Multi-agent formation maneuvering control method based on pigeon flock dynamic leader mechanism

Abstract

The invention relates to a multi-agent formation maneuvering control method based on a pigeon flock dynamic leader mechanism, which is used for solving the problem of multi-agent formation obstacle avoidance, and can automatically select the most suitable agent as the leader in the multi-agent formation by referring to the dynamic replacement phenomenon of the leader in the pigeon flock according to the environment change and task demand. When encountering a complex terrain or an obstacle, the system can quickly identify and switch to a leader most suitable for the current situation, and ensures that the formation smoothly passes through a difficult area. Besides, the characteristics of the stress matrix are utilized to ensure that the formation can execute any affine transformation, such as translation, rotation, zooming and shearing, and the integrity of the formation shape is kept at the same time. According to the invention, by combining the application of the pigeon flock dynamic leader mechanism and the stress matrix, the efficient cooperation and flexibility of the multi-agent system in complex tasks are realized, and the performance and adaptability of formation control are significantly improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of intelligent control, and particularly relates to a multi-agent formation maneuver control method based on a pigeon flock dynamic leadership mechanism. Background Art

[0002] The multi-agent formation control task usually consists of two key subtasks: formation shape control and formation maneuver control. The goal of formation shape control is to enable a group of mobile agents to start from an arbitrary initial configuration and form a predefined geometric pattern or structure by coordinating their movements. This process involves precise positioning of each agent in space and adjustment of their relative positions to ensure that the entire formation can finally present a specific shape as required by the design. On the other hand, formation maneuver control focuses on guiding these mobile agents as a unified whole to perform complex dynamic adjustments, enabling continuous changes in the centroid, direction, scale, and other geometric parameters of the formation. This means that the formation not only has to maintain the stability of its internal structure but also be able to perform actions such as moving forward, backward, turning, expanding, or contracting as a whole. This is crucial for the agent formation to complete a series of advanced navigation tasks, such as autonomous navigation in a complex environment, tracking a moving target, performing search and rescue operations, or dynamically responding to environmental changes (e.g., avoiding obstacles). Formation maneuver control must also consider the cooperation among different agents, ensuring the synchronization of actions of all members, so as to achieve smooth and efficient collective movement.

[0003] As a typical biological group, pigeon flocks demonstrate high swarm intelligence in behaviors such as foraging, homing, and avoiding natural enemies. This intelligence is mainly reflected in their self-organization and self-adaptability, and is driven by the interaction patterns among individuals. Existing research shows that the individual interaction patterns in pigeon flocks mainly include two mainstream views: equal interaction and hierarchical interaction. In the equal interaction mode, all individuals follow the same rules and interact indiscriminately according to the principle of a fixed area or a fixed number of neighbors. In contrast, the hierarchical interaction mode emphasizes the key influence of a few dominant individuals as leaders on collective behavior, forming a clear leadership structure, in which each individual plays the role of a leader or a follower according to its position in the hierarchical network. However, in fact, pigeon flocks have the ability to change the leadership structure in different situations, that is, to follow more dominant individuals in case of emergencies to obtain timely guidance. This strategy endows pigeon flocks with extremely high adaptability and flexibility, enabling them to maintain efficient group actions in complex environments. During the multi-agent formation process, the role of the leader should also be dynamically adjusted according to task requirements and environmental changes. For example, when encountering complex terrain or obstacles, the most suitable agent to act as the leader may change in order to better guide the entire formation through difficult areas.

[0004] In the field of multi-agent formation control, the stress matrix, as an emerging tool, has shown great potential in dealing with complex formation maneuver problems. It is a generalized Laplacian matrix, whose structure is determined by the underlying graph, but the element values are determined by the formation configuration, allowing edge weights to be positive, negative, or zero, thus more flexibly describing the formation geometry. Particularly importantly, the stress matrix is invariant under affine transformations, which means that it can perform operations such as translation, rotation, scaling, or shearing, either individually or in combination, while maintaining the formation shape. Due to this property, the stress matrix provides a powerful tool to achieve various formation maneuver behaviors. However, there are still some deficiencies in the existing technologies: when dealing with complex formation maneuvers, the stress matrix in the existing technologies lacks a quantifiable and implementable dynamic leadership switching model applicable to multi-agent systems, which results in the traditional formation system being unable to quickly adjust the leadership strategy to optimize path planning when facing sudden obstacles or task changes, significantly reducing the obstacle avoidance efficiency and environmental adaptability. Summary of the Invention

[0005] The present invention proposes a multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism, aiming to improve the cooperation efficiency and adaptability of multi-agent systems in complex tasks by simulating the intelligent behavior of pigeon flocks in natural environments. This method particularly focuses on the dynamic transformation of the leader role to achieve more flexible and efficient formation maneuver control.

[0006] This method draws on the phenomenon in pigeon flocks that the leader dynamically changes according to environmental changes and task requirements, and designs a multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism, which can automatically select the most suitable agent as the leader in a multi-agent formation. This not only improves the response speed of the formation to sudden situations but also enhances the robustness and flexibility of the overall system.

[0007] The present invention utilizes the characteristics of the stress matrix to ensure that the formation can perform arbitrary affine transformations, such as translation, rotation, scaling, and shearing, while maintaining the integrity of the formation shape. This invariance enables the formation to perform precise maneuvers in complex environments and adapt to different task requirements. A distributed control law based on the stress matrix is adopted to ensure that each agent only needs to communicate with several adjacent companions to achieve complex formation maneuvers. These control laws have global stability, are applicable to formation control in any dimension, and do not require dependence on a global reference frame, thereby improving the scalability and practicality of the system.

[0008] The specific technical solution is as follows:

[0009] The multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism proposed by the present invention has characteristics such as flexible formation structure and wide application scenarios. The specific application steps of this method are as follows:

[0010] Preparation: Agent p i The initial positions of the obstacles in the environment are all known, and the quantity is also known. Let the number of agents be n, the nominal configuration of the multi-agent be r, the incidence matrix of the multi-agent interaction be H, and the agent numbers be denoted as [1, 2,..., n].

[0011] Step 1: Determine the n agents participating in the formation and clarify their nominal configuration

[0012] Step 2: At initialization, set at least d + 1 agents out of the n agents as leaders, ensuring that these leaders are affine-spanned in . Denote the set of leaders as V L , and denote the set of followers as V F .

[0013] Step 3: Check whether the nominal configuration r satisfies the condition of affine span, that is, ensure that the formation can achieve the required geometric transformation. If this condition is met, the subsequent formation control process can be directly entered; if not, it is necessary to return to the preparation stage to reset the nominal configuration.

[0014] Step 4: Ensure that the nominal configuration r is universally rigid. This can be verified by checking the properties of the stress matrix Ω to ensure that its null space only contains translation, rotation, and scaling transformations.

[0015] Step 5: Define the incidence matrix of the graph G=(V, E) where H describes the connection relationship between agents.

[0016] Step 6: Construct the matrix according to the nominal configuration r and the incidence matrix for subsequent solving of the equilibrium stress.

[0017] Step 7: Calculate the basis vectors z1,..., z of the null space of q by singular value decomposition (SVD). These vectors are used for subsequent calculation of the equilibrium stress.

[0018] Step 8: Use SVD decomposition to construct the matrix and solve the equilibrium stress through linear matrix inequality (LMI)

[0019] Step 9: When the nominal configuration r satisfies the condition of Step 3, the control action of the followers can be calculated through the equilibrium stress in Step 8, so that when the positions of the leaders are determined, the followers can track the desired positions of the corresponding formation.

[0020] So far, a basic control framework of "when the leader's position is determined, the follower is guided to the target formation position based on the characteristics of the stress matrix" has been formed. To ensure the formation efficiency, a pigeon flock dynamic leadership mechanism is introduced to change the leader according to the environment.

[0021] Step ten: To simulate the natural phenomenon that active individuals in a pigeon flock act as leaders, each agent calculates the average cross product result between the acceleration and velocity vectors of its neighbors at the current time t0 and calculates the cross product result C i (t0) with its own velocity vector v i (t0), so as to obtain the state score F according to the difference state .

[0022] Step eleven: To simulate the situation where individuals near obstacles take over the guiding responsibility when the pigeon flock faces threats, the agent calculates the distance d between itself and the obstacle io , uses the negative exponential function to calculate and obtain the influence score F of the obstacle on itself obs .

[0023] Step twelve: Construct a comprehensive evaluation basis F that combines the motion state and environmental threats sum , and obtain the comprehensive score F by adding the state score F state in step ten and the obstacle influence score F obs in step eleven sum .

[0024] Step thirteen: The pigeon flock dynamic leadership mechanism is based on the comprehensive score F of each agent sum , and is realized through the method of distributed collaborative voting. Each agent will try to become a candidate and send a voting request to its neighbors. After receiving the request, the neighbors will vote for the candidate. If a candidate gets more than half of the votes in a round of election, it is elected as the leader. This process will continue to iterate until enough leaders are elected or the maximum number of iterations is reached.

[0025] Step fourteen: The leader evaluates whether obstacle avoidance operations are needed according to the geometric configuration of the current formation and the position of the obstacle. If obstacle avoidance is required, the leader calculates the affine transformation parameters required for obstacle avoidance and determines the target formation p target , and the follower completes the tracking of the desired position through step nine.

[0026] In step one, an initial position is assigned to each agent i These positions can be preset, or obtained by random generation or other means. Among them, n≥d + 1, and d is the spatial dimension (for example, d = 2 for a two-dimensional formation, and d = 3 for a three-dimensional formation). is the nominal configuration vector of the i-th agent, The nominal configuration vector for all agents, with agent numbers corresponding to [1, 2,..., n].

[0027] In step two, at least d + 1 of the n agents are initially set as leaders, denoted as V L , and the remaining n - (d + 1) agents are followers, denoted as V F . V L is the index set of the leaders, and V F is the index set of the followers. It should be noted that this initial leader setting is only the basic configuration for the formation startup. During the subsequent formation maneuver control process (step fourteen), based on the pigeon flock dynamic leadership mechanism, the adaptive replacement of leaders will be achieved through dynamic adjustment strategies such as voting elections, rather than always relying on the fixed leaders set initially by humans. The number of leaders and the number of followers in step two need to satisfy the following formula:

[0028] |V L | = n L ≥ d + 1 (1)

[0029] |V F | = n F = n - n L (2)

[0030] is the nominal configuration vector of the leaders. is the nominal configuration vector of the followers.

[0031] To ensure that the nominal configuration vector r of the leaders L is affine spanned in , that is:

[0032]

[0033] where, is the matrix related to the nominal configuration vector of the leaders, defined as:

[0034]

[0035] If then the leaders can uniquely determine the positions of the followers.

[0036] The formula for checking whether the nominal configuration r satisfies affine span described in step three is:

[0037]

[0038] In the formula (5), is defined as:

[0039]

[0040] Among them, each row represents the nominal configuration information of an agent, and the second column being 1 indicates translational invariance.

[0041] If has a rank of d + 1, then the nominal configuration is affinely spanned in and the formation can achieve the required geometric transformation.

[0042] In step four, first construct an undirected graph G = (V, E) that describes the interaction relationship between agents, where the vertex set V = {1,..., n} is the set of agent numbers, and n represents the number of agents. And is the set of edges, representing the communication or physical connection relationship between agents. Each edge (i, j) ∈ E indicates that there is a direct interaction or constraint between agent i and agent j.

[0043] For G = (V, E), the stress is a set of scalars, {ω ij} (i,j)∈E where ω ij = ω ji and is assigned to all edges. If the stress satisfies the following conditions, it is called balanced stress:

[0044]

[0045] where N i = {j ∈ V: (i, j) ∈ E}, representing the neighbor set of agent i.

[0046] Then the stress matrix satisfying formula (7) is defined as follows:

[0047]

[0048] To verify whether the nominal configuration r is universally rigid, it is necessary to check whether the stress matrix Ω satisfies the following condition:

[0049] rank(Ω) = n - d - 1 (9)

[0050] where n is the number of agents and d is the spatial dimension.

[0051] In step five, after orienting the undirected graph G = (V, E) in step four (that is, assigning a direction to each edge), an incidence matrix describing the interaction relationship between agents can be obtained.

[0052]

[0053] where Each row represents each edge of G=(V, E), each column corresponds to an agent, and m is the number of edges of G=(V, E).

[0054] The matrix described in Step Six can be calculated based on the nominal configuration r in Step One and the incidence matrix H in Step Five. The specific formula is as follows:

[0055]

[0056] where is given by formula (6), which is a matrix related to the nominal configuration r. diag(h i ) is a diagonal matrix, and the i-th diagonal element is the i-th element of h i , and h i is the row vector of the incidence matrix H.

[0057] Described in Step Seven The basis vectors z1,..., z q of the null space can be calculated by singular value decomposition (SVD). The calculation formula is as follows:

[0058] E = UΣV T (12)

[0059] where the basis vectors of the null space correspond to the columns of V corresponding to the singular values close to zero in Σ. These vectors will be used for subsequent calculation of the equilibrium stress.

[0060] For each basis vector z i of the null space described in Step Eight, first calculate the matrix M i for solving the equilibrium stress M i The calculation formula is:

[0061]

[0062] where U is the left singular vector matrix in the SVD of , and U2 is the last n-(d + 1) columns of U.

[0063] Then the equilibrium stress can be expressed as a linear combination of the basis vectors of the null space. The formula is:

[0064]

[0065] where c1,..., c q are undetermined coefficients. To determine these coefficients, the following linear matrix inequality needs to be solved:

[0066]

[0067] Among them, M i is obtained from formula (14).

[0068] The feasibility problem represented by formula (16) can be numerically solved to obtain the undetermined coefficients c1,..., c q and then substituted into formula (15) to solve for the equilibrium stress

[0069] In step nine, according to the equilibrium stress of formula (16) the following formula is used to obtain the follower control action of the follower:

[0070]

[0071] Among them, m is the number of edges of G=(V, E), ω k corresponds to the stress on the k-th undirected edge (k = 1,..., m). If agents i and j are connected by the k-th undirected edge, then ω ij = ω k .

[0072] In step ten, the acceleration a i (t0) of each agent i at time t0 is defined by the rate of change of velocity:

[0073]

[0074] Among them, v i (t0) is the velocity vector of agent i at time t0.

[0075] Next, calculate the and C i (t0) of agent i at time t0, and the formula is as follows:

[0076] C i (t0) = v i (t0) × a i (t0) (20)

[0077]

[0078] Among them, N i is the neighbor set of agent i and is defined by formula (7). v j (t0) and a j (t0) are the velocity vector and acceleration vector of agent j at time t0 respectively.

[0079] Then the state score can be calculated by a piecewise function:

[0080]

[0081] Among them, k, p, a, a1, and a2 are the influence coefficients for adjusting the change in the motion state of the agent on F state of.

[0082] The influence score F of the obstacle on agent i is calculated and obtained by the negative exponential function described in Step Eleven obs The specific formula is as follows:

[0083] d io = ||p i - p obs || (24)

[0084]

[0085] Among them, p obs is the position of the obstacle, and k and b in formula (25) are the gain coefficient and bias term for score calculation, respectively.

[0086] In Step Thirteen, the leader index is stored in vector L, the election status of each agent is recorded by boolean vector E. The candidate status is recorded by vector C, and the number of votes received by each candidate is recorded by vector V. The voting threshold T is defined as That is, the majority vote. The candidate list C list is used to store the index and priority of each candidate, where the priority is determined by the total score F sum = F state + F obs .

[0087] In each iteration, agents not selected as leaders attempt to become candidates. If they have not already become candidates, they vote for themselves and join the candidate list. Each agent also receives potential votes from its neighbors. After that, the candidate list is sorted according to the priority. The candidate with the highest priority is selected as the leader if the number of votes it receives is greater than or equal to the threshold T. Its status in E is updated to true and it is added to the leader index vector L. The iteration continues until a sufficient number of leaders are selected or the maximum number of iterations M is reached.

[0088] The calculation formula for the target formation p described in Step Fourteen target is as follows:

[0089]

[0090] Among them, p target represents the position of the target formation, I n is the n×n identity matrix, represents the Kronecker product, and A is a d×d matrix that controls the nominal configuration Rotation, scaling, shearing, or a combination thereof, and b is a d-dimensional vector that controls the translation of r. In a multi-agent formation, the leader serves as the decision-making core and is usually pre-given prior knowledge such as environmental parameters and task rules (such as matrix A containing transformation parameters such as rotation and scaling, and the setting rules for vector b that controls translation), so the present invention is not specifically designed.

[0091] The remarkable beneficial effects of the present invention are:

[0092] The present invention proposes a multi-agent formation maneuver control method based on a pigeon flock dynamic leadership mechanism and uses it to solve the problem of formation obstacle avoidance for multi-agents. By mimicking the high swarm intelligence demonstrated by pigeon flocks in behaviors such as foraging, homing, and avoiding natural enemies, especially their self-organization and self-adaptability, the adaptability and flexibility of multi-agent formations in complex environments are significantly enhanced. This method integrates the advantages of pigeon flocks dynamically adjusting the leadership structure according to the environment, not only ensuring that all agents follow unified rules for coordination, but also optimizing the overall performance by dynamically selecting leaders, enabling the formation to quickly respond to emergencies, such as encountering obstacles or needing to change paths, thereby improving the formation operation efficiency and success rate. In addition, using the stress matrix as a tool, the present invention can achieve complex affine transformations while maintaining the formation shape unchanged, including operations such as translation, rotation, scaling, and shearing. This not only simplifies the control method, reduces the need for real-time communication, lowers system complexity and energy consumption, but also ensures that the formation can flexibly maneuver without destroying the structure, and is applicable to various advanced navigation tasks. Description of the Drawings

[0093] Figure 1 is a schematic diagram of the nominal configuration of multi-agents.

[0094] Figure 2 is the initial scenario view of the multi-agent formation maneuver control based on the pigeon flock dynamic leadership mechanism.

[0095] Figures 3 - 8 is the process view of the multi-agent formation maneuver control based on the pigeon flock dynamic leadership mechanism.

[0096] Figure 9 is the curve graph of the formation obstacle avoidance trajectory of multiple agents.

[0097] Figure 10 is the curve graph of the formation obstacle avoidance speed of multiple agents.

[0098] Figure 11 is the list of leader numbers during the formation obstacle avoidance process. Detailed Implementation Manner

[0099] The following further describes the present invention in combination with the specific implementation process, in order to Figure 2Take the environment as an example for illustration:

[0100] Preparation: In the experiment of formation transformation and obstacle avoidance simulating the behavior of a pigeon flock, it is first necessary to determine the number of agents (i.e., simulated individuals, which can be analogized to pigeons in a pigeon flock). For multiple agents to perform formation transformation and obstacle avoidance simultaneously, the number of agents n = 8, and the scenario is set as a two-dimensional plane, that is, d = 2, which simplifies the calculation and is convenient for observing and analyzing the interaction and obstacle avoidance behavior among agents. The obstacles in the scenario are geometric bodies with different widths, lengths, and angles. Multiple agents with random initial positions need to form a formation and pass through different obstacles by transforming the formation.

[0101] The parameter values are as follows: In the nominal configuration The initial position of each agent is randomly assigned p i =[p x ,p y T , where -2 ≤ p x ≤ 2, -2 ≤ p y ≤ 2; The influence coefficient k = 0.1, p = 1, a = 1 of the change in the agent's motion state on the state score F state affecting leader election, and the difference weight factors a1 = 100, a2 = 2 of its own state and the neighbor state (simulating the trade-off between the individual's own motion state and the group information in a pigeon flock); The gain coefficient k = 1 and the bias term b = 3 for the F obs score calculation (simulating the "vigilance level" of pigeon flock individuals towards obstacles); From F state and F obs obtain the total score F sum of each agent, as the basis for the voting election of the leader, where the voting threshold T = 4; The maximum number of iterations M = 150 for the voting election; The adjacency matrix such as Figure 1 shown, there are 8 agents in total, and there are 15 edges representing connection relationships among the agents, specifically:

[0102]

[0103] Step 1: Randomly assign an initial position to each agent i and clarify the nominal configuration vectors of all agents corresponding to the numbers [1, 2,..., 8].

[0104] Step 2: Initially, set n L = 3 out of n = 8 agents as leaders, and denote the leader set V L = [1, 2, 3], and the remaining n F ​= Five agents are followers, and the set of followers is denoted as V F = [4, 5, 6, 7, 8]. Here, the leader is artificially selected to initialize the formation leadership structure, ensuring that the number of leaders and followers satisfies the following formula:

[0105] |V L | = n L ≥ d + 1 (27)

[0106] |V F | = n F = n - n L (28)

[0107] To ensure that the nominal configuration vector r of the leader L is affinely spanned in, the calculation formula is as follows:

[0108]

[0109] Step 3: Check whether the nominal configuration r satisfies affine span. The calculation formula is as follows:

[0110]

[0111] Step 4: Construct an undirected graph G = (V, E) that describes the interaction relationship between agents. For G = (V, E), the stress is a set of scalars, {ω ij} (i,j)∈E , where is assigned to all edges, and the balanced stress should satisfy the following conditions:

[0112]

[0113] The stress matrix that satisfies formula (32) is defined as follows:

[0114]

[0115] Step 5: According to the undirected graph G = (V, E), the incidence matrix that describes the interaction relationship between agents can be constructed The calculation formula is as follows:

[0116]

[0117] Step 6: It is calculated based on the nominal configuration r in Step 1 and the incidence matrix H in formula (35). The calculation formula is as follows:

[0118]

[0119] Step 7: Perform singular value decomposition on the matrix E in formula (36). The calculation formula is as follows:

[0120] E = UΣV T (37)

[0121] Step Eight: For each null space basis vector z i , first calculate the matrix M i for solving the equilibrium stress ω, M i has the following calculation formula:

[0122]

[0123] Then the equilibrium stress can be expressed as a linear combination of the null space basis vectors, and the calculation formula is as follows:

[0124]

[0125]

[0126] The undetermined coefficients of formula (41) are numerically solved by using the Matlab LMI Toolbox. After the solution is completed, substitute them into formula (41) to obtain the equilibrium stress

[0127] Step Nine: According to the equilibrium stress ω calculated by formula (40), obtain the following control action of the follower, and the calculation formula is as follows:

[0128]

[0129] When the leader's position is determined, based on the characteristics of the stress matrix, guide the follower to reach the target formation position. However, this framework has a significant limitation of a fixed leader and it is difficult to guarantee the formation efficiency. Therefore, a dynamic leadership mechanism is introduced.

[0130] Step Ten: Each agent calculates the average cross product result between the acceleration and velocity vectors of its neighbors at the current moment t0 The calculation formula is as follows:

[0131]

[0132] and calculates the cross product result C i (t0) of its own acceleration a i (t0) and its own velocity vector v i (t0), and the calculation formula is as follows:

[0133] C i (t0) = v i (t0) × a i (t0) (45)

[0134] The state score can be calculated through a piecewise function as follows:

[0135]

[0136] Step Eleven: The influence score F of the obstacle on agent i obs is calculated as follows:

[0137] d io = ||p i - p obs || (48)

[0138]

[0139] Step Twelve: By adding the state score F state and the obstacle influence score F obs together, the comprehensive score F sum is obtained.

[0140] Step Thirteen: The leader is selected through a voting method. Each agent will try to become a candidate and send a voting request to its neighbors. After receiving the request, the neighbors will vote for the candidate. If a candidate receives more than half of the votes in a round of election, it is elected as the leader. This process will continue to iterate until enough leaders are elected or the maximum number of iterations is reached.

[0141] Step Fourteen: The leader determines the target formation p target using the following formula:

[0142]

[0143] Figure 1 shows the nominal configuration r of multiple agents, while Figures 2 to 3 intuitively demonstrates the process of the formation evolving from the initial configuration to finally forming a predefined geometric pattern or structure. Specifically, Figures 3 to 8 schematically shows which agents are elected as leaders in different scenarios and how they guide other members to perform coordinated movements to ensure that the entire formation can execute complex dynamic adjustments as a unified whole. Taking Figure 5 and Figure 6 as an example, Figure 5 the leaders are Agent No. 3, Agent No. 5, and Agent No. 7 respectively. In the face of a scenario that requires turning, Agents No. 1, No. 2, and No. 3, which were originally followers in favorable positions, become new leaders after the election process and continue to safely pass through the narrow passage by adjusting the scaling size and shear coefficient, demonstrating the ability of the formation to perform operations such as translation, rotation, scaling, or shearing in different scenarios, and reflecting the invariance of the stress matrix to affine transformation. Figure 10Shows the trajectories of multiple agents when performing affine formation maneuver control tasks. As can be seen from the figure, the position changes of each agent at different time points clearly reflect the dynamic adjustment process of the formation in a complex environment. The agents can work together to keep the overall shape of the formation unchanged, while performing operations such as translation, rotation, and scaling to ensure that the entire formation can efficiently pass through obstacles or perform specific tasks. Figure 11 Shows the curve of the magnitude of the velocity of these agents changing with time. As can be seen from the figure, the velocities of the agents rapidly decrease in the initial stage, and then remain relatively stable during the task execution, with occasional velocity fluctuations. This indicates that the formation can effectively coordinate the velocities of its members during the dynamic adjustment process to ensure the smoothness and consistency of the overall movement. This velocity change curve further proves the effectiveness of the stress matrix in achieving complex maneuvers while maintaining the formation shape, enabling the formation to efficiently complete various obstacle avoidance tasks while keeping the internal structure stable. During the overall formation obstacle avoidance process, the change of the leader role is as Figure 11 shown.

Claims

1. A multi-agent formation maneuver control method based on a pigeon flock dynamic leadership mechanism, characterized in that, It includes the following steps: Step 1: Determine the n agents participating in the formation and clarify their nominal configurations Step 2: At initialization, set at least d + 1 out of n agents as leaders, ensuring that these leaders are affinely spanned in ; Denote the set of leaders as V L , and denote the set of followers as V F ; Step 3: Check whether the nominal configuration r satisfies the condition of affine span, that is Ensure that the formation can achieve the required geometric transformation; If this condition is met, it can directly enter the subsequent formation control process; if not, it needs to return to the preparation stage to reset the nominal configuration; Step 4: Ensure that the nominal configuration r is generally rigid; this can be verified by checking the properties of the stress matrix Ω to ensure that its null space only contains translation, rotation, and scaling transformations; Step Five: Define the incidence matrix of graph G=(V, E) where H describes the connection relationship between agents; Step 6: Configure r according to the nominal value and the incidence matrix Construct a matrix for subsequent solution of the equilibrium stress; Step 7: Calculate by singular value decomposition (SVD) of the null space basis vectors z1,..., z q , and these vectors are used for subsequent calculation of the balanced stress; Step 8: Use SVD decomposition Construct a matrix and solve for the balanced stress through linear matrix inequalities (LMI) Step Nine: When the nominal configuration r meets the conditions of Step Three, the control action of the follower can be calculated through the balanced stress in Step Eight, so that when the leader's position is determined, the follower can track the expected position of the corresponding formation; ​ So far, a basic control framework of "when the leader's position is determined, guiding the followers to the target formation position based on the characteristics of the stress matrix" has been formed. To ensure the formation efficiency, a pigeon flock dynamic leadership mechanism is introduced to change the leader according to the environment; Step 10: To simulate the natural phenomenon where active individuals in the pigeon flock act as leaders, each agent calculates the average cross - product result between the acceleration of its neighbors and the velocity vector at the current moment t0 and calculates the cross - product result C i (t0) with its own velocity vector v i (t0), so as to obtain the state score F based on the difference state ; Step Eleven: To simulate the situation where individuals near obstacles take over the guiding responsibility when the pigeon flock faces threats, the agent calculates the distance d between itself and the obstacle io , calculates and obtains the influence score F of the obstacle on itself using the negative exponential function obs ; Step Twelve: Construct a comprehensive evaluation basis F that integrates the motion state and environmental threats sum , through the state score F in Step Ten state and the obstacle impact score F in Step Eleven obs add them together to obtain the comprehensive score F sum ; Step 13: The pigeon flock dynamic leadership mechanism is based on the comprehensive score F of each agent sum and is achieved through a distributed collaborative voting method. Each agent will try to become a candidate and send a voting request to its neighbors. After receiving the request, the neighbors will vote for the candidate. If a candidate receives more than half of the votes in a round of election, it is elected as the leader. This process will continue to iterate until a sufficient number of leaders are elected or the maximum number of iterations is reached; Step 14: The leader evaluates whether obstacle avoidance operation is required based on the geometric configuration of the current formation and the position of the obstacle. If obstacle avoidance is needed, the leader calculates the affine transformation parameters required for obstacle avoidance and determines the target formation p target , and the followers complete the tracking of the desired positions through Step 9.

2. A multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism according to claim 1, characterized in that: In step one, an initial position is assigned to each agent \(i\). These positions can be preset, or obtained by random generation or other means; where \(n\geq d + 1\), and \(d\) is the spatial dimension (for example, \(d = 2\) for two-dimensional formation, and \(d = 3\) for three-dimensional formation). is the nominal configuration vector of the \(i\)-th agent. is the nominal configuration vector of all agents, and the agent numbers correspond to \([1, 2, \ldots, n]\).

3. A multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism according to claim 2, characterized in that: In step two, at least d + 1 of the n agents are initially set as leaders, denoted as V L , and the remaining n - (d + 1) agents are followers, denoted as V F ; V L is the index set of the leaders, and V F is the index set of the followers; it should be noted that this initial leader setting is only the basic configuration for the formation startup. During the subsequent formation maneuver control process (step fourteen), based on the pigeon flock dynamic leadership mechanism, the adaptive replacement of the leaders will be achieved through dynamic adjustment strategies such as voting, rather than always relying on the fixed leaders set artificially initially; the number of leaders and the number of followers in step two need to satisfy the following formula: |V L | = n L ≥ d + 1(1) |V F | = n F = n - n L (2) The nominal configuration vector for the leader; The nominal configuration vector for the follower; To ensure the nominal configuration vector r of the leader L In the affine span, i.e.: Among them, is a matrix related to the leader's nominal configuration vector, defined as: If then the leader can uniquely determine the position of the follower.

4. A multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism according to claim 3, characterized in that: The formula for checking whether the nominal configuration r satisfies affine span in step 3 is: In formula (5), is defined as: Among them, each row represents the nominal configuration information of an agent, and the second column is 1, indicating translational invariance; If has rank d + 1, then the nominal configuration affinely spans in and the formation can achieve the desired geometric transformation.

5. A multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism according to claim 4, characterized in that: In step four, first construct an undirected graph G=(V, E) that describes the interaction relationship between agents, where V={1,..., n} is the set of agents, and n represents the number of agents; and E is the set of edges, representing the communication or physical connection relationship between agents. Each edge (i, j) ∈ E indicates that there is a direct interaction or constraint between agent i and agent j; For \(G=(V, E)\), the stress is a set of scalars, \(\{\omega ij \} (i,j)∈E \), where \(\omega ij =\omega ji \), assigned to all edges; If the stress satisfies the following conditions, it is called balanced stress: where N i = {j ∈ V: (i, j) ∈ E} represents the set of neighbors of agent i; The stress matrix that satisfies Equation (7) is defined as follows: To verify whether the nominal configuration r is generally rigid, it is necessary to check whether the stress matrix Ω satisfies the following conditions: rank(Ω) = n - d - 1(9) Among them, n is the number of agents, and d is the spatial dimension; In Step 5, after orienting the undirected graph G=(V, E) in Step 4 (i.e., assigning a direction to each edge), an incidence matrix describing the interaction relationship between agents can be obtained. Its elements are {0, ±1} and are defined as follows: Among them, Each row represents each edge of G = (V, E), and each column corresponds to an agent, where m is the number of edges of G = (V, E).

6. A multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism according to claim 5, characterized in that: The matrix described in Step 6 can be calculated based on the nominal configuration r in Step 1 and the correlation matrix H in Step 5. The specific formula is as follows: Among them, is given by formula (6), which is a matrix related to the nominal configuration r, and diag(h i ) is a diagonal matrix, and the i-th diagonal element is the i-th element of h i , and h i is the row vector of the incidence matrix H.

7. A multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism according to claim 6, characterized in that: As described in Step Seven The null space basis vectors z1,..., z q can be calculated by singular value decomposition (SVD), and the calculation formula is as follows: E = UΣV T (12) Among them, the basis vectors of the null space correspond to the columns of V corresponding to the singular values close to zero in Σ; these vectors will be used for subsequent calculation of balanced stress; For each null space basis vector z i , first calculate the matrix M i for solving the equilibrium stress M i The calculation formula is: where i = 1,..., q (14) where U is the left singular vector matrix in the SVD of, and U2 is the last n-(d+1) columns of U; The equilibrium stress can be expressed as a linear combination of the null space basis vectors of E with the formula: where \(c_1,\cdots,c\) q are undetermined coefficients. To determine these coefficients, the following linear matrix inequality needs to be solved: Among them, M i is obtained from formula (14); The feasibility problem represented by Equation (16) can be solved numerically to obtain the undetermined coefficients c1,..., c q and then substitute them into Equation (15) to solve for the equilibrium stress 8. A multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism according to claim 7, characterized in that: In Step 9, according to the equilibrium stress of formula (16) the follower's following control action is obtained, and the calculation formula is as follows: Among them, m is the number of edges of G = (V, E), ω k corresponds to the stress on the k-th undirected edge (k = 1,..., m). If the current agents i and j are connected by the k-th undirected edge, then ω ij = ω k ; In step ten, the acceleration a of each agent i at time t0 i (t0) is defined by the rate of change of velocity: where v i (t0) is the velocity vector of agent i at time t0; Next, calculate for agent i at time t0 the and C i (t0), the formula is as follows: C i (t0) = v i (t0) × a i (t0)(20) where, N i is the neighbor set of agent i and is defined by formula (7), v j (t0) and a j (t0) are the velocity vector and acceleration vector of agent j at time t0, respectively; Then the state score can be calculated by a piecewise function: Among them, k, p, a, a1, and a2 are the influence coefficients for adjusting the change in the motion state of the agent on F state .

9. A multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism according to claim 8, characterized in that: The negative exponential function described in Step Eleven calculates and obtains the influence score F of the obstacle on agent i obs The specific formula is as follows: d io = ||p i -p obs || (24) where p obs is the position of the obstacle, and k and b in formula (25) are the gain coefficient and bias term for score calculation, respectively.

10. A multi-agent formation maneuver control method based on the pigeon flock dynamic leadership mechanism according to claim 9, characterized in that: In Step 13, the leader index is stored in vector L, and the election status of each agent is recorded by a boolean vector E; the candidate status is recorded by vector C, and the number of votes received by each candidate is recorded by vector V; the voting threshold T is defined as i.e., the majority vote; the candidate list C list is used to store the index and priority of each candidate, where the priority is determined by the total score F sum = F state + F obs decide; In each iteration, the agents not selected as leaders try to become candidates. If they have not become candidates yet, they vote for themselves and join the candidate list; each agent also gets potential votes from its neighbors; then, the candidate list is sorted according to the priority. The candidate with the highest priority is selected as the leader if the number of votes it gets is greater than or equal to the threshold T. Its state in E is updated to true and it is added to the leader index vector L; the iteration continues until enough leaders are selected or the maximum number of iterations M is reached; The target formation p described in Step 14 target has the following calculation formula: where p target represents the position of the target formation, I n is the n×n identity matrix, represents the Kronecker product, A is a d×d matrix that controls the rotation, scaling, shearing, or a combination thereof of the nominal configuration ; b is a d-dimensional vector that controls the translation of r. In a multi-agent formation, the leader serves as the decision-making core and is usually pre-given prior knowledge such as environmental parameters and task rules (such as the matrix A containing transformation parameters such as rotation and scaling, and the setting rules of the vector b that controls translation). Therefore, the present invention is not specifically designed.

Images