CN113868780B - Unmanned aerial vehicle intensive formation safety envelope construction method - Google Patents
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Abstract
The invention provides a method for constructing an intensive formation safety envelope of an unmanned aerial vehicle, which comprises the following steps: s1: collect all of the data in the formationnFlight state parameters of the unmanned aerial vehicle comprise the maximum length, width and height of the unmanned aerial vehicle body, the maximum forward flight speed, the maximum backward flight speed, the maximum climbing speed, the maximum descending speed and the maximum lateral movement speed; s2: solving the sphere center position and the sphere radius of the minimum enveloping sphere S which can envelop all the spatial position points of the unmanned aerial vehicles in the formation in the space; s3: calculating the maximum flight distance which can be reached by the unmanned aerial vehicle formation in each direction within the given safety response time; s4: solving a diagonal matrix according to the maximum length, width and height of the fuselage, the radius of the minimum envelope ball S and the maximum flight distance in each directionM j (ii) a S5: and constructing an unmanned aerial vehicle safety envelope model. The method provided by the invention comprehensively considers the spatial position distribution, the flight state and the response speed of the unmanned aerial vehicles in the formation, has low model complexity, and can be effectively used for low-altitude conflict prediction and obstacle avoidance planning of the intensive formation formed by a plurality of unmanned aerial vehicles.
Description
Technical Field
The invention relates to the technical field of modeling of safety regions of aircrafts, in particular to a method for constructing an intensive formation safety envelope of unmanned aerial vehicles.
Background
Compared with individual unmanned aerial vehicles, the intensive formation formed by multiple unmanned aerial vehicles occupies a larger flight airspace, and the formation form is dynamically adjusted in real time according to the change of tasks and environments in the flight process. This results in the formation of drones facing a greater probability of flight conflicts, with the flight safety being greatly compromised.
The research on the flight safety area of unmanned aerial vehicle formation is less developed internationally. The existing research mainly focuses on a single unmanned aerial vehicle, and most of the thought is close to the research of a civil aircraft safety region. The concept of maneuvering space is provided in unmanned plane situation awareness research by the Dutch national defense research institute and the Dutch Delft university, and dynamic threat constraints and flight response time are not considered in the research. The united states air force research laboratory has proposed the concept of 3-D collision avoidance uncertainty corridor, and its basic idea is to capture the intention of invading the aircraft and establish a safe airway for the local machine. The unmanned aerial vehicle safety zones in these studies are typically replaced with simple enclosures such as cuboids or cylinders to avoid the complications that arise from using actual aircraft shapes and sizes.
The above-described secure enclave construction method has some disadvantages. Firstly, the modeling object is an individual aircraft which is difficult to be used for the safety situation research of a dense formation formed by a plurality of unmanned aerial vehicles; second, the safety separation standard for aircraft is often used to analyze open airspace and fixed trajectories, while the formation flight routes of drones are variable and may not be in a complex airspace environment sufficient to accommodate such traditional safety zones. Thirdly, collision detection and collision probability calculation of unmanned aerial vehicle formation cannot be accurately performed based on a traditional simple safe region model, an accurate reachable region of unmanned aerial vehicle formation is obtained based on high-fidelity dynamics model calculation, and too high calculation complexity is generated. Therefore, it is necessary to provide a low-order and high-confidence-level security envelope modeling method for intensive formation of unmanned aerial vehicles.
Disclosure of Invention
The method solves the technical problem of how to construct a safe region model of the intensive formation formed by a plurality of unmanned aerial vehicles, comprehensively considers the space position, flight state and response speed of the unmanned aerial vehicles in the formation, has low model complexity, and can be effectively used for low-altitude conflict prediction and obstacle avoidance planning of the unmanned aerial vehicle formation.
In order to solve the technical problem, the invention provides
An unmanned aerial vehicle intensive formation safety envelope construction method mainly comprises the following steps:
the method comprises the following steps: collect all of the data in the formationnFlight state parameters of Unmanned Aerial Vehicle (UAV) including a length of a fuselageL i Great width of machine bodyW i Great at the fuselage heightH i Great forward flying speedv fi Great back flying speedv bi Great vertical climbing speedv ui Great vertical descending speedv di Great side moving speedv li Therein ofiNumbering of drones in a cluster: (1≤i≤ n）；
The maximum value of each group of parameters is obtained to obtain the maximum body length of the whole formationLMaximum fuselage widthWMaximum fuselage heightHAnd the maximum flying speeds in all directions are respectively as follows: maximum forward flight speedV f Maximum rear flying speedV b Maximum vertical climbing speedV u Maximum vertical descent velocityV d Maximum lateral moving speedV l ；
Step two: collect all of the data in the formationnErecting unmanned aerial vehicle in inertial coordinate systemO xyz In (2) a spatial position pointx i ,y i , z i ]Solving to obtain the station in space which can be enveloped and formedMinimum envelope sphere with unmanned aerial vehicle spatial location pointsSSpatial position of the center of sphereXs=[x S ,y S ,z S ]Radius of sphereR；
Step three: setting safe response time of unmanned aerial vehicles in formationτ(ii) a Calculating the safety response time of the unmanned aerial vehicle formation in the given direction by combining the maximum flight speed of the unmanned aerial vehicle formation in each direction acquired in the step oneτWithin range, the maximum flight distance achievable in each direction: in turn, the maximum forward flight distanceF=V f ×τMaximum rear flight distanceB=V b ×τMaximum climbing distanceU=V u ×τMaximum distance of descentD=V d ×τAnd maximum left-right lateral movement distanceE=G=V l ×τ；
Step four: according to the maximum fuselage length of the unmanned aerial vehicle formation acquired in the step oneLMaximum fuselage widthWMaximum fuselage heightHAnd step two, solving the minimum envelope sphere obtainedSRadius of (2)RAnd the maximum flight distance of the unmanned aerial vehicle formation obtained in the step three in all directions is calculated, and a diagonal matrix is constructed
Wherein
RepresentsmLine ofnMatrix space of columns:
in a matrix ofR+F+L/2，R+B+L/2，R+U+H/2，R+D+H/2，R+E+W/2On the premise of comprehensively considering the size of the unmanned aerial vehicle body and the space distribution of the unmanned aerial vehicle formation, the whole unmanned aerial vehicle formation is in safe response timeτThe maximum distance that can be achieved in each spatial direction;
step five: according to the minimum envelope sphere obtained by solving in the step twoSSpatial position of the center of sphereXs=[x S ,y S ,z S ]And the diagonal matrix obtained by solving in the step four
And constructing a safety envelope model for unmanned aerial vehicle formationE (X s )Wherein
Preferably, the number of unmanned aerial vehicles in the formation is 2 and above.
Preferably, in the second step, the minimum envelope sphere is solvedSSpatial position of the center of sphereXs=[x S ,y S ,z S ]Radius of sphereRThe method comprises the following specific steps:
[1]with first unmanned aerial vehicle's spatial position pointX 1 =[x 1 ,y 1 ,z 1 ]And a second unmanned aerial vehicleX 2 = [x 2 ,y 2 ,z 2 ]The distance between the two is the diameter,X 1 andX 2 the midpoint of the connecting line is the center of the sphere to establish an initial sphereS 0 ；
[2]Whether all the other unmanned aerial vehicle space position points are on the ball or not is judged in sequenceS 0 Internal; if it is not goodX i On-ballS 0 Inner (3)≤i≤ n) Skipping to continuously judge the next spatial position point; if it is not goodX i Out of the ballS 0 In the interior, the followingX 1 AndX 2 the distance between the two is the diameter, the midpoint of the connecting line is the sphere center to establish the sphereS 1 ；
[3]Before the sequential judgmentiWhether the point is on the ballS 1 Internal; if it is not goodX j On-ballS 1 Inner layer (A)j<i) Skipping to continuously judge the next spatial position point; if it is not goodX j Out of the ballS 1 Inner, then solveX 1 、X i AndX j a triangle circumscribed circle formed by three points, and a sphere is established by taking the radius of the circle as the radius of the sphere and the center of the circle as the center of the sphereS 2 ；
[4]Before the sequential judgmentjWhether the point is on the ballS 2 Internal; if it is not goodX k On-ballS 2 Inner layer (A)k<j) Skipping to continuously judge the next spatial position point; if it is not goodX k Out of the ballS 2 Inner, then solveX i 、X j AndX k a triangle circumscribed circle formed by three points, and a sphere is established by taking the radius of the circle as the radius of the sphere and the center of the circle as the center of the sphereS 3 ；
[5]Before the sequential judgmentkWhether the point is on the ballS 3 Internal; if it is not goodX l On-ballS 3 Inner layer (A)l<k) Skipping to continuously judge the next spatial position point; if it is not goodX l Out of the ballS 3 Inner, then solveX i 、X j 、X k AndX l minimum external sphere of tetrahedron composed of four pointsS 4 (ii) a Are compared to obtainS 1 、S 2 、S 3 、S 4 Largest ball in (1)S m ；
[6]Will be provided withS m As a new ballS 3 Repeating the step [5 ]]Until beforekAll points are detected; using the finally obtained ballS 3 As a new ballS 2 Heavy and heavyRepeating the step [4]And [5 ]]Until beforejAll points are detected; using the finally obtained ballS 2 As a new ballS 1 Repeating the step [3 ]]、[4]And [5 ]]Until beforeiAll points are detected; using the finally obtained ballS 1 As a new ballS 0 Repeating the step [2 ]]、[3]、[4]And [5 ]]Up to allnAll the spatial position points are detected; the ball finally obtainedS 0 I.e. the minimum enveloping sphere of all the spatial position points of the unmanned aerial vehicles in the formationSOutputting the spatial position of the center of the ballXs=[x S ,y S , z S ]Radius of sphereR。
Compared with the prior art, the invention has the advantages that:
(1) the method for constructing the safety envelope of the dense formation of the unmanned aerial vehicles is suitable for the dense formation formed by a plurality of unmanned aerial vehicles, not only aiming at a single unmanned aerial vehicle, but also approximating the complexity of the model to the safety envelope model of the single unmanned aerial vehicle.
(2) The method for constructing the dense formation safety envelope of the unmanned aerial vehicles has no limit on the number and the spatial positions of the unmanned aerial vehicles in the formation, and is strong in universality and convenient to operate.
Drawings
FIG. 1 is a three-dimensional schematic diagram of a minimum envelope sphere of spatial location points of drones in a formation;
FIG. 2 is a front view of a minimum envelope sphere of spatial location points of drones in a formation;
FIG. 3 is a side view of a minimum envelope sphere of spatial location points of drones in a formation;
FIG. 4 is a top view of a minimum envelope sphere of spatial location points of drones in a formation;
FIG. 5 is a three-dimensional schematic diagram of a safety envelope model of a sample dense formation of unmanned aerial vehicles;
FIG. 6 is a front view of a safety envelope model of a sample dense formation of unmanned aerial vehicles;
FIG. 7 is a side view of a safety envelope model of a sample dense formation of drones;
fig. 8 is a top view of a safety envelope model of a sample dense formation of drones.
Detailed Description
In order to clearly explain the technical scheme and contents of the invention, the invention is further described in detail with reference to the accompanying drawings.
The invention provides a method for constructing the safety envelope of the intensive formation of unmanned aerial vehicles, wherein the established safety envelope is a closed curved surface which is formed by surrounding the minimum envelope sphere center of the formation of the unmanned aerial vehicles and can reach the farthest range in each direction according to the spatial distribution and the flight performance in a certain response time. The construction process mainly comprises the following steps:
the method comprises the following steps: collect all of the data in the formationnFlight state parameters of Unmanned Aerial Vehicle (UAV) including a length of a fuselageL i Great width of machine bodyW i Great at the fuselage heightH i Great forward flying speedv fi Great back flying speedv bi Great vertical climbing speedv ui Great vertical descending speedv di Great side moving speedv li Therein ofiNumbering of drones in a cluster: (1≤i≤ n）；
The maximum value of each group of parameters is obtained to obtain the maximum body length of the whole formationLMaximum fuselage widthWMaximum fuselage heightHAnd the maximum flying speeds in all directions are respectively as follows: maximum forward flight speedV f Maximum rear flying speedV b Maximum vertical climbing speedV u Maximum vertical descent velocityV d Maximum lateral moving speedV l ；
Step two: collect all of the data in the formationnErecting unmanned aerial vehicle in inertial coordinate systemO xyz In (2) a spatial position pointx i ,y i ,z i ]Solving to obtain in spaceWith the smallest envelope sphere enveloping all the spatial location points of the drones in the formationSSpatial position of the center of sphereXs =[x S ,y S ,z S ]Radius of sphereRThe method comprises the following specific steps:
[1]with first unmanned aerial vehicle's spatial position pointX 1 =[x 1 ,y 1 ,z 1 ]And a second unmanned aerial vehicleX 2 = [x 2 ,y 2 ,z 2 ]The distance between the two is the diameter,X 1 andX 2 the midpoint of the connecting line is the center of the sphere to establish an initial sphereS 0 ；
[2]Whether all the other unmanned aerial vehicle space position points are on the ball or not is judged in sequenceS 0 Internal; if it is not goodX i On-ballS 0 Inner (3)≤i≤ n) Skipping to continuously judge the next spatial position point; if it is not goodX i Out of the ballS 0 In the interior, the followingX 1 AndX 2 the distance between the two is the diameter, the midpoint of the connecting line is the sphere center to establish the sphereS 1 ；
[3]Before the sequential judgmentiWhether the point is on the ballS 1 Internal; if it is not goodX j On-ballS 1 Inner layer (A)j<i) Skipping to continuously judge the next spatial position point; if it is not goodX j Out of the ballS 1 Inner, then solveX 1 、X i AndX j a triangle circumscribed circle formed by three points, and a sphere is established by taking the radius of the circle as the radius of the sphere and the center of the circle as the center of the sphereS 2 ；
[4]Before the sequential judgmentjWhether the point is on the ballS 2 Internal; if it is not goodX k On-ballS 2 Inner layer (A)k<j) Then skip and continue to determine the next space bitPlacing points; if it is not goodX k Out of the ballS 2 Inner, then solveX i 、X j AndX k a triangle circumscribed circle formed by three points, and a sphere is established by taking the radius of the circle as the radius of the sphere and the center of the circle as the center of the sphereS 3 ；
[5]Before the sequential judgmentkWhether the point is on the ballS 3 Internal; if it is not goodX l On-ballS 3 Inner layer (A)l<k) Skipping to continuously judge the next spatial position point; if it is not goodX l Out of the ballS 3 Inner, then solveX i 、X j 、X k AndX l minimum external sphere of tetrahedron composed of four pointsS 4 (ii) a Are compared to obtainS 1 、S 2 、S 3 、S 4 Largest ball in (1)S m ；
[6]Will be provided withS m As a new ballS 3 Repeating the step [5 ]]Until beforekAll points are detected; using the finally obtained ballS 3 As a new ballS 2 Repeating the step [4 ]]And [5 ]]Until beforejAll points are detected; using the finally obtained ballS 2 As a new ballS 1 Repeating the step [3 ]]、[4]And [5 ]]Until beforeiAll points are detected; using the finally obtained ballS 1 As a new ballS 0 Repeating the step [2 ]]、[3]、[4]And [5 ]]Up to allnAll the spatial position points are detected; the ball finally obtainedS 0 I.e. the minimum enveloping sphere of all the spatial position points of the unmanned aerial vehicles in the formationSOutputting the spatial position of the center of the ballXs=[x S ,y S , z S ]Radius of sphereR；
Step three: setting nobody in formationSafe response time of machineτ(ii) a Calculating the safety response time of the unmanned aerial vehicle formation in the given direction by combining the maximum flight speed of the unmanned aerial vehicle formation in each direction acquired in the step oneτWithin range, the maximum flight distance achievable in each direction: in turn, the maximum forward flight distanceF=V f ×τMaximum rear flight distanceB=V b ×τMaximum climbing distanceU=V u ×τMaximum distance of descentD=V d ×τAnd maximum left-right lateral movement distanceE=G=V l ×τ；
Step four: the maximum fuselage length of the unmanned aerial vehicle formation is statistically obtained in the step oneLWidth, widthWHeight, heightHAnd step two, solving the minimum envelope sphere obtainedSRadius of (2)RAnd the maximum flight distance of the unmanned aerial vehicle formation obtained in the step three in all directions is calculated, and a diagonal matrix is constructed
Wherein
RepresentsmLine ofnMatrix space of columns:
in a matrix ofR+F+L/2，R+B+L/2，R+U+H/2，R+D+H/2，R+E+W/2On the premise of comprehensively considering the size of the unmanned aerial vehicle body and the space distribution of the unmanned aerial vehicle formation, the whole unmanned aerial vehicle formation is in safe response timeτThe maximum distance that can be achieved in each spatial direction;
step five: according to the minimum envelope sphere obtained by solving in the step twoSSpatial position of the center of sphereXs=[x S ,y S ,z S ]And the diagonal matrix obtained by solving in the step four
And constructing a safety envelope model for unmanned aerial vehicle formationE (X s )Wherein
The shape of the safety envelope is determined by the position and the size of the minimum envelope circle of the unmanned aerial vehicle formation, the maximum body length, the width and the height of the unmanned aerial vehicle formation and the maximum flying speed of the unmanned aerial vehicle formation in all directions, and the obtained safety response time determines the size of the safety envelope of the unmanned aerial vehicle formation.
Example 1:
the method is adopted to carry out safe envelope modeling on the sample unmanned aerial vehicle dense formation. Assuming that there are 7 drones in the formation, the fuselage sizes and performance parameters of all the drones are the same, and the fuselage lengthL=8mWidth, widthW=16mHeight, heightH=2.4m. Maximum forward flight speedV f =100m/sMaximum rear flying speedV b =0m/sMaximum vertical climbing speedV u =20m/sMaximum vertical descent velocityV d = 25m/sMaximum lateral moving speedV l =10m/s。
If 7 unmanned aerial vehicles center in inertial coordinate systemO xyz Respectively, areX 1 =[520,110,225]，X 2 =[500,130,220]，X 3 =[545,145,225]，X 4 =[510,115,205]，X 5 =[540,135,205]，X 6 =[510,135,235]，X 7 =[530,125,250]. According to the method for solving the minimum envelope circle in the second step, the minimum envelope sphere of all the spatial position points of the unmanned aerial vehicles in the space envelope formation can be obtainedSSpatial position of the center of sphereX s =[525.692,126.412,223.545]Radius of sphereR=26.840m。
FIGS. 1-4 show all 7 nobody frames in a formationThe spatial distribution position of the unmanned aerial vehicle and the minimum enveloping sphere enveloping the 7 unmanned aerial vehicle spatial position pointsS. Fig. 1, fig. 2, fig. 3, and fig. 4 are a three-dimensional view, a front view, a side view, and a top view, respectively.
Response time of formation of unmanned aerial vehicles is taken asτ=0.6s. According to the method provided by the invention, the maximum flight distances which can be reached by the unmanned aerial vehicles in all directions in the range of the given safety response time are sequentially the maximum forward flight distancesF=V f ×τ= 60mMaximum rear flight distanceB=V b ×τ=0mMaximum climbing distanceU=V u ×τ=12mMaximum distance of descentD=V d ×τ=15mAnd maximum left-right lateral movement distanceE=G=V l ×τ=6m. From this, four diagonal matrices are obtained
Are respectively as
Finally, the safety envelope model of the unmanned aerial vehicle intensive formation is
The inner sphere in the views shown in fig. 5-8 is the minimum enveloping sphere of all spatial position points of the unmanned aerial vehicles in the formation, and the outer curved surface is the safety envelope of the intensive formation of the unmanned aerial vehicles calculated according to the application. Fig. 5, 6, 7 and 8 are a three-dimensional view, a front view, a side view and a top view, respectively.
As can be seen from the specific embodiments, the invention has the advantages over the prior art that:
(1) the method for constructing the safety envelope of the dense formation of the unmanned aerial vehicles is suitable for the dense formation formed by a plurality of unmanned aerial vehicles, not only aiming at a single unmanned aerial vehicle, but also approximating the complexity of the model to the safety envelope model of the single unmanned aerial vehicle.
(2) The method for constructing the dense formation safety envelope of the unmanned aerial vehicles has no limit on the number and the spatial positions of the unmanned aerial vehicles in the formation, and is strong in universality and convenient to operate.
The above-described embodiments are merely preferred embodiments of the present invention, which is not intended to limit the present invention in any way. Those skilled in the art can make many changes, modifications, and equivalents to the embodiments of the invention without departing from the scope of the invention as set forth in the claims below. Therefore, the protection scope of the present invention should be covered by the equivalents and changes made according to the spirit of the present invention without departing from the contents of the technical solutions of the present invention.
Claims (3)
1. An unmanned aerial vehicle intensive formation safety envelope construction method is characterized in that the construction process mainly comprises the following steps:
the method comprises the following steps: collect all of the data in the formationnFlight state parameters of Unmanned Aerial Vehicle (UAV) including a length of a fuselageL i Great width of machine bodyW i Great at the fuselage heightH i Great forward flying speedv fi Great back flying speedv bi Great vertical climbing speedv ui Great vertical descending speedv di Great side moving speedv li Therein ofiNumbering for unmanned aerial vehicles in the cluster, which is not more than 1i≤n；
Taking the maximum value of the flight state parameters to obtain the maximum fuselage length of the whole formationLMaximum fuselage widthWMaximum fuselage heightHAnd the maximum flying speeds in all directions are respectively as follows: maximum forward flight speedV f Maximum rear flying speedV b Maximum vertical climbing speedV u Maximum vertical descent velocityV d Maximum lateral moving speedV l ；
Step two: collect all of the data in the formationnErecting unmanned aerial vehicle in inertial coordinate systemO xyz In (2) a spatial position pointx i ,y i ,z i ]And solving to obtain the minimum enveloping sphere capable of enveloping all the space position points of the unmanned aerial vehicles in the formation in the spaceSSpatial position of the center of sphereXs= [x S ,y S ,z S ]Radius of sphereR；
Step three: setting safe response time of unmanned aerial vehicles in formationτ(ii) a Calculating the safety response time of the unmanned aerial vehicle formation in the given direction by combining the maximum flight speed of the unmanned aerial vehicle formation in each direction acquired in the step oneτWithin range, the maximum flight distance achievable in each direction: in turn, the maximum forward flight distanceF=V f ×τMaximum rear flight distanceB=V b ×τMaximum climbing distanceU= V u ×τMaximum distance of descentD=V d ×τAnd maximum left-right lateral movement distanceE=G=V l ×τ；
Step four: according to the maximum fuselage length of the unmanned aerial vehicle formation acquired in the step oneLMaximum fuselage widthWMaximum fuselage heightHAnd step two, solving the minimum envelope sphere obtainedSRadius of (2)RAnd the maximum flight distance of the unmanned aerial vehicle formation obtained in the step three in all directions is calculated, and a diagonal matrix is constructed
Wherein
RepresentsmLine ofnMatrix space of columns:
in a matrix ofR+F+L/2，R+B+L/2，R+U+H/2，R+D+H/2，R+E+W/2On the premise of comprehensively considering the size of the unmanned aerial vehicle body and the space distribution of the unmanned aerial vehicle formation, the whole unmanned aerial vehicle formation is in safe response timeτThe maximum distance that can be achieved in each spatial direction;
step five: according to the minimum envelope sphere obtained by solving in the step twoSSpatial position of the center of sphereXs=[x S ,y S ,z S ]And the diagonal matrix obtained by solving in the step four
And constructing a safety envelope model for unmanned aerial vehicle formationE(X s )Wherein
2. The method for constructing the safety envelope of the dense formation of unmanned aerial vehicles according to claim 1, wherein: unmanned aerial vehicle quantity is 2 and above in the formation.
3. The method for constructing the safety envelope of the dense formation of unmanned aerial vehicles according to any one of claims 1-2, wherein: in the second step, the minimum envelope sphere is solvedSSpatial position of the center of sphereXs=[x S ,y S ,z S ]Radius of sphereRThe method comprises the following specific steps:
[1]with first unmanned aerial vehicle's spatial position pointX 1 =[x 1 ,y 1 ,z 1 ]And a second unmanned aerial vehicleX 2 =[x 2 , y 2 ,z 2 ]The distance between the two is the diameter,X 1 andX 2 the midpoint of the connecting line is the center of the sphere to establish an initial sphereS 0 ；
[2]Whether all the other unmanned aerial vehicle space position points are on the ball or not is judged in sequenceS 0 Internal; if it is not goodX i On-ballS 0 In the interior of said container body,3≤i≤nskipping to continuously judge the next spatial position point; if it is not goodX i Out of the ballS 0 In the interior, the followingX 1 AndX 2 the distance between the two is the diameter, the midpoint of the connecting line is the sphere center to establish the sphereS 1 ；
[3]Before the sequential judgmentiWhether the point is on the ballS 1 Internal; if it is not goodX j On-ballS 1 In the interior of said container body,j<iskipping to continuously judge the next spatial position point; if it is not goodX j Out of the ballS 1 Inner, then solveX 1 、X i AndX j a triangle circumscribed circle formed by three points, and a sphere is established by taking the radius of the circle as the radius of the sphere and the center of the circle as the center of the sphereS 2 ；
[4]Before the sequential judgmentjWhether the point is on the ballS 2 Internal; if it is not goodX k On-ballS 2 In the interior of said container body,k<jskipping to continuously judge the next spatial position point; if it is not goodX k Out of the ballS 2 Inner, then solveX i 、X j AndX k a triangle circumscribed circle formed by three points, and a sphere is established by taking the radius of the circle as the radius of the sphere and the center of the circle as the center of the sphereS 3 ；
[5]Before the sequential judgmentkWhether the point is on the ballS 3 Internal; if it is not goodX l On-ballS 3 In the interior of said container body,l<kskipping to continuously judge the next spatial position point; if it is not goodX l Out of the ballS 3 Inner, then solveX i 、X j 、X k AndX l minimum external sphere of tetrahedron composed of four pointsS 4 (ii) a Are compared to obtainS 1 、S 2 、S 3 、S 4 Largest ball in (1)S m ；
[6]Will be provided withS m As a new ballS 3 Repeating the step [5 ]]Until beforekAll points are detected; using the finally obtained ballS 3 As a new ballS 2 Repeating the step [4 ]]And [5 ]]Until beforejAll points are detected; using the finally obtained ballS 2 As a new ballS 1 Repeating the step [3 ]]、[4]And [5 ]]Until beforeiAll points are detected; using the finally obtained ballS 1 As a new ballS 0 Repeating the step [2 ]]、[3]、[4]And [5 ]]Up to allnAll the spatial position points are detected; the ball finally obtainedS 0 I.e. the minimum enveloping sphere of all the spatial position points of the unmanned aerial vehicles in the formationSOutputting the spatial position of the center of the ballXs=[x S ,y S ,z S ]Radius of sphereR。
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