Fast finite time two-way formation obstacle avoidance control method for time-delay mobile robot cluster

Abstract

The invention relates to a fast finite time two-way formation obstacle avoidance control method for a time-delay mobile robot cluster, and belongs to the technical field of control engineering. In order to improve the convergence speed of the formation pattern, the invention provides a fast finite time formation control method, which can enable a mobile robot cluster to realize an expected two-way formation pattern within fast finite time; considering that the output of an actuator in an actual system is usually limited and is easily influenced by faults, an anti-saturation technology and a fault-tolerant control strategy are fused in the design of the control method; because the prior map knowledge of the operation environment does not exist, an obstacle avoidance mechanism is embedded in the control method by using an artificial potential field technology so as to ensure the safe operation of the robot formation; an improved Lyapunov-Crasovsky functional is constructed to eliminate the influence of time delay on the system. According to the method, the artificial potential field technology is effectively utilized to construct the virtual potential field, the influence of time lag on the system is eliminated on the premise that saturation constraint conditions are not violated and an actuator has a deviation fault, it is ensured that a mobile robot cluster can achieve expected two-way formation within a fast finite time and avoid collision with obstacles, and the system performance is improved. And the response speed and the security of the system are improved.

Description

Technical Field

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

[0002] Industrial modernization has promoted the rapid development of mobile robot research. The research trend at home and abroad has also developed from single robots to multi-robot cluster collaborative control. Through the mutual cooperation between robots, many complex and dangerous tasks can be completed. Formation control is a fundamental problem in the field of multi-robot collaborative control. Its goal is to enable the entire robot cluster to achieve the desired formation through distributed information perception between robots.

[0003] When mobile robots perform formation tasks, they usually have no prior knowledge of the map, which means that they may be affected by obstacles during operation. In order 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, efficiency, and strong real-time performance. It simulates potential fields such as gravity fields and electromagnetic fields, sets each obstacle as a high potential point, and then constructs a virtual potential field function. When the robot enters the potential field, a repulsive force will be generated to make it bypass the obstacle.

[0004] Most of the current research on formation control assumes that there is only one cooperative relationship between robots, that is, all communication links between robots are non-negative. However, in reality, robots often have adversarial relationships in addition to cooperative relationships, similar to the interaction between opponents and teammates in multiplayer games. Therefore, robots are divided into two groups according to the positive and negative weights in the signed topological graph, and bipartite formation control is introduced to describe the formation control of robot clusters running in cooperative adversarial interaction networks.

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

[0006] The convergence speed of formation error is one of the key indicators for evaluating its performance. In obstacle avoidance problems and actual formation construction, the convergence speed of infinite time cannot guarantee good transient performance. In order 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 practical mobile robot swarm systems under the influence of time delay. Summary of the invention

[0008] The main purpose of the present invention is to: in view of the shortcomings and gaps in the prior art, the present invention provides a fast finite-time binary formation obstacle avoidance control method for a mobile robot cluster, while considering the communication topology of coexistence of cooperative and confrontational interactions, and utilizing the advantages of the artificial potential field method in terms of efficiency and convenience to construct a virtual potential field function to achieve the desired obstacle avoidance behavior. In the case of paranoid faults, saturation constraints in the actuator and the influence of time lag in the system, the fast finite-time binary formation formation of the mobile robot cluster can be constructed, thereby improving the response speed and robustness of the system.

[0009] In order to achieve the above objectives, a fast finite-time binary formation obstacle avoidance control method for a time-delay mobile robot cluster studied in the present invention comprises the following steps:

[0010] 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;

[0011] 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;

[0012] 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;

[0013] 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;

[0014] 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.

[0015] The fast finite time binary formation obstacle avoidance control method for a time-delay mobile robot cluster as described above is characterized in that the operation of S1 is specifically as follows:

[0016] 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:

[0017]

[0018] where p i (t), q i (t),u i (t) represents the position, velocity and control input of the ith robot respectively; 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:

[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] The fast finite time binary formation obstacle avoidance control method for a time-delay mobile robot cluster as described above is characterized in that the operation of S2 is specifically as follows:

[0022] 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 picture 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 i and jBelong 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. If for any bounded initial value r i (t0) there is a distributed controller u i (t) enables each follower robot to maintain an offset vector with the leader after a bounded stabilization time T0, then the mobile robot cluster can achieve the desired fast finite-time bisection formation control, that is,

[0023]

[0024] in represents the offset vector between the ith robot and the leader, and σ>0 is the allowable formation error limit.

[0025] The fast finite time binary formation obstacle avoidance control method for a time-delay mobile robot cluster as described above is characterized in that the operation of S3 is specifically as follows:

[0026] The actual actuator output is represented as:

[0027]

[0028] where μ i (t) is the original control input signal; ε 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 i-th robot and the s-th obstacle is expressed as:

[0029]

[0030] Where s = 1, 2, ..., M. The repulsive potential function generated by the sth obstacle on the ith 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 Satisfies: 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, where is the maximum distance threshold of the repulsive force.

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

[0032] 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).

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

[0034]

[0035] According to the defined bisection formation error, the first Lyapunov function candidate is designed as follows:

[0036]

[0037] in And satisfy e=e p +e q . Using fuzzy logic system to approximate nonlinear functions, we have For the considered time delay function g i (p i (t-τ i ), q i (t-τ i ), assuming that 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, δ0(·) satisfies 0<δ0(·)≤δ m The Lyapunov-Krasovsky functional is designed as follows:

[0040]

[0041] The fast finite time binary formation obstacle avoidance control method for a time-delay mobile robot cluster as described above is characterized in that the operation of S5 is specifically as follows:

[0042] 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:

[0043]

[0044] The specific expressions of each element in the formula are:

[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 strength parameter; 0<β<1 is the design parameter; is the actuator fault compensation term, which satisfies To estimate unknown constants For update The adaptive law is:

[0047]

[0048] Among them, γ2>0 is the design parameter.

[0049] The control strategy designed based on the Lyapunov function is quickly verified for finite-time stability and ensures the obstacle avoidance capability of the designed formation.

[0050] Compared with the prior art, the present invention has the following beneficial effects:

[0051] 1) Aiming at the mobile robot system affected by time lag, the present invention considers the situation where there is cooperative confrontation interaction between robots at the same time, and constructs a fast finite-time binary formation control strategy with obstacle avoidance capability, which effectively improves the response speed of the system;

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

[0053] 3) The present invention avoids the influence of actuator failure and saturation constraints on controller design and system operation. At the same time, based on the improved Lyapunov-Krasovsky functional, it eliminates the influence of time delay on system performance, can avoid singularity problems, and improve the robustness of the system.

[0054] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the embodiments of the present invention are specifically cited below and described in detail with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for use in the embodiments are briefly introduced below. It should be understood that the following drawings only show certain embodiments of the present invention and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other related drawings can be obtained based on these drawings without creative work.

[0056] Figure 1 Flow chart of the control method in the embodiment.

[0057] Figure 2 Schematic diagram of the structure of the wheeled mobile robot in the embodiment.

[0058] Figure 3 Schematic diagram of the communication topology within the mobile robot cluster in the embodiment.

[0059] Figure 4 Snapshots of the motion trajectory and position 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 is the evolution law of the position vector of the robot in the embodiment.

[0062] Figure 7 : is the speed error evolution law of the robot in the embodiment. DETAILED DESCRIPTION

[0063] The present invention will be further described in detail below in conjunction with the accompanying drawings.

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

[0065] The present application embodiment provides a fast finite time binary formation obstacle avoidance control method for a time-delay mobile robot cluster, see Figure 1 , including the following steps:

[0066] 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;

[0067] 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;

[0068] 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;

[0069] 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;

[0070] 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.

[0071] 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. The specific steps are as follows.

[0072] Step 1-1: Give a schematic diagram of the wheeled mobile robot. Figure 2 As shown, the kinematic and dynamic equations of the wheeled mobile robot are established as follows:

[0073]

[0074] In the formula (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; i =[ι l，i , ι r，i ] T , ι l，i and r，ι are the control torques of the left and right wheels respectively; J i (η i ) is the rotation matrix; M i is a symmetric positive definite inertia matrix; is the Coriolis centripetal acceleration matrix; D i is the damping matrix. The specific expression of the matrix mentioned in the robot model (12) is as follows:

[0075]

[0076]

[0077] Where R i is the wheel radius; L i Half the distance between the two wheels; m 1，i and m 2，i is the inertial mass; c i is the Coriolis centripetal coefficient; d 1，i and d 2，i is the damping coefficient.

[0078] Step 1-2: Before performing error conversion, it should be noted that the wheeled mobile robot satisfies the following no-slip kinematic constraints:

[0079]

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

[0081]

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

[0083]

[0084] in 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] Among them (p x，i , p y，i ) is the position coordinate of the robot center point, d i Represents a point (x i ,y i ) to 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 reformulated as:

[0088]

[0089] in

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

[0091]

[0092] where p i (t), q i (t),u i (t) represents the position, velocity and control input of the ith robot respectively; 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:

[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 artificially set as the reference signal.

[0095] S2: Use topological graph theory to establish the communication relationship between the follower robot and the leader signal, and define the desired binary formation control goal based on the constructed mobile robot state space expression. The specific steps are as follows.

[0096] Step 2-1: The information exchange between robot clusters can be represented by a signed topology graph To indicate that Represents a set of vertices; represents the edge set; represents the adjacency matrix. If the i-th robot can receive information from the j-th robot, then there is a ij ≠0, otherwise a ij = 0. In addition, if a ij > 0 means that the relationship between the ith robot and the jth robot is cooperative. ij <0 means that the relationship between the ith robot and the jth robot is confrontational and there is no information transmission between them, that is, a ii = 0. The degree matrix associated with 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 means that the i-th robot and the leader are in a cooperative relationship. i <0 means that the i-th robot is in a confrontational relationship with the leader. The Laplace matrix is ​​defined as in The information interaction matrix is ​​expressed as in picture There is a spanning tree with the leader as the root node.

[0097] Step 2-2: If the vertex set can be decomposed into satisfying and Two subsets of vertices and Topology is structurally balanced. If 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.

[0098] Step 2-3: The desired control goal is to design a distributed control strategy so that: i) all follower robots can achieve the desired binary formation based on the leader's reference trajectory in a fast finite time; ii) the effects of actuator failure, saturation constraints and time delay on the system can be eliminated; iii) and each robot can be prevented from colliding with any obstacles in the operating environment. If for any bounded initial value r i (t0) there is a distributed controller u i (t) enables each follower robot to maintain a set offset vector with the leader after a bounded stabilization time T0, then the mobile robot cluster can achieve the desired fast finite-time bisection formation control, that is,

[0099]

[0100] in represents the offset vector between the ith robot and the leader, and σ>0 is the allowable formation error limit. It should be noted that h i (t) and r i (t) have the same dimensions, which means that r i Each term in (t) has its corresponding offset vector.

[0101] 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. The specific steps are as follows.

[0102] Step 3-1: In engineering practice, due to power supply and technical limitations, the output of the actuator is usually limited to a certain extent, and the actuator may fail during operation due to factors such as wear and aging. Therefore, the output of the actuator affected by saturation constraints and failures is modeled as follows:

[0103]

[0104] where μ i (t) is the original control input signal; is the actuator bias fault, which 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 associated with physical limitations, respectively. To handle actuator saturation, the saturation constraint is refactored as:

[0107]

[0108] in or ε i (t) is the approximate error. Using the mean value theorem we can get:

[0109]

[0110] in make Then the actual actuator output can be reformulated as:

[0111]

[0112] And the following results hold:

[0113]

[0114] So we know that Γ i The maximum eigenvalue of is less than 0.712.

[0115] Step 3-2: Using the artificial potential field method, all obstacles in the operating environment are considered as high potential points. If the robot runs near an obstacle, it will be affected by the repulsive force and move away from the obstacle. The position of the sth 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 location information of the obstacle, but the specific location of the obstacle is known only when the robot is close enough to the obstacle.

[0118] Step 3-3: Define the repulsive potential function generated by the sth 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 potential function The negative gradient of Repulsive potential function Satisfies: 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, where is the maximum distance threshold of the repulsive force.

[0119] 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;

[0120] Step 4-1: Define the binary 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 binary consensus error is defined as:

[0121]

[0122] According to the defined bisection formation error, the first Lyapunov function candidate is designed as follows:

[0123]

[0124] in And satisfy e=e p +e q .

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

[0126] Lemma 1: For the definition on the compact set Ω r For any continuous function r(x) on *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] By using Lemma 1, we can get Then, calculating the derivative of V1(t) yields:

[0130]

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

[0132]

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

[0134] Step 4-3: For the time lag function g considered i (p i (t-τ i ), q i (t-τ i ), assuming that 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, δ0(·) satisfies 0<δ0(·)≤δ m The Lyapunov-Krasovsky functional is designed 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) we get:

[0139]

[0140] 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.

[0141] Step 5-1: 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:

[0142]

[0143] The specific expressions of each element 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 strength parameter; 0<β<1 is the design parameter; is the actuator fault compensation term, which satisfies To estimate unknown constants For update The adaptive law is:

[0146]

[0147] Among them, γ2>0 is the design parameter.

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

[0149]

[0150] Through calculation, we can know that the derivative of V(t) satisfies:

[0151]

[0152] To handle the saturation constraints imposed on the actuator, the following relationship can be established:

[0153]

[0154] In addition, there are and Established. With the help of Young's inequality, we can get:

[0155]

[0156] Substituting the above inequalities into (40), we can obtain:

[0157]

[0158] in

[0159] Step 5-3: Give the following lemma to deal with power terms:

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

[0161]

[0162] in are all positive design constants.

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

[0164]

[0165] Apply Lemma 2, let Then the following inequality holds:

[0166]

[0167] in Applying Lemma 3, we can get In addition, the following results can 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: Give the following lemma to show 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 such that:

[0172]

[0173] in System It is actually fast finite time stable. In addition, the system's stable time T0 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, the designed control strategy can make the error system achieve fast finite time stability, which means that fast finite time binary formation control of mobile robot clusters can be achieved. 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 the obstacle, that is, reaches the range of the repulsive force, the following dynamics can be obtained based on the defined relative displacement vector:

[0180]

[0181] The energy function between the i-th robot and the θ-th obstacle is designed as Taking its derivative, we can get:

[0182]

[0183] make It can be seen that Δ i,θ All signals in are bounded. In addition, it can be concluded that is also bounded. When the i-th robot approaches the θ-th obstacle along the gradient direction of the repulsive potential, that is, when When , we can get:

[0184]

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

[0186]

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

[0188]

[0189] Integrating the above inequality from t to t0 yields:

[0190] E i，θ (t)>exp(ω i，θ t-ω i，θ t0)E i，θ (t0) (60)

[0191] This means that the following holds true:

[0192]

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

[0194] The present invention is further described below with a preferred embodiment.

[0195] Figure 2 The mobile robot system shown has the following parameters: R i =0.15m, L i =0.75m,d i =0.2m,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 to f0(t) = [0.1sin(0.15t), 0.3cos(0·15t)] T The time lag function is given by 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 to 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 , initial velocity q i (0) = [0, 0] T , i = 1, 2, 3, 4, 5, 6. The initial position of the leader is set to p0(0) = [5.5, 4.5] T , the leader’s initial velocity is set to q0(0) = [1, 1] T The position of obstacles in the operating environment is set to and The formation offset vectors of mobile robots numbered 1, 2, and 3 are given as:

[0196]

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

[0198]

[0199] The robot is affected by the actuator paranoid fault given by The artificial potential field function is set as:

[0200]

[0201] The repulsive force function is given by:

[0202]

[0203] In order to verify the fast finite time two-division formation obstacle avoidance control strategy proposed in this 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 system's formation performance and obstacle avoidance capabilities.

[0205] Figure 1 Flow chart of the control method of the present invention. Figure 2 This is a schematic diagram of the structure of a wheeled mobile robot for verifying the algorithm of the present invention. Figure 3 This is the communication topology diagram within the mobile robot cluster used to verify the algorithm of the present invention. Figure 4 The following are the motion trajectories of the six follower robots and the leader robot from 0s to 50s, as well as the position snapshots of the mobile robots at 0s, 8s, 18s, and 40s under the fast finite time binary 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 realize a triangular formation according to the designed formation offset vector and avoid obstacles. Figure 5 This is the running trajectory of the robot when it approaches an obstacle. When it approaches a certain distance from the obstacle, the embedded artificial potential field obstacle avoidance mechanism will generate a repulsive force to keep the robot away from the obstacle. Figure 6 The evolution trajectory of the position vectors of the follower robot and the leader in the X and Y directions. It can be seen that the two groups of queues can exhibit a complete binary confrontation phenomenon. Figure 7 This is the velocity error evolution trajectory of the follower robot. Due to the obstacle avoidance behavior, the error still has two mutations after stabilization. However, when the obstacle avoidance behavior is completed, the velocity errors of all follower robots are finally stabilized near the origin.

[0206] The above results show that the fast finite-time binary formation obstacle avoidance control method for a time-delay mobile robot cluster proposed in the present invention can enable the robots to operate safely in an obstacle environment without colliding with obstacles, and can achieve fast finite-time stability, has good transient performance, and is highly robust to time delays, faults, and system uncertainties.

[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 technicians in this technical field within the essential scope 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.

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 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 respectively; 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.

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 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 picture 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 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. 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: where represents the offset vector between the i-th robot and the leader, and 口>0 is the allowable formation error bound.

4. 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 S3 is specifically as follows: The actual actuator output is represented as: where μ i (t) is the original control input signal; or ε 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 i-th robot and the s-th obstacle is expressed as: Where s = 1, 2, ..., M. The repulsive potential function generated by the sth obstacle on the ith 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.

5. 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 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 . Using fuzzy logic system to approximate nonlinear functions, 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-t i ),q i (t-t i ))‖ 2 ≤‖ψ i (t-t i )‖ 2 ‖d i (p i (t-t i ),q i (t-t i ))‖+δ0(t-τ i ) (8) where δ i (·) is a known function, δ0(·) satisfies 0<δ0(·)≤δ m The Lyapunov-Krasovsky functional is designed as follows:

6. 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 = [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 strength parameter;< / x> < / x> 0<β<1 is the design parameter; is the actuator fault compensation term, which satisfies Used to estimate unknown constants For update The adaptive law is: Among them, 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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