A Method for Modeling Intelligent Cooperative Behavior of Unmanned Swarms Based on Riemannian Manifolds

Abstract

The present application relates to a kind of unmanned cluster intelligence cooperative behavior modeling method based on riemannian manifold, group manifold and environment manifold are established;Riemannian metric functional on synthetic manifold is constructed, and the equation group that its variation obtains the riemannian metric of synthetic manifold satisfies, realize measurement fusion and generate synthetic manifold;Continuous group model is converted into discrete group model;Optimization solution is carried out to path manifold, and discrete group model is converted into individual motion model.For the intelligent cooperative path planning scene of unmanned cluster in complex local obstacle environment, time-varying riemannian manifold model is established to solve for each individual of cluster, solve the technical problems such as depending on specific central processing unit, depending on large range of environmental information, easily falling into local optimum in the obstacle avoidance path planning of unmanned cluster, realize the decentralized intelligent cooperative optimal motion path planning of unmanned cluster in local obstacle environment, to guide the actual cooperative obstacle avoidance flight trajectory of unmanned cluster.

Description

Technical Field

[0001] This invention belongs to the field of unmanned swarm intelligent cooperative behavior modeling technology, and particularly relates to a method for modeling unmanned swarm intelligent cooperative behavior based on Riemannian manifolds. Background Technology

[0002] Swarm intelligence, characterized by robust, highly adaptive, and scalable self-organizing behaviors formed through simple rules and localized interactions, manifests as emergent intelligence at the system level. To date, however, a well-defined, clearly described, and effectively applied model of swarm intelligence has yet to be found. This not only hinders research and application of the properties and performance of swarm intelligence itself but also impedes the deep and broad development of its integration with transformative AI methods and technologies, such as "formal abstraction" first exemplified by deep learning. Existing modeling methods have three main drawbacks: First, the abstract definition and formal description of swarm intelligence itself, which form the basis of modeling, have not met the requirements, lacking sufficient logical rigor and completeness. Second, given the ineffective definition, existing modeling attempts have also suffered from insufficient formalization, either resulting in limited connotations due to insufficient formal abstraction or disorganized extensions due to low logical levels. Third, there is a lack of sufficient awareness of the mathematical and physical essence of swarm intelligence and the relationships of integration and coupling among its members; existing methods, considering only a single angle and path, inevitably neglect the integration and coupling relationships and their effects.

[0003] Taking the existing Vicsek model as an example:

[0004] In the Vicsek model, a discrete-time system consists of N autonomous individuals or particles, denoted by the set ∑(1, 2, 3, ..., N). The initial positions and directions of motion of the individuals are randomly distributed. All individuals move freely within a defined planar region. The magnitude of an individual's velocity is constant, and its direction of motion is determined by the average of the velocity vectors of its neighbors, plus a noise perturbation with zero mean.

[0005] Individuals possess the same perceptual abilities; each individual can perceive information about its neighbors within its neighborhood. An individual's perceptual range is defined by its location. The perception range is defined by the circular region centered at R with radius R. All individuals have the same velocity v0, and the direction of motion of individual i at time t is θ(t)∈(-π,π], then its velocity vector is... Individual location updates follow The individual's motion direction is updated according to the following formula: θ(t+1)=θ(t)+δ(t), where δ(t) is a uniformly distributed noise signal. Therefore, the position and motion direction update formulas of the Vicsek model are as follows:

[0006]

[0007] In the Vicsek model's research on intelligent collaboration in unmanned swarms, starting from the individual elements, dynamic equations are directly established and treated as a multi-objective, multi-constraint optimization problem. The differences lie primarily in the selection and setting of objectives and constraints, and the optimization algorithm used—whether it's based on mathematical methods or modern artificial intelligence methods such as deep learning, or a combination of both. Simply combining individual, separate elements without considering the interrelationships and interactions between conditions fails to create an internally coordinated and consistent holistic mathematical structure suitable for the optimization algorithm.

[0008] Therefore, how to improve the existing unmanned swarm intelligent collaborative modeling method to solve the problem that the existing unmanned swarm intelligent collaborative modeling method directly combines the various separate specific elements without considering the interrelationships and interactions between conditions, and thus does not form an internally coordinated and consistent whole mathematical structure to make it suitable for optimization algorithms in a holistic and structural way. Summary of the Invention

[0009] The purpose of this invention is to provide a modeling method for intelligent cooperative behavior of unmanned swarms based on Riemannian manifolds. This method is designed for intelligent cooperative path planning scenarios of unmanned swarms in complex local obstacle environments. Based on the perception range of each individual in the swarm, a time-varying Riemannian manifold model is established and solved using the above method. This solves the technical problems in obstacle avoidance path planning of unmanned swarms, such as reliance on a specific central processing unit, reliance on large-scale environmental information, and easy getting trapped in local optima. It realizes decentralized intelligent cooperative optimal motion path planning for unmanned swarms in local obstacle environments, thereby guiding the actual cooperative obstacle avoidance flight trajectory of unmanned swarms.

[0010] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0011] A method for modeling the intelligent cooperative behavior of unmanned swarms based on Riemannian manifolds includes the following steps:

[0012] S1: The Lagrange method is used to establish a local coordinate system for the sensing neighborhood, and different fluid particles are identified to establish a collective manifold;

[0013] S2: Based on the attraction of the target point in the environment to each individual in the group and the repulsive effect of obstacles in the neighborhood perceived by each individual, establish the environmental manifold;

[0014] S3: Construct a Riemannian metric functional on the composite manifold by adding the Riemannian metric functional on the group manifold and the Riemannian metric functional on the environment manifold. Variationally evaluate the composite manifold to obtain the set of equations satisfied by the Riemannian metric, thus achieving metric fusion to generate the composite manifold.

[0015] S4: Transform the continuous population model into a discrete population model;

[0016] S5: Optimize the solution for the path manifold;

[0017] S6: Transform the discrete group model into an individual motion model.

[0018] Preferably, the specific process of step S1 is as follows:

[0019] S11: Establish the initial local coordinate system for individual A. in At the same time, choose another coordinate system. As a local coordinate system describing individual motion, the initial local coordinates are: The spatial coordinates of the individual at time t are:

[0020] Let it be u A (x A ,t);

[0021] Will Represented as A function of time t:

[0022] Let it be x A (u A ,t), indicating that the coordinates at time t are The point corresponds to the initial coordinates of Individuals;

[0023] S12: Construct the corresponding Ricci curvature tensor and establish the Riemannian metric functional based on the effects of other individuals and the environment on each individual in the neighborhood;

[0024] S13: Integrating on the group manifold after metric splicing and smoothing yields the Riemannian metric functional:

[0025]

[0026] Among them, R ij It is the Ricci curvature tensor on the Riemannian manifold, which describes the curvature of Riemannian space:

[0027]

[0028] Where K ij =v i v j It is the kinetic energy tensor on the Riemannian manifold, and its contraction It is a quantity used to describe the distance traveled by a group;

[0029]

[0030] Vij It is the volume tensor on the Riemannian manifold, whose contraction g ij V ij It is a quantity used to describe the interactions within a group;

[0031]

[0032] Riemann volume element;

[0033]

[0034] J(t) is the determinant of the Jacobian transformation. This represents the volume element after coordinate transformation;

[0035] S14: Variation of the Riemannian metric functional of the swarm action yields the optimal Riemannian metric describing the swarm intelligence model. Variation of the Riemannian metric functional yields the set of equations satisfied by the Riemannian metric of the swarm manifold.

[0036] Preferably, the specific process of step S14 is as follows:

[0037] S141: The Riemannian metric functional of the group action is as follows:

[0038]

[0039] S142: Variational analysis of the Riemannian metric functional of the group action yields:

[0040]

[0041] When g ij When it is the critical point of the Riemannian metric functional,

[0042]

[0043] That is equivalent to:

[0044]

[0045] Substituting the Riemannian metric functional, we obtain:

[0046]

[0047] Using the divergence formula on a Riemannian manifold:

[0048]

[0049] get:

[0050] Therefore, we get:

[0051] The following integral is equal to 0:

[0052]

[0053] Preferably, step S1 also includes coordinate transformation on different perceptual neighborhoods, Riemannian metric splicing, and smoothing of the collective manifold, as detailed below:

[0054] Coordinate transformation across different perceptual neighborhoods:

[0055] Establish a local coordinate system within the perceptual neighborhood of each individual. For each point located at the intersection of different perceptual neighborhoods, i.e. Its coordinates u in different coordinate systems A (x,t)u B The difference between (x,t) is a translation and rotation transformation φ. AB :

[0056]

[0057] A coordinate system is established with a point at the initial moment of the group as the origin. The local coordinates of each individual in the perceptual neighborhood are transformed into coordinates u(x,t) in the global coordinate system through a rotation and translation transformation.

[0058] Sequential envelope smoothing of the group manifold:

[0059] The geometry formed by piecing together the various neighborhoods is smoothed to form a unified and continuous whole:

[0060] N A (t)=O(u(x A ,t),r(v A ,t))

[0061] Where u(x) A r(v) represents the position of the Ath individual at time t. A ,t) represents the individual's perception and radius of influence along direction v, r(v A ,t) changes smoothly with the direction vector, where v is the unit vector representing the direction;

[0062] The perceived neighborhood generates k disconnected subgroups L1, L2, ..., L k For each disconnected subgroup, assume it to be L i ,|L i |=m i Perceptive neighborhood The order-wise envelope of a group manifold is essentially a smoothing of the mapping:

[0063]

[0064] Where M is the environmental manifold, and the desired smooth mapping is... That is to make

[0065]

[0066] The order of the minimal smooth mapping is determined by the order of the Fourier series of the set to which f0 belongs; the smoothed population manifold S can be obtained through order-wise envelope smoothing. t ,

[0067] Riemannian metric splicing:

[0068] Individuals located in the intersection of different perceptual neighborhoods are considered as points within different neighborhoods, and their Riemannian metrics will have different values. When solving for the metric of the intersection, the influence of all its neighborhoods is considered. For a point in the population manifold: Its Riemannian metric is derived from The Riemannian metric functional determines:

[0069]

[0070] For F u (g ij Variational equations yield g ij The system of differential equations satisfied:

[0071]

[0072] Solving the system of equations satisfied by the Riemann metric yields the Riemann metric of points in the intersection of different neighborhoods.

[0073] Preferably, the specific process in step S2 is as follows:

[0074] S21: Based on the attraction of the target point in the environment to each individual in the group and the repulsive effect of obstacles in the perceived neighborhood on each individual, construct the target attraction tensor and repulsion tensor:

[0075] The target attraction tensor is defined as:

[0076] T ij =(u i (x,t)-target i (u) j (x,t)-target j );

[0077] Its contraction g ij T ij =g ij (u i (x,t)-target i (u) j(x,t)-target j Used to depict the distance between a group and a target;

[0078] The repulsion tensor of the βth obstacle is defined as:

[0079]

[0080] Its contraction Used to depict the distance between the group and the βth obstacle;

[0081] S22: By superimposing the effects of the target and the obstacles, the Riemannian metric functional corresponding to the external environment is obtained as follows:

[0082]

[0083] The variational equation for the Riemannian metric corresponding to the environmental manifold is obtained as follows:

[0084]

[0085] Right now:

[0086]

[0087] S23: The above variational pair for any δg ij Since it is always zero, the system of differential equations satisfied by the Riemann metric is as follows:

[0088]

[0089] Solving the system of differential equations satisfied by the Riemannian metric yields the Riemannian metric of the surrounding manifold.

[0090] Preferably, the specific process of step S3 is as follows:

[0091] S31: Tensors inside and outside the group

[0092]

[0093] The superposition yields the combined action, which, when integrated, gives the metric functional as follows:

[0094]

[0095] S32: Perform variational analysis on the metric functional:

[0096]

[0097] The above variational expression applies to any δg ij Since it is always zero, the system of differential equations satisfied by the Riemann metric is as follows:

[0098]

[0099] Preferably, the specific process of step S4 is as follows:

[0100] S41: Select a local neighborhood N for the α-th individual according to a certain selection method. α (t),

[0101] N α (t)=O(u(x α ,t),r(v α ,t));

[0102] Where u(x) α r(v) represents the position of the α-th member at time t. α ,t) represents the individual along the direction The perception and radius of action, r(v) α ,t) as the direction vector changes smoothly, It is a unit vector representing direction;

[0103] S42: Discretizing the Riemann metric functional yields the Riemann metric functional satisfied by the αth individual:

[0104]

[0105] The distance tensor between individuals within the neighborhood is:

[0106]

[0107] The distance tensor between an individual and an obstacle within the neighborhood is:

[0108]

[0109] S43: Variational analysis of the Riemannian metric functional for each individual yields:

[0110]

[0111] Variational solution of the Riemannian metric functional yields:

[0112]

[0113] From δg ij Given the arbitrariness of the α-th individual, the Riemann metric on its trajectory points satisfies the following equation:

[0114]

[0115] The equation that the Riemann metric of each individual satisfies is:

[0116]

[0117] S44: Solve the equation satisfied by the Riemann metric of each individual to obtain the Riemann metric on the motion path of each individual;

[0118] S45: For a finite population, the motion of each individual in the composite manifold satisfies the principle of minimum energy. Construct a finite number of population energy functionals:

[0119]

[0120] Variational analysis of the energy functional for the finite number of populations yields a set of equations of motion for each individual:

[0121]

[0122] For each individual, the variation of its trajectory satisfies the equation: δu α :

[0123]

[0124] Based on the equations satisfied by the variation of the motion trajectory, we obtain the set of motion equations satisfied by individuals in a finite population.

[0125]

[0126] Among them, Γ 1 ij (u α () indicates that the position at time t is u α The Riemannian connection corresponding to the individual;

[0127] S47: Solve the set of motion equations satisfied by the individuals in the finite population to obtain the motion equations of the population.

[0128] Preferably, the specific process of step S5 is as follows:

[0129] We choose the energy-based Lagrange action as the energy functional for the population:

[0130]

[0131] Where u(x,t) is the trajectory of the group, which is the variational object, and J represents the Jacobian matrix of the coordinate transformation; Represents the Riemann volume element;

[0132] The kinetic energy term in the energy-based Lagrange action is:

[0133]

[0134] ρ(x,t) is the population density, satisfying:

[0135]

[0136] Where p represents the density function at each point; This is the group internal energy term in the Lagrange action, taken as:

[0137]

[0138] The external potential energy term in the Lagrange action is V(u(x,t)), which is taken as:

[0139]

[0140] The variational equation for the Lagrange action is:

[0141] δL=L(u+δu)-L(u(x,t))=0, that is:

[0142]

[0143] Taking variational steps, we get:

[0144]

[0145] By equation:

[0146]

[0147] The system of equations satisfied by the group motion is obtained as follows:

[0148]

[0149] in, The Riemann contact coefficient represents the effect of the external environment on an individual.

[0150]

[0151] Taking the diagonalization measure:

[0152]

[0153] Preferably, the specific process of step S6 is as follows:

[0154] The functional satisfied by the Riemannian metric of the environmental manifold:

[0155]

[0156] At this point, the Riemannian metric of the environmental manifold is the critical point of the Riemannian metric functional in the above equation:

[0157]

[0158] The Riemannian metric of the environmental manifold satisfies the following set of differential equations:

[0159]

[0160] Based on the motion of an individual in a Riemannian manifold, which satisfies the principle of minimum energy, the energy functional variation of the individual's motion yields a set of geodesic equations. Solving the set of equations yields the trajectory of the individual's motion.

[0161] The energy functional satisfied by individual movement:

[0162]

[0163] δL(u)=0,

[0164] Taking variational calculus with respect to δL(u) = 0, we obtain the following system of differential equations satisfied by the individual's trajectory u(x,t):

[0165]

[0166] The beneficial effects of this invention include:

[0167] This invention provides a method for modeling intelligent cooperative behavior of unmanned swarms based on Riemannian manifolds. It employs the Lagrange method to establish a local coordinate system of the perception neighborhood to create a swarm manifold; it then establishes an environmental manifold; it constructs a Riemannian metric functional on the composite manifold by adding the Riemannian metric functionals on the swarm manifold and the environmental manifold, and obtains the equations satisfied by the Riemannian metric of the composite manifold through variational analysis, thus achieving metric fusion to generate the composite manifold; it transforms the continuous swarm model into a discrete swarm model; and it optimizes the path manifold to transform the discrete swarm model into an individual motion model. For intelligent cooperative path planning scenarios of unmanned swarms in complex local obstacle environments, it establishes a time-varying Riemannian manifold model based on the perception range of each individual in the swarm, solving technical problems such as reliance on a specific central processing unit, reliance on large-scale environmental information, and susceptibility to local optima in unmanned swarm obstacle avoidance path planning. This achieves decentralized intelligent cooperative optimal motion path planning for unmanned swarms in local obstacle environments, thereby guiding the actual cooperative obstacle avoidance flight trajectory of the unmanned swarm. Attached Figure Description

[0168] Figure 1 This is a flowchart illustrating the intelligent collaborative behavior modeling method for unmanned swarms based on Riemannian manifolds according to the present invention.

[0169] Figure 2 This is a simulation of the unmanned swarm in mountainous terrain according to the present invention.

[0170] Figure 3 This is a simulation of the unmanned swarm in canyon terrain according to the present invention.

[0171] Figure 4 This is a simulation of the unmanned swarm in a basin terrain according to the present invention. Detailed Implementation

[0172] Example 1

[0173] The following is in conjunction with the appendix Figures 1-4 The present invention will be further described in detail below:

[0174] See appendix Figure 1 As shown, a method for modeling the intelligent cooperative behavior of unmanned swarms based on Riemannian manifolds includes the following steps:

[0175] S1: The Lagrange method is used to establish a local coordinate system for the sensing neighborhood, and different fluid particles are identified to establish a collective manifold;

[0176] S2: Based on the attraction of the target point in the environment to each individual in the group and the repulsive effect of obstacles in the neighborhood perceived by each individual, establish the environmental manifold;

[0177] S3: Construct a Riemannian metric functional on the composite manifold by adding the Riemannian metric functional on the group manifold and the Riemannian metric functional on the environment manifold. Variationally evaluate the composite manifold to obtain the set of equations satisfied by the Riemannian metric, thus achieving metric fusion to generate the composite manifold.

[0178] S4: Transform the continuous population model into a discrete population model;

[0179] S5: Optimize the solution for the path manifold;

[0180] S6: Transform the discrete group model into an individual motion model.

[0181] Each individual in an unmanned intelligent swarm naturally possesses a certain perception and range of action. Mathematically, this range corresponds to a neighborhood centered on that individual, within which the individual can only perceive and transmit information and actions. Furthermore, there is overlap between individual neighborhoods, allowing information and actions to be transmitted and synthesized between them, thus enabling the swarm to exhibit certain intelligent behaviors. The aforementioned swarm behavior pattern is mathematically highly consistent with the method of constructing a manifold; therefore, this invention mathematically models the intelligent swarm as a swarm manifold. For the intelligent cooperative path planning scenario of unmanned swarms in complex local obstacle environments, a time-varying Riemannian manifold model is established and solved based on the perception and range of action of each individual in the swarm, using the aforementioned method. This solves the technical problems in unmanned swarm obstacle avoidance path planning, such as reliance on a specific central processing unit, reliance on large-scale environmental information, and susceptibility to local optima. It achieves decentralized intelligent cooperative optimal motion path planning for unmanned swarms in local obstacle environments, thereby guiding the actual cooperative obstacle avoidance flight trajectory of the unmanned swarm.

[0182] Because the perception range of each individual in a group is limited in reality, a local coordinate system can only be established within a finite area to describe the external information of other individuals and the environment. The Lagrange method in fluid mechanics is suitable for describing the actual motion of a group; therefore, the Lagrange method is used to establish a local coordinate system in the perception neighborhood. The Lagrange method focuses on fluid particles. Since particle systems in fluid mechanics are easily deformed during motion, it is necessary to first identify different fluid particles to track the specific fluid particle being studied. Based on the attraction of the target point in the environment to each individual in the group and the repulsive effect of obstacles within each individual's perception neighborhood, the following attraction and repulsion tensors can be constructed.

[0183] The Riemannian metric of a swarm manifold can only reflect the interactions within the swarm, while the Riemannian metric of the environment manifold can only reflect the external effects of the environment on the swarm. Therefore, the Riemannian metric functionals on the swarm manifold and the environment manifold are added to construct the Riemannian metric functional on the composite manifold. Variational analysis of this composite manifold yields the equations satisfied by the Riemannian metric. In this case, the curvature of the Riemann space originates from the combined effects of the swarm's internal and external environments. This superposition variational process is called metric fusion. Since the number of individuals in a swarm is finite in reality, and only the positions and velocities of a finite number of individuals are known, and the perception range of each individual is limited, a local neighborhood N is selected for the α-th individual according to a certain selection method to reflect the locality of the UAV's perception and actions. α (t), each individual can only perceive the effects of other individuals in its neighborhood and the environment on itself.

[0184] In the same Riemannian space, a group of manifolds can have different motion paths. These different paths reflect different group motion objectives, such as minimizing energy, minimizing path length, and minimizing path manifold volume. Based on the Lagrange principle of least action, an energy-based Lagrange action can be constructed to represent the group's motion objectives. By minimizing this Lagrange action, the set of motion equations satisfied by the group motion can be obtained. Since the group motion objective of minimizing path length is already represented by the Riemannian metric of the Riemannian space in which the group motion occurs, the corresponding Lagrange action and trajectory functional are constructed by minimizing the group energy (the sum of the group's kinetic energy, internal energy, and potential energy).

[0185] Due to the mutual influence among individuals in a group, the motion of each individual at every moment is affected by the motion of other individuals. Essentially, this is a multi-body problem on a Riemannian manifold, which is difficult to solve or simulate. Therefore, for each individual, we can assume that the positions of other individuals do not change within a very short time, thus treating other individuals as equivalent to the environment. We can then establish an environmental manifold and solve for the motion equations of the individual within that manifold.

[0186] Example 2

[0187] Based on Example 1, the specific process of step S1 is as follows:

[0188] Because the perceptual range of each individual in a group is limited in reality, a local coordinate system can only be established within a finite area to describe external information about other individuals and the environment. The Lagrange method in fluid mechanics, which derives geodesic equations from the energy functional variational analysis of individual motion, is suitable for describing the actual motion of a group. Therefore, this invention employs the Lagrange method to establish a local coordinate system in the perceptual neighborhood. The Lagrange method focuses on fluid particles. Since particle systems in fluid mechanics are easily deformed during motion, it is first necessary to identify different fluid particles to track the specific fluid particle to be studied.

[0189] S11: Establish the initial local coordinate system for individual A. in At the same time, choose another coordinate system. As a local coordinate system describing individual motion, the initial local coordinates are: The spatial coordinates of the individual at time t are:

[0190] Let it be u A (x A ,t);

[0191] Will Represented as A function of time t:

[0192] Let it be x A (u A ,t), indicating that the coordinates at time t are The point corresponds to the initial coordinates of Individuals;

[0193] S12: Construct the corresponding Ricci curvature tensor and establish the Riemannian metric functional based on the effects of other individuals and the environment on each individual in the neighborhood. At this time, the Riemannian metric of the group manifold is the critical point of the Riemannian metric functional. When the Riemannian metric is the critical point of the functional, the effects of other individuals and the environment on the individual are equivalent to the curvature of the Riemannian space.

[0194] S13: Integrating on the group manifold after metric splicing and smoothing (at which point the action on each point of the group manifold is determined) yields the Riemannian metric functional:

[0195]

[0196] Among them, R ijIt is the Ricci curvature tensor on the Riemannian manifold, which describes the curvature of Riemannian space:

[0197]

[0198] Where K ij =v i v j It is the kinetic energy tensor on the Riemannian manifold, and its contraction It is a quantity used to describe the distance traveled by a group;

[0199]

[0200] V ij It is the volume tensor on the Riemannian manifold, whose contraction g ij V ij It is a quantity used to describe the interactions within a group;

[0201]

[0202] Riemann volume element;

[0203]

[0204] J(t) is the determinant of the Jacobian transformation. This represents the volume element after coordinate transformation. For a given Riemannian metric, this invention provides the mathematical descriptions of some key physical quantities in swarm intelligence within a Riemannian manifold. By taking the variational form of the equation, we can obtain the optimal Riemann metric that describes the swarm intelligence model.

[0205] S14: Variation of the Riemannian metric functional of the swarm action yields the optimal Riemannian metric describing the swarm intelligence model. Variation of the Riemannian metric functional yields the set of equations satisfied by the Riemannian metric of the swarm manifold.

[0206] In this embodiment, the specific process of step S14 is as follows:

[0207] S141: The Riemannian metric functional of the group action is as follows:

[0208]

[0209] S142: Variational analysis of the Riemannian metric functional of the group action yields:

[0210]

[0211] When g ij When it is the critical point of the Riemannian metric functional,

[0212]

[0213] That is equivalent to:

[0214]

[0215] Substituting the Riemannian metric functional, we obtain:

[0216]

[0217] Using the divergence formula on a Riemannian manifold:

[0218]

[0219] get:

[0220] Therefore, we get:

[0221] The following integral is equal to 0:

[0222]

[0223] Example 3

[0224] Based on Example 1 or Example 2, step S1 further includes coordinate transformation over different perceptual neighborhoods, Riemannian metric splicing, and smoothing of the collective manifold, as detailed below:

[0225] Coordinate transformation across different perceptual neighborhoods:

[0226] Since each individual's perceptual ability is limited, a local coordinate system is established within each individual's perceptual neighborhood. To unify the coordinate representation on the group manifold, it is necessary to determine the relationship and transformation rules of these local coordinate systems. For each point located at the intersection of different perceptual neighborhoods... Its coordinates u in different coordinate systems A (x,t)u B The difference between (x,t) is a translation and rotation transformation φ. AB :

[0227]

[0228] A coordinate system is established with a point at the initial moment of the group as the origin. The local coordinates of each individual in the perceptual neighborhood are transformed into coordinates u(x,t) in the global coordinate system through a rotation and translation transformation.

[0229] Sequential envelope smoothing of the group manifold:

[0230] Since the collective manifold formed by directly piecing together perceptual neighborhoods is not smooth in terms of geometry, for the needs of subsequent calculations, the geometry formed by piecing together each neighborhood is smoothed to form a unified and continuous whole:

[0231] N A (t)=O(u(x A ,t),r(v A ,t))

[0232] Where u(x) A r(v) represents the position of the Ath individual at time t. A ,t) represents the individual's perception and radius of influence along direction v, r(v A ,t) changes smoothly with the direction vector, where v is the unit vector representing the direction;

[0233] The perceived neighborhood generates k disconnected subgroups L1, L2, ..., L k For each disconnected subgroup, assume it to be L i ,|L i |=m i Perceptive neighborhood The order-wise envelope of a group manifold is essentially a smoothing of the mapping:

[0234]

[0235] Where M is the environmental manifold, and the desired smooth mapping is... That is to make

[0236]

[0237] The order of the minimal smooth mapping is determined by the order of the Fourier series of the set to which f0 belongs; the smoothed population manifold S can be obtained through order-wise envelope smoothing. t ,

[0238] Riemannian metric splicing:

[0239] Individuals located in the intersection of different perceptual neighborhoods are considered as points within different neighborhoods, and their Riemannian metrics will have different values. When solving for the metric of the intersection, the influence of all its neighborhoods is considered. For a point in the population manifold: Its Riemannian metric is derived from The Riemannian metric functional determines:

[0240]

[0241] For F u (g ij Variational equations yield g ijThe system of differential equations satisfied:

[0242]

[0243] Solving the system of equations satisfied by the Riemann metric yields the Riemann metric of points in the intersection of different neighborhoods.

[0244] Example 4

[0245] Based on Example 1, Example 2, or Example 3, the specific process in step S2 is as follows:

[0246] S21: Based on the attraction of the target point in the environment to each individual in the group and the repulsive effect of obstacles in the perceived neighborhood on each individual, construct the target attraction tensor and repulsion tensor:

[0247] The target attraction tensor is defined as:

[0248] T ij =(u i (x,t)-target i (u) j (x,t)-target j );

[0249] Its contraction g ij T ij =g ij (u i (x,t)-target i (u) j (x,t)-target j Used to depict the distance between a group and a target;

[0250] The repulsion tensor of the βth obstacle is defined as:

[0251]

[0252] Its contraction Used to depict the distance between the group and the βth obstacle;

[0253] S22: By superimposing the effects of the target and the obstacles, the Riemannian metric functional corresponding to the external environment is obtained as follows:

[0254]

[0255] The variational equation for the Riemannian metric corresponding to the environmental manifold is obtained as follows:

[0256]

[0257] Right now:

[0258]

[0259] S23: The above variational pair for any δg ij Since it is always zero, the system of differential equations satisfied by the Riemann metric is as follows:

[0260]

[0261] Solving the system of differential equations satisfied by the Riemannian metric yields the Riemannian metric of the surrounding manifold.

[0262] The specific process of step S3 is as follows:

[0263] Since the Riemannian metric of a population manifold can only reflect the interactions within the population, and the Riemannian metric of the environment manifold can only reflect the external effects of the environment on the population, this invention constructs a Riemannian metric functional on the composite manifold by adding the Riemannian metric functionals on the population manifold and the environment manifold. Variational analysis of this functional yields the set of equations satisfied by the Riemannian metric of the composite manifold. In this case, the curvature of the Riemannian space originates from the combined effects of the population's internal and external environments. This superposition variational process is called metric fusion.

[0264] S31: Tensors inside and outside the group

[0265]

[0266] The superposition yields the combined action, which, when integrated, gives the metric functional as follows:

[0267]

[0268] S32: Perform variational analysis on the metric functional:

[0269]

[0270] The above variational expression applies to any δg ij Since it is always zero, the system of differential equations satisfied by the Riemann metric is as follows:

[0271]

[0272] Example 5

[0273] Based on Example 1, Example 2, Example 3, or Example 4, the specific process of step S4 is as follows:

[0274] Since the number of individuals in a real-world group is limited, and only the positions and velocities of a finite number of individuals are known, and the perception range of each individual is limited, a local neighborhood N is selected for the α-th individual according to a certain selection method to reflect the locality of the perception and action of each individual UAV. α(t), each individual can only perceive the effects of other individuals in its neighborhood and the environment on itself.

[0275] S41: Select a local neighborhood N for the α-th individual according to a certain selection method. α (t),

[0276] N α (t)=O(u(x α ,t),r(v α ,t));

[0277] Where u(x) α r(v) represents the position of the α-th member at time t. α ,t) represents the individual along the direction The perception and radius of action, r(v) α ,t) as the direction vector changes smoothly, It is a unit vector representing direction;

[0278] S42: Discretizing the Riemann metric functional yields the Riemann metric functional satisfied by the αth individual:

[0279]

[0280] The distance tensor between individuals within the neighborhood is:

[0281]

[0282] The distance tensor between an individual and an obstacle within the neighborhood is:

[0283]

[0284] S43: Variational analysis of the Riemannian metric functional for each individual yields:

[0285]

[0286] Variational solution of the Riemannian metric functional yields:

[0287]

[0288] From δg ij Given the arbitrariness of the α-th individual, the Riemann metric on its trajectory points satisfies the following equation:

[0289]

[0290] The equation that the Riemann metric of each individual satisfies is:

[0291]

[0292] S44: Solve the equation satisfied by the Riemann metric of each individual to obtain the Riemann metric on the motion path of each individual;

[0293] S45: For a finite population, the motion of each individual in the composite manifold satisfies the principle of minimum energy. Construct a finite number of population energy functionals:

[0294]

[0295] Variational analysis of the energy functional for the finite number of populations yields a set of equations of motion for each individual:

[0296]

[0297] For each individual, the variation of its trajectory satisfies the equation: δu α :

[0298]

[0299] Based on the equations satisfied by the variation of the motion trajectory, we obtain the set of motion equations satisfied by individuals in a finite population.

[0300]

[0301] Among them, Γ 1 ij (u α () indicates that the position at time t is u α The Riemannian connection corresponding to the individual;

[0302] S47: Solve the set of motion equations satisfied by the individuals in the finite population to obtain the motion equations of the population.

[0303] Example 6

[0304] Based on Example 1, Example 2, Example 3, Example 4, or Example 5, the specific process of step S5 is as follows:

[0305] In the same Riemannian space, a group of manifolds can have different motion paths. These different paths reflect different group motion objectives, such as minimizing energy, minimizing path length, and minimizing path manifold volume. Based on the Lagrange principle of least action, an energy-based Lagrange action is constructed to represent the group motion objective. By minimizing this Lagrange action, the set of motion equations satisfied by the group motion can be obtained. Since the group motion objective of minimizing path length is already represented by the Riemannian metric of the Riemannian space in which the group motion occurs, the corresponding Lagrange action and trajectory functional are constructed by minimizing the group energy (the sum of the group's kinetic energy, internal energy, and potential energy).

[0306] We choose the energy-based Lagrange action as the energy functional for the population:

[0307]

[0308] Where u(x,t) is the trajectory of the group, which is the variational object, and J represents the Jacobian matrix of the coordinate transformation; Represents the Riemann volume element;

[0309] The kinetic energy term in the energy-based Lagrange action is:

[0310]

[0311] ρ(x,t) is the population density, satisfying:

[0312]

[0313] Where p represents the density function at each point; This is the group internal energy term in the Lagrange action, taken as:

[0314]

[0315] The external potential energy term in the Lagrange action is V(u(x,t)), which is taken as:

[0316]

[0317] The variational equation for the Lagrange action is:

[0318] δL=L(u+δu)-L(u(x,t))=0, that is:

[0319]

[0320] Taking variational steps, we get:

[0321]

[0322] By equation:

[0323]

[0324] The system of equations satisfied by the group motion is obtained as follows:

[0325]

[0326] in, The Riemann contact coefficient represents the effect of the external environment on an individual.

[0327]

[0328] Taking the diagonalization measure:

[0329]

[0330] Example 7

[0331] Based on Example 1, Example 2, Example 3, Example 4, Example 5, or Example 6,

[0332] Due to the mutual influence among individuals in a group, the motion of each individual at every moment is affected by the motion of other individuals. Essentially, this is a many-body problem on a Riemannian manifold, which is difficult to solve or simulate. Therefore, we assume that the positions of other individuals do not change within a very short time for each individual, thus treating other individuals as equivalent to the environment. We then establish an environmental manifold and solve for the equations of motion of the individual within it.

[0333] The specific process of step S6 is as follows:

[0334] The functional satisfied by the Riemannian metric of the environmental manifold:

[0335]

[0336] At this point, the Riemannian metric of the environmental manifold is the critical point of the Riemannian metric functional in the above equation:

[0337]

[0338] The Riemannian metric of the environmental manifold satisfies the following set of differential equations:

[0339]

[0340] Based on the motion of an individual in a Riemannian manifold, which satisfies the principle of minimum energy, the energy functional variation of the individual's motion yields a set of geodesic equations. Solving the set of equations yields the trajectory of the individual's motion.

[0341] The energy functional satisfied by individual movement:

[0342]

[0343] δL(u)=0,

[0344] Taking variational calculus with respect to δL(u) = 0, we obtain the following system of differential equations satisfied by the individual's trajectory u(x,t):

[0345]

[0346] See Figures 2-4As shown, numerical simulations were conducted using the model constructed through the above process to observe the obstacle avoidance and flight to the target point performance of unmanned swarms in three simulated terrain scenarios: mountains, canyons, and basins. The results showed that in terrain-like environments, the time-varying Riemannian manifold model excelled in basic intelligent evaluation indicators of swarm behavior, such as consistent convergence function, convergence rate, clustering degree, and average path. Furthermore, under different terrain environments and swarm sizes, the divergence between the individual path and the average path depicted by the time-varying Riemannian manifold model was small, and the swarm's average path fluctuated little. This indicates that unmanned swarms guided by the time-varying Riemannian manifold model can reach and effectively cluster at the target point faster, better, and more stably, with superior individual path performance and more stable overall swarm behavior. This implies that the swarm will have advantages in terms of time, distance, and energy consumption for completing specific tasks, demonstrating the superiority and intelligence of the Riemannian manifold model in guiding unmanned swarms. Simultaneously, the simulation results in terrain-like environments also provide a positive preliminary conclusion regarding the feasibility of the time-varying Riemannian manifold-based swarm behavior intelligent theoretical model in practical applications.

Claims

1. A method for modeling swarm intelligence cooperative behavior of unmanned vehicles based on Riemannian manifold, characterized in that, Comprising the following steps: S1: using the Lagrange method to establish the local coordinate system of the perception neighborhood, identifying different fluid particles, and establishing the group manifold; S2: according to the attraction of each individual in the group to the target point in the environment and the repulsion of each individual in the group to the obstacles in the perception neighborhood, an environmental manifold is established; S3: the Riemannian metric functional on the group manifold and the Riemannian metric functional on the environmental manifold are added to construct the Riemannian metric functional on the synthetic manifold, and the variation of the Riemannian metric functional is obtained to satisfy the equation group of the Riemannian metric of the synthetic manifold, realizing the metric fusion to generate the synthetic manifold; S4: converting the continuous group model into a discrete group model; S5: optimizing and solving the path manifold; S6: converting the discrete group model into an individual motion model; In the overview of the unmanned cluster intelligent cooperative behavior modeling method based on Riemann manifold, the specific process in S1 is as follows: S11: Establish the initial local coordinate system of the A-th individual wherein Meanwhile, another coordinate system is selected As a local coordinate system for describing the motion of the individual, the initial local coordinate of the individual at time t is The space coordinate of the individual at time t is denoted by u A (x A ,t) The is represented as and time t: denoted by x A (u A , t) represents the individual whose coordinate at time t is corresponds to the individual whose initial coordinate is x S12: according to the action of each individual from the neighborhood and the environment, the corresponding Ricci curvature tensor is constructed and the Riemannian metric functional is established; S13: integrating the group manifold after metric splicing and smoothing to obtain the Riemannian metric functional: Among them, R ij It is the Ricci curvature tensor on the Riemannian manifold, which describes the curvature of Riemannian space: where K ij = v i v j is the kinetic tensor on a Riemannian manifold, whose contraction is a quantity used to characterize the motion path of a group; V ij is the volume tensor on a Riemannian manifold, whose contraction with g ij V ij is a quantity used to describe the internal action of a group; is the Riemann volume element; J(t) is the Jacobian determinant, represents the volume element after coordinate transformation; S14: taking the variation of the Riemannian metric functional of the group action to obtain the optimal Riemannian metric of the group intelligence model, and the equation group satisfied by the Riemannian metric of the group manifold is obtained by taking the variation of the Riemannian metric functional; In the overview of the unmanned cluster intelligent cooperative behavior modeling method based on Riemann manifold, the specific process in S2 is as follows: S21: according to the attraction of each individual in the group to the target point in the environment and the repulsion of each individual in the group to the obstacles in the perception neighborhood, a target attraction tensor and a repulsion tensor are constructed: The target attraction tensor is defined as: T ij = (u i (x,t)-target i )(u j (x,t)-target j ); Its condensed g ij T ij = g ij (u i (x,t)-target i )(u j (x,t)-target j ) for depicting the distance between the group and the target; The βth obstacle repulsion tensor is defined as: a condensed form thereof for delineating the proximity of the group to the βth obstacle; S22: superimposing the effects of the target and the obstacles to obtain the corresponding Riemannian metric functional of the external environment as follows: The variation equation of the Riemannian metric of the environmental manifold is: That is: S23: The above variation pair is arbitrary δg ij The differential equations satisfied by the Riemann metric are obtained as follows: Solving the differential equation group satisfied by the Riemannian metric obtains the Riemannian metric of the environmental manifold.

2. The method of claim 1, wherein, The specific process of step S14 is as follows: S141: the Riemannian metric functional of the group action is as follows: S142: taking the variation of the Riemannian metric functional of the group action to obtain: When g ij is a critical point of the Riemannian metric functional, That is equivalent to: The Riemannian metric functional is brought in to obtain: Using the divergence formula on the Riemann manifold: obtained: Further obtaining: The following integral is zero:

3. The method of claim 1, wherein, Step S1 further includes coordinate transformation on different perception neighborhoods, Riemannian metric splicing, and smoothing of the group manifold, and the specific process is as follows: Coordinate transformation on different perception neighborhoods: A local coordinate system is established in each individual's perception neighborhood. For each point in the intersection of different perception neighborhoods, the coordinates u in different coordinate systems A (x,t)u B (x,t) differ by a translation, rotation transformation φ AB : A coordinate system is established with a certain point at the initial time of the group as the origin, and the local coordinates of the individuals in each perception neighborhood are converted into coordinates u(x, t) in the global coordinate system through a rotation and translation transformation; Smooth smoothing of the group manifold envelope: The geometric bodies spliced by the neighborhoods are smoothed to form a unified and continuous whole: N A (t) = O(u(x A ,t),r(v A ,t)) where u(x A ,t) denotes the position of the A-th individual at time t, r(v A ,t) denotes the sensing and action radius of the individual in direction v, r(v A ,t) varies smoothly with the direction vector v, v being a unit vector denoting the direction. The perception neighborhood yields k disconnected subgroups L1, L2,..., L k For each disconnected subgroup assume L i |L i | = m i There is a perception neighborhood The partial order envelope of the group manifold is the smoothing of the map: where M is an ambient manifold, the sought smooth map is such that The smallest smooth mapping, whose order is determined by the order of the Fourier series of the set to which f0 belongs; The smooth group manifold S is obtained by smoothing the step envelope t , Riemannian metric splicing: Individuals located in the intersection of different perceptual neighborhoods are considered as points within different neighborhoods, and their Riemannian metrics will have different values. When solving for the metric of the intersection, the influence of all its neighborhoods is considered. For a point in the population manifold: Its Riemannian metric is derived from The Riemannian metric functional determines: F u (g ij ) the variational equation ij satisfied by g Solving the equation group satisfied by the Riemannian metric obtains the Riemannian metric of the points in the intersection part of different neighborhoods.

4. The method of claim 1, wherein, The specific process of step S3 is as follows: S31: the tensors of the group inside and outside are The synthetic action is obtained after superposition, and the metric functional is obtained by integration as follows: S32: Variation is performed on the metric functional: The above variation is for any δg ij It is always zero, and the differential equation group that satisfies the Riemann metric is obtained:

5. The method of claim 1, wherein, The specific process of step S4 is as follows: S41: select a local neighborhood N for the a-th individual according to a selection method α (t), N α (t) = O(u(x α ,t), r(v α ,t)); where u(x α ,t) denotes the position of the a-th member at time t, r(v α ,t) denotes the sensing and action radius of the individual in direction v, r(v α ,t) varies smoothly with the direction vector, is a unit vector representing the direction. S42: The Riemannian metric functional satisfied by the αth individual is obtained after discretization of the Riemannian metric functional: The distance tensor between individuals in the neighborhood is: The distance tensor between the individual in the neighborhood and the obstacle is: S43: The Riemannian metric functional of each individual is varied to obtain: The Riemannian metric functional is varied to obtain: By δg ij For the ath individual, the equation satisfied by the Riemannian metric on the trajectory point is: The equation satisfied by the Riemannian metric of each individual is: S44: The equation satisfied by the Riemannian metric of each individual is solved to obtain the Riemannian metric on the motion path of each individual; S45: For a finite group, the motion of each individual in the synthetic manifold satisfies the principle of minimum energy, and a finite group energy functional is constructed: Variation of the finite group energy functional obtains the motion equation group satisfied by each individual: For each individual, the variation of its motion trajectory satisfies the equation: According to the equation satisfied by the variation of the motion trajectory, the motion equation group satisfied by the individual in the finite group is obtained where Γ 1 ij (u α ) denotes the Riemannian connection of the individual at position u α at time t; S47: The motion equation group satisfied by the individual in the finite group is solved to obtain the motion equation of the group.

6. The method of claim 1, wherein, The specific process of step S5 is as follows: The energy-based Lagrangian action is selected as the energy functional of the group: where u(x, t) is the motion trajectory of the group, is the variation object, and J represents the Jacobian matrix of the coordinate transformation; denotes the Riemann volume element; The kinetic energy term in the energy-based Lagrangian action is: ρ(x, t) is the density of the group, satisfying: where p represents the density function at each point; is the internal energy of the group in the Lagrangian action, taken as: is the external potential term in the Lagrangian action, V(u(x, t)) is taken as: The variation equation of the Lagrangian action is: δL = L(u + δu) - L(u(x, t)) = 0, That is: Variation is performed to obtain: Through the equation: The equation group satisfied by the group motion is obtained: wherein denotes the effect of the external environment on the individual, and the Riemannian connection coefficients are: Take the diagonalized metric, and then:

7. The method of claim 1, wherein, The specific process of step S6 is as follows: The Riemannian metric of the environment manifold satisfies the functional: At this time, the Riemannian metric of the environment manifold is the critical point of the Riemannian metric functional: The differential equation group satisfied by the Riemannian metric of the environment manifold is obtained: According to the motion of the individual in the Riemannian manifold, the principle of minimum energy is satisfied, the energy functional of the individual motion is varied to obtain the geodesic equation group, and the equation group is solved to obtain the motion trajectory of the individual; The energy functional satisfied by the individual motion is: δL(u) = 0, Variation is performed on δL(u) = 0 to obtain the differential equation group satisfied by the motion trajectory u(x, t) of the individual as follows:

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