U-k theory-based multi-mobile robot system formation trajectory analysis method

Abstract

The present application relates to the field of intelligent robot control, in particular to a multi-mobile robot system formation trajectory analysis method based on U-K theory, comprising the following steps: S1. using Newton-Euler method to establish a dynamic model of a single mobile robot; S2. establishing a communication topology graph between each mobile robot, and the given communication topology graph is a directed weighted graph with a directed spanning tree; S3. rewriting an expected formation control target into a constraint equation form, and combining a U-K equation to obtain a required servo control force of a nonholonomic constraint mobile robot system for completing the expected formation control target; S4. substituting the obtained servo control force into the dynamic model of the mobile robot, and deducing a formation trajectory of the multi-mobile robot system under a leaderless condition. The present application analyzes and verifies that under the leaderless condition, the movement trajectory of the multi-mobile robot system after realizing the formation control has three possible conditions: stabilization at a point, straight line movement, and circular movement.

Description

Technical Field

[0001] The present 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 multiple mobile robot systems has been a hot topic for many researchers. Compared to individual robots, multi-mobile robot systems have gained widespread application in both civilian and military fields due to their high fault tolerance, low cost, and high flexibility. However, mobile robots are constrained systems subject to nonholonomic constraints, limiting their lateral motion. Therefore, studying the formation control problem of multiple mobile robot systems subject to nonholonomic constraints has become a hot topic.

[0003] Currently, there are numerous methods for studying the formation control problem of multiple mobile robots, such as leader-follower methods, artificial potential field methods, and behavior-based methods. 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 discuss the leaderless case. However, since the trajectory movement of the formation in the leaderless case is restricted and cannot move according to any desired motion trajectory, it cannot meet the diverse task requirements. Therefore, many researchers have expanded the formation control problem in the leaderless case to the leader-follower case. In this case, it is only necessary to assign an arbitrary desired motion trajectory to the movement of the leader. After the followers and the leader achieve the desired formation shape, the entire robot system can move together according to any desired trajectory, thereby better meeting 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 movement trajectory. This area 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. Based on the motion of the mobile robot system in two-dimensional space and the system motion parameters of the mobile robot system, a dynamic model of a single mobile robot is established using the Newton-Euler method.

[0008] S2. Using algebraic graph theory, establish a communication topology graph between mobile robots in a leaderless situation, where the given communication topology graph is a directed weighted graph with a directed spanning tree.

[0009] S3. Rewrite the desired formation control objective into a constraint equation form and combine it with the UK equation to obtain the servo control force required by the mobile robot system subject to nonholonomic constraints to achieve the desired formation control objective;

[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, 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 represented by q = (x, y, θ) T Indicates that u r and u l The direction in which the vehicle is driven forward is considered positive;

[0013] In the robot's forward direction, Newton's law of motion gives:

[0014]

[0015] Which meets

[0016] The velocity 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 coordinate; y is the y-axis component of the robot's center of mass coordinate; θ is the angle between the robot's velocity direction and the positive direction of the x-axis; m is the robot's mass; 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 wheels; 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, 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 node i and node 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 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 represented by The directed communication topology of the robot system is represented by Indicates 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 i-th 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 i-th 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 i-th robot, respectively; ηj represents the sum of the elements in the j-th column of the Laplace matrix;

[0070] After the formation of the desired formation, the following should be satisfied:

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

[0072]

[0073] Then, the above formula can be obtained,

[0074]

[0075] Since Then

[0076]

[0077] According to the specific form of M, F matrix, combined with matrix operation, the following can be obtained:

[0078]

[0079] Therefore,

[0080]

[0081] The coordinates of the formation center point of the multi-mobile robot system are defined as q cen (t) = [x cen (t), y cen (t), θ cen (t)] T Then, the following is satisfied Wherein, x i (t), y i (t), θ i (t) respectively represent the x direction coordinate, y direction coordinate, and the angle between the velocity direction and the positive direction of the x axis of the i th robot with respect to time;

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

[0083] As a further improvement of the present application, when δ = 0, ∈ = 0, γ = 0, the following is satisfied That is , and the following is derived 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 a 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 we substitute and solve to get For the center point of the formation, Performing first-order integration yields The center point of the formation is a circle with q0 as the center. The circular motion with radius 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 further includes the following steps:

[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 this invention is that, through theoretical analysis and simulation verification, a multi-mobile robot system without a leader can only follow three possible trajectories after achieving formation control: settling to a point, linear motion, and circular motion. The specific trajectory is determined by the initial state of the multi-mobile robot system. Compared with existing technologies, this method of multi-mobile robot system formation trajectory analysis based on UK theory provides theoretical derivation and simulation verification, filling a gap in this field. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

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

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

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

[0094] Figure 6 : is a numerical simulation experiment result diagram of the present invention under the condition of δ=∈=γ=1;

[0095] Figure 7 : is a numerical simulation experiment result diagram of the present invention under the condition of δ=∈=0.5, 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 objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to 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 a 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, while the front wheels support and guide the robot without any control effect. Figure 2As shown, the generalized coordinates of the mobile robot are q = (x, y, θ) T Indicates that (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 law of motion gives:

[0106]

[0107] Which meets

[0108] The velocity 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 equation (1.1) yields:

[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] Where, represents the velocity in the x-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 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] Building a robot's dynamics model using Newton-Euler mechanics not only reflects the robot's dynamic characteristics but also takes into account its fully integrated nonholonomic properties. Compared to a robot's kinematic model, designing a formation controller based on this dynamic model not only better reflects the robot's dynamic characteristics but also achieves better control objectives, leading to widespread practical application.

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

[0131] Algebraic graph theory is a method for representing and analyzing multi-agent systems using graphs. In this technical field, robots are considered nodes in a graph, and communication links are considered edges. Using algebraic graph theory, it is easy to define and describe the communication relationships between robots and the impact of topology on global information dissemination and control. A typical communication topology graph should be well-connected and capable of efficient information dissemination.

[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 the directed weighted graph should satisfy the following condition: there exists at least one directed spanning tree in the graph. This ensures that the communication links between all robots are connected, guaranteeing that information can be propagated from any one robot to all other robots. The weight of an 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 functions: The main components of the communication topology graph include nodes and edges. Nodes: Represent individual robots in the networked robot system. Edges: Represent information transmission channels between robots, with directionality and weight attributes.

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

[0134] Suppose a multi-mobile robot system is composed of n robots, and the communication topology between robots is represented by a directed graph , 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 node i and node j are neighbors, and node j can obtain the state information of node i. ij is the element in the adjacency matrix, if (j, i) ∈ Θ, ij > 0, otherwise a ij = 0. The degree matrix in the element satisfies i = 1, 2,..., n If i ≠ j, d ij = 0, the Laplacian matrix associated with the communication topology graph is defined as where l ij = -a ij , i ≠ j, satisfying

[0135] In the formula, the values ​​of symbols i and j range from 1 to n, n is the number of mobile robots, and node i and node j are equivalent to the 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 Elements in .

[0136] Lemma 1. Graph The corresponding Laplace matrix The rank of is N-1, that is If and only if the graph Has a 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 objective 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 nonholonomic constraints to achieve the desired formation control objective. 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 Indicates 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 i-th 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 i-th 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 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 Where 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, directly using the original Baumgarte correction form results in the error system having difficulty converging to zero during the subsequent stability derivation process. Therefore, an improved and innovative Baumgarte correction form is proposed to implement formation control of robotic systems based on directed, weighted graphs. Using the improved and innovative Baumgarte correction form, the constraint equations can be rewritten as:

[0149]

[0150] Where 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 i-th robot is It can be expressed as:

[0157]

[0158] in

[0159]

[0160] Based on the relationships between the robots in the communication topology, constraint equations are used to express the formation objectives. These constraints are then combined with the dynamic model of the robot system and an innovative Baumgarte correction form. Since the original Baumgarte correction form only applies to undirected and unweighted graphs, it cannot ensure the stability of the formation controller when applied to directed and weighted communication topologies. Therefore, an innovative Baumgarte correction form is proposed.

[0161] The innovative Baumgarte correction form allows for improvements in adaptive parameter adjustment, the introduction of nonlinear correction terms, observer-based disturbance compensation, and dynamic target correction. This enhances the flexibility and robustness of the correction method, making it more suitable for complex multi-robot systems, particularly in formation control or highly dynamic environments. These improvements not only address the shortcomings of traditional correction forms but also better adapt to the nonlinearities and uncertainties of complex multi-robot systems. In practical applications, all robots exchange data in a communication topology. After receiving status information from neighboring robots, each robot calculates its own control force based on the set formation goal and constraint equations. The servo control force is designed based on a feedback control algorithm, ensuring that each robot can achieve the global formation goal based on local information. This approach enables stable and efficient formation control of networked robot systems under nonholonomic constraints.

[0162] S4. Substitute the calculated servo control forces into the robot dynamics model and theoretically derive the formation trajectories of a leaderless multi-robot system. This theoretical deduction concludes that the formation trajectories of a leaderless multi-robot system have only three typical geometric features: point, line, and circle, and are related to the initial state of the multi-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] Where, τ i 、M i ,q i 、F i Respectively represent the control force, mass, generalized coordinates, and external force of the i-th 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 formation center point 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) 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 i-th 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 Where δ, ε, and γ represent the average x-direction velocity, average y-direction velocity, and 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. Therefore, after the multi-mobile robot system achieves the desired formation, the center of the formation becomes stationary. At this point, the multi-mobile robot system will settle to a point after achieving the desired formation.

[0182] (2) When at least one of δ and ∈ is not 0, γ = 0, then it satisfies but Then we can deduce q cen =[δt,εt,0] T +q0, the 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 the multi-mobile robot system achieves the desired formation, it will move together in a straight line.

[0183] (3) When δ≠0, ∈≠0, γ≠0, then By solving these differential equations, we can obtain Considering that when t=0, holds, so we substitute and solve to get For the center point of the formation, Performing first-order integration yields 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 theoretical analysis and derivation above, we find that in the absence of a leader, a multi-robot system will exhibit three typical geometric features after forming a desired formation: point, line, and circle. The specific trajectory depends on the system's initial conditions, 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 As can be seen, initially, the six robots were aligned in a straight vertical line, equally spaced from each other. The different colored lines represent the movement paths of the different robots, and the black dashed line represents the final formation, a regular hexagon. When the six robots achieved the desired regular hexagonal formation, they all remained 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 when δ = ∈ = 1, γ = 0, which is shown in Fig. 6. Figure 5 It can be seen that the six robots are initially on a vertical straight line, and the distance between each other is equal. The straight lines of different colors represent the movement trajectories of different robots, and the black dotted line represents the final formation shape of the regular hexagon. At this time, when the six robots achieve the desired formation shape of the regular hexagon, the six robots are moving in a straight line, and since δ = ∈ = 1 in this experiment, the slope of the straight line motion is 1.

[0198] Experiment Three:

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

[0200] Table 4: Initial state of mobile robots in Experiment Three

[0201]

[0202] At this time, after calculation, δ = ∈ = γ = 1, as shown in Figure 6 the numerical simulation result: 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 Tables 5 and 6, corresponding to δ = 0.5, ∈ = 0.5, γ = 1 and δ = 1, ∈ = 1, γ = 0.5, respectively, and two groups of numerical simulation experiments are performed. The simulation results are shown in Figure 7 , Figure 8

[0204] Table 5: Initial state of mobile robots in Experiment Three

[0205]

[0206] Table 6: Initial state of mobile robots in Experiment Three

[0207]

[0208] As shown in the numerical simulation result of Figure 7 : when δ = ∈ = 0.5, γ = 1, the robot moves in a circular motion. As shown in the numerical simulation result of Figure 8 : when δ = ∈ = 1, γ = 0.5, the robot moves in a circular motion.

[0209] Figure 6 is the numerical simulation result when δ = ∈ = γ = 1, which is shown in Fig. 6. Figure 6 ​As you can see, initially, the six robots are all aligned in a straight vertical line, equidistant from each other. The different colored lines represent the motion paths of the different robots, and the black dashed line represents the final formation, a regular hexagon. Once the six robots have achieved the desired hexagonal formation, they are all moving in a 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, thus verifying the above conclusion.

[0211] This paper first uses the Newton-Euler method to model the dynamics of individual robots. It then applies algebraic graph theory to establish the communication topology between the robots. The control objectives are rewritten as constraint equations and substituted into the UK equations to obtain the servo control forces of the multi-robot system. This servo control force is then substituted into the robots' dynamic model. Through a series of theoretical derivations, the motion trajectory of a leaderless multi-robot formation is analyzed. Finally, multiple numerical simulation experiments are conducted to verify the correctness of the theoretical derivations.

[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 adopted 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 the 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 is a further detailed description of the present invention in conjunction with specific preferred embodiments, and the specific implementation of the present invention should not be considered to be limited to these descriptions. For those skilled in the art of the present invention, without departing from the concept of the present invention, several simple deductions or substitutions can be made, which should be considered to fall within the scope of protection of the present invention.

Claims

1. A multi-mobile robot system formation trajectory analysis method based on UK theory, characterized by: The following steps are involved: S1. Based on the motion of the mobile robot system in two-dimensional space and the system motion parameters of the mobile robot system, a dynamic model of a single mobile robot is established using the Newton-Euler method. S2. Establish a communication topology graph between each mobile robot using algebraic graph theory, where the given communication topology graph is a directed weighted graph with a directed spanning tree; S3. Rewrite the desired formation control objective into a constraint equation form and combine it with the UK equation to obtain the servo control force required by the mobile robot system subject to nonholonomic constraints to achieve the desired formation control objective; S4. Substitute the obtained servo control force into the dynamic model of the mobile robot to derive the formation trajectory of the multi-mobile robot system in the absence of a leader; 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 obtains the state 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 The associated Laplacian matrix is ​​defined as in satisfy 2. The multi-mobile robot system formation trajectory analysis method based on UK theory according to claim 1 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 represented by q = (x, y, θ) T Indicates that u r and u l The direction in which the vehicle is driven forward is considered positive; In the robot's forward direction, Newton's law of motion gives: Which meets The velocity 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 coordinate; y is the y-axis component of the robot's center of mass coordinate; θ is the angle between the robot's velocity direction and the positive direction of the x-axis; m is the robot's mass; 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 wheels; 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 1 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 represented 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 i-th 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 is 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 i-th robot is It can be expressed as: in 4. The multi-mobile robot system formation trajectory analysis method based on UK theory according to claim 3 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 Where, τ i 、M i ,q i 、F i Respectively represent the control force, mass, generalized coordinates, and external force of the i-th robot; η j represents the sum of the elements in the jth column of the Laplace matrix; After the desired formation is achieved, it should be satisfied: q i -q j =z i -z j Combining the above formula, we can get: because but According to the specific forms of the M and F matrices, combined with matrix operations, we can get: so, Define the coordinates of the formation center point 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 i-th 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 Where δ, ε, and γ represent the average x-direction velocity, average y-direction velocity, and 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.

5. The multi-mobile robot system formation trajectory analysis method based on UK theory according to claim 4 is characterized in that: When δ=0,∈=0,γ=0, then it satisfies 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.

6. The multi-mobile robot system formation trajectory analysis method based on UK theory according to claim 4 is characterized in that: When at least one of δ and ∈ is not 0, γ=0, then it satisfies 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 a straight line motion together.

7. The multi-mobile robot system formation trajectory analysis method based on UK theory according to claim 4 is characterized in that: When δ≠0,∈≠0,γ≠0, then it satisfies By solving these differential equations, we can obtain Considering that when t=0, holds, so we substitute and solve to get For the center point of the formation, Performing first-order integration yields The center point of the formation is a circle with q0 as the center. The circular motion with radius is proportional to δ,∈ and inversely proportional to γ. At this time, the multi-mobile robot system performs circular motion after forming a formation.

8. 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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