A simplified visual map modeling and planning method based on delaunay triangulation

Abstract

The application discloses a simplified visual graph modeling and planning method based on Delaunay triangulation, and first acquires polygonal obstacles in an environment, constructs a limited Delaunay triangulation network by using a centroid point set of the polygonal obstacles; obtains neighbor obstacles of each obstacle from the triangulation network to construct neighbor pairs; then constructs a global environment simplified visual graph by using a visibility relationship between adjacent obstacle vertices; maintains a local region where a starting point and an ending point are located into the global visual graph, and plans a global path by using a Dijkstra algorithm with fusion pruning; generates an elliptical window associated with the path, divides boundary points of obstacles in the elliptical window and elliptical boundary points into two types of samples by using the global path, and finally trains a support vector machine based on a radial basis function to obtain a safe and smooth path, so that the path planning method is more advanced and faster.

Description

Technical Field

[0001] This invention belongs to the field of environmental modeling and path planning, specifically involving a simplified visual modeling and planning method based on Delaunay triangulation. Background Technology

[0002] Path planning is a key topic and a hot research area in robot navigation technology. The task of robot path planning is to plan the best or near-optimal path from a given starting point to a given target point in a perceived environment map, based on certain optimization criteria, while avoiding collisions with obstacles in the environment. In recent years, many path planning methods based on environment modeling have been proposed.

[0003] Before path planning, the environment needs to be modeled. This typically involves creating geometric models, such as grid maps, topological maps, or occupied grid maps. These models represent obstacles and traversable areas in the environment. After environment modeling, a suitable path search algorithm needs to be selected. Commonly used path search algorithms include Dijkstra's algorithm, A* algorithm, and RRT (Rapidly-exploring Random Trees) algorithm. These algorithms find the optimal path based on different problems and constraints. After finding the initial path, it usually needs to be optimized and smoothed to meet certain constraints (such as robot dynamics constraints). Common smoothing techniques include Bézier curves, spline curves, and support vector machines (SVM).

[0004] Traditional visibility-based environment modeling methods use convex polygons to represent obstacles in the environment. The most basic and intuitive algorithm has an O(n) time complexity. 3 The time complexity is O(n). 3 Time complexity means that the algorithm's time consumption increases significantly with the number of vertices. Therefore, more advanced and faster algorithms have been developed to address this problem. This research focuses on further optimizing the problem by introducing Delaunay triangulation to simplify visual image modeling planning and employing a support vector machine based on radial basis functions to further optimize the path, improving the speed, safety, and smoothness of path planning. Summary of the Invention

[0005] To address the aforementioned issues, this invention discloses a simplified visual image modeling and planning method based on Delaunay triangulation. By introducing Delaunay triangulation to simplify visual image modeling, it flexibly maintains the starting and ending points in the visual image, enabling rapid planning. Finally, through support vector machine optimization based on radial basis functions, a safe and smooth path is obtained.

[0006] To achieve the above objectives, the technical solution of the present invention is as follows:

[0007] A simplified visualization modeling and planning method based on Delaunay triangulation includes the following steps:

[0008] S1: Model obstacles in the environment map as polygons and extract centroids, boundary vertices, discrete edges, and discrete boundary point sets;

[0009] S2: Construct a restricted Deloitte triangulation on the set of obstacle material cores obtained in step S1;

[0010] S3: Based on the triangulation obtained in step S2, obtain the neighboring obstacles of each obstacle to construct neighbor pairs;

[0011] S4: Based on the visibility relationships between adjacent obstacle vertices obtained in step S3, construct a simplified global environment view;

[0012] S5: Based on the triangulation of step S2, triangulate the local areas where the starting point and the ending point are located;

[0013] S6: Maintain the local triangulation region obtained in step S5 to the global view to obtain the final simplified view. Then, use the Dijkstra algorithm with fused pruning to search the global path between the starting point and the ending point according to the shortest path principle.

[0014] S7: Based on the global path obtained in step S6, generate an elliptical window associated with the path on the map;

[0015] S8: Based on the elliptical window region obtained in step S7, determine which side of the global path the obstacle boundary points and elliptical boundary points within the elliptical window are on, and divide the point set into positive and negative samples.

[0016] S9: Based on the positive and negative samples obtained in step S8, a safe and smooth path is obtained by training a support vector machine based on radial basis functions;

[0017] Furthermore, step S1 specifically includes the following process:

[0018] (1-1) Identify global environmental obstacles through semantic segmentation, threshold segmentation or other image processing techniques;

[0019] (1-2) The identified obstacle regions are fitted with polygons using the convex hull and Douglas-Peucker algorithm to obtain obstacles represented by polygons. The core of the polygonal obstacle, C = {O}, is then calculated. i |i=1,2,3,…,n}, boundary vertex V i ={v ij |j=1,2,3,…,num(B i Discrete edge E i ={(v i1 ;vi2 ),(v i2 ;v i3 ),…,(v inum(Bi) ;v i1 Discrete boundary point set PE i ={pe1,pe2,...,pe co_i ,}, where v ij Let B represent the j-th vertex of the i-th obstacle. i Let i represent the obstacle numbered i, n represent the number of obstacles, num represent the function for calculating the number of obstacle vertices, and co_i represent the number of points on the boundary of the discrete obstacle i.

[0020] Further, the specific process of step S2 is as follows: based on the obstacle material core point set C = {O} i Construct a restricted Delaunay triangulation for |i=1,2,3,…,n} such that the circumcircle of any triangle does not contain any other vertices, and maximize the minimum angle;

[0021] Furthermore, step S3 specifically involves: using the relationship between the centroids and the sides of the triangles to obtain the set of neighbor pairs of the current polygonal obstacle. i ={(B i B nj )∣i≠nj,i=1,2,3,…,n,nj=1,2,3…,n}, B i B represents obstacle numbered i. nj B i The neighbor elements of obstacle i are determined, and the set of all edges E of the neighboring obstacles of obstacle i is determined. ineibor .

[0022] Furthermore, step S4 specifically includes the following process:

[0023] (4-1) Create a graph G = graph();

[0024] (4-2) Maintain the edge and edge length of each obstacle in the graph;

[0025] (4-3) For the j-th vertex v of the i-th obstacle ij Determine the r-th vertex v of the k-th neighboring obstacle of the i-th obstacle. ikr Connecting these two points forms edge e. jr =[v ij ;v kr ];

[0026] (4-4) Determine edge e jr Is it related to the set of edges E of the i-th obstacle? i and the set E of all edges of the neighboring obstacles of the i-th obstacle.ineibor intersect;

[0027] (4-5) If they do not intersect, then edge e jr and the length of the side norm(e) jr Maintain G = addedge(G,e) in the graph jr ,norm(e jr ), where norm is the length function;

[0028] (4-6) Repeat steps (4-3) through (4-5) until all obstacles have been traversed, resulting in a simplified view of the environment.

[0029] Furthermore, step S5 specifically includes the following process:

[0030] (5-1) Based on the triangular mesh obtained in step S2, determine which triangle the starting point S is located in, and denote that triangle as T. i ;

[0031] (5-2) Find T i The set T of adjacent triangles of a triangle i_neibor Determine the set of vertices V of the triangle and its neighboring triangles. T ;

[0032] (5-3) Regarding V T Triangulation of point sets, determination of neighbor pairs, and maintenance of the local region visibility after adding the starting point into the global visibility;

[0033] (5-4) Maintain the endpoint in the global view according to steps (5-1)-(5-3).

[0034] Furthermore, step S6 specifically involves: on graph G, with the goal of finding the shortest path, using Dijkstra's algorithm to plan the path from the starting point to the ending point, and then using a pruning method to obtain the global path P. sg ={p s ,p2,...,p g}

[0035] Furthermore, step S7 specifically includes the following process:

[0036] (7-1) With the starting point s and the ending point g as the foci of the ellipse, and the path P sg The length of the ellipse is twice the length of its major axis, 2a. The ellipse E is determined, and the point set VE is obtained by discretizing the ellipse boundary according to the equal step size.

[0037] (7-2) Determine the set of boundary points V of the polygonal obstacles within the ellipse on the map. er Set of VE and V er Constitutes the sample point set V sample={q1,q2,...q ns}, where ns represents the number of samples.

[0038] Furthermore, step S8 specifically involves the following process: for the sample point set V... sample Each point q in i (i∈ns), find q i to path point sequence P sg ={p s ,p2,...,p g The coordinates of the point on the shortest path in the chain are denoted as p. o Define vector Vec1 = [q i -p o [,0]; if p o Not the endpoint coordinates, define Vec2 = [p o+1 -p o [,0]; if p o These are the endpoint coordinates, defined as Vec2 = [p o -p o-1 The classification category is determined by the sign of the z-axis value of the cross product of Vec1 and Vec2. If the z-axis value is greater than 0, the point is defined as a positive sample point; if it is less than 0, the point is defined as a negative sample point.

[0039] Furthermore, the specific process of step S9 is to define Sp = (q i ,γ i ) represents the SVM training data, q i γ represents the coordinates of the i-th sample point. i ∈{-1,+1}, where is the label of the i-th sample point. A nonlinear SVM classifier is constructed using the radial basis function as the kernel function, defined as:

[0040] K(q i ,q j )=exp(-||q i -q j || 2 / δ 2 (1)

[0041] The hyperplane equation for the nonlinear SVM is:

[0042]

[0043] Where n is the number of support vectors, α i These are the Lagrange multipliers.

[0044] The normalized sample points and their labels are then input into the radial basis function support vector machine model for training to obtain the boundary decision function. After further restoring the scale, a smooth path between the start and end points is determined at the boundary points.

[0045] The beneficial effects of this invention are:

[0046] This invention proposes a simplified visualization modeling and planning method based on Delaunay triangulation. Firstly, in visualization modeling, a constrained Delaunay triangulation is constructed using the centroid set of polygonal obstacles to build neighbor pairs, and the visualization is built using these neighbor relationships, significantly reducing redundancy and saving time. Secondly, based on the simplified visualization, a Dijkstra's search algorithm with fused pruning is used, improving planning efficiency while ensuring the shortest possible planned path. Finally, an elliptical window envelope method is introduced, and a safe and smooth path is obtained through training a support vector machine based on radial basis functions. Attached Figure Description

[0047] Figure 1 A schematic diagram of the method flow of this invention;

[0048] Figure 2 A schematic diagram of obstacle neighbor pairs constructed based on a constrained Deloitte triangulation;

[0049] Figure 3 A simplified visual diagram of the global environment based on neighbor relationships;

[0050] Figure 4 Maintain the start and end points in a globally visible diagram;

[0051] Figure 5 A schematic diagram of global path planning using Dijkstra's algorithm with pruning.

[0052] Figure 6 A schematic diagram of an elliptical window associated with a global path;

[0053] Figure 7 A schematic diagram of training sample points is generated within the elliptical frame;

[0054] Figure 8 A safe and smooth path diagram is obtained by training and solving a support vector machine based on radial basis functions;

[0055] Figure 9 Comparison of the number of sides between traditional and simplified visual view construction;

[0056] Figure 10 A comparison chart showing the time consumption of traditional visualization construction and simplified visualization construction;

[0057] Figure 11A comparison chart of time consumption and path length for planning on traditional and simplified visualizations. Detailed Implementation

[0058] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0059] A Delaunay triangulation is a set of connected but non-overlapping triangles, the circumcircles of which do not contain any other points in the region. It possesses two unique properties:

[0060] (1) The circumcircle of each Delaunay triangle does not contain any other points in the face;

[0061] (2) When the diagonals of a convex quadrilateral formed by two adjacent triangles are exchanged, the smallest of the six interior angles no longer increases.

[0062] Example 1: As Figure 1 As shown, a simplified visualization modeling and planning method based on Delaunay triangulation is presented, which includes the following steps:

[0063] S1: Model obstacles in the environment map as polygons and extract centroids, boundary vertices, discrete edges, and discrete boundary point sets;

[0064] (1-1) Identify global environmental obstacles through semantic segmentation, threshold segmentation or other image processing techniques;

[0065] (1-2) The identified obstacle regions are fitted with polygons using the convex hull and Douglas-Peucker algorithm to obtain obstacles represented by polygons. The core of the polygonal obstacle, C = {O}, is then calculated. i |i=1,2,3,…,n}, boundary vertex V i ={v ij |j=1,2,3,…,num(B i Discrete edge E i ={(v i1 ;v i2 ),(v i2 ;v i3 ),…,(v inum(Bi) ;v i1 Discrete boundary point set PE i ={pe i ,pe2,...,pe co_i ,}。 Where, v ij Let B represent the j-th vertex of the i-th obstacle. iLet i represent the obstacle numbered i, n represent the number of obstacles, num represent the function for calculating the number of obstacle vertices, and co_i represent the number of points on the boundary of the discrete obstacle i.

[0066] S2: As Figure 2 As shown, a restricted Deloitte triangulation is constructed on the obstacle material core point set obtained in step S1.

[0067] The specific process is as follows: Based on the obstacle material mind set C = {O} i Construct a restricted Delaunay triangulation for |i=1,2,3,…,n} such that the circumcircle of any triangle does not contain any other vertices, and maximize the minimum angle.

[0068] S3: As Figure 2 As shown, based on the triangulation obtained in step S2, the neighboring obstacles of each obstacle are obtained to construct neighbor pairs;

[0069] The specific process is as follows: using the relationship between the centroids and the sides of the triangles, we obtain the set of neighbor pairs of the current polygonal obstacle. i ={(B i B nj )∣i≠nj,i=1,2,3,…,n,nj=1,2,3…,n}, B i B represents obstacle numbered i. nj B i Neighbor elements, such as Figure 2 The neighbor set of obstacle number 17 is Pair. 17 ={(17,23),(17,6),(17,26),(17,16),(17,12),(17,29),(17,10)}, and determine the set of all edges of the neighboring obstacles of obstacle i.

[0070] S4: As Figure 3 As shown, a simplified global environment visibility view is constructed based on the visibility relationships between adjacent obstacle vertices obtained in step S3;

[0071] (4-1) Create a graph G = graph();

[0072] (4-2) Maintain the edge and edge length of each obstacle in the graph;

[0073] (4-3) For the j-th vertex v of the i-th obstacle ij Determine the r-th vertex v of the k-th neighboring obstacle of the i-th obstacle. ikr Connecting these two points forms edge e. jr =[v ij ;v kr ];

[0074] (4-4) Determine edge e jr Is it related to the set of edges E of the i-th obstacle? i and the set of all edges of the neighboring obstacles of the i-th obstacle. intersect;

[0075] (4-5) If they do not intersect, then edge e jr and the length of the side norm(e) jr Maintain G = addedge(G,e) in the graph jr ,norm(e jr ), where norm is the length function;

[0076] (4-6) Repeat steps (4-3) through (4-5) until all obstacles have been traversed, resulting in a simplified view of the environment.

[0077] S5: As Figure 4 As shown, based on the triangular mesh of step S2, the local regions where the starting point and the ending point are located are triangulated.

[0078] (5-1) Based on the triangular mesh obtained in step S2, determine which triangle the starting point S is located in, and denote that triangle as T. i ;

[0079] (5-2) Find T i The set T of adjacent triangles of a triangle i_neibor Determine the set of vertices V of the triangle and its neighboring triangles. T ;

[0080] (5-3) Regarding V T Triangulation of point sets, determination of neighbor pairs, and maintenance of the local region visibility after adding the starting point into the global visibility;

[0081] (5-4) The steps for maintaining the endpoint in the global view are the same as (5-1)-(5-3).

[0082] S6: Maintain the local triangulation region obtained in step S5 into the global visible view to obtain the final simplified visible view G. For example... Figure 5 As shown, on graph G, with the goal of finding the shortest path, Dijkstra's algorithm is used to plan the path from the starting point to the ending point, and then a pruning method is used to obtain the global path P. sg ={p s ,p2,...,p g}

[0083] The pruning process is as follows: Let there be k path points {P} in the path. m Given a list of nodes |m=1,2,3,…,k}, iterate through all path points and connect each path point P in turn.i (1≤i≤k-2) and other path points P j (i+2≤j≤k). If P i P j If a line segment does not pass through any obstacles on the map, then the path point P is considered to be a path point. i and P j Path points between the two paths can be removed, and the remaining path points are the pruned paths.

[0084] S7: As Figure 6 As shown, based on the global path obtained in step S6, an elliptical window associated with the path is generated on the map.

[0085] (7-1) With the starting point s and the ending point g as the foci of the ellipse, and the path P sg The length of the ellipse is twice the length of its major axis, 2a. The ellipse E is determined, and the point set VE is obtained by discretizing the ellipse boundary according to the equal step size.

[0086] (7-2) Determine the set of boundary points V of the polygonal obstacles within the ellipse on the map. er Set of VE and V er Constitutes the sample point set V sample ={q1,q2,...q ns}, where ns represents the number of samples.

[0087] S8: As Figure 7 As shown, based on the elliptical window region obtained in step S7, it is determined which side of the global path the obstacle boundary point and the elliptical boundary point are on, and the point set is divided into positive and negative samples.

[0088] For the sample point set V sample Each point q in i (i∈ns), find q i to path point sequence P sg ={p s ,p2,...,p g The coordinates of the point on the shortest path in the chain are denoted as p. o Define vector Vec1 = [q i -p o [,0]; if p o Not the endpoint coordinates, define Vec2 = [p o+1 -p o [,0]; if p o These are the endpoint coordinates, defined as Vec2 = [p o -p o-1 The classification category is determined by the sign of the z-axis value of the cross product of Vec1 and Vec2. If the z-axis value is greater than 0, the point is defined as a positive sample point; if it is less than 0, the point is defined as a negative sample point.

[0089] S9: As Figure 8 As shown, based on the positive and negative samples obtained in step S8, a safe and smooth path is obtained by training a support vector machine based on radial basis functions.

[0090] Define Sp = (q i ,γ i ) represents the SVM training data, q i γ represents the coordinates of the i-th sample point. i ∈{-1,+1}, where is the label of the i-th sample point. A nonlinear SVM classifier is constructed using the radial basis function as the kernel function, defined as:

[0091] K(q i ,q j )=exp(-||q i -q j || 2 / δ 2 )

[0092] The equation of the nonlinear SVM hyperplane is:

[0093]

[0094] Where n is the number of support vectors, α i These are the Lagrange multipliers.

[0095] The normalized sample points and their labels are then input into the radial basis function support vector machine model for training to obtain the boundary decision function. After further restoring the scale, a safe and smooth path between the start and end points is determined at the boundary points.

[0096] The advantages of this invention are verified through the following experiments:

[0097] (1) Comparison of traditional visualization and simplified visualization in terms of the number of edges and time consumption.

[0098] like Figure 9 and Figure 10 As shown, to fully demonstrate the advantages of the Delaunay triangulation-based view creation algorithm over traditional view creation methods, 20 environmental maps each with polygon obstacle occupancy rates of 0.1, 0.2, 0.3, and 0.4 were randomly generated. Views were then constructed using both the proposed method and traditional view creation methods, and the average number of edges and construction time were compared. It can be seen that the proposed Delaunay triangulation-based view creation method reduces the number of edges by more than 40% and saves more than 50% of the time.

[0099] (2) Comparison of Path Planning Results

[0100] like Figure 11As shown, 50 sets of start and end points are randomly generated on the same map (the distance between the start and end points is specified to be greater than two-thirds of the map slope distance). The proposed method based on Delaunay triangulation to establish a visual plan and the traditional visual plan are used to search for paths. It can be seen that the method proposed in this invention saves more time without reducing the effectiveness of path planning.

[0101] It should be noted that the above content merely illustrates the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. For those skilled in the art, various improvements and modifications can be made without departing from the principle of the present invention, and all such improvements and modifications fall within the scope of protection of the claims of the present invention.

Claims

1. A simplified visualization modeling and planning method based on Delaunay triangulation, characterized in that, The method includes the following steps: S1: Model obstacles in the environment map as polygons and extract centroids, boundary vertices, discrete edges, and discrete boundary point sets; Specifically, the process includes the following: (1-1) Identify global environmental obstacles through semantic segmentation, threshold segmentation or other image processing techniques; (1-2) The identified obstacle regions are fitted with polygons using the convex hull and Douglas-Peucker algorithm to obtain obstacles represented by polygons. The core of the polygon obstacle is then calculated. Boundary vertices Discrete edge Discrete boundary point set ,in, This represents the j-th vertex of the i-th obstacle. Indicates the number is The obstacles are denoted by n, where n represents the number of obstacles, and num represents a function to calculate the number of obstacle vertices. The number of points representing the boundary of discrete obstacle i; S2: Construct a restricted Deloitte triangulation for the set of obstacle material centers obtained in step S1, such that the circumcircle of any triangle does not contain any other vertices, and maximizes the minimum angle; S3: Based on the triangulation obtained in step S2, obtain the neighboring obstacles of each obstacle to construct neighbor pairs; The specific process is as follows: using the relationship between the sides of the triangles connected by the centroids, we obtain the set of neighbor pairs of the current polygonal obstacle. ,in express Find the neighbors of obstacle i, and determine the set of all edges of the neighboring obstacles of obstacle i. ; S4: Based on the visibility relationships between adjacent obstacle vertices obtained in step S3, construct a simplified global environment view; Specifically, the process includes the following: (4-1) Create a graph G = graph(); (4-2) Maintain the edge and edge length of each obstacle in the graph; (4-3) For the j-th vertex of the i-th obstacle Determine the r-th vertex of its neighboring obstacles. Connecting these two points forms an edge. ; (4-4) Determine the edge Is it related to the set of edges of the i-th obstacle? and the set of all edges of the neighboring obstacles of the i-th obstacle. intersect; (4-5) If they do not intersect, then... and the length of the side Maintain in the diagram ,in It is a function for finding length; (4-6) Repeat steps (4-3) through (4-5) until all obstacles have been traversed, resulting in a simplified view of the environment; S5: Based on the triangulation of step S2, triangulate the local areas where the starting point and the ending point are located; S6: Maintain the local triangulation region obtained in step S5 to the global view to obtain the final simplified view. Then, use the Dijkstra algorithm with fused pruning to search the global path between the starting point and the ending point according to the shortest path principle. S7: Based on the global path obtained in step S6, generate an elliptical window associated with the path on the map; S8: Based on the elliptical window region obtained in step S7, determine which side of the global path the obstacle boundary points and elliptical boundary points within the elliptical window are on, and divide the point set into positive and negative samples. S9: Based on the positive and negative samples obtained in step S8, a safe and smooth path is obtained by training a support vector machine based on radial basis functions.

2. The simplified visual image modeling and planning method based on Delaunay triangulation according to claim 1, characterized in that, In step S5, the specific details are as follows: Includes the following processes: (5-1) Based on the triangular mesh obtained in step S2, determine which triangle the starting point S is located in, and denote that triangle as . ; (5-2) Find The set of adjacent triangles of a triangle Determine the set of vertices of the triangle and its neighboring triangles. ; (5-3) Triangulation of point sets, determination of neighbor pairs, and maintenance of the local region visibility after adding the starting point into the global visibility; (5-4) Maintain the endpoint in the global view according to steps (5-1)-(5-3).

3. The simplified visual image modeling and planning method based on Delaunay triangulation according to claim 1, characterized in that, Step S6 specifically involves using Dijkstra's algorithm to plan the path from the starting point to the ending point on graph G, with the goal of finding the shortest path, and then using a pruning method to obtain the global path. .

4. The simplified visual image modeling and planning method based on Delaunay triangulation according to claim 3, characterized in that, In step S7, the specific details are as follows: Includes the following processes: (7-1) With the starting point s and the ending point g as the foci of the ellipse, and the path... The length is twice the length of the major axis of the ellipse. Determine the ellipse The point set is obtained by discretizing the elliptical boundary according to the constant step size. ; (7-2) Determine the set of boundary points of the polygonal obstacles within the ellipse on the map. ,gather and Constructing the sample point set ,in Indicates the number of samples.

5. The simplified visual image modeling and planning method based on Delaunay triangulation according to claim 4, characterized in that, The specific process in step S8 is as follows: for the sample point set Each point in ,in ,Sure With path point sequence The coordinates of the points that form the shortest path are denoted as . Define vector =[ [,0]; if Not the endpoint coordinates, defined =[ [,0]; if These are the endpoint coordinates, defined. =[ [,0] ;through Cross product The sign of the z-axis value determines the classification category. If it is greater than 0, the point is defined as a positive sample point; if it is less than 0, the point is defined as a negative sample point.

6. The simplified visual image modeling and planning method based on Delaunay triangulation according to claim 1, characterized in that, The specific process in step S9 is as follows: definition For SVM training data, Let the coordinates be those of the e-th sample point; Let be the classification label of the e-th sample point; a nonlinear SVM classifier is constructed using the radial basis function as the kernel function, defined as: ; The hyperplane equation for the nonlinear SVM is: ; Where ns is the number of samples. These are the Lagrange multipliers; The normalized sample points and their labels are then input into the radial basis function support vector machine model for training to obtain the boundary decision function. After further restoring the scale, a smooth path between the start and end points is determined at the boundary points.

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