Simulation method for incorporating leader level into agent cluster flight motion mechanism

Abstract

The invention discloses a simulation method for incorporating a leader level into an agent cluster flight motion mechanism. The method comprises the following steps: carrying out preparation work before an experiment and an agent flight experiment; extracting motion features of the double-layer intelligent agent clusters of the local layer and the global layer; the cross-level mediation effect of leadership is ascertained; by establishing a leader force cross-level dynamic model, a nest-surrounding flight movement mechanism of the homing pigeon group under the leader force cross-level mediation effect is disclosed; according to the invention, the motion characteristics of the intelligent agent cluster can be simulated more truly.

Description

Technical Field

[0001] The present invention relates to the technical field of swarm behavior simulation, and in particular to a simulation method that incorporates leadership levels into the flight motion mechanism of intelligent body swarms. Background Art

[0002] The simulation and prediction of animal flock movement patterns has always been an important research issue in the fields of behavioral ecology and physical biology.

[0003] Reynolds (1987) first proposed the Boid model, defining three rules: separation, alignment, and cohesion. This model formed the basis for an agent-based model for simulating the movement of animal flocks. However, the Boid model did not fully consider the speed constraints and spontaneous behavior of individual animals during movement, limiting its applicability.

[0004] Subsequently, Vicsek et al. (1995) proposed the classic Vicsek model, which achieved a preliminary simulation of the swarm emergence process by strictly limiting the spatial range of particles and the conditions for completely equal individual speeds. However, because the Vicsek model ignores the individual heterogeneity that is common in real animal groups (such as leadership hierarchy and speed differences), it is difficult to accurately describe the movement characteristics of complex animal groups.

[0005] To overcome the limitations of the Vicsek model, researchers have proposed several improved models, such as using parameter fitting to the movement patterns of specific species to verify the model's applicability (Ling et al., 2019). Furthermore, data-driven recurrent neural networks (RNNs) and long-short-term memory (LSTM) networks have been used to capture the "leader-follower" patterns in animal group movements and predict future movement states (Pérez-Escudero and Polavieja, 2011). However, these methods lack a clear description of the leadership hierarchy within animal groups, making it difficult to systematically reveal how leadership hierarchies function in group movements.

[0006] Therefore, existing technologies have not yet effectively incorporated differences in leadership hierarchies within a group, nor have they revealed the mechanism by which leadership hierarchies influence the movement patterns of animal groups.

[0007] The traditional methods include the following:

[0008] (1) Construction and basic principles of the Vicsek model:

[0009] The Vicsek model is a classic self-organizing swarm motion model, proposed by Vicsek et al. based on the Reynolds model. The model consists of a large number of self-propelled particles, each moving at a constant velocity in a two-dimensional plane. The model's sole evolutionary rule is the "velocity alignment" principle: in discrete time steps, each particle adjusts its own motion direction to the average velocity of all neighboring particles within its perception range (a certain radius), while maintaining a constant velocity. Furthermore, the model typically introduces a small random perturbation factor (noise) to simulate the uncertainty inherent in real environments, ensuring a more versatile and stable simulation.

[0010] (2) The formation mechanism of cluster emergence process:

[0011] Leveraging this local alignment interaction mechanism, the particle swarms of the Vicsek model emerge with ordered collective motion on a macroscopic scale. Specifically, the model demonstrates the transition from disordered motion to large-scale ordered motion: when the system noise is high or the particle density is low, the motion directions of individual particles are chaotic, and the swarm exhibits a disordered state similar to that of a gas. However, as the noise intensity decreases or the particle density increases, the directions of the particles gradually converge, and the entire swarm eventually moves in a coordinated direction, forming an ordered cluster state similar to that of a liquid. Under certain conditions, local alignment can trigger "symmetry breaking," causing the swarm to spontaneously choose a common direction of motion. The resulting local groups of aligned pairs within a small area can further influence particles on a larger scale, ultimately expanding through repeated iterations into a globally coordinated motion. Thus, the Vicsek model successfully explains the emergence of swarm behavior through a simple local interaction mechanism: individual microscopic rules can generate macroscopic ordered structures at the group level.

[0012] (3) Improved version of Vicsek model:

[0013] To make the model more realistic for real-world group behavior, researchers have proposed various improved and extended versions of the Vicsek model. While retaining the basic alignment mechanism, these models incorporate additional interaction factors, such as repulsion for collision avoidance and attraction for group cohesion, to simulate the collision avoidance and aggregation behaviors of biological groups. For example, the Cucker-Smale model improves on the original Vicsek model by introducing the concept of weighted average velocity based on distance to establish a group consensus model. Similarly, the social force model of Couzin et al. divides the individual perception area into repulsion, alignment, and attraction zones, enabling particles to simultaneously avoid collisions and maintain cohesion. These improved models build on the principles of the Vicsek model by incorporating richer interaction rules, further enhancing the model's ability to depict real-world clustering phenomena. They also confirm, from different perspectives, the effectiveness of the group self-organization and coordination mechanisms revealed by the Vicsek model.

[0014] Existing technologies and models for animal group movement still have significant deficiencies, especially in terms of reflecting the differences in leadership hierarchies within animal groups and their impact on group movement mechanisms.

[0015] The classic Vicsek model simulates the cluster emergence process under strict spatial constraints and the assumption that individual particle speeds are consistent. However, the model ignores the hierarchical structure of animal groups in their natural state and the heterogeneity of movement speeds between individuals. The model conditions are quite different from the actual movement laws of animals in their natural state. The particles in the model are all abstract, inanimate basic units. Although random perturbations are introduced, they cannot effectively reflect the initiative and decision-making process of the autonomous movement of individual animals. In addition, the existing improved versions of the Vicsek model mostly improve the fit of the movement trajectory of specific species through parameter adjustment and fitting (Ling et al., 2019), but these models still do not explicitly incorporate key biological characteristics such as leadership hierarchy in animal groups. As a result, the model has weak explanatory power for the movement mechanism of animal clusters and is difficult to reveal the interaction laws and hierarchical driving effects between individuals in the real movement process (Yu Huiyang, 2021).

[0016] The reason for the above problems is that most of the current animal cluster movement models are abstract particle models constructed based on physical or computational theories. They lack deep integration at the ecological and behavioral biology levels and fail to effectively depict the complex individual hierarchy, movement heterogeneity and interaction mechanisms within real animal clusters. Therefore, it is difficult to fully reveal the true mechanism of animal cluster movement. Summary of the Invention

[0017] In order to solve the problems existing in the prior art, the purpose of the present invention is to provide a simulation method that incorporates leadership level into the flight motion mechanism of an intelligent cluster. The present invention can simulate the motion characteristics of an intelligent cluster more realistically.

[0018] To achieve the above-mentioned purpose, the present invention adopts a technical solution: a simulation method for incorporating leadership hierarchy into the flight motion mechanism of intelligent clusters, comprising the following steps:

[0019] Step 1: Carry out pre-experimental preparations and intelligent agent flight experiments;

[0020] Step 2: Extract the local layer and global layer dual-level intelligent cluster motion features;

[0021] Step 3: Explore the cross-level mediation effect of leadership;

[0022] Step 4: By establishing a cross-level dynamic model of leadership, the movement mechanism of pigeons flying around the nest under the cross-level mediation of leadership is revealed.

[0023] As a further improvement of the present invention, in step 2, the local layer extracts individual velocity, acceleration, motion trajectory, and adjacent distance; the global layer extracts cluster center of mass velocity, acceleration, cluster polarization value, entropy, and anisotropy characteristics; and the intelligent body flight obstacles in the environmental characteristics within the simulated experimental site provide a basis for the subsequent construction of the dynamic model.

[0024] As a further improvement of the present invention, in step 3, the local layer focuses on the leadership-follower interaction between individuals; the global layer focuses on the emergence process of the dynamic leadership network; and then the cross-level mediation principle of leadership is refined through the characteristics of leaders in the two levels, and the mathematical expression of the leadership traction force of the local level and the leadership perturbation force of the global level are characterized.

[0025] As a further improvement of the present invention, the local layer focuses on the leader-follower interaction between individuals as follows:

[0026] During the cluster movement of agents, if the speed direction of agent i at time slice t is copied by agent j after a delay of t, then agent i is said to lead agent j at time slice t. To determine whether there is a leader-follower condition, the similarity of the speed directions of the two pigeons is measured as follows:

[0027]

[0028] Where: · represents the inner product operation; C ij (t,τ) evaluates the similarity of the movement direction of the i-th agent at time slice t and the j-th agent at time slice t+t, and its value range is [-1,1];

[0029] If there is a leadership or replication relationship between agents i and j, the following τ * To determine the direction of movement between them, copy the delay:

[0030]

[0031] If t * > 0, then the j-th pigeon passes t * The delayed copy of the movement direction of the i-th pigeon at time slice t, that is, at time slice t, i leads j; if t * <0, the conclusion is the opposite.

[0032] As a further improvement of the present invention, the local level focuses on the dynamic leadership network emergence process as follows:

[0033] At each time slice t, a structural diagram of the intelligent agent leadership network is constructed; each node in the network represents each intelligent agent, and if there is a leader-follower relationship between every two nodes, a directed edge is formed, the edge length represents the delay corresponding to the leader-follower relationship, and the arrow points to the follower; the superposition of the leadership networks of all time slices constitutes the emergence of a dynamic leadership network throughout the entire cluster flight process; and based on the network characteristics, the leader-follower ratio, network density, and clustering coefficient cluster characteristics are further characterized, and its emergence process is analyzed through leadership stability, follower switching frequency, and network evolution pattern.

[0034] As a further improvement of the present invention, the mathematical expression of the leadership traction force of the local layer is specifically as follows:

[0035]

[0036] in: is the acceleration of agent i caused by the leader’s traction force at time slice t; is the set of leaders at time slice t; φ ik (t) is the influence attenuation coefficient of leader k on agent i; v k (t) is the velocity of leader k at time slice t; v i (t) is the velocity of agent i at time slice t; λ is the weight balance index between leadership and Vicsek alignment; t * is the response time constant;

[0037] The mathematical expression of the global layer leadership perturbation force is as follows:

[0038]

[0039] Where: v k (t) is the velocity of leader k at time slice t; v i(t) is the velocity of agent i at time slice t; is the set of leaders at time slice t; β is the perturbation strength coefficient; and n is the size of the agent cluster.

[0040] As a further improvement of the present invention, in step 4, a Vicsek coupled leadership dynamic model is established at the local layer, including the Vicsek alignment force term, the leadership traction force term, and the individual obstacle avoidance term; a Kepler-like system coupled leadership dynamic model is established at the global layer, including the ideal circular gravity term, the leadership perturbation force term, and the cluster obstacle avoidance force term; after completing the parameter calibration and model verification of the two-level dynamic models, the intelligent body cluster flight motion mechanism is revealed through modeling.

[0041] As a further improvement of the present invention, the acceleration of the Vicsek alignment force is mathematically expressed as follows:

[0042]

[0043] in: is the acceleration of agent i caused by the Vicsek alignment force at time slice t; is the neighbor set of agent i at time slice t; v j (t) is the velocity of agent j at time slice t; v i (t) is the velocity of agent i at time slice t; t * is the response time constant; j is the neighboring individual of the i-th agent;

[0044] The mathematical expression of obstacle avoidance force acceleration is as follows:

[0045]

[0046] in: is the acceleration of agent i caused by the obstacle avoidance force at time slice t; is the set of leaders at time slice t; φ ik· (t) is the influence attenuation coefficient of leader k on individual i; v obs,k (t) is the target velocity of leader k at time slice t, generated to avoid the obstacle; v i (t) is the velocity of pigeon i at time slice t; t * is the response time constant; μ is the relative weight coefficient of the obstacle avoidance acceleration term; k is the leader of the i-th pigeon;

[0047] The individual accelerations of the local layer are as follows:

[0048]

[0049] Among them: a i(t) is the acceleration of agent i at time slice t when it is affected by the above-mentioned resultant force;

[0050] The update rule of individual motion features at the local layer is expressed as follows:

[0051] v i (t+Δt)=v i (t)+a i (t)·Δt

[0052] r i (t+Δt)=r i (t)+v i (t)·Δt

[0053] Among them: a i (t) is the acceleration of agent i at time slice t; v i (t) is the velocity of agent i at time slice t; r i (t) is the spatial position of agent i at time slice t; Δt represents the time step.

[0054] As a further improvement of the present invention, the ideal circular gravitational force is as follows:

[0055]

[0056] Where: v0(t) is the nominal velocity of the centroid of the cluster of agents; R(t) is the position of the centroid of the cluster of agents at time t; is the unit direction vector of the centroid of the agent cluster at time t;

[0057] The cluster's obstacle avoidance capabilities are as follows:

[0058]

[0059] Where: R(t) is the position of the centroid of the agent cluster at time t; m is the position of the mth obstacle; γ is the obstacle avoidance strength coefficient; n is the total number of individuals in the agent cluster; M is the number of obstacles;

[0060] The acceleration of the global layer is expressed by a second-order differential equation to express the source of the center of mass acceleration as follows:

[0061]

[0062] Among them: a R (t) is the acceleration of the center of mass of the agent cluster;

[0063] The global layer cluster centroid motion feature update rule is as follows:

[0064] Since the derivative of position is equal to velocity, and the derivative of velocity is equal to acceleration, we have: and Where: v R (t) is the velocity of the centroid of the agent cluster at time slice t;

[0065] The global layer cluster centroid motion feature update rule is as follows:

[0066] v R (t+Δt)=v R (t)+a R (t)·Δt

[0067] R(t+Δt)=R(t)+v R (t)·Δt

[0068] Among them: a R (t) is the acceleration of the center of mass of the agent cluster at time slice t; v R (t) is the velocity of the centroid of the agent cluster at time slice t; R(t) is the spatial position of the centroid of the agent cluster at time slice t; Δt represents the time step.

[0069] As a further improvement of the present invention, the intelligent agent cluster is a bird or drone cluster.

[0070] The beneficial effects of the present invention are:

[0071] (1) The dynamic leadership network structure of the pigeon flock flying around the nest emerges:

[0072] Previous studies have shown that homing pigeons are flocking animals with a clear leadership structure, and this structure changes with each stage of flocking. This is similar to the changes in leadership structure observed in some migratory birds, such as geese and ducks, during long-distance migration. However, due to the limited payload and volume of bird monitoring equipment, it is difficult to collect high-frequency, high-precision positioning data. Consequently, current research has struggled to accurately characterize the dynamic leadership network structure during flocking, and can only make preliminary assessments of the static leadership network over the overall flocking process based on less accurate data. This project, utilizing self-developed lightweight GNSS-RTK monitoring equipment and using homing pigeon flocks as experimental subjects, is able to acquire high-frequency, high-precision movement trajectories. This allows the analysis of the instantaneous local interaction characteristics and the dynamic leadership network structure of the flock based on the velocity, velocity direction, spatial position, and topological relationships between individual homing pigeons during flocking. Furthermore, the dynamic leadership network structure of homing pigeon flocking can be precisely characterized, allowing for a clear understanding of the global emergence of flocking movement from the perspective of leadership structure.

[0073] (2) The mechanism of pigeon flocks flying around the nest under the cross-level mediation of leadership:

[0074] Leaders not only dominate the "leader-follower" interactions between individuals at the local level but also form dynamic leadership networks, driving global emergence and playing a key role in the movement of homing pigeon flocks. Therefore, the key to exploring the dynamics of homing pigeon flocks' flight patterns lies in revealing the cross-level mediation of leadership. To this end, we employed a Vicsek agent-based model to couple leadership traction and individual obstacle avoidance at the local level, and a Kepler-like system to couple leadership perturbation and cluster obstacle avoidance at the global level. Together, we constructed a dynamic model of homing pigeon flocks' flight patterns under the cross-level mediation of leadership and revealed their movement mechanisms. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] Figure 1 This is a conceptual diagram of the dynamic model of pigeons flying around their nests, which mediates the cross-level role of leadership in an embodiment of the present invention. The light gray arrows represent a schematic diagram of the dynamic leadership network of a pigeon flock at a certain time slice. Figure a shows the individual dynamic mechanism at the local level, and Figure b shows the global cluster center of mass dynamic mechanism.

[0076] Figure 2 The data of the flight trajectory of a flock of homing pigeons around their nests obtained in the embodiment of the present invention;

[0077] Figure 3 The three-view dynamic image of the pigeon flock motion trajectory in the embodiment of the present invention;

[0078] Figure 4 Schematic diagram of the dynamic change process of the global layer leader network during the T191-T220 period in a flight according to an embodiment of the present invention. DETAILED DESCRIPTION

[0079] The embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0080] Example

[0081] A simulation method that incorporates leadership hierarchy into the flight motion mechanism of intelligent swarms, such as Figure 1 As shown, this embodiment uses homing pigeons as an example for specific explanation. The bound state exhibited by Newtonian gravity is used as a framework. At the same time, a leadership force that can reflect the active movement of individuals within the flock is introduced (both pulling individuals to change their movement characteristics and perturbing the flock to break away from the original gravitational system) (Couzin et al., 2005). Furthermore, the obstacle avoidance term of the animal flock in the environment (reflected as a random perturbation term in the Vicsek model; in real animal flock movement, there will also be an intention and behavior to actively avoid obstacles or predators in the environment) is added. This allows the simulation of animal flock movement to be consistent with reality. Specifically, the following steps are included:

[0082] Step 1: Conduct pre-experimental preparations. This includes the development and production of lightweight GNSS-RTK monitoring equipment, calculating the wind resistance of the pigeon-carrying equipment, training the pigeons with weights, assessing the impact of the equipment load on the pigeons' normal flight, and determining the number of individual pigeons in the flock.

[0083] The second step is to conduct a flock of homing pigeons flying around the nest. The purpose of this experiment is to obtain the motion characteristic data of the pigeon flock (and its individual) at a frequency of 10 Hz. The experiment is conducted in an open field in the northern plains rural area with a flock of 10 homing pigeons. Figure 2 As shown, from Figure 2 It can be seen that the flock of homing pigeons has a relatively obvious Keplerian orbital motion when flying around the nest, and will avoid or cross obstacles (poplar trees); through 10-15 experiments, 5 optimal data were selected, and pre-processing measures such as denoising, smoothing, and conversion to the projection coordinate system were performed, and finally a three-dimensional trajectory time series table of each individual in the cluster was obtained, as shown in the figure below. Figure 3 As shown, Figure 3 The black dot in the middle is the center of mass, and the other colored points with arrows are the individuals in the pigeon flock and their speed directions; the XY section, YZ section and XZ section respectively represent the motion feature projections of the individuals on different planes.

[0084] The third step is to extract the movement characteristics of the pigeon flock at both the local and global levels. The local level extracts individual velocity, acceleration, trajectory, and proximity, while the global level extracts features such as the velocity, acceleration, polarization value, entropy, and anisotropy of the flock's center of mass. The environmental characteristics of the simulated experimental site, including flight obstacles, provide a foundation for the subsequent construction of a dynamic model.

[0085] The fourth step is to explore the cross-level mediation of leadership. The local level focuses on the "leader-follower" interaction between individuals, while the global level focuses on the emergence of dynamic leadership networks. Furthermore, the characteristics of leaders at both levels are used to refine the principles of cross-level mediation of leadership and to mathematically express the local leadership traction and global leadership perturbation.

[0086] In this step, within the local layer, the leader's traction term converts the leader's velocity deviation into center-of-mass acceleration by combining velocity difference with a time constant. This reflects the leader's ability to dynamically control the movement of the cluster's center of mass, serving as a critical bridge between the local leader's individual behavior and the trajectory of the global center of mass. The position of the global cluster's center of mass determines the strength of the repulsive force exerted by obstacles, reflecting the influence of the global potential field on the local layer. Furthermore, the weight of the local leader's obstacle avoidance decision is also determined by the change in the global cluster's center of mass, representing feedback from the global layer to the local layer. These two factors together demonstrate the cross-level mediating role of leadership.

[0087] Step 5: By establishing a cross-level leadership dynamic model, we reveal the circling pigeon movement mechanism under the cross-level mediation of leadership. At the local level, we establish a Vicsek-coupled leadership dynamic model, which includes Vicsek alignment, leadership traction, and individual obstacle avoidance. At the global level, we establish a Kepler-like system-coupled leadership dynamic model, which includes ideal circular gravity, leadership perturbation, and cluster obstacle avoidance. After completing parameter calibration and model verification for these two dynamic models, we reveal the circling pigeon movement mechanism through modeling.

[0088] The present embodiment will be further described below:

[0089] 1. The cross-level mediating role of leadership

[0090] 1.1 Local-level “leader-follower” interaction between individuals

[0091] Definition of a "leader-follower" interaction: During the movement of flocking pigeons, if the velocity direction of pigeon i at time slice t is "copied" by pigeon j after a delay of t, we say that pigeon i leads pigeon j at time slice t. Based on this definition, to determine whether a "leader-follower" event exists, we need a measure of the similarity of the velocity directions of the two pigeons, as shown in the following formula.

[0092]

[0093] in:

[0094] g represents the inner product operation;

[0095] C ij (t,τ) evaluates the similarity between the movement directions of the i-th pigeon at time slice t and the j-th pigeon at time slice t+t, and its value range is [-1,1].

[0096] If there is a leadership or replication relationship between pigeons i and j, we can use τ in formula 2 * To determine the direction of movement between them and the replication delay.

[0097]

[0098] It should be noted that the above formula can be solved for each time slice t. If t * >0, indicating that the j-th pigeon has passed t * The delayed copy of the movement direction of the i-th pigeon at time slice t, that is, at time slice t, i leads j; if t * <0, the conclusion is the opposite. * In the process of solving , t and C ij The value range of needs to be appropriately defined based on the data.

[0099] 1.2 Emergence of a dynamic leadership network at the global level

[0100] At each time slice t, a pigeon group leader network structure diagram is constructed, such as Figure 4 As shown, Figure 4 This is the dynamic change process of the global layer leadership network in the T191-T220 period of a flight. The circled individual is the absolute parent node in the network at this time slice (defined as: the individual has no parent node). Each node of the network represents each pigeon. If there is a "leader-follower" relationship between every two nodes, a directed edge is formed. The length of the edge represents the delay corresponding to the "leader-follower" relationship, and the arrow points to the follower. The superposition of the leadership networks of all time slices constitutes the emergence of the dynamic leadership network during the entire cluster flight process. Based on the above network characteristics, the cluster characteristics such as the "leader-follower" ratio, network density, and clustering coefficient are further characterized, and the emergence process is analyzed through leadership stability, follower switching frequency, network evolution pattern, etc. 1.3 Mathematical expression of the cross-level mediating effect of leadership

[0101] Local leader-follower interactions contribute to the emergence of a dynamic leadership network at the global level. The characteristics of a dynamic leadership network demonstrate a continuous shift in leaders over time. Network characteristics revealing leader-follower interactions among individuals indicate that, in a global dynamic leadership network, the leader is not simply a single individual at the forefront of the network, but rather a collection of leaders at the forefront of the network structure. Due to the dynamic nature of the network, the leadership collection also exhibits dynamic characteristics over time. Therefore, leadership can be defined as a virtual force generated by the collective action of a collection of leaders, exerting a pulling effect on follower individuals at the local level and a perturbing effect on the cluster's center of mass at the global level.

[0102] (1) Mathematical expression of leadership traction at the local level

[0103] Leadership is manifested at the local level as a traction force in addition to the Vicsek alignment force. Taking into account the "leader-follower" interaction at the local level, and following the basic form of individual movement changes in the Vicsek model, the mathematical expression of the local leadership traction force acting on individual i is shown in Equation 3:

[0104]

[0105] in:

[0106] is the acceleration of pigeon i caused by the leader's pulling force at time slice t;

[0107] is the set of leaders at time slice t;

[0108] φ ik·(t) is the influence decay coefficient of leader k on individual i (decreases with increasing distance);

[0109] v k (t) is the velocity of leader k at time slice t;

[0110] v i (t) is the velocity of pigeon i at time slice t;

[0111] λ is the weighted balance index of leadership and Vicsek alignment;

[0112] t * is the response time constant (corresponding to the replication delay in Equation 2), which is used to measure the degree of “leader-follower” response between individuals in the local layer and has the dimension [s].

[0113] (2) Mathematical expression of global layer leadership perturbation

[0114] At the global level, the center of mass of the flock performs a Keplerian-like motion around the nest. In the absence of any other internal or external driving forces, the center of mass of the flock is driven solely by the ideal circular gravitational force, moving in a circular motion around the nest at an initial velocity v0. Leadership, as the internal driving force of the flock, manifests itself at the global level as a perturbation-like force that can shift the trajectory of the center of mass of the flock. Therefore, at the global level, the mathematical expression of the leadership perturbation term is shown in Equation 4:

[0115]

[0116] in:

[0117] v k (t) is the velocity of leader k at time slice t;

[0118] v i (t) is the velocity of pigeon i at time slice t;

[0119] is the set of leaders at time slice t;

[0120] β is the perturbation intensity coefficient, with [s -1 ] dimension, ensuring that the perturbation term is consistent with the acceleration [m / s 2 ] consistent;

[0121] n is the size of the pigeon flock, which is used to normalize the micro-level effects and ensure that the model is applicable to pigeon flocks of different sizes.

[0122] (3) The cross-level mediating role of leadership

[0123] From a global perspective, the leader's velocity deviation can be converted into center-of-mass acceleration, reflecting the leader's ability to dynamically control the movement of the cluster's center of mass. This serves as a critical bridge through which the individual behavior of the local leader influences the trajectory of the global center of mass. Furthermore, from a local perspective, the position of the cluster's center of mass determines the strength of the repulsive force exerted by obstacles, reflecting the influence of the global potential field on the local layer. Furthermore, the weight of the local leader's obstacle avoidance decision is also determined by changes in the global cluster's center of mass, representing feedback from the global layer to the local layer. These two factors together demonstrate the cross-level mediating role of leadership.

[0124] 2. The mechanism of pigeon flocks flying around their nests under the cross-level mediation of leadership

[0125] 2.1 Local-Level Vicsek Coupled Leadership Dynamics Model

[0126] The interactions between individuals within a flock still follow the Vicsek principle, but individuals are also influenced by the pull of the leader within the flock, causing them to change their movement characteristics. Therefore, the speed change of an individual within a local layer is influenced by the Vicsek alignment force, the leader's pull, and the obstacle avoidance force triggered by the leader.

[0127] However, the Vicsek model is a physical model that only specifies the iterative changes in speed and position. Therefore, when introducing leadership to drive changes in individual movement characteristics, we need to improve the Vicsek model, convert the above-mentioned main driving force into acceleration, and then incorporate it into the mathematical expression of individual speed and position.

[0128] (1) Vicsek alignment force

[0129] Vicsek mathematically expresses the acceleration of the alignment force as shown in Equation 5:

[0130]

[0131] in:

[0132] is the acceleration of pigeon i caused by the Vicsek alignment force at time slice t;

[0133] is the set of neighbors of pigeon i at time slice t;

[0134] v j (t) is the velocity of neighbor j at time slice t;

[0135] v i (t) is the velocity of pigeon i at time slice t;

[0136] t *is the response time constant, which is used for the degree of “leader-follower” response between individuals at the local level, with the dimension [s];

[0137] j is the neighboring individual of the i-th pigeon.

[0138] (2) Leadership traction

[0139] See section 1.3(2).

[0140] (3) Obstacle avoidance

[0141] When an obstacle appears in the environment, the leader in the swarm initiates a dynamic response, leading its followers to change their motion characteristics. This process is essential for describing the realism of local-level motion, as flight obstacles are ubiquitous and cannot be ignored. Therefore, considering the above factors, the obstacle avoidance force triggered by the leader must also be incorporated into the local-level modeling process. The mathematical expression of the obstacle avoidance force acceleration is shown in Equation 6:

[0142]

[0143] in:

[0144] is the acceleration of pigeon i caused by the obstacle avoidance force at time slice t;

[0145] is the set of leaders at time slice t;

[0146] φ ik· (t) is the influence decay coefficient of leader k on individual i (decreases with increasing distance);

[0147] v obs,k (t) is the target speed of leader k at time slice t in the direction away from the obstacle to avoid the obstacle;

[0148] v i (t) is the velocity of pigeon i at time slice t;

[0149] t * is the response time constant, which is used for the degree of “leader-follower” response between individuals at the local level, with the dimension [s];

[0150] μ is the relative weight coefficient of the obstacle avoidance acceleration term, which is used to adjust the contribution intensity of the leader's obstacle avoidance behavior to the follower's acceleration. Its dimension is [s -1 ];

[0151] k is the leader of the i-th pigeon.

[0152] (4) Local layer individual acceleration

[0153] The acceleration of the local layer is composed of the three acceleration components mentioned above, as shown in Equation 7:

[0154]

[0155] in:

[0156] a i (t) is the acceleration of pigeon i at time slice t caused by the above-mentioned resultant force;

[0157] For the remaining items, please refer to the notes in the above three sub-items.

[0158] (5) Update rules for individual motion features at the local level

[0159] Referring to the expression of the Vicsek agent model, the individual motion characteristics still need to be reflected through changes in speed and spatial position. Therefore, the update rule of the individual motion characteristics is expressed as shown in Equation 8:

[0160] v i (t+Δt)=v i (t)+a i (t)·Δt

[0161] r i (t+Δt)=r i (t)+v i (t)·Δt#(8)

[0162] in:

[0163] a i (t) is the acceleration of pigeon i at time slice t;

[0164] v i (t) is the velocity of pigeon i at time slice t;

[0165] r i (t) is the spatial position of pigeon i at time slice t;

[0166] Δt represents the time step.

[0167] 2.2 Global Kepler-like system coupled leadership dynamics model:

[0168] Because the pigeon nest has a natural attraction for homing pigeons, when the flock flies around the nest, its center of mass will initially move in a Keplerian orbit around the dovecote. When the leader of the flock exerts leadership force (gravitational perturbation), the center of mass of the flock will be influenced to change its orbit. In this simulation, the movement of the center of mass of the flock is driven by the combined force of ideal circular gravity, the perturbation force of the leader, and the obstacle avoidance force of the flock.

[0169] (1) Ideal circular gravity

[0170] This term simulates the "gravitational force" of the pigeonhole on the cluster at the global level, maintaining the circular motion of the center of mass. Its expression is as follows

[0171] As shown in formula 9:

[0172]

[0173] in:

[0174] v0(t) is the nominal speed of the center of mass of the flock circling the nest, which can be set as a constant;

[0175] R(t) is the position of the center of mass of the pigeon flock at time t;

[0176] is the unit direction vector of the center of mass of the flock at time t, providing the direction of the centripetal acceleration, always pointing towards the pigeon nest; it is also used to convert scalar acceleration into vector form.

[0177] (2) Leadership power

[0178] The mathematical expression of the leadership perturbation force at the global level can be found in part (2) of Section 1.3.

[0179] (3) Cluster obstacle avoidance

[0180] The global cluster obstacle avoidance force represents the overall avoidance behavior of the pigeon flock when encountering an obstacle. When the center of mass position R(t) approaches an obstacle, a repulsive force is generated, causing the center of mass to deviate from the original path. Therefore, the mathematical expression of the cluster obstacle avoidance force is shown in Equation 10:

[0181]

[0182] in:

[0183] R(t) is the position of the center of mass of the pigeon flock at time t;

[0184] o m is the position of the mth obstacle;

[0185] γ is the obstacle avoidance strength coefficient, the dimension is [m 3 / s 2 ];

[0186] n: total number of individuals in the pigeon flock (dimensionless);

[0187] M: number of obstacles (dimensionless).

[0188] (4) Global layer center of mass acceleration

[0189] The acceleration of the global layer is composed of the above three items. The source of the center of mass acceleration is expressed by a second-order differential equation, as shown in Equation 11:

[0190]

[0191] in:

[0192] a R (t) is the acceleration of the cluster center of mass;

[0193] For the remaining items, please refer to the notes in the above three sub-items.

[0194] (5) Global layer cluster centroid motion feature update rules

[0195] Since the derivative of position is equal to velocity, and the derivative of velocity is equal to acceleration, we have: and Where: v R (t) is the velocity of the cluster centroid at time slice t.

[0196] Then, the global layer cluster centroid motion feature update rule can be expressed as formula 12:

[0197] v R (t+Δt)=v R (t)+a R (t)·Δt

[0198] R(t+Δt)=R(t)+v R (t)·Δt#(12)

[0199] in:

[0200] a R (t) is the acceleration of the cluster center of mass at time slice t;

[0201] v R (t) is the velocity of the cluster centroid at time slice t;

[0202] R(t) is the spatial position of the cluster centroid at time slice t;

[0203] Δt represents the time step.

[0204] This example aims to address the disconnect between local interactions and global emergent mechanisms in existing animal flocking models, which prevents them from reflecting leadership hierarchies. By proposing a method for simulating flocking movements of homing pigeons orbiting their nests based on the influence of leadership hierarchies, this method maintains the individual interaction characteristics of the Vicsek model at the local level while constructing Keplerian-like orbits based on the centroid of the flock at the global level, enabling a more realistic simulation of the flocking movement characteristics of homing pigeons.

[0205] The above-described embodiments merely represent specific implementations of the present invention. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that a person skilled in the art would be able to make numerous variations and improvements without departing from the spirit of the present invention, and all such variations and improvements fall within the scope of protection of the present invention.

Claims

1. A simulation method that incorporates leadership hierarchy into the flight motion mechanism of intelligent swarms, characterized in that: The following steps are involved: Step 1: Carry out pre-experimental preparations and intelligent agent flight experiments; Step 2: Extract the local layer and global layer dual-level intelligent cluster motion features; Step 3: Explore the cross-level mediation effect of leadership; Step 4: By establishing a cross-level dynamic model of leadership, the movement mechanism of pigeons flying around the nest under the cross-level mediation of leadership is revealed.

2. The simulation method for incorporating leadership levels into the flight motion mechanism of intelligent swarms according to claim 1, characterized in that: In step 2, the local layer extracts individual velocity, acceleration, motion trajectory, and adjacent distance; the global layer extracts cluster centroid velocity, acceleration, cluster polarization value, entropy, and anisotropy characteristics; The flight obstacles of the intelligent agent in the environmental characteristics of the simulated experimental site provide a basis for the subsequent construction of the dynamic model.

3. The simulation method of incorporating leadership level into the flight motion mechanism of intelligent swarm according to claim 1, characterized in that: In step 3, the local layer focuses on the leadership-follower interaction between individuals; the global layer focuses on the emergence process of the dynamic leadership network; then, through the characteristics of leaders at the two levels, the principle of cross-level mediation of leadership is refined, and the mathematical expression of the leadership traction force at the local level and the leadership perturbation force at the global level are characterized.

4. The simulation method for incorporating leadership levels into the flight motion mechanism of intelligent swarms according to claim 3, characterized in that: The local level focuses on the leader-follower interaction between individuals as follows: During the cluster movement of agents, if the speed direction of agent i at time slice t is copied by agent j after a delay of t, then agent i is said to lead agent j at time slice t. To determine whether there is a leader-follower condition, the similarity of the speed directions of the two pigeons is measured as follows: Where: · represents the inner product operation; C ij (t,τ) evaluates the similarity of the movement direction of the i-th agent at time slice t and the j-th agent at time slice t+t, and its value range is [-1,1]; If there is a leadership or replication relationship between agents i and j, the following τ * To determine the direction of movement between them, copy the delay: If t * > 0, then the j-th pigeon passes t * The delayed copy of the movement direction of the i-th pigeon at time slice t, that is, at time slice t, i leads j; if t * <0, the conclusion is the opposite.

5. The simulation method of incorporating leadership level into the flight motion mechanism of intelligent swarm according to claim 4, characterized in that: The global level focuses on the emergence process of dynamic leadership networks as follows: At each time slice t, a structural diagram of the intelligent agent leadership network is constructed; each node in the network represents each intelligent agent, and if there is a leader-follower relationship between every two nodes, a directed edge is formed, the edge length represents the delay corresponding to the leader-follower relationship, and the arrow points to the follower; the superposition of the leadership networks of all time slices constitutes the emergence of a dynamic leadership network throughout the entire cluster flight process; and based on the network characteristics, the leader-follower ratio, network density, and clustering coefficient cluster characteristics are further characterized, and its emergence process is analyzed through leadership stability, follower switching frequency, and network evolution pattern.

6. The simulation method for incorporating leadership levels into the flight motion mechanism of intelligent swarms according to claim 5, characterized in that: The mathematical expression of leadership traction at the local level is as follows: in: is the acceleration of agent i caused by the leader’s traction force at time slice t; is the set of leaders at time slice t; φ ik (t) is the influence decay coefficient of leader k on agent i; v k (t) is the velocity of leader k at time slice t; v i (t) is the velocity of agent i at time slice t; λ is the weight balance index between leadership and Vicsek alignment; t * is the response time constant; The mathematical expression of the global layer leadership perturbation force is as follows: Where: v k (t) is the velocity of leader k at time slice t; v i (t) is the velocity of agent i at time slice t; is the set of leaders at time slice t; β is the perturbation strength coefficient; and n is the size of the agent cluster.

7. The simulation method for incorporating leadership levels into the flight motion mechanism of intelligent swarms according to claim 6, characterized in that: In step 4, a Vicsek coupled leadership dynamic model is established at the local level, including the Vicsek alignment force term, the leadership traction force term, and the individual obstacle avoidance term; a Kepler-like system coupled leadership dynamic model is established at the global level, including the ideal circular gravity term, the leadership perturbation force term, and the cluster obstacle avoidance term; after completing the parameter calibration and model verification of the two-level dynamic models, the flight motion mechanism of the intelligent body cluster is revealed through modeling.

8. The simulation method for incorporating leadership levels into the flight motion mechanism of intelligent swarms according to claim 7, characterized in that: Vicsek's mathematical expression for the acceleration of the alignment force is as follows: in: is the acceleration of agent i caused by the Vicsek alignment force at time slice t; is the neighbor set of agent i at time slice t; v j (t) is the velocity of agent j at time slice t; v i (t) is the velocity of agent i at time slice t; t * is the response time constant; j is the neighboring individual of the i-th agent; The mathematical expression of obstacle avoidance force acceleration is as follows: in: is the acceleration of agent i caused by the obstacle avoidance force at time slice t; is the set of leaders at time slice t; φ ik (t) is the influence decay coefficient of leader k on individual i; v obs,k (t) is the target velocity of leader k at time slice t, generated to avoid the obstacle; v i (t) is the velocity of pigeon i at time slice t; t * is the response time constant; μ is the relative weight coefficient of the obstacle avoidance acceleration term; k is the leader of the i-th pigeon; The individual accelerations of the local layer are as follows: Among them: a i (t) is the acceleration of agent i at time slice t when it is affected by the above-mentioned resultant force; The update rule of individual motion features at the local layer is expressed as follows: v i (t+Δt)=v i (t)+a i (t)·Δt r i (t+Δt)=r i (t)+v i (t)·Δt Among them: a i (t) is the acceleration of agent i at time slice t; v i (t) is the velocity of agent i at time slice t; r i (t) is the spatial position of agent i at time slice t; Δt represents the time step.

9. The simulation method for incorporating leadership levels into the flight motion mechanism of intelligent swarms according to claim 8, characterized in that: The ideal circular gravitational force is shown below: Where: v0(t) is the nominal velocity of the centroid of the cluster of agents; R(t) is the position of the centroid of the cluster of agents at time t; is the unit direction vector of the centroid of the agent cluster at time t; The cluster's obstacle avoidance capabilities are as follows: Where: R(t) is the position of the centroid of the agent cluster at time t; m is the position of the mth obstacle; γ is the obstacle avoidance strength coefficient; n is the total number of individuals in the agent cluster; M is the number of obstacles; The acceleration of the global layer is expressed by a second-order differential equation to express the source of the center of mass acceleration as follows: Among them: a R (t) is the acceleration of the center of mass of the agent cluster; The global layer cluster centroid motion feature update rule is as follows: Since the derivative of position is equal to velocity, and the derivative of velocity is equal to acceleration, we have: and Where: v R (t) is the velocity of the centroid of the agent cluster at time slice t; The global layer cluster centroid motion feature update rule is as follows: v R (t+Δt)=v R (t)+a R (t)·Δt R(t+Δt)=R(t)+v R (t)·Δt Among them: a R (t) is the acceleration of the center of mass of the agent cluster at time slice t; v R (t) is the velocity of the centroid of the agent cluster at time slice t; R(t) is the spatial position of the centroid of the agent cluster at time slice t; Δt represents the time step.

10. The simulation method for incorporating leadership levels into the flight motion mechanism of intelligent swarms according to any one of claims 1 to 9, characterized in that: The intelligent agent cluster is a bird or drone cluster.

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