Multi-mobile robot system formation trajectory analysis method based on U-K theory

Abstract

The invention relates to the field of intelligent robot control, in particular to a multi-mobile-robot system formation trajectory analysis method based on a U-K theory, and the method comprises the steps: S1, building a dynamic model of a single mobile robot through employing a Newton-Euler method; s2, establishing a communication topological graph among the mobile robots, wherein the given communication topological graph is a directed weighted graph with a directed spanning tree; s3, an expected formation control target is rewritten into a constraint equation form, and servo control force needed by the mobile robot system under incomplete constraint to complete the expected formation control target is obtained in combination with a U-K equation; and S4, substituting the obtained servo control force into the dynamic model of the mobile robot, and deducing the formation track of the multi-mobile robot system under the condition of no leader. The method analyzes and verifies that the motion trail of the multi-mobile-robot system after formation control has three possible conditions, namely stabilization to one point, linear motion and circular motion, under the condition of no leader.

Description

Technical Field

[0001] The invention relates to the field of intelligent robot control, and in particular to a multi-mobile robot system formation trajectory analysis method based on UK theory. Background Art

[0002] In recent years, the formation control problem of multi-mobile robot systems has been a hot topic for many researchers. Compared with single robots, multi-mobile robot systems have been widely used in civil and military fields due to their high fault tolerance, low cost, and high flexibility. However, mobile robots are a type of constrained system subject to non-holonomic constraints, and the lateral movement of the robot is restricted. Therefore, how to study the formation control problem of multi-mobile robot systems subject to non-holonomic constraints has gradually become a hot issue.

[0003] At present, there are many methods to study the formation control problem of multiple mobile robots, such as the leader-follower method, the artificial potential field method, the behavior-based method, etc. Among them, the most commonly used is the leader-follower method, which is divided into the leaderless case and the leader-follower case. Many papers and research inventions are discussed based on the leaderless case, but because the trajectory movement of the formation in the leaderless case is restricted, it cannot move according to any expected motion trajectory and cannot meet the diverse task requirements. Therefore, many researchers have expanded the formation control problem in the leaderless case to the formation control problem in the leader-follower case. At this time, it is only necessary to assign any expected motion trajectory to the movement of the leader. After the followers and the leader achieve the expected formation shape, the entire robot system can move together according to any expected trajectory, so as to better meet the actual task requirements.

[0004] However, in the currently published papers and patents, no researchers have studied the trajectories that a multi-robot system can follow when achieving formation control without a leader, and what factors will affect the formation's movement trajectory. Currently, this field still needs to be studied and developed. Summary of the invention

[0005] The present invention provides a multi-mobile robot system formation trajectory analysis method based on UK theory, aiming to solve the technical gap problem existing in the current multi-mobile robot formation control research.

[0006] The present invention provides a multi-mobile robot system formation trajectory analysis method based on UK theory, comprising the following steps:

[0007] S1. According to the motion of the mobile robot system in two-dimensional space and the system motion parameters of the mobile robot system, the dynamic model of a single mobile robot is established using the Newton-Euler method;

[0008] S2. Combine algebraic graph theory to establish a communication topology graph between mobile robots in the absence of a leader, and the given communication topology graph is a directed weighted graph with a directed spanning tree;

[0009] S3. Rewrite the expected formation control goal into the form of constraint equations, and combine the UK equations to obtain the servo control force required by the mobile robot system subject to nonholonomic constraints to achieve the expected formation control goal;

[0010] S4. Substitute the obtained servo control force into the dynamic model of the mobile robot and derive the formation trajectory of the multi-mobile robot system in the absence of a leader.

[0011] As a further improvement of the present invention, the step S1 specifically includes:

[0012] According to a multi-mobile robot system consisting of n four-wheeled mobile robots, each four-wheeled mobile robot consists of two rear wheels independently driven by motors and two front wheels that support and guide the movement of the robot. The generalized coordinates of the mobile robot are q = (x, y, θ) T Indicates that u r and u l The direction that drives the vehicle forward is positive;

[0013] In the robot's forward direction, Newton's laws of motion give:

[0014]

[0015] Which meets

[0016] The speed of the robot is expressed as:

[0017]

[0018] Taking the derivative of the robot's velocity with respect to time, we get:

[0019]

[0020] The above calculation formula can be obtained:

[0021]

[0022] The robot has nonholonomic constraints, which can be expressed as:

[0023]

[0024] Taking the first-order differential with respect to time, we get:

[0025]

[0026] From the angular momentum theorem we can get:

[0027]

[0028] Right now

[0029]

[0030] The above calculation formula can be written into matrix form:

[0031]

[0032] The dynamic equation of the system is expressed as:

[0033]

[0034] in

[0035]

[0036] Where x is the x-axis component of the robot's center of mass coordinates; y is the y-axis component of the robot's center of mass coordinates; θ is the angle between the robot's velocity direction and the positive direction of the x-axis; m is the mass of the robot; d is the radius of the robot's wheels; J is the robot's moment of inertia around the center of mass; l is the lateral distance between the robot's center of mass and the center of the wheel; v is the robot's velocity; u r is the driving torque of the motor acting on the right rear wheel; u l is the driving torque of the motor acting on the left rear wheel; f r is the driving force generated by the right rear wheel of the robot; f l is the driving force generated by the left rear wheel of the robot; M is the mass matrix of the robot, F is the external force matrix of the robot, and F c is the robot’s control force matrix.

[0037] As a further improvement of the present invention, the step S2 specifically includes:

[0038] The communication topology between mobile robots is composed of a directed graph Represents, where Ω = {1, 2, ..., n} represents the set of mobile robot nodes in the network, n represents the number of mobile robots, represents the set of edges between mobile robot nodes in the network, is the adjacency matrix; if (i, j)∈Θ, it means that nodes i and j are neighbors, and node j can obtain the status information of node i; a ij is an element in the adjacency matrix. If (j, i)∈Θ, then a ij > 0, otherwise a ij =0; degree matrix The elements in i=1,2,…,n,if i≠j,then d ij =0, and the communication topology diagram The associated Laplacian matrix is ​​defined as in l ij =-a ij , i≠j, satisfying

[0039] As a further improvement of the present invention, step S3 specifically includes:

[0040] According to a networked multi-robot system containing n identical wheeled robots, the robots are labeled as numbers 1 to n; the dynamic model of each robot is given by The directed communication topology of the robot system is represented by It means that Ω={1,2,…,n}, The corresponding Laplace matrix is

[0041] Let q i =(x i ,y i ,θ i ) T represents the generalized coordinates of the i-th wheeled robot in the system, and the dynamic model of the system is expressed as:

[0042]

[0043] in:

[0044]

[0045] make represents the desired formation of the system, where symbol z represents the desired formation matrix of the robot system, and symbol z i represents the position information of the ith robot in the desired formation; the formation control problem is to enable multiple robots to form the desired formation shape under the action of the controller, and the control objective is expressed as:

[0046]

[0047] Rewrite the above calculation formula as follows:

[0048]

[0049] Among them, a ij Is the communication topology diagram The corresponding adjacency matrix Elements in

[0050] The control objective is further expressed as a constraint equation:

[0051]

[0052] Using the Baumgarte modified form, the constraint equation can be rewritten as:

[0053]

[0054] Wherein, α and β represent coupling strength parameters, and α>0, β>0;

[0055] Substitute into the UK equation and solve the expression of the control force:

[0056]

[0057] Among them, I n represents the n-dimensional identity matrix;

[0058] Usually N=M -2 , then the constraint expression can be rewritten as:

[0059]

[0060] Then the constraint force of the ith robot is It can be expressed as:

[0061]

[0062] in

[0063]

[0064] As a further improvement of the present invention, step S4 specifically includes:

[0065] The control force F c Substituting the expression into the dynamic model of the robot, we can get:

[0066]

[0067] Where, η is defined as (η1, η2, ..., η n ) T satisfy j=1,2,…,n

[0068]

[0069] Where τi, Mi, qi, and Fi represent the control force, mass, generalized coordinates, and external force of the ith robot, respectively; ηj represents the sum of the elements in the jth column of the Laplace matrix;

[0070] After the desired formation is achieved, it should be satisfied:

[0071] q i -q j =z i -z j

[0072]

[0073] Combining the above formula, we can get:

[0074]

[0075] because but

[0076]

[0077] According to the specific form of M and F matrices, combined with matrix operations, we can get:

[0078]

[0079] so,

[0080]

[0081] Define the coordinates of the center point of the formation of the multi-mobile robot system as q cen (t) = [x cen (t), y cen (t), θ cen (t)] T , then it satisfies Among them, x i (t), y i (t),θ i (t) represents the change of the x-direction coordinate, y-direction coordinate, and the angle between the velocity direction and the positive direction of the x-axis of the ith robot over time;

[0082] Define the coordinates of the center point when the multi-mobile robot system just forms the desired formation as q0 = [x0, y0, θ0] T ,but in Wherein, δ, ε, and γ represent the average speed in the x direction, the average speed in the y direction, and the average angular velocity of the n mobile robots at time t = 0, respectively; They represent the x-direction velocity, y-direction velocity, and angular velocity of the i-th mobile robot at time t respectively.

[0083] As a further improvement of the present invention, when δ=0, ∈=0, γ=0, then Right now , we can derive qcen =q0; after the multi-mobile robot system realizes the desired formation, the center point of the formation remains stationary. At this time, the multi-mobile robot system realizes the desired formation and settles to a point.

[0084] As a further improvement of the present invention, when at least one of δ and ∈ is not 0, γ = 0, then but Derived q cen =[δt,εt,0] T +q0; at this time, the motion trajectory of the center point of the formation is a straight line motion with q0 as the starting point, and the slope of the straight line motion is in the same direction as the vector (δ, ∈). After achieving the desired formation, the multi-mobile robot system performs straight line motion together.

[0085] As a further improvement of the present invention, when δ≠0, ∈≠0, γ≠0, then By solving these differential equations, we can obtain Considering that when t = 0, holds, so substituting into the solution we get For the center point of the formation, Performing first-order integration gives The center point of the formation is a circle with q0 as the center. It is a circular motion with a radius of , and the radius of the circular motion is proportional to δ,∈, and inversely proportional to γ. At this time, the multi-mobile robot system performs circular motion after forming a formation.

[0086] As a further improvement of the present invention, the multi-mobile robot system formation trajectory analysis method based on UK theory also includes the steps of:

[0087] S5. Conduct multiple sets of numerical simulation experiments to verify the correctness of the motion trajectory of the multi-mobile robot system formation when achieving the desired formation shape.

[0088] The beneficial effect of the present invention is that, through theoretical analysis and simulation verification, in the absence of a leader, the motion trajectory of the multi-mobile robot system after achieving formation control has only three possible situations: settling at one point, linear motion, and circular motion; the specific motion trajectory is determined by the initial state of the multi-mobile robot system. Compared with the existing technology, the multi-mobile robot system formation trajectory analysis method based on the UK theory of the present invention provides theoretical derivation and simulation verification, filling the gap in this field. BRIEF DESCRIPTION OF THE DRAWINGS

[0089] Figure 1 It is a flow chart of the multi-mobile robot system formation trajectory analysis method based on UK theory of the present invention;

[0090] Figure 2It is a schematic diagram of the geometric model of the unmanned vehicle in the present invention;

[0091] Figure 3 is the communication topology diagram in the present invention;

[0092] Figure 4 It is a numerical simulation experiment result diagram under the condition of δ=0,∈=0,γ=0 in the present invention;

[0093] Figure 5 It is a numerical simulation experiment result diagram under the condition of δ=1, ∈=1, γ=0 in the present invention;

[0094] Figure 6 It is a numerical simulation experiment result diagram when δ=∈=γ=1 in the present invention;

[0095] Figure 7 It is a numerical simulation experiment result diagram of the present invention when δ=∈=0.5 and γ=1;

[0096] Figure 8 This is a numerical simulation experiment result diagram for the case where δ=∈=1, γ=0.5 in the present invention. DETAILED DESCRIPTION

[0097] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments.

[0098] like Figure 1 As shown, the present invention provides a formation trajectory analysis method for a multi-mobile robot system in the absence of a leader based on the Udwadia-Kalaba (UK) theory. Through theoretical analysis and simulation verification, in the absence of a leader, the motion trajectory of the multi-mobile robot system after achieving formation control has only three possible situations: settling to a point, linear motion, and circular motion; the specific motion trajectory is determined by the initial state of the multi-mobile robot system.

[0099] The formation trajectory analysis method of a multi-mobile robot system in the absence of a leader based on the Udwadia-Kalaba theory specifically includes the following steps.

[0100] S1. In order to study the motion of the mobile robot system in two-dimensional space, the dynamic model of the robot was established using the Newton-Euler mechanics method in combination with the system motion parameters of the mobile robot system.

[0101] Consider a multi-mobile robot system consisting of n four-wheeled mobile robots. Each four-wheeled mobile robot consists of two rear wheels driven independently by motors and two front wheels that support and guide the robot's movement. The rear wheels are driven independently, and the front wheels support and guide, without any control effect. Figure 2As shown, the generalized coordinates of the mobile robot are q = (x, y, θ) T Indicates, where (x, y) is the position coordinate of the center of mass of the mobile robot, and θ represents the attitude angle of the mobile robot in the inertial coordinate system.

[0102] Table 1 Variable definitions of the unmanned vehicle model

[0103]

[0104] The system parameters are shown in Table 1, where u r and u l The direction that drives the vehicle forward is positive.

[0105] In the robot's forward direction, Newton's laws of motion give:

[0106]

[0107] Which meets

[0108] The speed of the robot is expressed as:

[0109]

[0110] Taking the derivative of the robot's velocity with respect to time, we get:

[0111]

[0112] Substituting equation (1.3) into (1.1) we get:

[0113]

[0114] The robot has nonholonomic constraints, which can be expressed as:

[0115]

[0116] Taking the first-order differential of equation (1.5) with respect to time, we can obtain:

[0117]

[0118] From the angular momentum theorem we can get:

[0119]

[0120] Right now

[0121]

[0122] Writing equations (1.4), (1.6), and (1.8) in matrix form yields:

[0123]

[0124] The dynamic equation of the system is expressed as:

[0125]

[0126] in

[0127]

[0128] In the formula, represents the velocity in the x direction, that is, the first-order derivative of the x-axis component of the robot's center of mass coordinate; represents the acceleration in the x-direction, that is, the second-order derivative of the x-axis component of the robot's center of mass coordinate; represents the velocity in the y direction, i.e., the first-order derivative of the x-axis component of the robot's center of mass coordinate; represents the acceleration in the y direction, that is, the second-order derivative of the x-axis component of the robot's center of mass coordinate; It represents the velocity corresponding to the generalized coordinate q, that is, the first-order derivative of the generalized coordinate q with respect to time; It represents the acceleration corresponding to the generalized coordinate q, that is, the second-order derivative of the generalized coordinate q with respect to time; It represents the angular velocity of the robot's velocity direction and the positive direction of the x-axis, that is, the first-order derivative of the angle θ between the robot's velocity direction and the positive direction of the x-axis with respect to time. M is the robot's mass matrix, F is the robot's external force matrix, and F c is the robot’s control force matrix.

[0129] The dynamic model of the robot is established through Newton-Euler mechanics, which can not only reflect the dynamic characteristics of the robot, but also has the non-holonomic characteristics of full integration. Compared with the kinematic model of the robot, the formation controller designed based on the dynamic model can not only better reflect the dynamic characteristics of the robot, but also better achieve the control objectives, and has been widely used in practice.

[0130] S2. In order to describe the communication topology relationship of the multi-mobile robot system, the communication topology graph between the robots in the robot system without a leader is established by combining the knowledge of algebraic graph theory, and the given communication topology graph is a directed weighted graph with a directed spanning tree.

[0131] The algebraic graph theory strategy is a method to represent and analyze multi-agent systems using graph structures. In this technical field, robots are regarded as nodes of a graph, and communication links are regarded as edges of the graph. Through algebraic graph theory, it is easy to define and describe the communication relationship between robots, as well as the impact of the topological structure on the propagation and control of global information. A typical communication topology graph should have connectivity and effective information dissemination capabilities.

[0132] The communication topology graph is in the form of a directed weighted graph, where each node represents a robot and each directed edge represents a communication link between robots. The construction of a directed weighted graph should meet the following conditions: There is at least one directed spanning tree in the graph. This ensures that the communication links between all robots are connected and that information can be transmitted from any robot to all other robots. The weight of the edge is used to represent the communication strength or importance between robots. This can be set according to the actual communication distance, signal quality or control weight. Component function: The main components of the communication topology graph include nodes and edges. Node: represents each robot in the networked robot system. Edge: represents the information transmission channel between robots, which has directionality and weight properties.

[0133] The Laplace matrix is ​​generated based on the structure of the communication topology graph. It contains the degree of each node in the graph and the connectivity relationship between adjacent nodes. In an undirected or directed graph, the Laplace matrix can characterize the direct connection relationship between nodes and the connectivity of the overall topology. The eigenvalues ​​and eigenvectors of the Laplace matrix can reflect certain properties of the topology graph, such as whether the graph is connected. In other words, the role of the Laplace matrix in the communication topology graph is mainly reflected in representing the topology structure and connectivity, measuring consistency, stability analysis, describing information flow, and detecting faults.

[0134] Assume that a multi-mobile robot system consists of n robots, and the communication topology between robots is a directed graph Represents, where Ω = {1, 2, ..., n} represents the set of mobile robot nodes in the network, represents the set of edges between mobile robot nodes in the network, is the adjacency matrix. If (i, j)∈Θ, it means that nodes i and j are neighbors, and node j can obtain the status information of node i. ij is an element in the adjacency matrix. If (j, i)∈Θ, then a ij > 0, otherwise a ij = 0. Degree matrix The elements in i=1,2,…,nIf i≠j, then d ij =0, and the communication topology diagram The associated Laplacian matrix is ​​defined as in l ij =-a ij , i≠j, satisfying

[0135] In the formula, the values ​​of symbols i and j are from 1 to n, n is the number of mobile robots, and nodes i and j are equivalent to mobile robots corresponding to the i-th and j-th nodes in n nodes. Nodes are used to represent mobile robots. ij is the adjacency matrix The elements in d ij is an element in the degree matrix D, l ij is the Laplace matrix The elements in .

[0136] Lemma 1. Graph The corresponding Laplace matrix The rank of is N-1, that is If and only if the graph With spanning tree.

[0137] Directed Graph has a directed spanning tree if and only if At least one node has a directed path to all other nodes. In the present invention, it is assumed that the directed graph A directed weighted graph with a directed spanning tree, the communication topology diagram established is as follows Figure 3 As shown. The corresponding Laplace matrix is: S3. Rewrite the desired formation control goal into the form of constraint equations and substitute them into the UK equation to solve the servo control force required by the robot system subject to non-holonomic constraints to achieve the desired formation control goal. The formation control task is described as making the robots in the multi-mobile robot system form the desired formation shape and maintain a fixed distance between each other. Consider a networked multi-robot system containing n identical wheeled robots, where the robots are numbered 1 to n. The dynamic model of each robot is represented by (1.10). The directed communication topology of the robot system is given by It means that Ω={1,2,…,n}, The corresponding Laplace matrix is

[0138] Let q i =(x i ,y i ,θ i ) T represents the generalized coordinates of the i-th wheeled robot in the system, where xi represents the x-coordinate of the i-th robot, y i represents the y-coordinate of the i-th robot, θ i Represents the angle between the velocity direction of the ith robot and the positive direction of the x-axis. The dynamic model of the system is expressed as:

[0139] in

[0140] M=diag{M1,…,M n}, make represents the desired formation of the system, where symbol z represents the desired formation matrix of the robot system, and symbol z i Represents the position information of the ith robot in the desired formation. The formation control problem is to enable multiple robots to form the desired formation shape under the action of the controller. The control objective is expressed as:

[0141]

[0142] Rewrite (1.12) as:

[0143]

[0144] Among them, a ij Is the communication topology diagram The corresponding adjacency matrix The elements in .

[0145] The control objective is further expressed as a constraint equation:

[0146]

[0147] For the constrained equation For a mechanical system, the control force obtained by solving the UK equation requires that the initial conditions must satisfy the constraint equation. However, in actual situations, the initial conditions of the robot system are difficult to satisfy the constraint equation. For this reason, Baumgarte proposed a constraint correction method to convert the constraint equation Rewrite as Among them, α>0, β>0 are coupling strength parameters.

[0148] For undirected unweighted graphs, the Baumgarte correction form can be used directly; however, for directed weighted graphs, the error system is difficult to converge to 0 when the original Baumgarte correction form is used directly in the subsequent stability derivation process. Therefore, in order to realize the formation control of the robot system based on the directed weighted graph, an improved innovative Baumgarte correction form is proposed. Through the improved innovative Baumgarte correction form, the constraint equation can be rewritten as:

[0149]

[0150] Among them, α and β represent the coupling strength parameters.

[0151] Substitute into the UK equation and solve the expression of the control force

[0152]

[0153] Among them, I n Represents the n-dimensional identity matrix.

[0154] Usually N=M -2 , then the constraint expression can be rewritten as:

[0155]

[0156] Then the constraint force of the ith robot is It can be expressed as:

[0157]

[0158] in

[0159]

[0160] According to the relationship between the robots in the communication topology, the constraint equations are used to represent the formation goal, and these constraints are combined with the dynamic model of the robot system and the innovative Baumgarte correction form. Since the original Baumgarte correction form is only applicable to undirected and unweighted graphs, when it is used in directed and weighted communication topologies, it cannot satisfy the stability of the formation controller, so an innovative Baumgarte correction form is proposed.

[0161] The innovative Baumgarte correction form can be improved in terms of adaptive parameter adjustment, introduction of nonlinear correction terms, observer-based disturbance compensation, and dynamic target correction, thereby enhancing the flexibility and robustness of the correction method and making it more suitable for complex multi-robot systems, especially in formation control or high-dynamic environments. These improved methods not only solve the shortcomings of traditional correction forms, but also better adapt to the nonlinearity and uncertainty of complex multi-robot systems. In practical applications, all robots exchange data in the communication topology graph. After each robot receives the status information of its neighboring robots, it calculates the control force it needs to apply according to the set formation goals and constraint equations. The design of the servo control force is based on the feedback control algorithm to ensure that each robot can achieve the global formation goal based on local information. In this way, the networked robot system can achieve stable and efficient formation control under non-holonomic constraints.

[0162] S4. Substitute the calculated servo control force into the robot's dynamic model and theoretically derive the formation trajectory of the multi-mobile robot system without a leader. Theoretical derivation leads to the conclusion that the formation trajectory of the multi-mobile robot system without a leader has only three typical geometric features: point, straight line and circle, and is related to the initial state of the multi-mobile robot system.

[0163] The control force F c Substituting the expression into the robot's dynamic model (1.11), we can get:

[0164]

[0165] Where, η is defined as (η1, η2, ..., η n ) T satisfy j=1,2,…,n

[0166]

[0167] In the formula, τ i 、M i ,q i 、F i Respectively represent the control force, mass, generalized coordinates, and external force of the ith robot; η j represents the sum of the elements in the jth column of the Laplacian matrix.

[0168] After the desired formation is achieved, it should be satisfied:

[0169]

[0170] Then, combining (1.19) and (1.20), we can obtain

[0171]

[0172] because but

[0173]

[0174] From the specific forms of the M and F matrices in (1.10), combined with matrix operations, we can obtain:

[0175]

[0176] so,

[0177]

[0178] Define the coordinates of the center point of the formation of the multi-mobile robot system as q cen (t) = [x cen (t), ycen (t), θ cen (t)] T , then it satisfies Among them, x i (t), y i (t),θ i (t) respectively represents the changes of the x-direction coordinate, y-direction coordinate, and the angle between the velocity direction and the positive direction of the x-axis of the ith robot over time.

[0179] Define the coordinates of the center point when the multi-mobile robot system just forms the desired formation as q0 = [x0, y0, θ0] T ,but in Wherein, δ, ε, and γ represent the average speed in the x direction, the average speed in the y direction, and the average angular velocity of the n mobile robots at time t = 0, respectively; They represent the x-direction velocity, y-direction velocity, and angular velocity of the i-th mobile robot at time t respectively.

[0180] From (1.24), we can see that The following classification discussion is based on the specific values ​​of δ, ∈, and γ:

[0181] (1) When δ = 0, ∈ = 0, γ = 0, then Right now Then we can deduce q cen =q0. So after the multi-mobile robot system realizes the desired formation, the center point of the formation will be stationary. At this time, the multi-mobile robot system will be calm as a point after realizing the desired formation.

[0182] (2) When at least one of δ and ∈ is not 0, γ = 0, then but Then we can deduce q cen =[δt,εt,0] T +q0, at this time, the motion trajectory of the center point of the formation is a straight line motion starting from q0, and the slope of the straight line motion is in the same direction as the vector (δ, ∈). Therefore, after achieving the desired formation, the multi-mobile robot system will move in a straight line together.

[0183] (3) When δ≠0, ∈≠0, γ≠0, then By solving these differential equations, we can obtain Considering that when t = 0, holds, so substituting into the solution we get For the center point of the formation, Performing first-order integration gives Through this formula, we can see that the center point of the formation is making a circle with q0 as the center. The radius of the circular motion is proportional to δ,∈, and inversely proportional to γ. Therefore, in this case, the multi-mobile robot system will perform circular motion after forming a formation.

[0184] Based on the above theoretical analysis and derivation, it is found that in the absence of a leader, the multi-mobile robot system will have three typical geometric features after forming the desired formation: point, straight line and circle. The specific trajectory depends on the initial conditions of the system δ,∈,γ.

[0185] S5. Conduct multiple sets of numerical simulation experiments to verify the correctness of the motion trajectory of the multi-mobile robot system formation when achieving the desired formation shape.

[0186] Given that the desired formation shape is a regular hexagon, the specific information of each robot in the desired formation shape is as follows:

[0187]

[0188] Experiment 1: δ = 0, ∈ = 0, γ = 0, the corresponding multi-robot system will calm down after achieving the desired formation.

[0189] The initial state of the given multi-mobile robot system is shown in Table 2.

[0190] Table 2 Experiment 1: Initial state of the mobile robot

[0191]

[0192] Figure 4 is the numerical simulation result under the condition of δ=0,∈=0,γ=0, Figure 4 It can be seen that at the beginning, the six robots are all on a vertical straight line with equal distances between them. The straight lines of different colors represent the movement trajectories of different robots, and the black dotted line represents the final formation of a regular hexagon. At this time, when the six robots achieve the desired regular hexagonal formation shape, the six robots are all calm.

[0193] Experiment 2: δ = 1, ∈ = 1, γ = 0. In this case, the multi-robot system will move in a straight line after achieving the desired formation.

[0194] The initial state of the given multi-mobile robot system is shown in Table 3.

[0195] Table 3 Experiment 2: Initial state of the mobile robot

[0196]

[0197] Figure 5is the numerical simulation result under the condition of δ=1,∈=1,γ=0, Figure 5 It can be seen that at the beginning, the six robots are all on a vertical straight line, and the distance between them is equal. The straight lines of different colors represent the movement trajectories of different robots, and the black dotted line represents the final formation of a regular hexagon. At this time, when the six robots achieve the desired regular hexagonal formation shape, the six robots are all moving in a straight line, and because δ=∈=1 in this experiment, the slope of the straight line motion is 1.

[0198] Experiment 3:

[0199] The initial state of the given multi-mobile robot system is shown in Table 4.

[0200] Table 4 Experiment 3: Initial state of the mobile robot

[0201]

[0202] At this time, after calculation, δ=∈=γ=1, such as Figure 6 Numerical simulation experimental results: When δ=∈=γ=1, the robot moves in a circular motion.

[0203] At this time, the initial state of the robot system is changed, as shown in Table 5 and Table 6, corresponding to δ = 0.5, ∈ = 0.5, γ = 1 and δ = 1, ∈ = 1, γ = 0.5, and two sets of numerical simulation experiments are performed. The simulation results are respectively Figure 7 , Figure 8 shown.

[0204] Table 5 Experiment 3: Initial state of the mobile robot

[0205]

[0206] Table 6 Experiment 3: Initial state of the mobile robot

[0207]

[0208] like Figure 7 The numerical simulation experimental results show that when δ=∈=0.5 and γ=1, the robot moves in a circle. Figure 8 Numerical simulation experimental results: When δ=∈=1 and γ=0.5, the robot moves in a circular motion.

[0209] Figure 6 is the numerical simulation result under the condition of δ=∈=γ=1, Figure 6It can be seen that at the beginning, the six robots are all on a vertical straight line with equal distances between them. The straight lines of different colors represent the movement trajectories of different robots, and the black dotted line represents the final formation of a regular hexagon. At this time, when the six robots achieve the desired regular hexagon formation shape, the six robots are all doing circular motion.

[0210] In order to verify that the radius of circular motion is proportional to δ,∈ and inversely proportional to γ, two more sets of experiments were conducted. The experimental results are as follows Figure 7 , Figure 8 As shown. Figure 6 , Figure 7 , Figure 8 By comparison, we can see that as δ,∈ decreases, the circular motion radius decreases; as γ decreases, the circular motion radius increases. This verifies the above conclusion.

[0211] The present invention first uses the Newton-Euler method to model the dynamics of a single robot, and then uses algebraic graph theory to establish the communication topology between robots. The control target is rewritten in the form of a constraint equation and substituted into the UK equation to solve the servo control force of the multi-robot system. The obtained servo control force is substituted into the dynamic model of the robot, and the formation motion trajectory of the multi-mobile robot system without a leader is analyzed through a series of theoretical derivations. Finally, multiple sets of numerical simulation experiments are carried out to verify the correctness of the theoretical derivation.

[0212] Through theoretical derivation and simulation experimental verification, it is confirmed that in the absence of a leader, the formation motion trajectory of a multi-mobile robot system has only three typical geometric characteristics: point, straight line and circle. The specific motion trajectory depends on the initial state of the multi-mobile robot system.

[0213] It is precisely because the formation motion trajectory of a multi-mobile robot system in a leaderless case has only three possible trajectories, and is restricted by initial conditions and cannot meet more requirements for the motion trajectory. Therefore, the researchers extended the formation control problem in the leaderless case to the formation control problem in the leader-follower case, assigning the desired motion trajectory to the leader's movement, so that the entire robot system can move according to any desired trajectory.

[0214] The above contents are further detailed descriptions of the present invention in combination with specific preferred embodiments, and it cannot be determined that the specific implementation of the present invention is limited to these descriptions. For ordinary technicians in the technical field to which the present invention belongs, several simple deductions or substitutions can be made without departing from the concept of the present invention, which should be regarded as falling within the protection scope of the present invention.

Claims

1. A multi-mobile robot system formation trajectory analysis method based on UK theory, characterized in that: The following steps are involved: S1. According to the motion of the mobile robot system in two-dimensional space and the system motion parameters of the mobile robot system, the dynamic model of a single mobile robot is established using the Newton-Euler method; S2. Establish a communication topology graph between each mobile robot by combining algebraic graph theory, and the given communication topology graph is a directed weighted graph with a directed spanning tree; S3. Rewrite the expected formation control goal into the form of constraint equations, and combine the UK equations to obtain the servo control force required by the mobile robot system subject to nonholonomic constraints to achieve the expected formation control goal; S4. Substitute the obtained servo control force into the dynamic model of the mobile robot and derive the formation trajectory of the multi-mobile robot system in the absence of a leader.

2. According to claim 1, the method for analyzing the formation trajectories of a multi-mobile robot system based on UK theory is characterized in that: The step S1 specifically includes: According to a multi-mobile robot system consisting of n four-wheeled mobile robots, each four-wheeled mobile robot consists of two rear wheels independently driven by motors and two front wheels that support and guide the movement of the robot. The generalized coordinates of the mobile robot are q = (x, y, θ) T Indicates that u r and u l The direction that drives the vehicle forward is positive; In the robot's forward direction, Newton's laws of motion give: Which meets The speed of the robot is expressed as: Taking the derivative of the robot's velocity with respect to time, we get: The above calculation formula can be obtained: The robot has nonholonomic constraints, which can be expressed as: Taking the first-order differential with respect to time, we get: From the angular momentum theorem we can get: Right now The above calculation formula can be written into matrix form: The dynamic equation of the system is expressed as: in Where x is the x-axis component of the robot's center of mass coordinates; y is the y-axis component of the robot's center of mass coordinates; θ is the angle between the robot's velocity direction and the positive direction of the x-axis; m is the mass of the robot; d is the radius of the robot's wheels; J is the robot's moment of inertia around the center of mass; l is the lateral distance between the robot's center of mass and the center of the wheel; v is the robot's velocity; u r is the driving torque of the motor acting on the right rear wheel; u l is the driving torque of the motor acting on the left rear wheel; f r is the driving force generated by the right rear wheel of the robot; f l is the driving force generated by the left rear wheel of the robot; M is the mass matrix of the robot, F is the external force matrix of the robot, and F c is the robot’s control force matrix.

3. The multi-mobile robot system formation trajectory analysis method based on UK theory according to claim 2 is characterized in that: The step S2 specifically includes: The communication topology between mobile robots is composed of a directed graph Represents, where Ω = {1, 2, ..., n} represents the set of mobile robot nodes in the network, n represents the number of mobile robots, represents the set of edges between mobile robot nodes in the network, is the adjacency matrix; if (i, j)∈Θ, it means that nodes i and j are neighbors, and node j can obtain the status information of node i; a ij is an element in the adjacency matrix. If (j, i)∈Θ, then a ij > 0, otherwise a ij =0; degree matrix The elements in If i≠j, then d ij =0, and the communication topology diagram The associated Laplacian matrix is ​​defined as in satisfy 4. The multi-mobile robot system formation trajectory analysis method based on UK theory according to claim 3 is characterized in that: The step S3 specifically includes: According to a networked multi-robot system containing n identical wheeled robots, the robots are labeled as numbers 1 to n; the dynamic model of each robot is given by The directed communication topology of the robot system is represented by Express, satisfy The corresponding Laplace matrix is Let q i =(x i ,y i ,θ i ) T represents the generalized coordinates of the i-th wheeled robot in the system, and the dynamic model of the system is expressed as: in: make represents the desired formation of the system, where symbol z represents the desired formation matrix of the robot system, and symbol z i represents the position information of the ith robot in the desired formation; the formation control problem is to enable multiple robots to form the desired formation shape under the action of the controller, and the control objective is expressed as: Rewrite the above calculation formula as follows: Among them, a ij Is the communication topology diagram The corresponding adjacency matrix Elements in The control objective is further expressed as a constraint equation: Using the Baumgarte modified form, the constraint equation can be rewritten as: Wherein, α and β represent coupling strength parameters, and α>0, β>0; Substitute into the UK equation and solve the expression of the control force: Among them, I n represents the n-dimensional identity matrix; Usually N=M -2 , then the constraint expression can be rewritten as: Then the constraint force of the ith robot is It can be expressed as: in 5. The multi-mobile robot system formation trajectory analysis method based on UK theory according to claim 4 is characterized in that: The step S4 specifically includes: The control force F c Substituting the expression into the dynamic model of the robot, we can get: Where, η is defined as (η1, η2, ..., η n ) T satisfy In the formula, τ i 、M i ,q i 、F i Respectively represent the control force, mass, generalized coordinates, and external force of the ith robot; η j represents the sum of the elements in the jth column of the Laplacian matrix; After the desired formation is achieved, it should be satisfied: Combining the above formula, we can get: because but According to the specific form of the MF matrix, combined with matrix operations, we can get: so, Define the coordinates of the center point of the formation of the multi-mobile robot system as q cen (t) = [x cen (t), y cen (t), θ cen (t)] T , then it satisfies Among them, x i (t), y i (t),θ i (t) represents the change of the x-direction coordinate, y-direction coordinate, and the angle between the velocity direction and the positive direction of the x-axis of the ith robot over time; Define the coordinates of the center point when the multi-mobile robot system just forms the desired formation as q0 = [x0, y0, θ0] T ,but satisfy Wherein, δ, ε, and γ represent the average speed in the x direction, the average speed in the y direction, and the average angular velocity of the n mobile robots at time t = 0, respectively; They represent the x-direction velocity, y-direction velocity, and angular velocity of the i-th mobile robot at time t respectively.

6. The multi-mobile robot system formation trajectory analysis method based on UK theory according to claim 5 is characterized in that: When δ=0,∈=0,γ=0, then Right now Derived q cen =q0; after the multi-mobile robot system realizes the desired formation, the center point of the formation remains stationary. At this time, the multi-mobile robot system realizes the desired formation and settles to a point.

7. The method for analyzing the formation trajectory of a multi-mobile robot system based on UK theory according to claim 5 is characterized in that: When at least one of δ and ∈ is not 0, γ=0, then but Derived q cen =[δt,εt,0] T +q0; at this time, the motion trajectory of the center point of the formation is a straight line motion with q0 as the starting point, and the slope of the straight line motion is in the same direction as the vector (δ, ∈). After achieving the desired formation, the multi-mobile robot system performs straight line motion together.

8. The method for analyzing the formation trajectories of a multi-mobile robot system based on UK theory according to claim 5 is characterized in that: When δ≠0, ∈≠0, γ≠0, then By solving these differential equations, we can obtain Considering that when t = 0, holds, so substituting into the solution we get For the center point of the formation, Performing first-order integration gives The center point of the formation is a circle with q0 as the center. It is a circular motion with a radius of , and the radius of the circular motion is proportional to δ,∈, and inversely proportional to γ. At this time, the multi-mobile robot system performs circular motion after forming a formation.

9. The multi-mobile robot system formation trajectory analysis method based on UK theory according to claim 1 is characterized in that: Also includes the steps: S5. Conduct multiple sets of numerical simulation experiments to verify the correctness of the motion trajectory of the multi-mobile robot system formation when achieving the desired formation shape.

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