i，θ (t) > \(\exp(\omega\) i，θ t - \(\omega\) i，θ t_0)E i，θ (t_0) (60)

[0191] This means the following results hold:

[0192]

[0193] That is to say, once the robot approaches the obstacle within a certain range, the designed obstacle avoidance strategy can make the robot move away from the obstacle.

[0194] The following is a further description of the present invention with a preferred embodiment.

[0195] Figure 2 The shown mobile robot system has the following parameters: R i = 0.15 m, L i = 0.75 m, d i = 0.2 m, m 1,i = 0.4, m 2，i = 0.02, c i = 1.2, d 1，i = d 2，i = 5. The leader reference signal is set as f0(t) = [0.1sin(0.15t), 0.3cos(0·15t)] T . The time-delay function is given as g i (p i (t - τ i ), q i (t - τ i )) = 0.25p i (t - τ i )cos(t - τ i ) + 0.3q i (t - τ i )sin(t - τ i ). The time-delay is set as τ1 = 0.75, τ1 = 0.86, τ1 = 0.89, τ1 = 0.92, τ1 = 0.95, τ6 = 1. The initial position of each mobile robot is given as p1(0) = [0, -30] T , p2(0) = [0, -20] T , p3(0) = [0, -10] T , p4(0) = [0, 10]T, p5(0) = [0, 20] T , p6(0) = [0, -30] T , the initial velocity q i (0) = [0, 0] T , i = 1, 2, 3, 4, 5, 6. The initial position of the leader is set as p0(0) = [5.5, 4.5] T , the initial velocity of the leader is set as q0(0) = [1, 1] T . The positions of the obstacles in the operating environment are set as and The formation offset vectors of the mobile robots numbered 1, 2, 3 are given as:

[0196]

[0197] The formation offset vectors of mobile robots numbered 4, 5, and 6 are given as:

[0198]

[0199] The influence of the actuator bias fault on the robot is given as The artificial potential field function is set as:

[0200]

[0201] The repulsive force function is given as:

[0202]

[0203] To verify the fast finite-time bipartite formation obstacle avoidance control strategy proposed by the present invention, the parameters are set as γ1 = 7, κ1 = 10, κ2 = 8, β = 0.4, σ = 1, ζ i = 1.

[0204] According to the above simulation conditions, the system is simulated to verify the formation performance and obstacle avoidance ability of the system.

[0205] Figure 1 This is the flowchart of the control method of the present invention. Figure 2 This is the schematic diagram of the structure of the wheeled mobile robot for the verification algorithm of the present invention. Figure 3 This is the communication topology diagram inside the mobile robot cluster for the verification algorithm of the present invention. Figure 4 These are the motion trajectories of six follower robots and the leader robot from 0s to 50s, as well as the position snapshots of the mobile robot at 0s, 8s, 18s, and 40s under fast finite-time bipartite formation control. It can be seen that the six follower robots can complete their respective formation tasks within a fast finite time and form two opposing groups. Each group of follower robots can achieve a triangular formation according to the designed formation offset vector and can avoid obstacles. Figure 5 These are the running trajectories of the robot when approaching an obstacle. When approaching the obstacle at a certain distance, the embedded artificial potential field obstacle avoidance mechanism will generate a repulsive force to make the robot move away from the obstacle. Figure 6 These are the evolution trajectories of the position vectors of the follower robot and the leader in the X and Y directions. It can be seen that the two queues can exhibit a complete bipartite confrontation phenomenon. Figure 7 These are the evolution trajectories of the speed errors of the follower robots. Due to the obstacle avoidance behavior, the errors have two mutations even after stabilization. But when the obstacle avoidance behavior is completed, the speed errors of all follower robots finally stabilize near the origin.

[0206] The above results show that the proposed time-delay mobile robot swarm fast finite-time bipartite formation obstacle avoidance control method can enable the robots to operate safely in the obstacle environment without colliding with obstacles, and can achieve fast finite-time stability, with good transient performance, and has strong robustness to time-delay, faults, system uncertainties, etc.

[0207] Of course, the above description is not a limitation of the present invention, and the present invention is not limited to the above examples. Changes, modifications, additions or substitutions made by those skilled in the art within the essence of the present invention should also fall within the protection scope of the present invention.< / x> ​< / x> < / x> ​

Claims

1. A fast finite-time binary formation obstacle avoidance control method for a time-delay mobile robot cluster, characterized in that: The following steps are involved: S1: Based on the kinematic and dynamic models of the mobile robot, combined with the variable transformation method, the state space expression of the mobile robot with time lag and unknown nonlinear terms is constructed, and the expected leader trajectory signal is given; S2: Establish the communication relationship between the follower robot and the leader signal with the help of topological graph theory, and define the desired bipartite formation control goal based on the constructed mobile robot state space expression; S3: Model the actuator in the actual scenario, consider the actuator model under saturation constraints and faults, and construct the virtual potential field obstacle avoidance function and obstacle avoidance repulsion based on the artificial potential field method; S4: Design the first Lyapunov function candidate and use fuzzy logic system to approximate the nonlinear terms in the dynamics; design the Lyapunov-Krasovsky functional to eliminate the influence of time delay on the controller design; S5: The designed control strategy and adaptive update law are given to achieve fast finite-time convergence of the error system and ensure the obstacle avoidance capability of the designed formation; The operation of S1 is specifically as follows: The kinematic and dynamic equations of the wheeled mobile robot are established, and the model conversion and coordinate transformation are performed. At the same time, the influence of time delay on the system is considered, and the state space expression of the mobile robot system is obtained as follows: where p i (t), q i (t),u i (t) represents the position, velocity and control input of the ith robot; f i (p i ,q i ,t) represents a heterogeneous nonlinear vector-valued function with uncertainty; g i (p i (t-τ i ),q i (t-τ i )) represents a smooth vector-valued function with time lag; τ i is an unknown bounded time delay, satisfying 0<τ i ≤τ max ; The reference trajectory or path of the leader is given as follows: where p0(t) and q0(t) are the position and velocity vectors of the leader, respectively, and f0(t) represents a bounded function set as the reference signal; The operation of S3 is specifically as follows: The actual actuator output is represented as: where μ i (t) is the original control input signal; or i (t) is the approximate error, is the actuator bias fault; using the artificial potential field method, the position of the sth obstacle is expressed as The relative displacement vector between the ith robot and the sth obstacle is expressed as: Where s = 1, 2, ..., M; the repulsive potential function generated by the s-th obstacle on the i-th robot close to it is defined as a non-negative differentiable function The repulsive force function of the obstacle on the robot is defined as the potential function The negative gradient of Repulsive potential function Satisfy: i) When When , the repulsive force gradually increases, that is in is the minimum obstacle avoidance distance of the sth obstacle; ii) when When , the repulsive force gradually weakens to the minimum value, is the maximum distance threshold of the repulsive force; The operation of S4 is specifically as follows: Define the binary formation error as e p,i (t) = p i -s i p0-h p,i , e q,i (t) = q i -s i q0-h q,i , e i (t) = e p,i (t)+e q,i (t); The distributed binary consensus error is defined as: According to the defined bisection formation error, the first Lyapunov function candidate is designed as follows: in And satisfy e=e p +e q ; Use fuzzy logic system to approximate nonlinear functions, and we have For the considered time delay function g i (p i (t-τ i ),q i (t-τ i ), assuming that it satisfies: ||g i (p i (t-τ i ),q i (t-τ i ))|| 2 ≤||ψ i (t-τ i )|| 2 ||δ i (p i (t-τ i ),q i (t-τ i ))||+δ0(t-τ i ) (7) where δ i (·) is a known function, δ0(·) satisfies 0<δ0(·)≤δ m The bounded function of ; the Lyapunov-Krasovsky functional is designed as follows:

2. The fast finite-time binary formation obstacle avoidance control method for a time-delay mobile robot cluster according to claim 1, characterized in that: The operation of S2 is specifically as follows: Represent the communication relationship between follower robots and leader signals as a signed topological graph It consists of a set of vertices Edge Collection and the adjacency matrix The degree matrix associated with the leader reference signal is defined as The Laplacian matrix is ​​defined as Among them l ij =-a ij ,i≠j and The information interaction matrix is ​​expressed as in Topology There exists a spanning tree with the leader as the root node; if the vertex set can be decomposed into satisfying and Two subsets of vertices and Topology is structurally balanced; if υ i and j Belong to the same subset of vertices or Then a ij > 0, otherwise a ij <0; In addition, define a diagonal matrix If the i-th robot is in a cooperative relationship with the leader, then s i =1, otherwise s i = -1; make If for any bounded initial value r i (t0) there is a distributed controller u i (t) enables each follower robot to maintain the displacement of the set offset vector with the leader after a bounded stable time T0, then the mobile robot cluster can achieve the desired fast finite-time bisection formation control, that is: in represents the offset vector between the ith robot and the leader, and σ>0 is the allowable formation error limit.

3. The fast finite-time binary formation obstacle avoidance control method for a time-delay mobile robot cluster according to claim 1, characterized in that: The operation of S5 is specifically as follows: In order to achieve the desired fast finite-time binary formation control and obstacle avoidance capabilities, the fuzzy adaptive control strategy is designed as follows: The specific expressions of each element in the formula are: where λ max (·) represents the maximum eigenvalue of the matrix; for a vector <x> a Defined as κ1>λ max (H)+σ e / 2+2 and κ2>0 are control gains; is the repulsive force strength parameter;< / x> 0<β<1 is the design parameter; is the actuator fault compensation term, which satisfies γ1>1; Used to estimate unknown constants For update The adaptive law is: Where γ2>0 is the design parameter; The control strategy designed based on the Lyapunov function is quickly verified for finite-time stability, and it is ensured that the designed formation has obstacle avoidance capability.

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