Unmanned cluster intelligent cooperative behavior modeling method based on Riemannian manifold

Abstract

The invention relates to an unmanned cluster intelligent cooperative behavior modeling method based on a Riemannian manifold. The method comprises the following steps: establishing a group manifold and an environment manifold; constructing a Riemannian metric functional on the synthetic manifold, performing variation on the Riemannian metric functional to obtain an equation set satisfied by the Riemannian metric of the synthetic manifold, and realizing metric fusion to generate the synthetic manifold; converting the continuous group model into a discrete group model; and carrying out optimization solution on the path manifold, and converting a discrete group model into an individual motion model. For an intelligent cooperative path planning scene of an unmanned cluster in a complex local obstacle environment, a time-varying Riemannian manifold model is established based on a sensing action range of each individual of the cluster for solution; the technical problems that unmanned cluster obstacle avoidance path planning depends on a specific central processing unit, depends on large-range environment information, is prone to falling into local optimum and the like are solved, decentralized intelligent collaborative optimal motion path planning of the unmanned cluster in the local obstacle environment is achieved, and therefore the actual collaborative obstacle avoidance flight path of the unmanned cluster is guided.

Description

Technical Field

[0001] The present invention belongs to the technical field of unmanned swarm intelligent collaborative behavior modeling, and in particular relates to an unmanned swarm intelligent collaborative behavior modeling method based on Riemann manifold. Background Art

[0002] Individuals in swarm intelligence form self-organizing behaviors with strong robustness, high adaptability and good scalability through simple rules and local interactions, which manifests as the emergence of intelligence at the system level. So far, there has not been a swarm intelligence model with a clear definition, clear description and effective application. This not only hinders the research and application of the nature and performance of swarm intelligence itself, but also hinders the deep and extensive development of the integration of swarm intelligence with transformative artificial intelligence methods and technologies such as "formal abstraction" first embodied in deep learning. The existing modeling methods have three main shortcomings: the abstract definition and formal description of swarm intelligence itself as the basis of modeling have not met the requirements, and the logical rigor and completeness are not enough; when the definition is not effective, the degree of formalization of existing modeling attempts is also insufficient, either the lack of formal abstraction leads to limited connotation, or the logical level is too low to cause discrete extension; there is no full awareness of the mathematical and physical essence of swarm intelligence and the fusion and coupling relationship between them. The existing methods consider a single angle and path, which inevitably leads to the neglect of the fusion, coupling relationship and its role.

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

[0004] In the Vicsek model, a discrete time system is composed of N autonomous individuals or particles. The set ∑(1.2.3…N) represents the set of all individuals. The initial positions and movement directions of the individuals are randomly distributed. All individuals move freely in a set plane area. The speed of individual movement is constant, and the direction of movement is determined by the average of the neighbor's speed vectors plus a noise disturbance with a mean of zero.

[0005] Individuals have the same perception ability, and each individual can perceive the information of neighboring individuals within its neighborhood. The individual perception range is based on the individual position. The circular area with R as the center and R as the radius is the perception range. All individuals have the same speed v0. The movement direction of individual i at time t is θ(t)∈(-π,π], then its velocity vector is Individual location updates follow The individual 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 formula of the Vicsek model is as follows:

[0006]

[0007] In the Vicsek model's research on unmanned swarm intelligent collaboration, starting from the individual, the dynamic equation is directly established and solved as a multi-objective and multi-constraint optimization problem. The difference is basically in the selection and setting of the objectives and constraints and whether the optimization algorithm used is based on mathematical methods or modern artificial intelligence methods such as deep learning or a combination of the two. The separate specific elements are directly combined without considering the mutual relationship and interaction between the conditions, so they are not formed into an internally coordinated overall mathematical structure to be applicable to the optimization algorithm in a holistic and structural manner.

[0008] Therefore, how to improve the existing unmanned swarm intelligent collaborative modeling methods to solve the problem that the existing unmanned swarm intelligent collaborative modeling methods directly combine the separate specific elements without considering the mutual relationship and interaction between the conditions, and thus fail to form an internally coordinated and consistent overall mathematical structure to be holistic and structurally applicable to the optimization algorithm. Summary of the invention

[0009] The purpose of the present invention is to provide a method for modeling the intelligent collaborative behavior of unmanned clusters based on Riemann manifolds, which is used for the intelligent collaborative path planning scenarios of unmanned clusters in complex local obstacle environments. Based on the perception range of each individual in the cluster, a time-varying Riemann manifold model is established and solved according to the above method, which solves the technical problems of unmanned cluster obstacle avoidance path planning relying on specific central processing units, relying on large-scale environmental information, and easily falling into local optimality. It realizes decentralized intelligent collaborative optimal motion path planning of unmanned clusters in local obstacle environments, thereby guiding the actual collaborative obstacle avoidance flight trajectory of the unmanned clusters.

[0010] In order to solve the above technical problems, the technical solution adopted by the present invention is as follows:

[0011] A method for modeling unmanned swarm intelligent collaborative behavior based on Riemann manifold, comprising the following steps:

[0012] S1: Use the Lagrangian method to establish the local coordinate system of the sensing neighborhood, identify different fluid particles, and establish the group manifold;

[0013] S2: Establish the environment manifold based on the attraction of the target point in the environment to each individual in the group and the repulsion of obstacles in the perceived neighborhood of each individual to the individual;

[0014] S3: Add the Riemann metric functional on the group manifold and the Riemann metric functional on the environment manifold to construct the Riemann metric functional on the composite manifold, and vary it to obtain the equation group satisfied by the Riemann metric of the composite manifold, so as to realize metric fusion and generate the composite manifold;

[0015] S4: Converting the continuous population model to a discrete population model;

[0016] S5: Optimize and solve the path manifold;

[0017] S6: Convert discrete group models into individual motion models.

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

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

[0020] Denoted as u A (x A ,t);

[0021] Will Expressed as And the function of time t:

[0022] Note that x A (u A ,t), indicating that the coordinates at time t are The point corresponding to the initial coordinates is individuals;

[0023] S12: Construct the corresponding Ricci curvature tensor and establish the Riemann metric functional according to the effects on each individual from other individuals and the environment in the neighborhood;

[0024] S13: The Riemann metric functional is obtained by integrating on the group manifold after metric splicing and smoothing:

[0025]

[0026] Among them, R ij is the Ricci curvature tensor on the Riemann manifold, describing the curvature of the Riemann space:

[0027]

[0028] Where K ij =v i v j is the kinetic energy tensor on the Riemann manifold, which is contracted It is a quantity used to describe the movement distance of a group;

[0029]

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

[0031]

[0032] is the Riemann volume element;

[0033]

[0034] J(t) is the Jacobi transform determinant, represents the volume element after coordinate transformation;

[0035] S14: The optimal Riemann metric describing the swarm intelligence model is obtained by varying the Riemann metric functional of the swarm action, and the set of equations satisfied by the Riemann metric of the swarm manifold is obtained by varying the Riemann metric functional.

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

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

[0038]

[0039] S142: Variation of the Riemann metric functional of the group action yields:

[0040]

[0041] When ij is the critical point of the Riemann metric functional,

[0042]

[0043] This is equivalent to:

[0044]

[0045] Substituting the Riemann metric functional into the equation, we get:

[0046]

[0047] Using the divergence formula on Riemann manifold:

[0048]

[0049] get:

[0050] Then we get:

[0051] The following integral is 0:

[0052]

[0053] Preferably, step S1 also includes coordinate transformation on different perceptual neighborhoods, Riemannian metric splicing, and smoothing of group manifolds. The specific process is as follows:

[0054] Coordinate transformation on different perceptual neighborhoods:

[0055] Establish a local coordinate system within each individual perception neighborhood, for each point in the intersection of different perception 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 :

[0056]

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

[0058] Step-by-step envelope smoothing of a swarm manifold:

[0059] Smooth the geometric bodies composed of various neighborhoods 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 ,t) represents the position of the Ath individual at time t, r(v A ,t) represents the perception and action radius of the individual along direction v, r(v A ,t) changes smoothly with the direction vector, v is the unit vector representing the direction;

[0062] The sensing neighborhood generates k disconnected subgroups L1, L2, ..., L k , for each disconnected subgroup, assume that L i ,|L i |=m i , with perceptual neighborhood The ordered envelope of the group manifold is the smoothing of the mapping:

[0063]

[0064] Where M is the environment manifold, the desired smooth mapping That is to make

[0065]

[0066] The smallest smooth mapping, whose order is determined by the order of the Fourier series of the set to which f0 belongs; the smoothed group manifold S can be obtained by order envelope smoothing t ,

[0067] Riemannian metric splicing:

[0068] If we regard individuals in the intersection of different perceptual neighborhoods as points in different neighborhoods, the Riemann metric on them will have different values. When solving the metric of the intersection, we consider the role of all the neighborhoods in which they are located. For a point in the group manifold: The Riemann metric on it is given by Determined by the Riemann metric functional on:

[0069]

[0070] F u (g ij ) Variation can be obtained g ij The differential equations satisfied are:

[0071]

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

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

[0074] S21: According to the attraction of the target point in the environment to each individual in the group and the repulsion of obstacles in the perceived neighborhood of each individual to the individual, the target attraction tensor and repulsion tensor are constructed:

[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 ij T ij =g ij (u i (x,t)-target i )(u j(x,t)-target j ) is used to describe the distance between a group and a target;

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

[0079]

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

[0081] S22: The effect of the target is superimposed on the effect of the obstacle, and the Riemann metric functional corresponding to the external environment is obtained as follows:

[0082]

[0083] The variational equation of the environmental manifold corresponding to the Riemann metric is obtained as:

[0084]

[0085] Right now:

[0086]

[0087] S23: The above variation for any δg ij Always equal to zero, the differential equations satisfied by the Riemann metric are:

[0088]

[0089] The Riemann metric of the environment manifold is obtained by solving the differential equation group satisfied by the Riemann metric.

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

[0091] S31: Tensors inside and outside the group

[0092]

[0093] After superposition, the composite action is obtained and the metric functional is obtained by integration as follows:

[0094]

[0095] S32: Perform variation on the metric functional:

[0096]

[0097] The above variational formula for any δg ij Always equal to zero, the differential equations satisfied by the Riemann metric are:

[0098]

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

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

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

[0102] Among them, u(x α ,t) represents the position of the αth member at time t, r(v α ,t) represents the direction of the individual The perception and radius of action, r(v α ,t) varies smoothly with the direction vector, is a unit vector representing direction;

[0103] S42: After discretizing the Riemann metric functional, the Riemann metric functional satisfied by the αth individual can be obtained:

[0104]

[0105] Among them, the distance tensor between individuals in the neighborhood is:

[0106]

[0107] The distance tensor between individuals and obstacles in the neighborhood is:

[0108]

[0109] S43: The Riemann metric functional variation of each individual is obtained:

[0110]

[0111] The variational solution of the Riemann metric functional is:

[0112]

[0113] By δg ij The arbitrariness of , for the αth individual, the equation satisfied by the Riemann metric on its trajectory point is:

[0114]

[0115] The equation satisfied by the Riemann metric of each individual is:

[0116]

[0117] S44: Solving 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 group, the motion of each individual in the composite manifold satisfies the minimum energy principle, and the energy functional of a finite group is constructed:

[0119]

[0120] The energy functional variation of the finite group is used to obtain the set of motion equations satisfied by each individual:

[0121]

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

[0123]

[0124] According to the equations satisfied by the variation of the motion trajectory, the motion equations satisfied by individuals in the finite group are obtained.

[0125]

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

[0127] S47: Solving the group of motion equations satisfied by the individuals in the finite group to obtain the motion equation of the group.

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

[0129] Select the energy-based Lagrangian action as the energy functional of the group:

[0130]

[0131] Among them, u(x,t) is the motion 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 Lagrangian action is:

[0133]

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

[0135]

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

[0137]

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

[0139]

[0140] The variational equation of the Lagrangian action is:

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

[0142]

[0143] By doing variation, we can get:

[0144]

[0145] By equation:

[0146]

[0147] The equations satisfied by the group motion are obtained:

[0148]

[0149] in, Represents the effect of the external environment on the individual, and the Riemann connection coefficient is:

[0150]

[0151] Take the diagonalization metric:

[0152]

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

[0154] The Riemannian metric of the ambient manifold satisfies the functional:

[0155]

[0156] At this time, the Riemann metric of the ambient manifold is the critical point of the Riemann metric functional above:

[0157]

[0158] The differential equations satisfied by the Riemann metric of the ambient manifold are obtained:

[0159]

[0160] According to the movement of individuals in the Riemann manifold, the minimum energy principle is satisfied, and the energy functional variation of the individual movement is obtained to obtain the geodesic equations. The equations are solved to obtain the individual's movement trajectory.

[0161] The energy functional satisfied by individual motion is:

[0162]

[0163] By varying δL(u) = 0, we can obtain the following differential equations satisfied by the individual motion trajectory u(x, t):

[0164]

[0165] The beneficial effects of the present invention include:

[0166] The method for modeling the intelligent collaborative behavior of unmanned swarms based on Riemann manifolds provided by the present invention adopts the Lagrangian method to establish a local coordinate system of the perception neighborhood to establish a swarm manifold; establish an environmental manifold; add the Riemann metric functional on the swarm manifold and the Riemann metric functional on the environmental manifold to construct the Riemann metric functional on the synthetic manifold, and obtain the equation group satisfied by the Riemann metric of the synthetic manifold by variation, so as to realize the metric fusion to generate the synthetic manifold; transform the continuous swarm model into a discrete swarm model; optimize and solve the path manifold, and transform the discrete swarm model into an individual motion model. For the intelligent collaborative path planning scenario of unmanned swarms in complex local obstacle environments, a time-varying Riemann manifold model is established and solved based on the perception range of each individual in the swarm, which solves the technical problems of relying on a specific central processing unit, relying on a large range of environmental information, and being prone to falling into local optimality in the obstacle avoidance path planning of unmanned swarms, and realizes the decentralized intelligent collaborative optimal motion path planning of unmanned swarms in local obstacle environments, thereby guiding the actual collaborative obstacle avoidance flight trajectory of unmanned swarms. BRIEF DESCRIPTION OF THE DRAWINGS

[0167] Figure 1 It is a flow chart of the unmanned swarm intelligent collaborative behavior modeling method based on Riemann manifold of the present invention.

[0168] Figure 2 This is the simulation effect of the unmanned swarm of the present invention in mountainous terrain.

[0169] Figure 3 This is the simulation effect of the unmanned swarm of the present invention in canyon terrain.

[0170] Figure 4This is the simulation effect of the unmanned cluster of the present invention in the basin terrain. DETAILED DESCRIPTION

[0171] Example 1

[0172] The following is combined with Figure 1 to Figure 4 The present invention is further described in detail:

[0173] See attached Figure 1 As shown, a method for modeling unmanned swarm intelligent collaborative behavior based on Riemann manifold includes the following steps:

[0174] S1: Use the Lagrangian method to establish the local coordinate system of the sensing neighborhood, identify different fluid particles, and establish the group manifold;

[0175] S2: Establish the environment manifold based on the attraction of the target point in the environment to each individual in the group and the repulsion of obstacles in the perceived neighborhood of each individual to the individual;

[0176] S3: Add the Riemann metric functional on the group manifold and the Riemann metric functional on the environment manifold to construct the Riemann metric functional on the composite manifold, and vary it to obtain the equation group satisfied by the Riemann metric of the composite manifold, so as to realize metric fusion and generate the composite manifold;

[0177] S4: Converting the continuous population model to a discrete population model;

[0178] S5: Optimize and solve the path manifold;

[0179] S6: Convert discrete group models into individual motion models.

[0180] Each individual in the unmanned intelligent group naturally has a certain range of perception and action, which can be mathematically corresponded to a certain neighborhood centered on the individual, and the individual can only perceive and transmit information and action within the neighborhood. On the other hand, there is overlap between individual neighborhoods, which enables information and action to be transmitted and synthesized between neighborhoods, so that the group can emerge with certain intelligent behaviors. The above group behavior pattern is highly consistent with the way of establishing manifolds mathematically. For this reason, the present invention mathematically models the intelligent group as a group manifold. For the intelligent collaborative path planning scenario of unmanned clusters in complex local obstacle environments, a time-varying Riemann manifold model is established and solved based on the perception range of each individual in the cluster according to the above method, which solves the technical problems of relying on specific central processing units, relying on large-scale environmental information, and being prone to local optimality in the obstacle avoidance path planning of unmanned clusters, and realizes the decentralized intelligent collaborative optimal motion path planning of unmanned clusters in local obstacle environments, thereby guiding the actual collaborative obstacle avoidance flight trajectory of unmanned clusters.

[0181] Since the perception range of each individual in the group is limited in actual situations, only a local coordinate system can be established within a limited range to describe the external information of other individuals and the environment. The Lagrange method in fluid mechanics is suitable for describing the actual movement of the group, so the Lagrange method is used to establish the local coordinate system of the perception neighborhood. The Lagrange method focuses on fluid particles. The particle system in fluid mechanics is very easy to deform during movement. Therefore, it is necessary to first identify different fluid particles in order to track the fluid particle that needs to be studied. According to the attraction of the target point in the environment to each individual in the group and the repulsion of obstacles in the perception neighborhood of each individual to the individual, the following attraction tensor and repulsion tensor can be constructed.

[0182] The Riemann metric of the group manifold can only reflect the interaction within the group, and the Riemann metric corresponding to the environmental manifold can only reflect the external effect of the environment on the group. To this end, the Riemann metric functional on the group manifold is added to the Riemann metric functional on the environmental manifold to construct the Riemann metric functional on the composite manifold. The variation of the Riemann metric on the composite manifold is used to obtain the set of equations satisfied by the Riemann metric of the composite manifold. At this time, the curvature of the Riemann space comes from the joint effect of the group inside and outside the group. This superposition variation process is called metric fusion. Since the number of individuals in the group is limited in practice, only the positions and velocities of a limited number of individuals are known, and the perception range of each individual is limited. Therefore, in order to reflect the locality of the perception and action of the drone individual, a local neighborhood N is selected for the αth individual according to a certain selection method. α (t), each individual can only perceive the effects of other individuals and the environment in its neighborhood on itself.

[0183] In the same Riemann space, group manifolds have different motion paths, and different motion paths of group manifolds reflect different group motion goals, such as minimum energy, shortest distance, minimum path manifold volume, etc. According to the Lagrangian minimum action principle, the energy-based Lagrangian action that reflects the group motion goal can be constructed. By minimizing the Lagrangian action, the motion equations satisfied by the group motion can be obtained. Since the group motion goal of the shortest distance has been reflected by the Riemannian metric of the Riemann space where the group motion is located, the corresponding Lagrangian action and motion trajectory functional are constructed by minimizing the group energy (the sum of the group's kinetic energy, internal energy and potential energy).

[0184] Due to the mutual influence between individuals in a group, the movement of each individual at every moment will be affected by other individuals. It is essentially a multi-body problem on a Riemann manifold, which is difficult to solve or simulate. Therefore, for each individual, it can be assumed that the positions of other individuals do not change in a very short time, so that other individuals are equivalent to the environment, and the environment manifold is established to solve the motion equation of the individual in the environment manifold.

[0185] Example 2

[0186] Based on Example 1, the specific process of step S1 is as follows:

[0187] Since the perception range of each individual in the group is limited in actual situations, only a local coordinate system can be established within a limited range to describe the external information of other individuals and the environment. The Lagrangian method in fluid mechanics obtains the geodesic equation by varying the energy functional of individual motion, which is suitable for describing the actual motion of the group. Therefore, the present invention adopts the Lagrangian method to establish a local coordinate system of the perception neighborhood. The Lagrangian method focuses on fluid particles. The particle system in fluid mechanics is very easy to deform during motion. Therefore, it is necessary to first identify different fluid particles in order to track the fluid particle to be studied.

[0188] S11: Establish the initial local coordinate system of individual A in Select another coordinate system at the same time As the local coordinate system describing the individual motion, the initial local coordinates are The spatial coordinates of the individual moment t are:

[0189] Denoted as u A (x A ,t);

[0190] Will Expressed as And the function of time t:

[0191] Note that x A (u A ,t), indicating that the coordinates at time t are The point corresponding to the initial coordinates is individuals;

[0192] S12: According to the influence of other individuals and the environment in the neighborhood on each individual, the corresponding Ricci curvature tensor is constructed and the Riemann metric functional is established. At this time, the Riemann metric of the group manifold is the critical point of the Riemann metric functional. When the Riemann metric is the critical point of the functional, the influence of other individuals and the environment on the individual is equivalent to the curvature of the Riemann space.

[0193] S13: The Riemann metric functional is obtained by integrating on the group manifold after metric splicing and smoothing (at this time, the mode of action on each point on the group manifold is determined):

[0194]

[0195] Among them, R ijis the Ricci curvature tensor on the Riemann manifold, describing the curvature of the Riemann space:

[0196]

[0197] Where K ij =v i v j is the kinetic energy tensor on the Riemann manifold, which is contracted It is a quantity used to describe the movement distance of a group;

[0198]

[0199] V ij is the volume tensor on the Riemannian manifold, whose contraction g ij V ij It is a quantity used to describe the internal effects of a group;

[0200]

[0201] is the Riemann volume element;

[0202]

[0203] J(t) is the Jacobi transform determinant, Represents the volume element after coordinate transformation. For a given Riemannian metric, the present invention provides the corresponding mathematical description of some key physical quantities in swarm intelligence in the Riemannian manifold. By solving the variational equation, we can obtain the optimal Riemann metric to describe the swarm intelligence model.

[0204] S14: The optimal Riemann metric describing the swarm intelligence model is obtained by varying the Riemann metric functional of the swarm action, and the set of equations satisfied by the Riemann metric of the swarm manifold is obtained by varying the Riemann metric functional.

[0205] In this embodiment, the specific process of step S14 is as follows:

[0206] S141: The Riemann metric functional of the group action is as follows:

[0207]

[0208] S142: Variation of the Riemann metric functional of the group action yields:

[0209]

[0210] When ij is the critical point of the Riemann metric functional,

[0211]

[0212] This is equivalent to:

[0213]

[0214] Substituting the Riemann metric functional into the equation, we get:

[0215] Using the divergence formula on Riemann manifold:

[0216]

[0217] get: Then we get:

[0218] The following integral is 0:

[0219]

[0220] Example 3

[0221] On the basis of Example 1 or Example 2, step S1 also includes coordinate transformation on different perceptual neighborhoods, Riemannian metric splicing, and smoothing of group manifolds. The specific process is as follows:

[0222] Coordinate transformation on different perceptual neighborhoods:

[0223] Since the perception ability of each individual is limited, a local coordinate system is established within the perception neighborhood of each individual. In order to unify the coordinate representation on the group manifold, it is necessary to determine the relationship and transformation rules of each local coordinate system. For each point in the intersection of different perception neighborhoods, that is, 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 :

[0224]

[0225] A coordinate system is established with a certain point at the initial moment of the group as the origin, and the local coordinates of each individual in the perception neighborhood are converted into coordinates u(x, t) in the global coordinate system through a rotation and translation transformation;

[0226] Step-by-step envelope smoothing of a swarm manifold:

[0227] Since the group manifold directly formed by splicing the perceptual neighborhood is not smooth in geometry, for the needs of subsequent calculations, the geometry formed by each neighborhood is smoothed to form a unified and continuous whole:

[0228] N A(t) = O (u (x A ,t),r(v A ,t)).

[0229] where u(x A ,t) represents the position of the Ath individual at time t, r(v A ,t) represents the perception and action radius of the individual along direction v, r(v A ,t) changes smoothly with the direction vector, v is the unit vector representing the direction;

[0230] The sensing neighborhood generates k disconnected subgroups L1, L2, ..., L k , for each disconnected subgroup, assume that L i ,|L i |=m i , with perceptual neighborhood The ordered envelope of the group manifold is the smoothing of the mapping:

[0231]

[0232] Where M is the environment manifold, the desired smooth mapping That is to make

[0233]

[0234] The smallest smooth mapping, whose order is determined by the order of the Fourier series of the set to which f0 belongs; the smoothed group manifold S can be obtained by order envelope smoothing t ,

[0235] Riemannian metric splicing:

[0236] If we regard individuals in the intersection of different perceptual neighborhoods as points in different neighborhoods, the Riemann metric on them will have different values. When solving the metric of the intersection, we consider the role of all the neighborhoods in which they are located. For a point in the group manifold: The Riemann metric on it is given by Determined by the Riemann metric functional on:

[0237]

[0238] F u (g ij ) Variation can be obtained g ij The differential equations satisfied are:

[0239]

[0240] Solving the system of equations satisfied by the Riemann metric obtains the Riemann metric of the points in the intersection of different neighborhoods.

[0241] Example 4

[0242] Based on Example 1 or Example 2 or Example 3, the specific process in step S2 is as follows:

[0243] S21: According to the attraction of the target point in the environment to each individual in the group and the repulsion of obstacles in the perceived neighborhood of each individual to the individual, the target attraction tensor and repulsion tensor are constructed:

[0244] The target attraction tensor is defined as:

[0245] T ij =(u i (x,t)-target i )(u j (x,t)-target j );

[0246] Its contraction ij T ij =g ij (u i (x,t)-target i )(u j (x,t)-target j ) is used to describe the distance between a group and a target;

[0247] The βth obstacle repulsion tensor is defined as:

[0248]

[0249] Its contraction Used to characterize the distance between the group and the βth obstacle;

[0250] S22: The effect of the target is superimposed on the effect of the obstacle, and the Riemann metric functional corresponding to the external environment is obtained as follows:

[0251]

[0252] The variational equation of the environmental manifold corresponding to the Riemann metric is obtained as:

[0253]

[0254] Right now:

[0255]

[0256] S23: The above variation for any δg ij Always equal to zero, the differential equations satisfied by the Riemann metric are:

[0257]

[0258] The Riemann metric of the environment manifold is obtained by solving the differential equation group satisfied by the Riemann metric.

[0259] The specific process of step S3 is as follows:

[0260] Since the Riemann metric of the group manifold can only reflect the interaction within the group, the Riemann metric corresponding to the environment manifold can only reflect the external effect of the environment on the group. For this reason, the present invention adds the Riemann metric functional on the group manifold and the Riemann metric functional on the environment manifold to construct the Riemann metric functional on the composite manifold, and varies it to obtain the equation group satisfied by the Riemann metric of the composite manifold. At this time, the curvature of the Riemann space comes from the joint effect of the group inside and outside the group. This superposition variation process is called metric fusion.

[0261] S31: Tensors inside and outside the group

[0262]

[0263] After superposition, the composite action is obtained and the metric functional is obtained by integration as follows:

[0264]

[0265] S32: Perform variation on the metric functional:

[0266]

[0267] The above variational formula for any δg ij Always equal to zero, the differential equations satisfied by the Riemann metric are:

[0268]

[0269] Example 5

[0270] On the basis of Example 1 or Example 2 or Example 3 or Example 4, the specific process of step S4 is as follows:

[0271] Since the number of individuals in the group is limited in practice, only the positions and velocities of a limited number of individuals are known, and the perception range of each individual is limited. Therefore, in order to reflect the locality of the perception and action of the individual drone, a local neighborhood N is selected for the αth individual according to a certain selection method. α (t), each individual can only perceive the effects of other individuals and the environment in its neighborhood on itself.

[0272] S41: Select a local neighborhood N for the αth individual according to a certain selection method α (t),

[0273] N α (t) = O (u (x α ,t),r(v α ,t)).;

[0274] Among them, u(x α ,t) represents the position of the αth member at time t, r(v α ,t) represents the direction of the individual The perception and radius of action, r(v α ,t) varies smoothly with the direction vector, is a unit vector representing direction;

[0275] S42: After discretizing the Riemann metric functional, the Riemann metric functional satisfied by the αth individual can be obtained:

[0276]

[0277] Among them, the distance tensor between individuals in the neighborhood is:

[0278]

[0279] The distance tensor between individuals and obstacles in the neighborhood is:

[0280]

[0281] S43: The Riemann metric functional variation of each individual is obtained:

[0282]

[0283] The variational solution of the Riemann metric functional is:

[0284]

[0285] By δg ij The arbitrariness of , for the αth individual, the equation satisfied by the Riemann metric on its trajectory point is:

[0286]

[0287] The equation satisfied by the Riemann metric of each individual is:

[0288]

[0289] S44: Solving the equation satisfied by the Riemann metric of each individual to obtain the Riemann metric on the motion path of each individual;

[0290] S45: For a finite group, the motion of each individual in the composite manifold satisfies the minimum energy principle, and the energy functional of a finite group is constructed:

[0291]

[0292] The energy functional variation of the finite group is used to obtain the set of motion equations satisfied by each individual:

[0293]

[0294] For each individual, the variation of its motion trajectory satisfies the equation: δu α :

[0295]

[0296] According to the equations satisfied by the variation of the motion trajectory, the motion equations satisfied by individuals in the finite group are obtained.

[0297]

[0298] Among them, Γ 1 ij (u α ) indicates that the position at time t is u α The Riemann connection corresponding to the individuals of ;

[0299] S47: Solving the group of motion equations satisfied by the individuals in the finite group to obtain the motion equation of the group.

[0300] Example 6

[0301] Based on Example 1 or Example 2 or Example 3 or Example 4 or Example 5, the specific process of step S5 is as follows:

[0302] In the same Riemann space, group manifolds have different motion paths, and different motion paths of group manifolds reflect different group motion goals, such as minimum energy, shortest distance, minimum path manifold volume, etc. According to the Lagrangian minimum action principle, the energy-based Lagrangian action that reflects the group motion goal is constructed. By minimizing the Lagrangian action, the motion equations satisfied by the group motion can be obtained. Since the group motion goal of the shortest distance has been reflected by the Riemannian metric of the Riemann space where the group motion is located, the corresponding Lagrangian action and motion trajectory functional are constructed by minimizing the group energy (the sum of the group's kinetic energy, internal energy and potential energy).

[0303] Select the energy-based Lagrangian action as the energy functional of the group:

[0304]

[0305] Among them, u(x,t) is the motion trajectory of the group, which is the variational object, and J represents the Jacobian matrix of the coordinate transformation; represents the Riemann volume element;

[0306] The kinetic energy term in the energy-based Lagrangian action is:

[0307]

[0308] ρ(x,t) is the density of the group, satisfying:

[0309]

[0310] Where p represents the density function at each point; is the group internal energy term in the Lagrangian action and is taken as:

[0311]

[0312] is the external potential energy term in the Lagrangian action, and V(u(x,t)) is taken as:

[0313]

[0314] The variational equation of the Lagrangian action is:

[0315] δL=L(u+δu)-L(u(x,t))=0,

[0316] Right now:

[0317]

[0318] By doing variation, we can get:

[0319]

[0320] By equation:

[0321]

[0322] The equations satisfied by the group motion are obtained:

[0323]

[0324] in, Represents the effect of the external environment on the individual, and the Riemann connection coefficient is:

[0325]

[0326] Take the diagonalization metric:

[0327]

[0328] Example 7

[0329] Based on Example 1 or Example 2 or Example 3 or Example 4 or Example 5 or Example 6,

[0330] Due to the mutual influence between individuals in the group, the movement of each individual at every moment will be affected by other individuals. It is essentially a multi-body problem on the Riemann manifold, which is difficult to solve or simulate. Therefore, for each individual, it is assumed that the positions of other individuals do not change in a very short time, so that other individuals are equivalent to the environment, and the environment manifold is established to solve the motion equation of the individual in the environment manifold.

[0331] The specific process of step S6 is as follows:

[0332] The Riemannian metric of the ambient manifold satisfies the functional:

[0333]

[0334] At this time, the Riemann metric of the ambient manifold is the critical point of the Riemann metric functional above:

[0335]

[0336] The differential equations satisfied by the Riemann metric of the ambient manifold are obtained:

[0337]

[0338] According to the movement of individuals in the Riemann manifold, the minimum energy principle is satisfied, and the energy functional variation of the individual movement is obtained to obtain the geodesic equations. The equations are solved to obtain the individual's movement trajectory.

[0339] The energy functional satisfied by individual motion is:

[0340]

[0341] δL(u)=0,

[0342] By varying δL(u) = 0, we can obtain the following differential equations satisfied by the individual motion trajectory u(x, t):

[0343]

[0344] See also Figure 2-Figure 4As shown in the figure, the model constructed by the above process is numerically simulated to observe the obstacle avoidance and flight to the target point of the unmanned swarm in three types of simulated terrain scenes: mountains, canyons, and basins. It is observed that in the terrain-like environment, the group behavior described by the time-varying Riemann manifold model is excellent in the basic group behavior intelligence evaluation indicators such as the consistent convergence function, convergence rate, aggregation degree, and average path. In addition, under different terrain environments and different group sizes, the individual distance described by the time-varying Riemann manifold model has a small divergence relative to the average distance, and the average distance of the cluster fluctuates little. It can be seen that the unmanned swarm guided by the time-varying Riemann manifold model can reach and effectively gather at the target point faster, better, and more stably, and the individual path performance is better, and the overall behavior of the cluster is more stable. This means that the time, distance, and energy consumption of the cluster to complete a specific task will be more advantageous, and the overall superiority and intelligence of the Riemann manifold model in guiding the unmanned swarm are presented. At the same time, the simulation results in the terrain-like environment also give a positive preliminary conclusion on the feasibility of the theoretical model of group behavior intelligence based on the time-varying Riemann manifold in practical applications.

Claims

1. A method for modeling unmanned swarm intelligent collaborative behavior based on Riemann manifold, characterized in that: The following steps are involved: S1: Use the Lagrangian method to establish the local coordinate system of the sensing neighborhood, identify different fluid particles, and establish the group manifold; S2: Establish the environment manifold based on the attraction of the target point in the environment to each individual in the group and the repulsion of obstacles in the perceived neighborhood of each individual to the individual; S3: Add the Riemann metric functional on the group manifold and the Riemann metric functional on the environment manifold to construct the Riemann metric functional on the composite manifold, and vary it to obtain the equation group satisfied by the Riemann metric of the composite manifold, so as to realize metric fusion and generate the composite manifold; S4: Converting the continuous population model to a discrete population model; S5: Optimize and solve the path manifold; S6: Convert discrete group models into individual motion models.

2. The method for modeling unmanned swarm intelligence collaborative behavior based on Riemann manifold according to claim 1, characterized in that: The specific process of step S1 is as follows: S11: Establish the initial local coordinate system of individual A in Select another coordinate system at the same time As the local coordinate system describing the individual motion, the initial local coordinates are The spatial coordinates of the individual moment t are: Denoted as u A (x A ,t); Will Expressed as And the function of time t: Note that x A (u A ,t), indicating that the coordinates at time t are The point corresponding to the initial coordinates is individuals; S12: Construct the corresponding Ricci curvature tensor and establish the Riemann metric functional according to the effects on each individual from other individuals and the environment in the neighborhood; S13: The Riemann metric functional is obtained by integrating on the group manifold after metric splicing and smoothing: Among them, R ij is the Ricci curvature tensor on the Riemann manifold, describing the curvature of the Riemann space: Where K ij =v i v j is the kinetic energy tensor on the Riemann manifold, which is contracted It is a quantity used to describe the movement distance of a group; V ij is the volume tensor on the Riemannian manifold, whose contraction g ij V ij It is a quantity used to describe the internal effects of a group; is the Riemann volume element; J(t) is the Jacobi transform determinant, represents the volume element after coordinate transformation; S14: The optimal Riemann metric describing the swarm intelligence model is obtained by varying the Riemann metric functional of the swarm action, and the set of equations satisfied by the Riemann metric of the swarm manifold is obtained by varying the Riemann metric functional.

3. The method for modeling unmanned swarm intelligence collaborative behavior based on Riemann manifold according to claim 2, characterized in that: The specific process of step S14 is as follows: S141: The Riemann metric functional of the group action is as follows: S142: Variation of the Riemann metric functional of the group action yields: When ij is the critical point of the Riemann metric functional, This is equivalent to: Substituting the Riemann metric functional into the equation, we get: Using the divergence formula on Riemann manifold: get: Then we get: The following integral is 0:

4. The method for modeling unmanned swarm intelligence collaborative behavior based on Riemann manifold according to claim 1, characterized in that: Step S1 also includes coordinate transformation on different perceptual neighborhoods, Riemannian metric splicing, and smoothing of the group manifold. The specific process is as follows: Coordinate transformation on different perceptual neighborhoods: Establish a local coordinate system within each individual perception neighborhood, for each point in the intersection of different perception 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 : A coordinate system is established with a certain point at the initial moment of the group as the origin, and the local coordinates of each individual in the perception neighborhood are converted into coordinates u(x, t) in the global coordinate system through a rotation and translation transformation; Ordered envelope smoothing of a swarm manifold: Smooth the geometric bodies composed of various neighborhoods to form a unified and continuous whole: N A (t)=O(u(x A ,t),r(v A ,t)). where u(x A ,t) represents the position of the Ath individual at time t, r(v A ,t) represents the perception and action radius of the individual along direction v, r(v A ,t) changes smoothly with the direction vector, v is the unit vector representing the direction; The sensing neighborhood generates k disconnected subgroups L1, L2, ..., L k , for each disconnected subgroup, assume that L i ,|L i |=m i , with perceptual neighborhood The ordered envelope of the group manifold is the smoothing of the mapping: Where M is the environment manifold, the desired smooth mapping That is to make The smallest smooth mapping, whose order is determined by the order of the Fourier series of the set to which f0 belongs; Through the step-by-step envelope smoothing, we can get the smoothed group manifold S t , Riemannian metric splicing: If we regard individuals in the intersection of different perceptual neighborhoods as points in different neighborhoods, the Riemann metric on them will have different values. When solving the metric of the intersection, we consider the role of all the neighborhoods in which they are located. For a point in the group manifold: The Riemann metric on it is given by Determined by the Riemann metric functional on: F u (g ij ) variational solution can be obtained as g ij The differential equations satisfied are: Solving the system of equations satisfied by the Riemann metric obtains the Riemann metric of the points in the intersection of different neighborhoods.

5. The method for modeling unmanned swarm intelligence collaborative behavior based on Riemann manifold according to claim 1, characterized in that: The specific process in step S2 is as follows: S21: According to the attraction of the target point in the environment to each individual in the group and the repulsion of obstacles in the perceived neighborhood of each individual to the individual, the target attraction tensor and 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 contraction ij T ij =g ij (u i (x,t)-target i )(u j (x,t)-target j ) is used to describe the distance between a group and a target; The βth obstacle repulsion tensor is defined as: Its contraction Used to characterize the distance between the group and the βth obstacle; S22: The effect of the target is superimposed on the effect of the obstacle, and the Riemann metric functional corresponding to the external environment is obtained as follows: The variational equation of the environmental manifold corresponding to the Riemann metric is obtained as: Right now: S23: The above variation for any δg ij Always equal to zero, the differential equations satisfied by the Riemann metric are: The Riemann metric of the environment manifold is obtained by solving the differential equation group satisfied by the Riemann metric.

6. The method for modeling unmanned swarm intelligence collaborative peer behavior based on Riemann manifold according to claim 1, characterized in that: The specific process of step S3 is as follows: S31: Tensors inside and outside the group After superposition, the composite action is obtained and the metric functional is obtained by integration as follows: S32: Perform variation on the metric functional: The above variational formula for any δg ij Always equal to zero, the differential equations satisfied by the Riemann metric are:

7. The method for modeling unmanned swarm intelligence collaborative behavior based on Riemann manifold according to claim 1, characterized in that: The specific process of step S4 is as follows: S41: Select a local neighborhood N for the αth individual according to a certain selection method α (t), N α (t)=O(u(x α ,t),r(v α ,t)).; Among them, u(x α ,t) represents the position of the αth member at time t, r(v α ,t) represents the direction of the individual The perception and radius of action, r(v α ,t) varies smoothly with the direction vector, is a unit vector representing direction; S42: After discretizing the Riemann metric functional, the Riemann metric functional satisfied by the αth individual can be obtained: Among them, the distance tensor between individuals in the neighborhood is: The distance tensor between individuals and obstacles in the neighborhood is: S43: The Riemann metric functional variation of each individual is obtained: The variational solution of the Riemann metric functional is: By δg ij The arbitrariness of , for the αth individual, the equation satisfied by the Riemann metric on its trajectory point is: The equation satisfied by the Riemann metric of each individual is: S44: Solving the equation satisfied by the Riemann metric of each individual to obtain the Riemann metric on the motion path of each individual; S45: For a finite group, the motion of each individual in the composite manifold satisfies the minimum energy principle, and the energy functional of a finite group is constructed: The energy functional variation of the finite group is used to obtain the set of motion equations satisfied by each individual: For each individual, the variation of its motion trajectory satisfies the equation: δu α : According to the equations satisfied by the variation of the motion trajectory, the motion equations satisfied by individuals in the finite group are obtained. Among them, Γ 1 ij (u α ) indicates that the position at time t is u α The Riemann connection corresponding to the individuals of ; S47: Solving the group of motion equations satisfied by the individuals in the finite group to obtain the motion equation of the group.

8. The method for modeling unmanned swarm intelligence collaborative behavior based on Riemann manifold according to claim 1, characterized in that: The specific process of step S5 is as follows: Select the energy-based Lagrangian action as the energy functional of the group: Among them, u(x,t) is the motion trajectory of the group, which is the variational object, and J represents the Jacobian matrix of the coordinate transformation; represents 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 group internal energy term in the Lagrangian action and is taken as: is the external potential energy term in the Lagrangian action, and V(u(x,t)) is taken as: The variational equation of the Lagrangian action is: δL=L(u+δu)-L(u(x,t))=0, Right now: By doing variation, we can get: By equation: The equations satisfied by the group motion are obtained: in, Represents the effect of the external environment on the individual, and the Riemann connection coefficient is: Take the diagonalization metric:

9. The method for modeling unmanned swarm intelligence collaborative behavior based on Riemann manifold according to claim 1, characterized in that: The specific process of step S6 is as follows: The Riemannian metric of the ambient manifold satisfies the functional: At this time, the Riemann metric of the ambient manifold is the critical point of the Riemann metric functional above: The differential equations satisfied by the Riemann metric of the ambient manifold are obtained: According to the movement of individuals in the Riemann manifold, the minimum energy principle is satisfied, and the energy functional variation of the individual movement is obtained to obtain the geodesic equations. The equations are solved to obtain the individual's movement trajectory. The energy functional satisfied by individual motion is: δL(u)=0, By varying δL(u) = 0, we can obtain the following differential equations satisfied by the individual motion trajectory u(x, t):

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