A multi-agent formation control method

Abstract

The present invention discloses a multi-agent formation control method. First, a communication connection network is established for agents according to the desired formation, and boundary constraint conditions are established for agents according to the limitations of physical conditions in practical applications. Then, each agent independently calculates the actions it needs to execute by using its own global position and the position information obtained by exchanging with adjacent agents through communication connections, so as to control multiple agents to converge to the desired formation. The present invention can control multiple agents to achieve any formation that satisfies infinitesimal rigidity on a two-dimensional plane, and ensures the uniqueness of the geometric structure of the convergent formation, ensures that the internal structures of the actual convergent formation and the desired formation are the same, and ensures that the relative positions and connection relationships among multiple agents are the same. At the same time, the method disclosed by the present invention adds boundary constraints to avoid collisions and communication connection interruptions among agents, and avoids the possible oscillations during the system switching process.

Description

Technical Field

[0001] The present invention relates to a multi-agent formation control method, belonging to the field of multi-agent cluster control. Background Art

[0002] Cluster phenomena are widespread in nature and human society. The cluster of multiple agents can complete tasks that cannot be completed by a single agent. Multi-agent formation means that during the movement of multiple agents, they can still maintain a certain formation among each other, that is, the geometric figure formed by multiple agents remains stable.

[0003] Multi-agent formation control technology can be applied in many fields. Underwater robots, ground robots, aerial robots, etc. can all complete specific tasks through formation. For example, in the field of logistics and warehousing, multiple robots can be used in formation to cooperate in handling objects with larger volume or heavier weight.

[0004] Currently, there have been many studies on multi-agent formation control methods. According to the different perception capabilities and interaction topologies of agents, the existing multi-agent formation control methods can be mainly divided into three categories: position-based control, displacement-based control, and distance-based control. Among them, the formation control method based on distance control has received extensive research and attention due to its advantages such as full distribution.

[0005] However, some existing formation control methods based on distance control may have the situation of non-unique formation. Although the controlled distances between agents meet the requirements of the desired formation, the geometric figures corresponding to the actual formation and the desired formation are not exactly the same. For example, the desired formation is a convex figure, while the actual formation is a concave figure. This non-unique formation situation may also cause collisions between agents. To avoid this situation, more communication connections or perceptions need to be introduced between agents, but this greatly increases the complexity of the interaction topology between agents.

[0006] In addition, the existing formation control methods often assume that agents are mass points without volume, while in actual applications, agents always have a certain volume, and this assumption may lead to collisions between agents. At the same time, the existing formation control methods often ignore discussing the communication connection maintenance range between agents. In actual applications, the communication connection between agents can only be maintained within a limited range. If the distance between agents is too far, it may cause the communication connection between agents to be interrupted. Summary of the Invention

[0007] The object of the present invention is to provide a multi-agent formation control technology to ensure the uniqueness of the formation and avoid collisions and communication connection interruptions between agents.

[0008] To achieve the above object, the technical solution of the present invention provides a multi-agent formation control method, which is characterized by including the following steps:

[0009] Step 1, establish the motion model of a single agent:

[0010] Establish a global X-Y rectangular coordinate system in the space where the agents move. Each agent is regarded as a particle obeying the single-integrator motion model. Then, for the i-th agent, i = 1,..., n, where n represents the total number of agents:

[0011]

[0012] In the formula: represents the coordinates of the i-th agent in the global coordinate system; u i is the control input of the i-th agent;

[0013] Step 2, describe the formation convergence target:

[0014] For any given desired formation containing n agents, represent the relative positions between the n agents as a set of scatter plots in the X-Y rectangular coordinate system. The number of scatter points is n, and the coordinates of the i-th scatter point are denoted as Denote col() represents a column vector;

[0015] Generate an undirected graph with n scatter points as vertices where, is the set of n vertices of the undirected graph ; is the set of l edges of the undirected graph ; Represent the desired formation as and q * such that the rank of the rigidity matrix B of is equal to 2n - 3. At this time, it is said that

[0016] satisfies infinitesimal rigidity; where, q i (t) is the position of the i-th agent in the global coordinate system, and the position of the agent changes with time;

[0017] Define a set to represent all formations obtained by translating, rotating, scaling, or combining them of the desired formation in the X-Y plane. Represent the convergence target of the actual formation as the following formula:

[0018]

[0019] The meaning of the above convergence objective is to make the actual formation eventually converge to one of the formations in. Since translation, rotation, and scaling are all linear transformations that do not change the structure of the graph, the geometric figures corresponding to the actual formation and the desired formation must be similar, and the internal structures of the actual formation and the desired formation are the same, that is, the relative positions and connection relationships among the n agents are the same. Generally speaking, due to the existence of translation, rotation, and scaling, the overall position, orientation, and size of the graph are not the same.

[0020] Step 3: Establish a communication connection network among multiple agents:

[0021] Determine the initial positions of multiple agents: Denote q i (0) = [x i (0), y i (0)] T as the initial position of the i-th agent, and denote It is required that the formation corresponding to the initial positions of the n agents satisfies infinitesimal rigidity;

[0022] Make the n agents in the multi-agent system correspond one-to-one with the n vertices of the undirected graph ; For the two vertices in the undirected graph connected by an edge, the corresponding agents are called adjacent agents. Establish a communication connection for each pair of adjacent agents. The adjacent agents exchange their global coordinates through wireless communication. The l edges of the undirected graph correspond to l communication links.

[0023] Step 4: Establish boundary constraint conditions:

[0024] Express the relative position between adjacent agents i and j as q ij , then there is:

[0025]

[0026] Express the distance between adjacent agents i and j as ||q ij ||;

[0027] Define the safety distance between adjacent agents i and j as where r si , r sj are the safety radii defined for the two adjacent agents i and j respectively, To avoid collisions between adjacent agents, during the entire control process, it is required that ||q ij (t)|| > rsij ;

[0028] Define the communication connection maintenance distance between adjacent agents i and j as where r ci and r cj are the communication connection maintenance radii defined for two adjacent agents i and j respectively, To maintain the communication connection between adjacent agents, throughout the control process, it is required that ||q ij (t)|| < r cij ;

[0029] Considering that in practical applications, the communication distance of general agents is much larger than their volume, therefore, it is reasonably assumed here that r sij << r cij ;

[0030] Then throughout the control process, the boundary constraint conditions are expressed as the following formula:

[0031]

[0032] Step 5: Design the controller:

[0033] For the formation convergence objective in Step 2 and the boundary constraint conditions in Step 4, design the controller u for the i-th agent as shown in the following formula i :

[0034]

[0035] is the set of all adjacent agents of agent i, including all agents with communication connections to the i-th agent;

[0036] In the controller u i A ij q ij is used to control the formation convergence, where:

[0037] A ij is the gain matrix, which controls the formation convergence by controlling the relative positions between adjacent agents, and its form is as follows:

[0038]

[0039] For A ij is a zero matrix; for A ij the calculation is transformed into an optimization problem;

[0040] In the controller u i fij (η ij )k ij ξ ij σ ij I2q ij used to ensure boundary constraints, where:

[0041] Let Find the derivative of η ij with respect to time, we get:

[0042]

[0043] According to the values of η ij and , the states of the agent are divided into the following six cases:

[0044]

[0045]

[0046]

[0047]

[0048]

[0049]

[0050] where c and δ are two positive constants and need to satisfy Their specific values are determined by experimental methods in engineering;

[0051] Define the switching signal as:

[0052]

[0053] To convert the constrained system into an equivalent unconstrained system, the following transformation is introduced:

[0054]

[0055] Find the derivative of σ ij with respect to time, we get:

[0056]

[0057] where,

[0058] k ij is a constant, and its specific value is determined by experimental methods in engineering;

[0059] I2 is a two-dimensional identity matrix;

[0060] Step 6: Each agent calculates the controller u using its own computing power i , thereby determining the action it needs to execute. After each refresh of the agent's position, the controller u is recalculated i .

[0061] Preferably, in step 2, the rigidity matrix R is given by the following formula:

[0062]

[0063] where is called and is the edge function associated with.

[0064] Preferably, the X-Y rectangular coordinate system is only used to describe the relative positions between agents in the desired formation and has nothing to do with the coordinate system of the agent's activity space, and can be arbitrarily selected.

[0065] Preferably, the scatter plot only represents the relative positions between individual agents and does not represent the global positions of the agents. There are infinitely many sets of scatter plots that meet the requirements of the desired formation pattern, and the coordinates of this set of scatter plots make the desired formation satisfy infinitesimal rigidity.

[0066] Preferably, in step 5, the calculation of A ij includes the following steps:

[0067] Define a matrix whose form is as follows:

[0068]

[0069] To achieve formation, the matrix A satisfies the following two conditions:

[0070] Condition 1: The vectors form a basis for ker(A), where ker() represents the kernel or null space of the matrix;

[0071] Condition 2: All non-zero eigenvalues of A have negative real parts;

[0072] In the above conditions: where is obtained by rotating by 90°, that is R π / 2 is the rotation matrix;

[0073] Let perform a singular value decomposition on N = USV T , denoted as For the last 2n - 4 columns of U, denote Matrix A and have the same non - zero eigenvalues. Thus, by solving the convex optimization problem, we have:

[0074]

[0075] subject to AN=0

[0076] Find the specific values of the elements in A, thereby determining A ij , where represents the matrix the largest eigenvalue.

[0077] Preferably, if a counter - clockwise rotation is taken, B π / 2 is expressed as:

[0078] A multi - agent formation control method disclosed by the present invention first establishes a communication connection network for agents according to the desired formation, and establishes boundary constraint conditions for agents according to the limitations of physical conditions in practical applications. Then each agent independently calculates the actions it needs to perform by using its own global position and the position information obtained by exchanging with adjacent agents through communication connections, so as to control multiple agents to converge to the desired formation.

[0079] Compared with the prior art, the present invention has the following beneficial effects: It provides a new multi - agent formation control method, which can control multiple agents to achieve any formation that satisfies infinitesimal rigidity in a two - dimensional plane. The method disclosed by the present invention ensures the uniqueness of the geometric structure of the convergent formation, ensures that the internal structures of the actual convergent formation and the desired formation are the same, and ensures that the relative positions and connection relationships among multiple agents are the same. At the same time, the method disclosed by the present invention fully considers the limitations of physical conditions in practical applications, adds boundary constraints to avoid collisions and communication connection interruptions among agents, and the implementation of boundary constraints adopts switching control with hysteresis. The addition of hysteresis avoids the possible oscillations during the switching process of the system. The method disclosed by the present invention introduces the idea of distance - based control to ensure boundary constraints, but overcomes the defect that the formation control method based on distance control may have non - unique formations. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Figure 1 is a scatter plot representation corresponding to the desired formation in the embodiment;

[0081] Figure 2 is a schematic diagram of an infinitesimal rigid structure corresponding to the desired formation in the embodiment;

[0082] Figure 3Schematic diagram of the initial positions of agents corresponding to the embodiments and the established communication connections;

[0083] Figure 4 Schematic diagram of the division of the hysteresis switching interval corresponding to the embodiments;

[0084] Figure 5 Schematic diagram of the formation simulation convergence trajectory corresponding to the embodiments;

[0085] Figure 6 Schematic diagram showing that the distances between adjacent agents are the same but the formations are not completely the same corresponding to the embodiments. Detailed implementation manners

[0086] The present invention will be further described below in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and not to limit the scope of the present invention. In addition, it should be understood that after reading the content taught by the present invention, those skilled in the art can make various changes or modifications to the present invention, and these equivalent forms also fall within the scope defined by the appended claims of this application.

[0087] In this embodiment, the formation of 5 robots on a horizontal ground is taken as an example to further illustrate and explain the present invention.

[0088] There are 5 robots with the same parameters on the horizontal ground. Each robot can obtain its own global coordinates from the global positioning system, and adjacent robots can exchange coordinates through their own wireless communication systems. The global positioning system can be a certain indoor positioning system (such as a motion capture system) or an outdoor positioning system (such as GPS).

[0089] Now it is desired that they can form a regular pentagon formation. To achieve this formation, the following steps need to be taken:

[0090] Step 1, establish a global X-Y rectangular coordinate system in the space where the 5 robots move, and regard each robot as a particle obeying the single integrator motion model, expressed as:

[0091]

[0092] where represents the coordinates of the i-th robot in the global coordinate system, and u i is the control input quantity, which represents the motion speed of the i-th robot here.

[0093] Step 2, describe the formation convergence target

[0094] For example Figure 1 , first represent the relative positions between these 5 robots as a set of scatter plots in the X-Y rectangular coordinate system. The number of scatter points is 5, and the coordinates of the 5 scatter points are respectively Record Using these 5 scatter points as vertices, generate an undirected graph Among them, is the set of 5 vertices of the undirected graph , and is the set of 7 edges of the undirected graph . Represent the desired formation as such as Figure 2 .

[0095] Next, verify whether it satisfies infinitesimal rigidity:

[0096] First, write out the edge functions associated with :

[0097]

[0098] Then, calculate the rigidity matrix:

[0099]

[0100] Substitute specific values to get:

[0101]

[0102] Transform the rigidity matrix into a row echelon form matrix, and obtain that the rank of is 7. Therefore, satisfies infinitesimal rigidity.

[0103] Represent the actual formation as where q i (t) is the position of the i-th robot in the global coordinate system, and the position of the robot changes with time.

[0104] Define a set to represent all formations obtained by translating, rotating, scaling, or combining them of the desired formation in the x-y plane. Represent the convergence target of the actual formation as:

[0105]

[0106] Step 3, establish a communication connection network among 5 robots

[0107] As Figure 3 shown, the 5 vertices correspond to 5 robots respectively, the connections between the 5 vertices are the communication connections among the 5 robots, and the initial positions of the 5 robots are q1(0) = [0, 0] Tq2(0)=[3,-3] T q3(0)=[0,-3] T q4(0) = [3, 0] T q5(0)=[-1, 1] T , let q(0) = [0, 0, 3, -3, 0, -3, 3, 0, -1, 1] T ,remember is the formation corresponding to the initial positions of the 5 robots.

[0108] Verify first Whether it satisfies infinitesimal rigidity:

[0109] First write Associated edge functions:

[0110]

[0111] Then calculate the stiffness matrix:

[0112]

[0113] Substitute specific values:

[0114]

[0115] The stiffness matrix R q(0) Transform into the simplest row matrix and obtain R q(0) The rank of is 7, so Satisfies infinitesimal rigidity.

[0116] Number the five robots as 1, 2, 3, 4, and 5, respectively, and The five vertices One-to-one correspondence. The robots corresponding to the two vertices connected by an edge are called adjacent robots. A communication connection is established for each pair of adjacent robots. The 7 edges of correspond to 7 communication links. Adjacent robots exchange their global coordinates through wireless communication.

[0117] Step 4: Establish boundary constraints

[0118] The relative position between adjacent robots i and j is expressed as:

[0119]

[0120] The distance between adjacent robots i and j is ||q ij ||.

[0121] Since the parameters of the five robots are the same, their safety radius is uniformly set to rs = 0.2. Based on this, the safety distance between adjacent robots i and j is defined as In order to avoid collisions between adjacent robots, it is required that ||q ij (t)||>r sij =0.4.

[0122] Since the parameters of the five robots are the same, their communication connection radius is uniformly set to r c = 5. Based on this, the communication connection between adjacent robots i and j is defined as maintaining a distance of r cij =r c =5, (i, j)∈ε. In order to maintain the communication connection between adjacent robots, it is required that ||q ij (t)||<r cij =5.

[0123] Therefore, in the entire control process, the boundary constraints can be expressed as:

[0124]

[0125] In particular, the safety distance and the communication connection maintaining distance are both measured in meters.

[0126] Step 5: Design the controller

[0127] Aiming at the formation convergence goal of step 2 and the boundary constraints of step 4, the following controller is designed:

[0128]

[0129] The controller is the sum of two parts, the first part A ij q ij Used to control the formation convergence, the second part f ij (η ij )k ij ξ ij σ ij I2Q ij Used to ensure boundary constraints.

[0130] In the first part, A ij is the gain matrix, which controls the formation convergence by controlling the relative positions between adjacent agents. Its form is as follows:

[0131]

[0132] is the set of all neighboring agents of agent i, including all agents that have communication connections with agent i. Aij is a zero matrix. For A ij the calculation is transformed into an optimization problem, which is described in detail as follows:

[0133] Define a matrix whose form is as follows:

[0134]

[0135] Two conditions that A needs to satisfy:

[0136] Condition 1: The vectors form a basis of ker(A).

[0137] Condition 2: All non-zero eigenvalues of A have negative real parts.

[0138] Among them, where is obtained by rotating by 90°, that is where R π / 2 is the rotation matrix. If the counterclockwise rotation is taken, then:

[0139]

[0140] Let perform the singular value decomposition on N, N = USV T , denote as the last 6 columns of U, denote then the matrix A and have the same non-zero eigenvalues. Thus, by solving the convex optimization problem:

[0141]

[0142] subject to AN = 0

[0143] the specific values of the elements in A are obtained as:

[0144]

[0145] Thereby determining A ij , where represents the matrix the largest eigenvalue.

[0146] Next, determine the parameters in the second part f ij (η ij )k ij ξ ij σ ij I2q ij in turn.

[0147] Let Find η ij The derivative with respect to time gives:

[0148]

[0149] As Figure 4 shown, according to the values of η ij and the control process is divided into the following six cases:

[0150]

[0151]

[0152]

[0153]

[0154]

[0155]

[0156] Among them, c and δ are constants, taken as 0.1 and 0.05 respectively.

[0157] Define the switching signal as:

[0158]

[0159] To transform the constrained system into an equivalent unconstrained system, the following transformation is introduced:

[0160]

[0161] Find the derivative of σ ij with respect to time, we get:

[0162]

[0163] Among them,

[0164] k ij is a constant, taken as 1 in this embodiment.

[0165] I2 is a two-dimensional identity matrix.

[0166] Step 6. Each agent calculates u i using its own computing power, thereby determining the action it needs to execute. After each refresh of the agent's position, u i is recalculated.

[0167] Figure 5 It is a simulation convergence trajectory diagram of five robots forming a regular pentagon formation. The dotted line is the convergence trajectory, Figure 5 The five outer points are the initial positions of the robots. The solid lines between the five outer points represent communication connections, and the inner regular pentagon is the final convergence formation.

[0168] In particular, the present invention guarantees the uniqueness of the formation, and this advantage can be demonstrated through Figure 6 to further illustrate. Figure 6 The left figure is the desired formation, but the formations obtained by some distance-based formation methods may be Figure 6 the right figure, Figure 6 The distances between the robots in the right figure meet the requirements, but it is not exactly the same as the desired formation. The formation control method proposed by the present invention will not have this situation.

[0169] The above embodiments are only partial embodiments of the present invention, rather than all embodiments. The described two-dimensional plane can be any plane in three-dimensional space, not limited to the horizontal ground. The described agents can be any machine or device with a certain degree of intelligence, including underwater robots, aerial robots, and so on.

Claims

1. A multi-agent formation control method, characterized in that, Including the following steps: Step 1, establish the motion model of a single agent: Establish a global X-Y rectangular coordinate system in the space where the agents are active. Each agent is regarded as a particle obeying the single-integrator motion model. Then for the i-th agent, i = 1, …, n, where n represents the total number of agents: In the formula: represents the coordinates of the i-th agent in the global coordinate system; u i is the control input of the i-th agent; Step 2, describe the formation convergence target: For any given desired formation containing n agents, represent the relative positions among the n agents as a set of scatter plots in the X-Y rectangular coordinate system. The number of scatter points is n, and the coordinates of the i-th scatter point are denoted as Denote col() represents a column vector; Generate an undirected graph with n scattered points as vertices where is the set of n vertices of the undirected graph , and is the set of l edges of the undirected graph ; represent the desired formation as and q * such that the rank of the rigidity matrix R of equals 2n - 3, and at this time is said to satisfy infinitesimal rigidity Represent the actual formation as where q i (t) is the position of the i-th agent in the global coordinate system, and the position of the agent changes with time; Define a set to represent the desired formation which is the set of all formations obtained by translation, rotation, scaling, or a combination of them in the X-Y plane, and express the convergence target of the actual formation as the following formula: Step 3, establish the communication connection network among multiple agents: Determine the initial positions of multiple agents: Denote \(q\) i (0) = [x i (0), y i (0)] T as the initial position of the \(i\)-th agent. Denote It is required that the formation corresponding to the initial positions of \(n\) agents satisfies infinitesimal rigidity; Enable the n agents in the multi-agent system to correspond one-to-one with the n vertices of the undirected graph ; Correspond the agents corresponding to the two vertices connected by an edge in the undirected graph as adjacent agents, establish a communication connection for each pair of adjacent agents, and the adjacent agents exchange their global coordinates through wireless communication. The l edges of the undirected graph correspond to l communication links; Step 4, establish the boundary constraint conditions: The relative position between adjacent agents i and j is denoted as q ij , then we have: q ij = q i -q j , where (i, j) ∈ ε Denote the distance between adjacent agents i and j as ||q ij ||; Define the safety distance between adjacent agents i and j as r sij =(r si +r sj )>0, (i, j) ∈ ε, where r si , r sj are the safety radii defined for two adjacent agents i and j respectively, During the entire control process, it is required that ||q ij (t)|| > r sij ; Define the communication connection maintenance distance between adjacent agents i and j as r cij = min{r ci , r cj}, (i, j) ∈ ε, where r ci , r cj are the communication connection maintenance radii defined for two adjacent agents i and j respectively, During the entire control process, it is required that ||q ij (t)|| < r cij ; and there is r sij <<r cij ; Then during the entire control process, the boundary constraint conditions are expressed as the following formula: r sij <||q ij (t)||<r cij ,(i,j)∈ε Step 5, design the controller: For the formation convergence objective in step 2 and the boundary constraint conditions in step 4, a controller u as shown in the following equation is designed for the i-th agent i : is the set of all neighboring agents of agent i, including all agents that have a communication connection with the i-th agent; Controller u i in A ij q ij used to control formation convergence, where: A ij is the gain matrix, which controls the formation convergence by controlling the relative positions between adjacent agents and has the following form: For A ij is a zero matrix; for A ij the calculation is transformed into an optimization problem; Controller u i in f ij (η ij )k ij ξ ij σ ij I2q ij used to ensure boundary constraints, where: Let η ij = ||q ij || 2 , (i, j) ∈ ε, Find the derivative of η ij with respect to time, we get: According to η ij and values, the states of the agent are divided into the following six cases: where c and δ are two positive constants and satisfy Define the switching signal as: To convert the constrained system into an equivalent unconstrained system, the following transformation is introduced: Find σ ij Taking the derivative with respect to time, we get: Among them, k ij is a constant; I2 is a two-dimensional identity matrix; Step 6: Each agent calculates the controller u using its own computing power i , thereby determining the action it needs to execute. After each refresh of the agent's position, the controller u is recalculated i .

2. The multi-agent formation control method according to claim 1, characterized in that, In Step 2, the stiffness matrix R is given by the following formula: In the formula, is called the edge function associated with 3. A multi-agent formation control method according to claim 1, characterized in that, The X-Y rectangular coordinate system is only used to describe the relative positions among the agents in the desired formation and has nothing to do with the coordinate system of the agent activity space and can be arbitrarily selected.

4. The multi-agent formation control method according to claim 1, wherein, The scatter plot only represents the relative positions between individual agents and does not represent the global positions of the agents. There are infinitely many sets of scatter plots that meet the requirements of the desired formation pattern, and the coordinates of this set of scatter plots make the desired formation satisfy infinitesimal rigidity.

5. The multi-agent formation control method according to claim 1, characterized in that, In step 5, A ij is calculated in the following steps: Define a matrix in the following form: To achieve formation, matrix A satisfies the following two conditions: Condition 1: Vector 1, q * , form a basis of ker(A), where ker() represents the kernel or null space of a matrix; Condition 2: All non-zero eigenvalues of A have negative real parts; Among the above conditions: wherein is obtained by rotating by 90°, that is R π / 2 is the rotation matrix; Let Perform singular value decomposition on N as N = USV T , and denote as the last 2n - 4 columns of U, and denote Matrix A and have the same non - zero eigenvalues. Thus, by solving the convex optimization problem, we have: subject to AN = 0 Obtain the specific values of the elements in A to determine A ij , where represents the matrix the largest eigenvalue 6. The multi-agent formation control method according to claim 5, characterized in that, If it is rotated counterclockwise, then R π / 2 is expressed as:

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