An aircraft off-line trajectory planning method based on spherical anya

Abstract

The application discloses a kind of aircraft off-line trajectory planning methods based on spherical Anya, comprising the following steps: step 1: the mathematical modeling of aircraft flyable area and threat area is carried out, and the Web Mercator coordinate of flyable area and the Web Mercator coordinate of threat area are obtained;Step 2: the mathematical modeling of any two points great circle route, and the mathematical modeling of great circle route and the intersection coordinates of certain meridian or parallel;Step 3: spherical Anya algorithm heuristic value f is designed using great circle route and the intersection coordinates of great circle route and certain meridian or parallel;Step 4: initialize map, input the Web Mercator coordinate of starting point and terminal point, and the Web Mercator coordinate of flyable area and the Web Mercator coordinate of threat area between starting point and terminal point, obtain the optimal path of spherical Anya using spherical Anya algorithm heuristic value f.

Description

Technical Field

[0001] This invention belongs to the field of trajectory planning application technology, specifically relating to an offline trajectory planning method for aircraft based on spherical Anya, and particularly to an offline trajectory planning method for long-distance, long-endurance unmanned aerial vehicles. Background Technology

[0002] Currently, offline trajectory planning for aircraft uses the classic A× pathfinding algorithm, where each node is represented by two-dimensional coordinates. When searching for successor nodes, it adopts an "8-neighbor system" (up, down, left, right, upper left, lower left, upper right, and lower right), so its pathfinding angle is essentially only a multiple of 45°, which shows that it cannot achieve arbitrary angles.

[0003] In existing trajectory planning algorithms, path optimization in Euclidean space is a commonly used strategy. However, for long-endurance navigation on the Earth's surface or in outer space, geometry approximating Euclidean space is unreliable for current navigation systems such as spacecraft, aircraft, long-endurance UAVs, and ships. Therefore, it is crucial to find a path optimization algorithm in non-Euclidean space that calculates the shortest path between two points on the Earth's surface at arbitrary angles. Based on this, this patent proposes an offline trajectory planning algorithm for aircraft based on the spherical Anya surface to solve the trajectory planning problem for long-endurance aircraft flying on the Earth's surface. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide an offline trajectory planning method for aircraft based on spherical Anya.

[0005] To solve the technical problem, the technical solution of this invention is: an offline trajectory planning method for aircraft based on spherical Anya, comprising the following steps:

[0006] Step 1: Perform mathematical modeling of the aircraft's flyable area and threat area to obtain the Web Mercator coordinates of the flyable area and the threat area;

[0007] Step 2: Mathematical modeling of the great circle route between any two points, and mathematical modeling of the coordinates of the intersection point of the great circle route with a certain meridian or parallel;

[0008] Step 3: Design the heuristic values ​​for the spherical Anya algorithm using the great circle route and the coordinates of the intersection points of the great circle route with a certain meridian or parallel. ;

[0009] Step 4: Initialize the map. Input the Web Mercator coordinates of the start and end points, as well as the Web Mercator coordinates of the flyable area and the threat area between the start and end points. Use the Anya spherical algorithm heuristics. Obtain the optimal path for the spherical Anya.

[0010] Preferably, the mathematical modeling of the aircraft's flyable area and threat area in step 1 specifically involves:

[0011] Step 1-1: Establish the transformation formula from WGS84 coordinates to Web Mercator coordinates;

[0012] Steps 1-2: Convert the threat area in WGS84 coordinates to the threat area in Web Mercator coordinates;

[0013] Steps 1-3: Convert the flyable area in WGS84 coordinates to the flyable area in Web Mercator coordinates.

[0014] Preferably, the conversion formula in step 1-1 is:

[0015]

[0016]

[0017] In the formula:

[0018] for Axis, the latitude of the Web Mercator coordinate system. ;

[0019] for Axis, longitude in Web Mercator coordinates. ;

[0020] The radius of the Earth is in meters (m).

[0021] Longitude, degrees, in WGS84 coordinates;

[0022] The coordinates are in latitude, degrees, of WGS84.

[0023] Let be the longitude, in degrees, of any central meridian.

[0024] Preferably, the conversion of the threat area in WGS84 coordinates to the threat area in Web Mercator coordinates in steps 1-2 specifically involves: converting the latitude and longitude coordinates of the center point of the threat area in WGS84 coordinates to the latitude and longitude coordinates of the center point of the threat area in Web Mercator coordinates; calculating the latitude and longitude coordinates of the boundary point of the threat area circle based on the known latitude and longitude coordinates and radius of the center point of the threat area; connecting these coordinates sequentially to form the initial threat area; and then expanding the initial threat area to the area bounded by the two meridians and parallels tangent to it. This area appears as a rectangle in Web Mercator coordinates, thus obtaining the threat area in Web Mercator coordinates.

[0025] Preferably, the mathematical modeling of the great circle route between any two points in step 2 is specifically as follows:

[0026] Assume the starting coordinates of any two points are The endpoint coordinates are ,

[0027]

[0028] In the formula:

[0029] The length of the great circle route is in meters (m).

[0030] Let be the Earth's radius, in meters (m).

[0031] Preferably, the mathematical modeling of the coordinates of the intersection point between the great circle route and a certain meridian or parallel in step 2 is specifically as follows:

[0032] Assuming the starting point is The coordinates are The destination is The coordinates are ;

[0033] The longitude of the intersection point of the great circle route and a certain meridian is: latitude of the intersection The calculation formula is:

[0034]

[0035] definition ;

[0036]

[0037]

[0038]

[0039] in:

[0040] ;

[0041] ;

[0042] In the formula:

[0043] Let be the Earth's radius, in meters (m).

[0044] Preferably, the latitude of the intersection point of the great circle route and a certain parallel is... Longitude of the intersection point The calculation formula is:

[0045]

[0046] definition ;

[0047]

[0048]

[0049]

[0050]

[0051] in:

[0052] ;

[0053] ;

[0054] ;

[0055] In the formula:

[0056] Let be the Earth's radius, in meters (m).

[0057] Preferably, the heuristic value of the Anya algorithm for spherical surfaces in step 3 is... The design formula is:

[0058]

[0059] in:

[0060] The sum of the great circle segments from the starting point to the left and right turning points of the current root node;

[0061] The great circle segment from the current root node to the end point needs to be calculated based on the type of the current node;

[0062] Let the starting point of the pathfinding be The destination is The search direction is defined as starting from... Pointing to the interval , The classification calculation method is as follows:

[0063] (1) If the current node is a flat node

[0064]

[0065] (2) If the current node is a standard cone node

[0066] ① The endpoint is on the same side as the search direction.

[0067] Connecting points using great circle routes With point The great circle route and the section The intersection of the meridians is denoted as ,

[0068] If point In the interval Inside:

[0069] ;

[0070] If point Not in the range Inside:

[0071] ;

[0072] ②The endpoint is on the opposite side of the search direction.

[0073] set up yes Regarding node intervals The symmetrical point is connected by a great circle route. With point The great circle route and the section The intersection point is denoted as ,

[0074] If point In the interval Inside:

[0075]

[0076] If point Not in the range Inside:

[0077]

[0078] (3) If the current node is a label-adjoint cone node

[0079] ① The endpoint is on the same side as the search direction.

[0080] Connecting points using great circle routes With point The great circle route and the section The intersection of the meridians is denoted as ,

[0081] If point In the interval Inside:

[0082]

[0083] If point Not in the range Inside:

[0084]

[0085] ②The endpoint is on the opposite side of the search direction.

[0086] set up yes Regarding node intervals The symmetrical point is connected by a great circle route. With point The great circle route and the section The intersection point is denoted as ,

[0087] If point In the interval Inside:

[0088]

[0089] If point Not in the range Inside

[0090]

[0091] In the formula:

[0092] This indicates the calculation of two points on the sphere ( Great circle routes between ) They are respectively , , , , , , ,m;

[0093] Indicates latitude segment The length, in meters;

[0094] Definition 1: Two points are said to be visible if they can be connected by a great circle route that does not pass through any obstacle or the intersection of two diagonally adjacent obstacles.

[0095] Definition 2: Grid Interval It is a set of continuous discrete points, taken from any row on a grid; each interval is defined by its endpoints. and Defined by; except and In addition, each interval only contains the middle non-corner point;

[0096] Definition 3: Standard Search Node It is a pair, where It is called the root node. It is an interval, and each point in the interval... Both are visible to r;

[0097] Definition 4: Accompanying search node It is a pair, where It is called the root node, located at the vertical edge of a certain obstacle zone; It is a reachable interval, but every point on the interval... It is not visible to r, and the connection is... and Great circle routes and The vertical edges of the obstacle intersect twice, once at point r;

[0098] Lemma 1: In spherical geometry, all turning points on the optimal path are obstacle corners. Or located within the interval defined by the two vertically aligned corners of the obstacle;

[0099] Definition 5: Standard Search Node Standard successor node It is a search node that meets the following conditions:

[0100] (1) It is a standard node;

[0101] (2) For all points There exists a point This makes the large circle segment Composition from arrive The local optimal path;

[0102] (3) It is all great circle segments The closest shared The only point;

[0103] (4) It is the largest interval that satisfies the standard node definition;

[0104] Definition 6: Search Node Accompanying successor node It is a search node that meets the following conditions:

[0105] (1) It is an accompanying node;

[0106] (2) It is the largest interval that satisfies the definition of an accompanying node;

[0107] (3) or ;

[0108] Definition 7: Search Node flat successor node It is to satisfy and Searching for nodes in the same row; the mechanism for generating subsequent nodes is called projection, which is divided into:

[0109] (1) Observable cone projection;

[0110] (2) Unobservable standard cone projection;

[0111] (3) Unobservable accompanying cone projection;

[0112] (4) Plane projection.

[0113] Preferably, the calculation of the optimal path for the spherical Anya in step 4 specifically involves:

[0114] Step 4-1: Initialize the map and input the Web Mercator coordinates of the flyable area and the threat area from Step 1, as well as the starting point. and the end point ,

[0115] Step 4-2: Starting from the beginning This is the root node of the current node;

[0116] Step 4-3: Calculate the longitude of the current root node minus the endpoint. Check if the longitude of the current root node is greater than zero. If it is, search down one row. Determine the longitude of the current root node minus the longitude of the endpoint. If the longitude is less than zero, search one row upwards; if it is not less than zero, generate a successor node and jump to steps 4-6.

[0117] Step 4-4: Generate all successor nodes;

[0118] Steps 4-5: Remove The successor node with the smallest value ;

[0119] Steps 4-6: Determine the endpoint Does the longitude belong to the interval? If yes, proceed to step 4-7; otherwise, skip to step 4-3.

[0120] Steps 4-7: Return and output the optimal path from the destination. Returns the coordinates of the intersections of the optimal path with each meridian and parallel, and uses the Anya algorithm for heuristic values. The optimal path is calculated.

[0121] Preferably, the distance between searching down one row or searching up one row is 0.1deg to 1deg.

[0122] Compared with the prior art, the advantages of the present invention are as follows:

[0123] (1) This invention discloses an offline trajectory planning method for aircraft based on spherical Anya, providing a method for finding the shortest path between any two points on a sphere at any angle, and providing a shortest trajectory planning method for long-distance aircraft flying on the Earth's surface. The spherical Anya path planning algorithm of this invention is a generalization of the Anya pathfinding algorithm in two-dimensional plane. Also in two-dimensional plane, the spherical Anya algorithm defines each node as a tuple. ,in It is the root node. From the root node The maximum reachable range is determined, thus allowing a path to be obtained at any angle;

[0124] (2) This invention converts latitude and longitude in WGS84 coordinates to latitude and longitude in Web Mercator coordinates, maps the three-dimensional Earth onto a two-dimensional plane using space, and uses great circle routes to express the shortest path. The heuristic value of the spherical Anya algorithm is designed through the great circle routes and the coordinates of the intersection points of the great circle routes with a certain meridian or parallel. Using the Anya algorithm for spherical heuristic values The optimal path for the spherical Anya is obtained. The optimal path is an arc in Euclidean space. The optimal route obtained by this method is relatively reliable and easy for the aircraft to execute.

[0125] (3) This invention transforms the three-dimensional Earth into a two-dimensional map with latitude on the horizontal axis and longitude on the vertical axis under the Web Mercator projection, expanding the original threat area to a range bounded by two lines of longitude and latitude. This range will appear as a rectangle on the Web Mercator projection, which facilitates the calculation of trajectory points, simplifies the calculation, and improves efficiency. At the same time, since the original threat area has been expanded, it can effectively avoid the threat area offline. Attached Figure Description

[0126] Figure 1 This is a perspective cylindrical projection of the present invention;

[0127] Figure 2 This is a schematic diagram of the threat zone conversion in this invention;

[0128] Figure 3 This is a schematic diagram of the current node being a flat node in this invention;

[0129] Figure 4 In this invention, the current node is a standard cone node, and the endpoint is on the same side of the search direction.

[0130] Figure 5 The current node in this invention is a standard cone node whose endpoint is on the opposite side of the search direction;

[0131] Figure 6 In this invention, the current node is the endpoint of the accompanying cone node, which is on the same side of the search direction.

[0132] Figure 7 In this invention, the current node is the endpoint of the accompanying cone node on the opposite side of the search direction;

[0133] Figure 8 This is a flowchart of the trajectory planning process of the present invention;

[0134] Figure 9 This is a schematic diagram illustrating the path optimization effect in scenario 1 of the present invention;

[0135] Figure 10 This is a schematic diagram illustrating the path optimization effect in scenario 2 of the present invention;

[0136] Figure 11 This is a schematic diagram illustrating the path optimization effect in scenario 3 of the present invention. Detailed Implementation

[0137] The specific implementation of the present invention is described below with reference to embodiments:

[0138] This invention discloses an offline trajectory planning method for aircraft based on spherical Anya, comprising the following steps:

[0139] Step 1: Perform mathematical modeling of the aircraft's flyable area and threat area to obtain the Web Mercator coordinates of the flyable area and the threat area;

[0140] Step 2: Mathematical modeling of the great circle route between any two points, and mathematical modeling of the coordinates of the intersection point of the great circle route with a certain meridian or parallel;

[0141] Step 3: Design the heuristic values ​​for the spherical Anya algorithm using the great circle route and the coordinates of the intersection points of the great circle route with a certain meridian or parallel. ;

[0142] Step 4: Initialize the map. Input the Web Mercator coordinates of the start and end points, as well as the Web Mercator coordinates of the flyable area and the threat area between the start and end points. Use the Anya spherical algorithm heuristics. Obtain the optimal path for the spherical Anya.

[0143] Preferably, the mathematical modeling of the aircraft's flyable area and threat area in step 1 specifically involves:

[0144] Step 1-1: Establish the transformation formula from WGS84 coordinates to Web Mercator coordinates;

[0145] Steps 1-2: Convert the threat area in WGS84 coordinates to the threat area in Web Mercator coordinates;

[0146] Steps 1-3: Convert the flyable area in WGS84 coordinates to the flyable area in Web Mercator coordinates.

[0147] Preferably, the conversion formula in step 1-1 is:

[0148]

[0149]

[0150] In the formula:

[0151] for Axis, the latitude of the Web Mercator coordinate system. ;

[0152] for Axis, longitude in Web Mercator coordinates. ;

[0153] The radius of the Earth is in meters (m).

[0154] Longitude, degrees, in WGS84 coordinates;

[0155] The coordinates are in latitude, degrees, of WGS84.

[0156] Let be the longitude, in degrees, of any central meridian.

[0157] Preferably, the conversion of the threat area in WGS84 coordinates to the threat area in Web Mercator coordinates in steps 1-2 specifically involves: converting the latitude and longitude coordinates of the center point of the threat area in WGS84 coordinates to the latitude and longitude coordinates of the center point of the threat area in Web Mercator coordinates, i.e., obstacle coordinates; given the latitude and longitude coordinates of the center point of the threat area and its radius, calculating the latitude and longitude coordinates of the boundary point of the threat area circle, connecting them sequentially to form the initial threat area; and then expanding the initial threat area to the range bounded by the two meridians and parallels tangent to it. This range appears as a rectangle in Web Mercator coordinates, thus obtaining the threat area in Web Mercator coordinates.

[0158] Preferably, the mathematical modeling of the great circle route between any two points in step 2 is specifically as follows:

[0159] Assume the starting coordinates of any two points are The endpoint coordinates are ,

[0160]

[0161] In the formula:

[0162] The length of the great circle route is in meters (m).

[0163] Let be the Earth's radius, in meters (m).

[0164] Preferably, the mathematical modeling of the coordinates of the intersection point between the great circle route and a certain meridian or parallel in step 2 is specifically as follows:

[0165] Assuming the starting point is The coordinates are The destination is The coordinates are ;

[0166] The longitude of the intersection point of the great circle route and a certain meridian is: latitude of the intersection The calculation formula is:

[0167]

[0168] definition ;

[0169]

[0170]

[0171]

[0172] in:

[0173] ;

[0174] ;

[0175] In the formula:

[0176] Let be the Earth's radius, in meters (m).

[0177] Preferably, the latitude of the intersection point of the great circle route and a certain parallel is... Longitude of the intersection point The calculation formula is:

[0178]

[0179] definition ;

[0180]

[0181]

[0182]

[0183]

[0184] in:

[0185] ;

[0186] ;

[0187] ;

[0188] In the formula:

[0189] Let be the Earth's radius, in meters (m).

[0190] Preferably, the heuristic value of the Anya algorithm for spherical surfaces in step 3 is... The design formula is:

[0191]

[0192] in:

[0193] The sum of the great circle segments from the starting point to the left and right turning points of the current root node;

[0194] The great circle segment from the current root node to the end point needs to be calculated based on the type of the current node;

[0195] Let the starting point of the pathfinding be The destination is The search direction is defined as starting from... Pointing to the interval , The classification calculation method is as follows:

[0196] (1) If the current node is a flat node

[0197]

[0198] (2) If the current node is a standard cone node

[0199] ① The endpoint is on the same side as the search direction.

[0200] Connecting points using great circle routes With point The great circle route and the section The intersection of the meridians is denoted as ,

[0201] If point In the interval Inside:

[0202] ;

[0203] If point Not in the range Inside:

[0204] ;

[0205] ②The endpoint is on the opposite side of the search direction.

[0206] set up yes Regarding node intervals The symmetrical point is connected by a great circle route. With point The great circle route and the section The intersection point is denoted as ,

[0207] If point In the interval Inside:

[0208]

[0209] If point Not in the range Inside:

[0210]

[0211] (3) If the current node is a label-adjoint cone node

[0212] ① The endpoint is on the same side as the search direction.

[0213] Connecting points using great circle routes With point The great circle route and the section The intersection of the meridians is denoted as ,

[0214] If point In the interval Inside:

[0215]

[0216] If point Not in the range Inside:

[0217]

[0218] ②The endpoint is on the opposite side of the search direction.

[0219] set up yes Regarding node intervals The symmetrical point is connected by a great circle route. With point The great circle route and the section The intersection point is denoted as ,

[0220] If point In the interval Inside:

[0221]

[0222] If point Not in the range Inside

[0223]

[0224] In the formula:

[0225] This indicates the calculation of two points on the sphere ( Great circle routes between ) They are respectively , , , , , , ,m;

[0226] Indicates latitude segment The length, in meters;

[0227] Definition 1: Two points are said to be visible if they can be connected by a great circle route that does not pass through any obstacle or the intersection of two diagonally adjacent obstacles.

[0228] Definition 2: Grid Interval It is a set of continuous discrete points, taken from any row on a grid; each interval is defined by its endpoints. and Defined by; except and In addition, each interval only contains the middle non-corner point;

[0229] Definition 3: Standard Search Node It is a pair, where It is called the root node. It is an interval, and each point in the interval... Both are visible to r;

[0230] Definition 4: Accompanying search node It is a pair, where It is called the root node, located at the vertical edge of a certain obstacle zone; It is a reachable interval, but every point on the interval... It is not visible to r, and the connection is... and Great circle routes and The vertical edges of the obstacle intersect twice, once at point r;

[0231] Lemma 1: In spherical geometry, all turning points on the optimal path are obstacle corners. Or located within the interval defined by the two vertically aligned corners of the obstacle;

[0232] Definition 5: Standard Search Node Standard successor node It is a search node that meets the following conditions:

[0233] (1) It is a standard node;

[0234] (2) For all points There exists a point This makes the large circle segment Composition from arrive The local optimal path;

[0235] (3) It is all great circle segments The closest shared The only point;

[0236] (4) It is the largest interval that satisfies the standard node definition;

[0237] Definition 6: Search Node Accompanying successor node It is a search node that meets the following conditions:

[0238] (1) It is an accompanying node;

[0239] (2) It is the largest interval that satisfies the definition of an accompanying node;

[0240] (3) or ;

[0241] Definition 7: Search Node flat successor node It is to satisfy and Searching for nodes in the same row; the mechanism for generating subsequent nodes is called projection, which is divided into:

[0242] (1) Observable cone projection;

[0243] (2) Unobservable standard cone projection;

[0244] (3) Unobservable accompanying cone projection;

[0245] (4) Plane projection.

[0246] Preferably, the calculation of the optimal path for the spherical Anya in step 4 specifically involves:

[0247] Step 4-1: Initialize the map and input the Web Mercator coordinates of the flyable area and the threat area from Step 1, as well as the starting point. and the end point ,

[0248] Step 4-2: Starting from the beginning This is the root node of the current node;

[0249] Step 4-3: Calculate the longitude of the current root node minus the endpoint. Check if the longitude of the current root node is greater than zero. If it is, search down one row. Determine the longitude of the current root node minus the longitude of the endpoint. If the longitude is less than zero, search one row up; if it is not less than zero, generate a successor node and jump to steps 4-6.

[0250] Step 4-4: Generate all successor nodes;

[0251] Steps 4-5: Remove The successor node with the smallest value ;

[0252] Steps 4-6: Determine the endpoint Does the longitude belong to the interval? If yes, proceed to step 4-7; otherwise, skip to step 4-3.

[0253] Steps 4-7: Return and output the optimal path from the destination. Returns the coordinates of the intersections of the optimal path with each meridian and parallel, and uses the Anya algorithm for heuristic values. The optimal path is calculated.

[0254] Preferably, the distance between searching down one row or searching up one row is 0.1deg to 1deg. Example 1

[0255] Mathematical modeling of the flyable zone and threat zone of an aircraft

[0256] In order to accurately establish mathematical models of the flyable and threat zones of aircraft, it is first necessary to define the coordinate system describing the aircraft's motion and introduce the angle transformation relationships.

[0257] ① Mercator projection method

[0258] The Mercator projection is an "equiangular tangent cylindrical projection" created by the Dutch cartographer G. Mercator in 1569. It imagines that the Earth is enclosed in a hollow cylinder with its equator in contact with the cylinder. It also imagines that there is a lamp at the center of the Earth, projecting the shape on the sphere onto the cylinder. When the cylinder is unfolded, this is a world map drawn using the Mercator projection with the standard latitude line at zero degrees (i.e., the equator).

[0259] After projection, the meridians are a set of vertical, equidistant parallel lines, and the parallels of latitude are a set of parallel lines perpendicular to the meridians. The interval between adjacent parallels of latitude increases from the equator to the poles. The length ratio in any direction at a point is equal, meaning there is no angular distortion, but the area distortion is significant and increases with distance from the reference parallel of latitude. This projection has the characteristic that conformal routes are represented as straight lines, so it is widely used in the compilation of nautical charts and aeronautical charts.

[0260] like Figure 1 As shown: The Mercator projection is a projection obtained by modifying the perspective cylindrical projection according to the condition of conformity. The point of modification of the perspective cylindrical projection by the Mercator projection is: to make the cylindrical projection have the property of conformity, the factor by which the meridians gradually extend from the equator to the poles must be the same as the factor by which the latitude of each point on the meridian expands. Let the circumference of the Earth's equator be... ( (where the radius is the equatorial radius), and the circumference of each latitude circle is... ( (This represents the corresponding latitude), therefore, the ratio of the length of a parallel on a Mercator projection map to the actual length of a parallel on Earth is:

[0261]

[0262] Since the parallels of latitude at each degree have expanded To maintain the same angle, the meridians at the points where parallels of latitude pass must also be correspondingly enlarged. Only then can the length ratio along the meridian be equal to the length ratio along the parallel. If we assume the length ratios along the meridian and parallel are respectively... Then at this time .

[0263] Specifically, the Mercator projection will project latitude as... The point is projected onto:

[0264]

[0265] For symbolic functions, With constant Logarithm with base 0, The number of rows for the tangent. For dimensions.

[0266] in

[0267]

[0268] This projection algorithm results in denser latitude lines near the equator and sparser latitude lines near the poles, projecting the poles to infinity. Therefore, this projection is unsuitable for high-latitude regions. Google Maps selects the area as follows: This approximates .

[0269] In summary, Mercator projection has the following characteristics:

[0270] The original meridians all converge at the poles, and the higher the latitude, the smaller the interval between the meridians. However, in the Mercator projection, all meridians are drawn as parallel straight lines with equal intervals to the equator.

[0271] On the Earth's surface, the imaginary intervals between parallels are basically equal, but in the Mercator projection, the intervals between parallels gradually increase from the equator to the poles, and the intervals between longitude and latitude increase by the same multiple.

[0272] In the Mercator projection, both longitude and latitude lines are parallel straight lines, and they are orthogonal to each other.

[0273] Conformal flight paths on Earth appear as straight lines on Mercator projection maps.

[0274] The map accurately represents the angle between the flight path and the meridian. For example, if the angle between the flight path and the meridian on Earth is 15°, it will also be represented as 15° on the map.

[0275] Map scale varies with latitude; on a single map, the scale for the same distance will differ at different latitudes, making it impossible to compare areas. Therefore, the same scale cannot be used when measuring distances.

[0276] Mercator projection maps use the equator as the standard parallel of latitude and the Prime Meridian as the central meridian. The intersection of these two meridians is the origin of the coordinate system. East and north are positive, while west and south are negative. The North and South Poles are directly below and above the map, while east and west are directly to the right and left. Because the Mercator projection tends towards infinity near the poles, it does not fully represent the entire world. The highest latitude on the map is 85.05 degrees (which can be calculated by inverse calculation using the range of latitude values, yielding a value of 85.05112877980659).

[0277] Note that the Mercator projection is not a coordinate system, but rather a spatial mapping used to represent the three-dimensional Earth on a two-dimensional plane. Therefore, in GIS maps and internet maps, although the map seen by the user has undergone Mercator projection, latitude and longitude coordinates are still used to represent the location of points on Earth. Map creation and visualization require presenting map data using a projection method.

[0278] The Web Mercator projection (also known as the spherical Mercator projection) is a variant of the Mercator projection. It receives latitude and longitude with a Datum of WGS84 as input, but in order to simplify calculations during projection, it no longer treats the Earth as an ellipsoid but as a standard sphere with a radius of 6,378,137 meters.

[0279] Under the Web Mercator projection, the range of projected coordinates (in meters) is:

[0280] ,

[0281] .

[0282] In the corresponding geographic coordinate system, the longitude range is The latitude range is

[0283] .

[0284] Figure 3 shows a spherical offline trajectory planning map based on Web Mercator projection. The coordinates of the axis are in latitude, and the maximum range is [missing value]. , The coordinates of the axis are longitude, with the largest possible range being... The difference between adjacent coordinates of the two axes depends on the required map accuracy in the actual application. Given a point in the WGS84 coordinate system (represented by latitude and longitude), its Mercator value can be obtained using the Web Mercator projection method. A detailed explanation follows. As we know from the previous introduction to the Mercator projection method, to perform the conversion between the two, we only need to know the conversion ratio when the Earth is transformed into a flat map; then we can calculate the Mercator value using latitude and longitude. Assume there is a point Q that is infinitely close to point P, and the Earth's radius is R, as shown in the diagram below. The latitude and longitude of point P are... ( and (Both are in radians), the latitude and longitude of the adjacent point Q are... So, parallel to the equator The radius of the circle containing the parallel of latitude:

[0285]

[0286] Radius PM:

[0287]

[0288] So, in the image above...

[0289]

[0290]

[0291] Parallelism conversion ratio:

[0292]

[0293] Parallelism conversion ratio with meridian:

[0294]

[0295] Since no deformation occurred after the meridian transformation, based on the comparison diagram of the globe and projection:

[0296]

[0297] Conclusion:

[0298]

[0299] Then, using the concept of differentiation, that is...

[0300]

[0301] for The first derivative, then the ratio of parallelism to latitude.

[0302]

[0303] Parallelism conversion ratio with meridian:

[0304]

[0305] in

[0306]

[0307] Because Web Mercator projection aims to achieve an isogonal projection, it can be considered that...

[0308]

[0309] So

[0310]

[0311] And it can be concluded that

[0312]

[0313] Through the Points can be earned and The value of, that is:

[0314]

[0315] Equation (19) above is the conversion relationship between WGS84 latitude and longitude and Web Mercator.

[0316] ② Mathematical description methods for threat areas

[0317] Since points on a map under the Web Mercator projection are represented by latitude and longitude coordinates, the latitude and longitude parameters of the original spherical threat area can be directly represented on the Mercator projection map. That is, any threat area represented by latitude and longitude (composed of multiple latitude and longitude points) can be converted into a projection under Web Mercator through a transformation relationship as shown in equation (19). The only difference is that the shape of the threat area will change after visualization.

[0318] like Figure 2 As shown: Taking a circular threat area on the original sphere as an example, given the latitude and longitude coordinates of the center point of the threat map and its threat radius, the latitude and longitude of the boundary points of the threat circle can be calculated. Then, the corresponding points can be found on the Mercator projection, and connecting them sequentially will yield the threat area under the Mercator projection. At this point, the shape of the threat area will appear distorted, but its position and actual size will not change. The same principle applies to threat areas of other shapes.

[0319] like Figure 2As shown: When using the trajectory planning algorithm in the future, since the map is a two-dimensional map based on the Mercator projection with latitude on the horizontal axis and longitude on the vertical axis, in order to facilitate the calculation of trajectory points, the original threat area is first expanded to the range bounded by two lines of longitude and latitude. This range will be presented as a rectangle on the Mercator projection. Example 2

[0320] Calculate the great circle route and its related coordinates

[0321] ① Calculate the great circle route

[0322] A great circle is the arc with the largest radius on the surface of a sphere, that is, an arc with a radius equal to the radius of the sphere. If the Earth is considered as a sphere, the shortest distance between two points on the ground is the arc length of a great circle less than 180° connecting the two points. All meridians are great circles, while only the equator is a great circle.

[0323] Assuming that a ship travels on a constant course throughout its journey, the ideal trajectory of the ship on the Earth's surface forms a curve known as the constant course or isotropic route.

[0324] When ships navigate at high latitudes, their course is closer to east-west, spanning a significant longitude difference. In this case, the great circle route is hundreds of nautical miles shorter than the heading route. However, strictly adhering to a great circle route requires constant course adjustments. A great circle route can be viewed as a series of vector waypoints, each containing information such as wind direction, wave height, course, and latitude / longitude coordinates. If we divide the great circle route into N equally spaced segments, each segment can be considered a heading line.

[0325] Given a great circle route between two points, it can be calculated as follows:

[0326] Assuming the starting point is The target point is Both the starting and destination points are expressed in latitude and longitude (in degrees). The distance S and initial heading C of the great circle route can be calculated based on the coordinates of the starting point.

[0327]

[0328] On a spherical projection map, the line connecting two points forms a great circle route. Calculate the geographical coordinates of the intersections of the great circle route with each meridian, then transfer these intersections to a Web Mercator projection map and connect them sequentially with smooth curves to obtain the great circle route on the Web Mercator projection map. If the great circle route on the Web Mercator projection map is divided into several segments, and the two endpoints of each segment are connected by straight lines, a navigation route approximating the great circle route from the starting point to the end point is obtained by connecting several equal-angle routes.

[0329] ② Formulas for calculating relevant coordinates

[0330] (1) Formula for calculating the length of a great circle

[0331] Assuming the starting point is The target point is If both the starting point and the target point are expressed in latitude and longitude (in degrees), then the shortest distance between two points on the sphere (i.e., the length of a great circle route) is... for:

[0332]

[0333] (2) Find the intersection of the great circle between two points and a certain meridian.

[0334] Assuming the starting point is The coordinates are The target point is The coordinates are The longitudes of the intersecting meridians are The latitude of the intersection point The calculation steps are as follows:

[0335] ① First, derive the equation of the great circle based on the coordinates of the two points:

[0336]

[0337] ②

[0338]

[0339] Substitute Solving for

[0340]

[0341] Due to the function The range of values ​​is Therefore, further calculations are needed:

[0342] remember The parameter expression is as follows:

[0343] ,

[0344] ,

[0345] but

[0346]

[0347] Similarly, we can obtain .

[0348] ③ Definition

[0349]

[0350] The latitude of the intersection point:

[0351]

[0352] (3) Find the intersection of the great circle between two points and a certain parallel of latitude.

[0353] Assuming the starting point is The coordinates are The target point is The coordinates are The latitude of the intersecting meridians is The longitude of the intersection point The calculation steps are as follows:

[0354] ① First, derive the equation of the great circle based on the coordinates of the two points: ;

[0355] ② , bring in Solution:

[0356]

[0357] Due to the function The range of values ​​is Therefore, further calculations are needed:

[0358] remember The parameter expression is as follows:

[0359]

[0360] but

[0361]

[0362] Similarly, we can obtain .

[0363] ③ Definition

[0364]

[0365]

[0366] The longitude of the intersection point:

[0367] Example 3

[0368] Designing heuristics for the Anya algorithm on a sphere

[0369] The spherical Anya algorithm is a generalization of the two-dimensional Anya pathfinding algorithm. The pathfinding environment is expanded from two dimensions to a three-dimensional sphere. The map uses an offline trajectory planning map based on the Web Mercator projection, and great circle routes are used instead of Euclidean distances when calculating the shortest distance between any two points. Each node in the pathfinding process remains a binary tuple. Starting from the origin, search layer by layer along the meridian towards the destination until the current pathfinding node is reached. The interval in If the destination coordinates are included, the pathfinding will stop, and the latitude and longitude coordinates of the critical path will be returned from the destination.

[0370] In Euclidean space, all turning points on the optimal path coincide with the corner points of obstacles. However, in spherical geometry, all turning points on the optimal path are not only corner points of obstacles but may also be points on the vertical boundaries of obstacles. Finally, connecting adjacent path points with great circle routes yields the optimal offline trajectory on the spherical surface.

[0371] Before performing path planning on a spherical Anya surface, we list the relevant theories and definitions for use during the path planning process, as follows:

[0372] ① Basic mathematical definition

[0373] Definition 1: Two points are said to be visible if they can be connected by a great circle that does not pass through any obstacle or the intersection of two diagonally adjacent obstacles.

[0374] Definition 2: Grid Interval It is a set of continuous discrete points, taken from any row on a grid. Each interval is defined by its endpoints. and It is defined by [the relevant authority]. Besides [the specific details], and In addition, each interval only contains the middle non-corner points.

[0375] Definition 3: Standard Search Node It is a pair, where It is called the root node. It is an interval, and each point in the interval... Both are visible to r.

[0376] Definition 4: Accompanying search node It is a pair, where It is called the root node and is located at the vertical edge of a certain obstacle zone. It is a reachable interval, but every point on the interval... It is not visible to r, and the connection is... and The great circle and The vertical edges of the obstacles will intersect twice (once at point r).

[0377] Lemma 1: In spherical geometry, all turning points on the optimal path are either corner points of an obstacle or lie within a region defined by two perpendicularly aligned corner points of an obstacle.

[0378] ② Definitions of successor nodes and mappings in the Anya algorithm for spherical surfaces

[0379] Definition 5: Search Node Standard successor node It is a search node that meets the following conditions:

[0380] (1) It is a standard node;

[0381] (2) For all points There exists a point This makes the large circle segment Composition from arrive The local optimal path;

[0382] (3) It is all great circle segments The closest shared The only point;

[0383] (4) It is the largest interval that satisfies the standard node definition.

[0384] Definition 6: Search Node Accompanying successor node It is a search node that meets the following conditions:

[0385] (1) It is an accompanying node;

[0386] (2) It is the largest interval that satisfies the definition of an accompanying node;

[0387] (3) or .

[0388] Definition 7: Search Node flat successor node It is to satisfy and Searching for nodes in the same row. The mechanism for generating subsequent nodes is called projection, similar to the Anya algorithm. These projections are divided into:

[0389] (1) Observable (visible) cone projection;

[0390] (2) Unobservable (invisible) standard cone projection;

[0391] (3) Unobservable accompanying cone projection;

[0392] (4) Plane projection.

[0393] ③ Heuristic value of the Anya algorithm for spherical surfaces Design

[0394]

[0395] in It is the sum of the great circles between the starting point and the left and right turning points of the current root node. The calculation depends on the type of the current node.

[0396] Let the destination of the pathfinding be The search direction is defined as starting from... Pointing to the interval , The method for calculating the value classification is as follows:

[0397] like Figure 3 As shown: (1) If the current node is a flat node

[0398]

[0399] In the above formula This indicates the calculation of two points on the sphere ( The shortest distance between (i.e., the length of the great circle), the same below.

[0400] like Figure 4 As shown: (2) If the current node is a standard cone node

[0401] ① The endpoint is on the same side as the search direction.

[0402] Connect the points with a large circle. With point The great circle line and the interval The intersection of the meridians is denoted as ,but

[0403] If point In the interval Inside: ;

[0404] If point Not in the range Inside: .

[0405] ②The endpoint is on the opposite side of the search direction.

[0406] like Figure 5 As shown: yes Regarding node intervals Connect the points of symmetry with a great circle. With point The great circle line and the interval The intersection point is denoted as ,but

[0407] If point In the interval Inside:

[0408]

[0409] If point Not in the range Inside:

[0410]

[0411] (3) If the current node is a label-adjoint cone node

[0412] ① The endpoint is on the same side as the search direction.

[0413] like Figure 6 As shown: Connect the points with a large circle. With point The great circle line and the interval The intersection of the meridians is denoted as ,but

[0414] If point In the interval Inside:

[0415]

[0416] If point Not in the range Inside:

[0417]

[0418] in Indicates latitude segment The length.

[0419] ②The endpoint is on the opposite side of the search direction.

[0420] like Figure 7 As shown: yes Regarding node intervals Connect the points of symmetry with a great circle. With point The great circle line and the interval The intersection point is denoted as ,but

[0421] If point In the interval Inside:

[0422]

[0423] If point Not in the range Inside:

[0424] Example 4

[0425] Calculate the optimal path for the spherical Anya

[0426] The specific steps for trajectory planning calculation are as follows:

[0427] ① Initialize the map and input the Web Mercator coordinates of the flyable zone and the threat zone, as well as the starting point. and the end point ;

[0428] ② Starting point This is the root node of the current node;

[0429] ③ Determine the longitude of the current root node minus the endpoint. Check if the longitude is greater than zero. If it is, search down one row (0.1deg to 1deg, depending on the specific situation); determine the longitude of the current root node minus the longitude of the endpoint. If the longitude is less than zero, search one row upwards; if it is not less than zero, generate a successor node and jump to step ⑥.

[0430] ④ Generate all successor nodes;

[0431] ⑤ Take out The successor node with the smallest value ;

[0432] ⑥ Determine the endpoint Does the longitude belong to the interval? If yes, proceed to step ⑦; otherwise, go to step ③.

[0433] ⑦ Return the optimal path and output it.

[0434] Application Example 1

[0435] Step 1: Perform mathematical modeling of the aircraft's flyable area and threat area to obtain the Web Mercator coordinates of the flyable area and the threat area;

[0436] Step 2: Mathematical modeling of the great circle route between any two points, and mathematical modeling of the coordinates of the intersection point of the great circle route with a certain meridian or parallel;

[0437] Step 3: Design the heuristic values ​​for the spherical Anya algorithm using the great circle route and the coordinates of the intersection points of the great circle route with a certain meridian or parallel. ;

[0438] Step 4: Initialize the map, input the Web Mercator coordinates of the start and end points, as well as the coordinates of obstacles between the start and end points. The trajectory planning process is as follows: Figure 8 As shown, the heuristic value is obtained using the Anya algorithm on a sphere. Obtain the optimal path for the spherical Anya.

[0439] This example includes the following scenarios:

[0440] Scenario 1: Trajectory planning in a multi-obstacle environment in the Northern Hemisphere

[0441] The simulation data are shown in Table 1.

[0442] Table 1 Simulation Data List

[0443]

[0444] The flyable area is a flight path composed of waypoints, also expressed in latitude and longitude. Similar to the threat area, it is mapped to Web Mercator coordinates via Mercator mapping, such as... Figure 9 As shown, three threat zones are formed on the Web Mercator coordinate system, and the path obtained by the trajectory planning method of this invention is the optimal path.

[0445] Scenario 2: Comparison of route finding results at different map longitudes

[0446] like Figure 10 As shown, three threat zones are formed on the Web Mercator coordinate system. Multiple paths are found through path optimization. Blue line A represents the optimal path with a latitude and longitude precision of 1°; black line B represents the optimal path with a precision of 0.8°; green line C represents the optimal path with a precision of 0.6°; and purple line D represents the optimal path with a precision of 0.5°. Clearly, lower precision results in shorter paths that are closer to the obstacle boundaries. However, this does not mean that lower precision is always better; the optimal precision should be determined based on the obstacle boundaries.

[0447] Furthermore, differences in accuracy and the number of obstacles also significantly affect the pathfinding performance and operating speed. Some test results are shown in Table 2.

[0448] Table 2. Pathfinding performance under different levels of accuracy and obstacles

[0449]

[0450] Clearly, the lower the accuracy, the shorter the optimal path, but the longer the running time.

[0451] Scenario 3: Temporarily add obstacles to the safe path in Scenario 1

[0452] like Figure 11 As shown, four threat zones are formed on the Web Mercator coordinate system. The path obtained by the trajectory planning method of this invention is the optimal path. In this scenario, the code execution speed is 422ms, which is still enough to quickly find the optimal safe path.

[0453] This invention discloses an offline trajectory planning method for aircraft based on spherical Anya, providing a method for finding the shortest path between any two points on a sphere at any angle. This offers a shortest trajectory planning method for long-distance aircraft flight on the Earth's surface. The spherical Anya path planning algorithm of this invention is a generalization of the two-dimensional Anya pathfinding algorithm. Also in a two-dimensional plane, the spherical Anya algorithm defines each node as a binary tuple. ,in It is the root node. From the root node The maximum reachable interval is determined, thus obtaining a path at any angle.

[0454] This invention transforms latitude and longitude in WGS84 coordinates to latitude and longitude in Web Mercator coordinates, mapping the three-dimensional Earth onto a two-dimensional plane. It uses great circle routes to express the shortest path and designs heuristic values ​​for the spherical Anya algorithm based on the great circle routes and the coordinates of their intersections with a meridian or parallel. Using the Anya algorithm for spherical heuristic values The optimal path for the spherical Anya is obtained. The optimal path is an arc in Euclidean space. The optimal route obtained by this method is relatively reliable and easy for the aircraft to execute.

[0455] This invention transforms a three-dimensional Earth into a two-dimensional map under Web Mercator projection, with latitude as the horizontal axis and longitude as the vertical axis. The original threat area is expanded to a range bounded by two lines of longitude and latitude. This range will appear as a rectangle on the Web Mercator projection, which facilitates the calculation of trajectory points, simplifies the calculation, and improves efficiency. At the same time, because the original threat area is expanded, it can effectively avoid the threat area offline.

[0456] The preferred embodiments of the present invention have been described in detail above. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

[0457] Many other changes and modifications can be made without departing from the concept and scope of this invention. It should be understood that this invention is not limited to the specific embodiments, and the scope of this invention is defined by the appended claims.

Claims

1. An offline trajectory planning method for aircraft based on spherical Anya, characterized in that, Includes the following steps: Step 1: Perform mathematical modeling of the aircraft's flyable area and threat area to obtain the Web Mercator coordinates of the flyable area and the threat area; Step 1-1: Establish the transformation formula from WGS84 coordinates to Web Mercator coordinates; The transformation relationship is as follows: In the formula: for Axis, the latitude of the Web Mercator coordinate system. ; for Axis, longitude in Web Mercator coordinates. ; The radius of the Earth is in meters (m). Longitude, degrees, in WGS84 coordinates; The coordinates are in latitude, degrees, of WGS84. Let be the longitude, in degrees, of any central meridian; Steps 1-2: Convert the threat area in WGS84 coordinates to the threat area in Web Mercator coordinates; Convert the latitude and longitude coordinates of the center point of the threat area in WGS84 coordinates to the latitude and longitude coordinates of the center point of the threat area in Web Mercator coordinates. Given the latitude and longitude coordinates of the center point of the threat area and its radius, calculate the latitude and longitude coordinates of the boundary point of the threat area circle and connect them in sequence to form the initial threat area. Then expand the initial threat area to the area bounded by the two meridians and parallels that are tangent to it. This area is presented as a rectangle in Web Mercator coordinates, and the threat area in Web Mercator coordinates can be obtained. Steps 1-3: Convert the flyable area in WGS84 coordinates to the flyable area in Web Mercator coordinates; Step 2: Mathematical modeling of the great circle route between any two points, and mathematical modeling of the coordinates of the intersection point of the great circle route with a certain meridian or parallel; Step 3: Design the heuristic values ​​for the spherical Anya algorithm using the great circle route and the coordinates of the intersection points of the great circle route with a certain meridian or parallel. ; Step 4: Initialize the map. Input the Web Mercator coordinates of the start and end points, as well as the Web Mercator coordinates of the flyable area and the threat area between the start and end points. Use the Anya spherical algorithm heuristics. Obtain the optimal path for the spherical Anya.

2. The offline trajectory planning method for aircraft based on spherical Anya as described in claim 1, characterized in that, The mathematical modeling of the great circle route between any two points in step 2 is specifically as follows: Assume the starting coordinates of any two points are The endpoint coordinates are , In the formula: The length of the great circle route is in meters (m). Let be the Earth's radius, in meters (m).

3. The offline trajectory planning method for aircraft based on spherical Anya as described in claim 2, characterized in that, The mathematical modeling of the intersection point of the great circle route with a meridian or parallel in step 2 is as follows: Assuming the starting point is The coordinates are The destination is The coordinates are ; The longitude of the intersection point of the great circle route and a certain meridian is: latitude of the intersection The calculation formula is: definition ; in: ； ； In the formula: Let be the Earth's radius, in meters (m).

4. The offline trajectory planning method for aircraft based on spherical Anya as described in claim 3, characterized in that, The latitude of the intersection point between the great circle route and a certain parallel is: Longitude of the intersection point The calculation formula is: definition ; in: ； ； ； In the formula: Let be the Earth's radius, in meters (m).

5. The offline trajectory planning method for aircraft based on spherical Anya as described in claim 1, characterized in that, Heuristic value of the Anya algorithm for spherical surface in step 3 The design formula is: in: The sum of the great circle segments from the starting point to the left and right turning points of the current root node; The great circle segment from the current root node to the end point needs to be calculated based on the type of the current node; Let the starting point of the pathfinding be The destination is The search direction is defined as starting from... Pointing to the interval , The classification calculation method is as follows: (1) If the current node is a flat node (2) If the current node is a standard cone node ① The endpoint is on the same side as the search direction. Connecting points using great circle routes With point The great circle route and the section The intersection of the meridians is denoted as , If point In the interval Inside: ； If point Not in the range Inside: ； ②The endpoint is on the opposite side of the search direction. set up yes Regarding node intervals The symmetrical point is connected by a great circle route. With point The great circle route and the section The intersection point is denoted as , If point In the interval Inside: If point Not in the range Inside: (3) If the current node is a label-adjoint cone node ① The endpoint is on the same side as the search direction. Connecting points using great circle routes With point The great circle route and the section The intersection of the meridians is denoted as , If point In the interval Inside: If point Not in the range Inside: ②The endpoint is on the opposite side of the search direction. set up yes Regarding node intervals The symmetrical point is connected by a great circle route. With point The great circle route and the section The intersection point is denoted as , If point In the interval Inside: If point Not in the range Inside In the formula: This indicates the calculation of two points on the sphere ( Great circle routes between ) They are respectively , , , , , , ,m; Indicates latitude segment The length, in meters; Definition 1: Two points are said to be visible if they can be connected by a great circle route that does not pass through any obstacle or the intersection of two diagonally adjacent obstacles. Definition 2: Grid Interval It is a set of continuous discrete points, taken from any row on a grid; each interval is defined by its endpoints. and Defined by; except and In addition, each interval only contains the middle non-corner point; Definition 3: Standard Search Node It is a pair, where It is called the root node. It is an interval, and each point in the interval... Both are visible to r; Definition 4: Accompanying search node It is a pair, where It is called the root node, located at the vertical edge of a certain obstacle zone; It is a reachable interval, but every point on the interval... It is not visible to r, and the connection is... and Great circle routes and The vertical edges of the obstacle intersect twice, once at point r; Lemma 1: In spherical geometry, all turning points on the optimal path are obstacle corners. Or located within the interval defined by the two vertically aligned corners of the obstacle; Definition 5: Standard Search Node Standard successor node It is a search node that meets the following conditions: (1) It is a standard node; (2) For all points There exists a point This makes the large circle segment Composition from arrive The local optimal path; (3) It is all great circle segments The closest shared The only point; (4) It is the largest interval that satisfies the standard node definition; Definition 6: Search Node Accompanying successor node It is a search node that meets the following conditions: (1) It is an accompanying node; (2) It is the largest interval that satisfies the definition of an accompanying node; (3) or ; Definition 7: Search Node flat successor node It is to satisfy and Searching for nodes in the same row; the mechanism for generating subsequent nodes is called projection, which is divided into: (1) Observable cone projection; (2) Unobservable standard cone projection; (3) Unobservable accompanying cone projection; (4) Plane projection.

6. The offline trajectory planning method for aircraft based on spherical Anya as described in claim 5, characterized in that, The specific steps for calculating the optimal path for the spherical Anya in step 4 are as follows: Step 4-1: Initialize the map and input the Web Mercator coordinates of the flyable area and the threat area from Step 1, as well as the starting point. and the end point , Step 4-2: Starting from the beginning This is the root node of the current node; Step 4-3: Calculate the longitude of the current root node minus the endpoint. Check if the longitude of the current root node is greater than zero. If it is, search down one row. Determine the longitude of the current root node minus the longitude of the endpoint. If the longitude is less than zero, search one row upwards; if it is not less than zero, generate a successor node and jump to steps 4-6. Step 4-4: Generate all successor nodes; Steps 4-5: Remove The successor node with the smallest value ; Steps 4-6: Determine the endpoint Does the longitude belong to the interval? If yes, proceed to step 4-7; otherwise, skip to step 4-3. Steps 4-7: Return and output the optimal path from the destination. Returns the coordinates of the intersections of the optimal path with each meridian and parallel, and uses the Anya algorithm for heuristic values. The optimal path is calculated.

7. The offline trajectory planning method for aircraft based on spherical Anya as described in claim 6, characterized in that, The distance between searching down one row or searching up one row is 0.1deg to 1deg.

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