CN114610065A - Graph theory-based cluster unmanned aerial vehicle formation flight trajectory optimization method - Google Patents
Graph theory-based cluster unmanned aerial vehicle formation flight trajectory optimization method Download PDFInfo
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Abstract
The invention discloses a graph theory-based cluster unmanned aerial vehicle formation flight track optimization method, which is characterized in that a differentiable formation similarity index based on the graph theory and a track optimization framework considering formation and obstacle avoidance simultaneously can realize a more stable formation keeping effect by means of effective overall geometric topological representation, and meanwhile, the similarity index also has invariance of translational rotation and size scaling, so that unmanned aerial vehicles can more flexibly deal with obstacles by rotating or scaling the overall formation when forming and flying in a complex scene, the success rate of the optimization-based method in the problems is greatly improved, and the method has great engineering application significance.
Description
Technical Field
The invention relates to the technical field of unmanned aerial vehicle navigation, in particular to a method for optimizing formation flight trajectories of cluster unmanned aerial vehicles based on graph theory.
Background
In recent years, with the increasing maturity of autonomous navigation technology and the high-speed development of embedded devices, unmanned aerial vehicles are deployed and applied in a large number in the civil and military fields. However, the single unmanned aerial vehicle still has the defects of small single load, low system fault tolerance rate and the like when executing tasks, and the cluster unmanned aerial vehicle can more robustly complete established tasks by increasing the system redundancy. Meanwhile, for scenes needing multi-individual cooperation, such as rapid search and rescue, intelligent logistics transportation, cooperative mapping and the like, the cluster unmanned aerial vehicle can provide a more systematic solution. As an important direction for the research of the cluster unmanned aerial vehicle, the problem of formation flight of multiple unmanned aerial vehicles is a key research field for improving the synergy and the scene adaptability of the cluster system.
When the unmanned aerial vehicle cluster carries out a complex formation navigation task, at least the following four points are required to be ensured: a. unmanned aerial vehicle flight safety in complex scenes; b. maintaining the expected formation during the flight process; c. safe flight intervals among all unmanned aerial vehicles in the cluster; d. the dynamics of the unmanned aerial vehicle body. For flight safety in a complex scene, an unmanned aerial vehicle is required to perform online sensing and construction on a surrounding environment map to obtain obstacle information, and common map types comprise a point cloud map, a Euclidean distance map or a probability grid map. For realizing the flight collision avoidance and the expected formation maintenance among the unmanned aerial vehicles in the cluster, the cluster system needs to be capable of carrying out communication among the unmanned aerial vehicles through a broadcast network, and the main communication content is the current position information and the planned track of each unmanned aerial vehicle. In order to ensure that the planned trajectory can be executed by the drone normally, the dynamic constraints of the body itself should also be met, typically including physical limitations such as maximum motor thrust, maximum flight inclination and maximum roll rate. For the three point requirements of acd, there are systematic solutions in the academic world at present. Literature reference[1]In the Zhejiang university, Zhou et al propose an unmanned aerial vehicle cluster autonomous navigation system, which adopts a time-space trajectory optimization method and can calculate and generate a high-quality trajectory which is safe, has no collision and meets the dynamic limitation within millisecond time by using an airborne computer of an unmanned aerial vehicle.
For the requirement b, much research work is focused on solving cooperative control and planning of unmanned aerial vehicle cluster formation in the current academic world, but most methods can only realize cluster unmanned aerial vehicle formation maintenance in an open scene, and formation flight in a complex environment with obstacles is still a difficult problem of no systematic solution. Because the safety requirement of each unmanned aerial vehicle for avoiding obstacles and the task requirement for maintaining the expected formation are often contradictory in a complex scene, how to systematically balance the two contradictory requirements becomes a core problem for solving the unmanned aerial vehicle formation obstacle avoidance flight. At present, most of formation flight schemes rely on local feedback control laws based on cluster consistency. The method has better convergence on the expected formation in an open scene, but the local feedback control law cannot consider the system state in a time window in the future, and the capability of carrying out advanced coping processing on a multi-obstacle scene is lacked, so that the method is difficult to adapt to the formation flight requirement in a complex environment. In contrast, the formation flight scheme based on the cluster trajectory optimization is more suitable for solving the obstacle avoidance problem frequently encountered in the complex scene. At present, most of track optimization schemes realize the maintenance of geometric formation by applying hard constraints on relative positions of all unmanned aerial vehicles, but the position constraints and collision avoidance constraints are difficult to be simultaneously satisfied, so that the optimization problem is unsolvable. Part of the optimization-based work guarantees the flight safety of the drones by passively abandoning the position constraints of the formation, which can greatly destroy the cooperativity of the drone cluster. In summary, there is a need in the academic world for an unmanned aerial vehicle cluster trajectory planning scheme capable of simultaneously solving formation obstacle avoidance flight in a complex environment.
At present, a plurality of methods are proposed in the field of cluster formation flight, such as a virtual structure method, a navigation function method, a reaction behavior method and a local feedback control method based on a consistency theory. But most of the above methods can only realize the formation flying of unmanned aerial vehicle clusters in an open scene without external obstacles.
In a complex environment, the difficulty of formation flight tasks is greatly improved, mainly because each unmanned aerial vehicle still satisfies the safety of cluster flight while keeping an expected formation. One method is to design a feedback control law to simultaneously realize obstacle avoidance and formation maintenance of the unmanned aerial vehicle. Literature documents[2]Han et al propose a formation flight controller with a complex laplace matrix as a feedback gain, the size of the formation during flight being determined by the pilot (long aircraft), who controls the size of the entire formation to make the cluster perform a specific flight mission, e.g. through a narrow slot. In the literature[3]Zhao proposes a control law based on a navigator-follower frame, and the control law can enable the formation of the unmanned aerial vehicle to perform affine transformation according to the change of the environment. Most feedback control methods adopt a pilot-follower framework, but the mode enables the parameters of the formation to be controlled only by a pilot, and once the pilot fails, the flight of the whole cluster fails.
Compared with the navigator-follower framework, the decentralization strategy can better deal with the condition that some unmanned aerial vehicles in the cluster have faults. Literature reference[4]Among them, Alonso-Mora et al divide the trajectory planning of a cluster drone into two phases, the first phase is to calculate the optimal formation parameters and the corresponding drone-target point assignment relationship according to the current map information, and the second phase is to arrive at the assigned target point by each drone independently performing local trajectory planning according to the calculation results of the first phase. The method can avoid obstacles by the unmanned aerial vehicle cluster, but the real-time performance of formation maintenance is poor because the requirement of formation is not considered in the second stage. Literature reference[5]In the middle, Zhou et al adopts a virtual structure method and combines an artificial potential field method to generate a collision-free formation flight trajectory, but because local optimal points are easily generated in a superimposed artificial potential field, an unmanned aerial vehicle cluster using the method is easy to generate trajectory oscillation and deadlock in the flight process, and meanwhile, the optimality of the overall trajectory cannot be guaranteed. Literature reference[6]In Pary et al, a method using distributed model predictive control is proposed to achieve formation of a cluster for flight, which method maintains a geometric formation primarily by imposing relative position constraints on the drones. In the flight process, once obstacles appear in the environment to cause the requirement of the relative position of the unmanned aerial vehicle to be not met, the framework gives up the requirement of formation cooperationThe safety of flight is satisfied. Although the passive mechanism can ensure obstacle avoidance, the synergy of the cluster system is greatly reduced.
Reference documents:
[1]X.Zhou，Z.Wang，X.Wen，J.Zhu，C.Xu，and F.Gao，“Decentralized spatial－temporal trajectory planning for multicopter swarms，”arXiv preprint arXiv：，2021
[2]Z.Han，L.Wang，and Z.Lin，“Local formation control strategies with undetermined and determined formation scales for co－leader vehicle networks，”in 52nd IEEE Conference on Decision and Control.IEEE，2013，pp.7339–7344.
[3]S.Zhao，“Affifine formation maneuver control of multiagent systems，”IEEE Transactions on Automatic Control， vol.63，no.12，pp.4140–4155，2018.
[4]J.Alonso－Mora，E.Montijano，M.Schwager，and D.Rus，“Distributed multi－robot formation control among obstacles：A geometric and optimization approach with consensus，”in 2016 IEEE international conference on robotics and automation (ICRA).IEEE，2016，pp.5356–5363.
[5]D.Zhou，Z.Wang，and M.Schwager，“Agile coordination and assistive collision avoidance for quadrotor swarms using virtual structures，”IEEE Transactions on Robotics，vol.34， no.4，pp.916–923，2018.
[6]R.Van Parys and G.Pipeleers，“Distributed model predictive formation control with inter－vehicle collision avoidance，”in 2017 11th Asian Control Conference(ASCC).IEEE， 2017，pp.2399–2404.
[7]Z.Wang，X.Zhou，C.Xu，and F.Gao，“Geometrically constrained trajectory optimization for multicopters，”arXiv preprint arXiv：，2021.
[8]M.Turpin，N.Michael，and V.Kumar，“Trajectory design and control for aggressive formation flflight with quadrotors，”Autonomous Robots，vol.33，no.1，pp.143–156， 2012.
disclosure of Invention
Aiming at the defects of the prior art, the invention aims to provide a method for optimizing the formation flight trajectory of the cluster unmanned aerial vehicle based on graph theory.
In order to achieve the purpose, the invention adopts the following technical scheme:
a method for optimizing formation flight trajectories of cluster unmanned aerial vehicles based on graph theory mainly comprises the following steps:
s1, modeling the cluster composed of N unmanned aerial vehicles and calculating the formation similarity index and the gradient thereof, wherein the specific process is as follows:
s1.1, modeling the topological relation of clusters consisting of N unmanned planes into a full-connection undirected graph G (V, E) according to the current formation form of the clusters, wherein V (1, 2.. multidot.N) represents a set of vertexes,
a set of representative edges; in graph G, vertex i represents the position coordinate p of the ith droneiIf and only if the ith unmanned aerial vehicle and the jth unmanned aerial vehicle can acquire mutual position information through network communication, the edge e connecting the vertex i and the vertex jijIs present;
s1.2, and regarding the edge e connecting the vertex i and the vertex j in the undirected graph constructed in the step S1.1ijGiving non-negative weight wijObtaining a weighted undirected graph;
s1.3, calculating a characterization matrix of the weighted undirected graph obtained in the step S1.2, wherein the characterization matrix comprises an adjacency matrix A, a degree matrix D and a Laplace matrix L; the calculation formula of the laplace matrix L is:
L＝D-A
the Laplace matrix L is normalized through the transition matrix D to obtain a symmetrical standard Laplace matrix S, and the formula is
S＝D-1/2LD-1/2＝I-D-1/2AD-1/2；
I is an identity matrix;
s1.4, specified for the taskThe desired formation form is also subjected to the processing of steps S1.1-S1.3 to obtain an adjacent matrix A of the desired formation formdesLaplace matrix LdesAnd a symmetric canonical Laplace matrix Sdes；
S1.5, calculating the similarity difference between the current formation form and the expected formation form; the similarity index of the formation is defined as follows:
for the ith unmanned aerial vehicle, constructing an adjacent weight vector of the ith unmanned aerial vehicle, writing an edge weight value adjacent to the node of the ith unmanned aerial vehicle into the vector to obtain the vector
wi＝[wi1，...，wij，...，win]T
The similarity index JfTo the ith drone position piThe gradient of (a) is calculated as:
wherein the similarity index JfThe gradient of the weights for the edges in graph G is calculated by:
then, the weight w is calculated according to the edge weight functionijPosition p of opposite vertexiThe gradient of the order of the final formation phase can be obtained by using the chain ruleSimilarity index JfFor ith drone position piA gradient of (a);
s2, after similarity indexes and gradients of the current formation form and the expected formation form are obtained through calculation, substituting the similarity indexes and the gradients into an optimization framework to solve to obtain a final high-quality unmanned aerial vehicle track, wherein the specific process is as follows:
s2.1, adopting a polynomial locus type MINCO with minimum control input to construct a time-space parameterized unmanned aerial vehicle flat output locus and using the time-space parameterized unmanned aerial vehicle flat output locus as an optimized object; this trajectory class characterizes the trajectory of the unmanned aerial vehicle being optimized by means of parameters c and T, where vector c represents the trajectory coefficients of the piecewise polynomial and vector T characterizes the temporal distribution of the piecewise trajectory; on the basis of the MINCO track representation, the track generation of formation flight is constructed into an unconstrained optimization problem:
j with subscripts represents a certain punishment item, subscripts { e, t, o, f, r, d } respectively represent an energy cost item, a total track time item, a safety obstacle avoidance constraint item, a formation similarity item, an inter-cluster mutual collision avoidance item and a dynamic feasibility item, and lambda represents weight;
s2.2, generating the ESDF in the current environment, acquiring the information of the adjacent obstacles and the corresponding numerical gradient, and constructing a safety obstacle avoidance constraint item Jo；JfThat is, the similarity index obtained in step S1; mutual collision avoidance item J between clusters is constructed by utilizing mutual position information obtained according to communication between unmanned aerial vehiclesr(ii) a Constructing a remaining penalty item J according to flight task parameters and unmanned aerial vehicle body parameterse，JtAnd Jd；
And S2.3, performing iterative optimization on the optimization problem in the step S2.1 by using an optimization algorithm based on gradient information, and finally generating a safe and collision-free high-quality track capable of maintaining the formation for the current unmanned aerial vehicle.
Further, in step S1.2, for edge e connecting vertex i and vertex jijWeight w ofijIs oneThe non-negative function of Euclidean distance between two unmanned aerial vehicles, namely:
wij＝||pi-pj||2
piposition coordinates for the ith drone, pjIs the position coordinate of the jth drone.
The invention has the beneficial effects that: compared with other methods at present, the differentiable formation form similarity index based on graph theory and the track optimization framework considering formation and obstacle avoidance simultaneously in the method can not only realize a more stable formation form keeping effect by means of effective overall geometric topological representation, but also have invariance of translational rotation and size scaling, so that the unmanned aerial vehicle can flexibly deal with the obstacle by rotating or scaling the overall formation form when the unmanned aerial vehicle is formed and flown in a complex scene, the success rate of the optimization-based method in the problems is greatly improved, and the method has great engineering application significance.
Drawings
Fig. 1 is a schematic diagram illustrating modeling of cluster formation of unmanned aerial vehicles in embodiment 1 of the present invention;
FIG. 2 is an exemplary diagram of similarity indicators and gradient fields thereof for formation forms in embodiment 1 of the present invention;
fig. 3 is a comparison graph of formation flight effects in the same scene of the methods in embodiment 2 of the present invention.
Detailed Description
The present invention will be further described with reference to the accompanying drawings, and it should be noted that the present embodiment is based on the technical scheme, and a detailed implementation manner and a specific operation process are provided, but the protection scope of the present invention is not limited to the present embodiment.
Example 1
The embodiment provides a graph theory-based clustered unmanned aerial vehicle formation flight trajectory optimization method, which is used for generating a trajectory of clustered unmanned aerial vehicle formation flight in a complex environment. The method mainly comprises the following steps:
firstly, initializing the starting position and the end position of a cluster unmanned aerial vehicle, setting an expected cluster geometric flight formation, and representing the topological relation of the expected formation by using graph theory knowledge;
in the flight process, each unmanned aerial vehicle in the cluster determines the current positions and local tracks of other unmanned aerial vehicles through a broadcast network, and further determines the formation formed by the current cluster and the representation of a topological graph thereof; generating an Euclidean Distance Field (ESDF) of the current environment through an airborne sensor, wherein the ESDF records the Distance value from any point in space to the nearest obstacle, and meanwhile, a first derivative value which can be used for numerical optimization can be obtained according to the change of the Distance value of the obstacle in the neighborhood of the indexed point;
an optimization algorithm is operated on airborne equipment of the unmanned aerial vehicle, meanwhile, the energy cost, the track time, the similarity of formation formations, the dynamics feasibility of the track and the safety obstacle avoidance constraint of the track of the unmanned aerial vehicle are considered, the track is generated and constructed into an optimization problem, and finally, a safe and collision-free high-quality track capable of maintaining the formation formations is subjected to iterative optimization and is used for the unmanned aerial vehicle to execute.
The present embodiment employs a polynomial locus class (mioc) of minimum control input defined in document [7] to construct a time-space parameterized drone flat output locus and to serve as an optimization target. This trajectory class characterizes the trajectory of the drone being optimized by means of parameters c and T, where vector c represents the trajectory coefficients of the piecewise polynomial and vector T characterizes the temporal distribution of the piecewise trajectory. On the basis of the MINCO track representation, the method constructs the track generation of formation flight into an unconstrained optimization problem
And c and T are MINCO parameters, namely optimization variables, J with subscripts represents a certain penalty term and weight thereof, subscripts { e, T, o, f, r and d } respectively represent an energy cost term, a total track time term, a safety obstacle avoidance constraint term, a formation similarity term, a mutual collision avoidance term among clusters and a dynamic feasibility term, and lambda represents the weight. As the method of the embodiment mainly provides the geometric team based on the graph theoryThe form similarity index and its model structure are described in the followingfThe structural mode of (1).
For a cluster consisting of N unmanned aerial vehicles, the method of the embodiment models the cluster and calculates the formation similarity index and the gradient thereof, and the specific process is as follows:
step 1: as shown in fig. 1, for a cluster composed of N drones, according to the current formation form thereof, the method of this embodiment models the mutual topological relationship thereof as a fully-connected undirected graph G ═ (V, E), where V: n represents a set of vertices,
representing a collection of edges. In G, vertex i represents the position coordinate p of the ith droneiIf and only if the ith unmanned aerial vehicle and the jth unmanned aerial vehicle can acquire mutual position information through network communication, the edge e connecting the vertex i and the vertex jijIs present. The geometry of the formation of drones is thus modeled as an undirected graph.
Step 2: endowing non-negative weight to the edge of the undirected graph constructed in the step 1 to obtain a weighted undirected graph; for edge e connecting vertex i and vertex jijWeight w ofijFor the nonnegative function of the euclidean distance between two drones, it is set to:
wij＝||pi-pj||2
and 3, step 3: and (3) calculating a characterization matrix of the weighted undirected graph obtained in the step 2, wherein the characterization matrix comprises an adjacency matrix A, a degree matrix D and a Laplace matrix L. The formula for the calculation of the Laplace matrix L is
Normalizing the Laplace matrix L through the degree matrix D of the graph to obtain a symmetrical standard Laplace matrix S, wherein the formula is
S＝D-1/2LD-1/2＝I-D-1/2AD-1/2.
I is an identity matrix;
and 4, step 4: processing the expected formation designated by the task in steps 1-3 to obtain the expectationAdjacency matrix A of formationdesLaplace matrix L, Laplace matrix LdesAnd a symmetric canonical Laplace matrix Sdes。
And 5, step 5: calculating the similarity difference between the current formation form and the formation form expected by the task; the similarity index proposed in this embodiment is defined as follows
The formation similarity index is a similarity penalty item J in the optimization problemf. The index quantitatively evaluates the difference degree between two geometric formations, and when the two formations are the same, the term value is zero, and the larger the difference between the two geometric formations is, the larger the term value is. It is noted that the similarity indicator J is provided in the present embodimentfThere are three very good properties. The first point is that it has the invariance of translational rotation, i.e. when the formation of the cluster performs a global translation or rotation in space, JfThe size does not change because the graph weight in step 2 is assigned as a function of euclidean distance, which can reasonably extract geometric representations based on relative distance rather than absolute position. The second point is that the index has size scaling invariance, because the geometric shape of the formation is not affected by the scaling of the cluster formation in equal proportion to the size, the embodiment achieves the effect of scaling invariance by normalizing the Laplace matrix by the utilization matrix. The third point is that the indicator is non-negative and differentiable, i.e. JfThe gradient values for the positions p of the individual drones in the cluster are analytically calculable. This property ensures that the optimization framework can perform gradient information-based optimization on the index, thereby generating an unmanned aerial vehicle flight trajectory capable of maintaining formation. For the ith unmanned aerial vehicle, constructing an adjacent weight vector of the ith unmanned aerial vehicle, writing an edge weight value adjacent to the unmanned aerial vehicle node as the vector, and obtaining:
wi＝[wi1，...，wij，...，win]T
then the index JfTo the ith drone position piCan be controlled byThe following formula is calculated:
wherein the gradient of the index to the graph edge weight is calculated by the following formula:
wherein the gradients of each part are respectively as follows:
then, the weight w is calculated according to the edge weight functionijPosition p of opposite vertexiThe gradient of the order can be used to obtain the final formation similarity index J by using the chain rulefFor ith drone position piOf the gradient of (c). Fig. 2 shows the changing effect of the formation similarity index caused by the position change of one of the drones when four drones form a square formation, and the corresponding gradient direction.
After similarity indexes and gradients of the current formation and the expected formation are obtained through calculation, the similarity indexes and the gradients are substituted into an optimization framework to solve to obtain a final high-quality unmanned aerial vehicle track, and the specific process is as follows:
step 1: generating ESDF in the current environment, acquiring the information of adjacent obstacles and corresponding numerical gradients, and constructing a safe obstacle avoidance constraint item Jo(ii) a The similarity item J of formation is constructed between unmanned aerial vehicles according to mutual position information obtained by communicationfAnd mutual collision avoidance items J among clustersr(ii) a Constructing a remaining penalty item J according to flight task parameters and unmanned aerial vehicle body parameterse，JtAnd JdExcept for JfOther thanConstruction of penalty terms expression referable [1]. Then constructing a final optimization problem
Step 2: and (3) performing iterative optimization on the optimization problem in the step 1 by using an optimization algorithm based on gradient information, and finally generating a safe and collision-free high-quality track capable of maintaining the formation for the current unmanned aerial vehicle.
Compared with other methods at present, the differentiable formation queue shape similarity index based on graph theory and the track optimization framework considering formation and obstacle avoidance simultaneously in the method can not only realize a more stable queue shape keeping effect by means of effective overall geometric topological representation, but also have invariance of translational rotation and size scaling, so that the unmanned aerial vehicle can flexibly deal with the obstacle by rotating or scaling the overall queue shape when the unmanned aerial vehicle is formed and flies in a complex scene, the success rate of the optimization-based method in the problems is greatly improved, and the method has great engineering application significance.
Example 2
In this embodiment, the method of embodiment 1 is compared with two currently best cluster formation flying methods in the industry to demonstrate the high efficiency and robustness of the method of embodiment 1. The two methods of comparison are respectively literature[5]Zhou method and literature[8]The Turpin method of (1). In the embodiment, formation flight experiments are carried out on the methods in three scenes with different barrier densities, and flight experiment results of the methods are subjected to the formation absolute position error edistSimilarity index e of formationsimAnd the success rate of the flight mission is compared in three dimensions, and the experimental data are shown in table 1.
TABLE 1
As can be seen from table 1, the method proposed in example 1 is superior to other methods in all respects. In terms of success rate, the method of example 1 can achieve a high success rate of 95% even in a scene with dense obstacles. In terms of the effect of formation keeping, under three density scenes, the method of embodiment 1 is no matter the absolute position error e of the formationdistOr the formation similarity index esimBoth methods are smaller than the other two methods, which shows that the method of embodiment 1 can make the cluster formation more stable.
Fig. 3 shows the comparison of the formation retention effect of the three methods in the same scene during the cluster flight, wherein (a) is the Zhou method, (b) is the Turpin method, and (c) is the method of example 1. It can be seen visually that the flight path generated by the method of embodiment 1 has the best formation keeping effect, and the minimum formation absolute position error e is obtaineddist。
Various corresponding changes and modifications can be made by those skilled in the art based on the above technical solutions and concepts, and all such changes and modifications should be included in the protection scope of the present invention.
Claims (2)
1. A method for optimizing formation flight trajectories of cluster unmanned aerial vehicles based on graph theory is characterized by mainly comprising the following steps:
s1, modeling the cluster composed of N unmanned aerial vehicles and calculating the formation similarity index and the gradient thereof, wherein the specific process is as follows:
s1.1, modeling the topological relation of clusters consisting of N unmanned planes into a full-connection undirected graph G (V, E) according to the current formation form of the clusters, wherein V (1, 2.. multidot.N) represents a set of vertexes,
a set of representative edges; in graph G, the vertex i represents the ithPosition coordinate p of individual unmanned aerial vehicleiIf and only if the ith unmanned aerial vehicle and the jth unmanned aerial vehicle can acquire mutual position information through network communication, the edge e connecting the vertex i and the vertex jijIs present;
s1.2, and regarding the edge e connecting the vertex i and the vertex j in the undirected graph constructed in the step S1.1ijGiving non-negative weight wijObtaining a weighted undirected graph;
s1.3, calculating a characterization matrix of the weighted undirected graph obtained in the step S1.2, wherein the characterization matrix comprises an adjacency matrix A, a degree matrix D and a Laplace matrix L; the calculation formula of the laplace matrix L is:
L＝D-A
normalizing the Laplace matrix L through the transition matrix D to obtain a symmetrical standard Laplace matrix S, wherein the formula is that S is equal to D-1/2LD-1/2＝I-D-1/2AD-1/2；
I is an identity matrix;
s1.4, the expected formation form designated by the task is also processed in the steps S1.1-S1.3, and the adjacent matrix A of the expected formation form is obtaineddesLaplace matrix LdesAnd a symmetric canonical Laplace matrix Sdes；
S1.5, calculating the similarity difference between the current formation form and the expected formation form; the similarity index of the formation is defined as follows:
for the ith unmanned aerial vehicle, constructing an adjacent weight vector of the ith unmanned aerial vehicle, writing an edge weight value adjacent to the ith unmanned aerial vehicle node into the vector to obtain the adjacent weight vector
wi＝[ωi1，...，ωij，...，win]T
The similarity index JfTo the ith drone position piThe gradient of (a) is calculated as:
wherein the similarity index JfThe gradient of the weights for the edges in graph G is calculated by:
then, the weight w is calculated according to the edge weight functionijPosition p of opposite vertexiThe gradient of the order can be used to obtain the final formation similarity index J by using the chain rulefFor ith drone position piA gradient of (a);
s2, after similarity indexes and gradients of the current formation form and the expected formation form are obtained through calculation, substituting the similarity indexes and the gradients into an optimization framework to solve to obtain a final high-quality unmanned aerial vehicle track, wherein the specific process is as follows:
s2.1, adopting a polynomial locus class MINC0 with minimum control input to construct a time-space parameterized unmanned aerial vehicle flat output locus and using the time-space parameterized unmanned aerial vehicle flat output locus as an optimization object; this trajectory class characterizes the trajectory of the unmanned aerial vehicle being optimized by means of parameters c and T, where vector c represents the trajectory coefficients of the piecewise polynomial and vector T characterizes the temporal distribution of the piecewise trajectory; on the basis of the MINC0 track representation, the track generation of formation flight is constructed into an unconstrained optimization problem:
j with subscripts represents a certain punishment item, subscripts { e, t, o, f, r, d } respectively represent an energy cost item, a total track time item, a safety obstacle avoidance constraint item, a formation similarity item, an inter-cluster mutual collision avoidance item and a dynamic feasibility item, and lambda represents weight;
s2.2, generating the ESDF in the current environment, acquiring the information of the adjacent obstacles and the corresponding numerical gradient, and constructing a safety obstacle avoidance constraint item Jo；JfThat is, the similarity index obtained in step S1; mutual collision avoidance item J between clusters is constructed by utilizing mutual position information obtained according to communication between unmanned aerial vehiclesr(ii) a Constructing a remaining penalty item J according to flight task parameters and unmanned aerial vehicle body parameterse，JtAnd Jd；
And S2.3, performing iterative optimization on the optimization problem in the step S2.1 by using an optimization algorithm based on gradient information, and finally generating a safe and collision-free high-quality track capable of maintaining the formation for the current unmanned aerial vehicle.
2. Method according to claim 1, characterized in that in step S1.2, for an edge e connecting vertex i and vertex jijWeight w ofijIs a non-negative function of the Euclidean distance between two unmanned planes, namely:
ωij＝||pi-pj||2
piposition coordinates for the ith drone, pjIs the position coordinate of the jth drone.
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