UAV Path Planning Method Based on Spatial Deformation

Abstract

The present invention discloses a UAV path planning algorithm based on spatial deformation in an urban environment, comprising the following steps: establishing a map model including the risk cost of the urban area and expanding the safety margin of obstacles; first using the gray wolf algorithm for global path planning. Obtain information such as the starting point and end point of the global path, the turning point of the global path, the vertices of obstacles, and the boundary points of the map; use the improved thin plate spline interpolation method to perform spatial deformation, perform secondary path modification, connect the shortest straight line between the starting point and the end point, and regard it as being on a deformable metal sheet independent of the working environment. Use the information of each point obtained previously to calculate the coefficient of spatial deformation, and then apply it to this metal sheet. The deformation of the sheet causes the path to deform. The path planned by the present invention has good smoothness and safety. Compared with the traditional gray wolf algorithm, the planned path is more in line with the actual flight of the UAV and more in line with the requirements of engineering ethics.

Description

Technical Field

[0001] The present invention belongs to the field of UAV path planning, and particularly relates to a UAV path planning method based on space deformation. Background Technique

[0002] With the development of intelligent technology, unmanned aerial vehicles (UAVs) have shown more diverse functions in the new era. They perform excellently in various applications such as agricultural plant protection, urban planning, commercial promotion, fire prevention and disaster relief. At the same time, along with this, there are frequent engineering ethics problems and safety accidents brought about by the complexity of application scenarios.

[0003] For the process of a UAV to complete a task, the core is the planning of the flight path. The so-called path planning of a UAV, simply speaking, aims to design a motion trajectory to complete the task of flying from the starting point to the target point without collision and try to avoid accidents. At present, there are many research results on path planning algorithms at home and abroad, and a large number of path planning algorithms have emerged. They can be classified differently from different perspectives. In the case of a completely known environment, based on the established global map model, there are the A* algorithm using heuristic strategies, random sampling method, and intelligent algorithms such as ant colony algorithm (ACO), particle swarm optimization (PSO), artificial fish swarm algorithm (AFSA), firefly algorithm (FA), etc.; in the case of an incompletely known environment, there are artificial potential field method originating from physical ideas, behavior-based methods, rolling window method, and combined algorithms combined with traditional algorithms.

[0004] However, when applying the path planned by the algorithm to the actual application scenario, the problems that are likely to occur are that the path planned by the algorithm often lacks continuity, is not smooth enough, and in the current situation of an increasingly complex environment, the personnel safety problem under engineering ethics is also becoming more and more important. This brings potential hazards to the actual flight process of the UAV. This article comprehensively considers the smoothness, feasibility and safety of the path, and on the basis of the space deformation algorithm, performs secondary deformation on the global path generated by the gray wolf algorithm to generate a better flight path, and at the same time minimizes the energy required for deformation. To achieve smooth flight of the UAV. Summary of the Invention

[0005] Object of the Invention: The present invention provides a UAV path planning method based on space deformation in an urban environment to achieve stable and safe flight of the UAV in a modern urban environment.

[0006] Technical Solution: The present invention provides a UAV path planning method based on space deformation, including the following steps:

[0007] Step 1: Establish a map model. Considering the complex situation of flight in urban areas, expand the safety margin of obstacles; add risk cost information to the map model.

[0008] Step 2: Considering the strong convergence and few parameters of the grey wolf algorithm, use the grey wolf algorithm to plan the global path L. Set the initial number of individuals in the grey wolf pack and the maximum number of iterations of the pack for hunting.

[0009] Step 3: Based on the thin plate spline interpolation algorithm, perform spatial deformation on the global path L, make secondary path modifications, connect the shortest straight line between the starting point and the ending point, and regard it as a deformable metal thin plate independent of the working environment. Calculate the coefficients of spatial deformation using the previously obtained point information, and then apply them to this metal thin plate. The deformation of the thin plate causes the path to deform, obtaining the optimal path L'.

[0010] Furthermore, in Step 1, establish a map model, expand the safety margin of obstacles, and add risk cost information to the map model; the specific steps are as follows:

[0011] Step 1.1: Establish a map model by expanding the boundary coordinates of obstacles to expand the safety margin of obstacles;

[0012] Establish an environmental map model, rasterize the environmental map, and the formula for expanding the safety margin of obstacles can be expressed as:

[0013]

[0014] where p represents the actual length of the obstacle, q represents the actual width of the obstacle, d represents the actual distance represented by the side length of a unit grid, g(a,b) represents whether this area is traversable, taking 1 as non-traversable and 0 as traversable; x and y represent the coordinates in the grid map.

[0015] Step 1.2: Add risk cost information to the map model

[0016] During actual flight, the risk brought by the UAV needs to be considered, so a risk assessment matrix R is established (n×m) , and each grid in the grid map is given the risk factor value R ij information.

[0017]

[0018] where n is the number of rows, m is the number of columns, and R ij represents the risk factor value after quantization of the area corresponding to the grid in the i-th row and j-th column of the matrix. For risk estimation, where 1 ≤ i ≤ n, 1 ≤ j ≤ m.

[0019] The calculation formula for the risk factor value R is as follows:

[0020] R = fMP

[0021] Wherein, f is the safety index of the drone, representing its crash probability. M is the number of people affected after a crash, which depends on the population density ρ and the ground impact area S. P is the casualty of the affected population.

[0022]

[0023] Wherein: E is the kinetic energy when the drone collides; d is the environmental shielding coefficient; p is the degree of casualty. e is the impact energy required for the mortality rate to reach 50% when d = 0.5; β is the impact energy required for the mortality rate to reach 100% without shielding;

[0024]

[0025]

[0026]

[0027] Wherein: v max is the maximum operating speed; v t is the real-time speed of the drone during the fall; v is the real-time speed of the drone, ρ 空 is the local actual air density, m 0 is the mass of the drone, S is the ground impact area, R t is the drag coefficient of the drone on people.

[0028] Therefore, the risk cost generated by the drone flight is as follows:

[0029] C = R 总 × 10 4 / R 标

[0030] Wherein, R 标 represents that the drone operation meets the appropriate safety standards; R 总 represents the total sum of the risk cost values accumulated during the actual operation of the drone, that is, the sum of the risk cost values in all grids passed by the drone.

[0031] Furthermore, step 1.1 also includes removing the grid of the safe area that is not adjacent to the obstacle, and obtaining the obstacle grid with a final safety distance.

[0032] Furthermore, step 2: Use the grey wolf algorithm to plan the global path L, and the specific steps are as follows:

[0033] Step 2.1, initialize the size of the wolf pack population and the positions of each grey wolf, and set the maximum number of iterations;

[0034] Step 2.2, update the current positions of all gray wolves and calculate the fitness of all gray wolves;

[0035] Step 2.3, select new α, β, and δ wolves;

[0036] Step 2.4, update the iteration number, and end if it exceeds;

[0037] Step 2.5, the optimal solution output is the position of the α wolf, representing the global path finally obtained by the algorithm.

[0038] Furthermore, in step 3, the improved thin plate spline interpolation method is used to perform spatial deformation on the global path L to obtain the optimal path L', and the specific steps are as follows:

[0039] Step 3.1, regard the straight line d between the starting point and the ending point of the global path L as a thin plate that can be deformed arbitrarily.

[0040] Step 3.2, calculate the inflection points in the global path L, regarded as the task points that the optimal path L' must pass through, that is, the target points (x t , y t ),

[0041] Step 3.2, according to the segmentation ratio of the target points (x t , y t ) in the global path L, determine the control points (x c , y c ) on the connection line d between the starting point and the ending point, and the control points correspond to the target points one by one.

[0042] Step 3.3, use the spatial deformation algorithm based on the TPS algorithm to deform the space where the thin plate is located.

[0043] According to the TPS algorithm, when the bending energy applied during the deformation process is the smallest, the deformation function of any point is

[0044]

[0045] Among them, z i represents the two-dimensional control point (x c , y c ), n is the number of control points; z is any point to be deformed; U is the radial basis function of ||z - z i ||. It represents the influence of other control points on any point during deformation.

[0046] The expression of the deformation coefficient M is as follows:

[0047] M = E -1 Y

[0048]

[0049] Among them, Y is a matrix composed of the target point coordinates (x t , y t ) and 0 padding; E consists of two parts. One part is the matrix P composed of the control point coordinates (x t , y t ) and 1 padding, and the other part is the radial basis K composed of the Euclidean distances between the control points.

[0050] Optimize and improve the thin plate spline interpolation algorithm. When performing actual path deformation modification, raise the dimensions of the Y matrix and the P matrix and add the constraints of the obstacle vertices.

[0051] In addition, the constraints of the working environment boundary are also considered. For the improved spatial deformation algorithm, the specific components of the Y matrix and the E matrix are as follows.

[0052]

[0053]

[0054] Among them: (x o , y o ) are the coordinates of each vertex of the obstacle. If the obstacle has four vertices in the map grid, they are respectively represented as (x o1 , y o1 ), (x o2 , y o2 ), (x o3 , y o3 ) and (x o4 , y o4 ), (x p , y p ) are the boundary points that make up the map model. The map model has four boundary points, which are respectively represented as (x p1 , y p1 ), (x p2 , y p2 ), (x p3 , y p3 ), (x p4 , y p4 ).

[0055] From this, the optimal path L' after spatial deformation can be obtained.

[0056]

[0057] Beneficial effects: Compared with the prior art, the beneficial effects of the present invention are as follows: 1. The hybrid path planning method based on spatial deformation of the present invention has better smoothing performance and is easier to implement compared with the traditional grey wolf algorithm; compared with the traditional thin plate spline interpolation method, it has better global convergence; 2. The present invention takes into account the safety and engineering ethics issues in the complex urban environment, quantifies the risks in different regions, and visualizes the risks of each path, which is beneficial for selection in actual flight. Description of the Drawings

[0058] Figure 1 is a schematic diagram of the grid representation of obstacles;

[0059] Figure 2 is a flowchart of the hybrid path planning algorithm;

[0060] Figure 3 is a flowchart of the grey wolf optimization algorithm;

[0061] Figure 4 is a schematic diagram of the influence of thin plate deformation on plane points;

[0062] Figure 5 is a schematic diagram of the influence of thin plate deformation on plane paths;

[0063] Figure 6 is a grid map model diagram considering risk with a safety distance;

[0064] Figure 7 is the initial global path planned by the grey wolf algorithm;

[0065] Figure 8 is the optimal path modified by spatial deformation for the second time;

[0066] Figure 9 is the curvature curve of the optimal path. Detailed Embodiment

[0067] The present invention will be further described in detail below with reference to the drawings.

[0068] The present invention provides a UAV path planning method based on spatial deformation, as Figure 2 shown, which specifically includes the following steps:

[0069] Step 1: Establish a map model containing risk cost information, and considering the complex situation of flight in the urban area, expand the safety margin of obstacles.

[0070] Establishing an environmental map model is the first step in UAV path planning. Rasterize the map, and the obstacles are represented as non-traversable black grids in the grid map. The center of the obstacle in the environmental map is represented by the coordinates (a, b), and expand its coordinates to obtain a map model containing a safety margin, asFigure 1 (a) and Figure 1 as shown in (b). The formula for expanding coordinates can be expressed as:

[0071]

[0072] where p represents the actual length of the obstacle, q represents the actual width of the obstacle, d represents the actual distance represented by the side length of a unit grid, g(a, b) represents whether this area is traversable, taking 1 as non-traversable and 0 as traversable; x and y represent the coordinates in the grid map.

[0073] For irregular obstacles, they are divided into regular obstacles and the above formula is used for coordinate expansion. To avoid waste of spatial resources during planning, the grid cells in the safe area that are not adjacent to the obstacles are removed, and the final grid representation of the obstacle with a safety distance can be obtained, as shown in Figure 1 (c).

[0074] During actual flight, the risks brought by the UAV need to be considered, so a risk assessment matrix R (n×m) is established. Each element R ij in the matrix corresponds to a grid in the grid map, and information on risk factor values is assigned to each grid in the grid map.

[0075]

[0076] where n is the number of rows, m is the number of columns, and R ij represents the risk factor value after quantization of the area corresponding to the grid in the i-th row and j-th column of the matrix. For risk estimation, where 1 ≤ i ≤ n, 1 ≤ j ≤ m.

[0077] There are open areas and areas with dense buildings in the map, and the population density is different in different areas. Set relevant risk parameters, including f, ρ 空 , v max , R 标 , etc. Calculate different risk factor values for different areas. Different areas correspond to different grid positions, and the risk factor values are assigned to the grid map.

[0078] The grid map model containing risk information and safety margin is as shown in Figure 6 . The black represents the obstacle, the gray adjacent to the obstacle represents the safety margin, and the remaining gray-scale grid colors represent the increase in risk factor values from light to dark. The formula for calculating the risk factor value R is as follows:

[0079] R = fMP

[0080] where f is the safety index of the UAV, representing its probability of crashing. M is the number of people affected after crashing, depending on the population density ρ and the ground impact area S. P is the casualty of the affected population.

[0081]

[0082] Where: E is the kinetic energy when the UAV collides; d is the environmental shielding coefficient; p is the degree of casualty. e is the impact energy required for the mortality rate to reach 50% when d = 0.5; β is the impact energy required for the mortality rate to reach 100% without shielding;

[0083]

[0084]

[0085]

[0086] Where: v max is the maximum operating speed; v t is the real-time speed of the UAV during falling; ρ 空 is the actual local air density, m 0 is the mass of the UAV, S is the ground impact area, R t is the drag coefficient of the UAV on people.

[0087] Therefore, the risk cost generated by the UAV flight is as follows:

[0088] C = R 总 × 10 4 / R 标

[0089] Where, R 标 represents that the UAV operation meets the appropriate safety standards; R 总 represents the total sum of the risk cost values accumulated during the actual operation of the UAV

[0090] Step 2: Use the Grey Wolf Algorithm to plan the global path L.

[0091] Set the initial number of individuals in the Grey Wolf pack and the maximum number of iterations of the pack, and conduct hunting. Obtain the start and end points of the global path and the information of the inflection points of the global path.

[0092] As Figure 3 shown, the specific description is as follows:

[0093] Step 1: Initialize the size of the wolf pack population and the positions of each grey wolf, and set the maximum number of iterations;

[0094] Step 2: Update the current positions of each grey wolf and calculate the fitness of all grey wolves;

[0095] Step 3: Select new α, β, and δ wolves;

[0096] Step 4: Update the number of iterations, and end if exceeded;

[0097] Step 5: The optimal solution output is the position of the alpha wolf, representing the global path finally obtained by the algorithm.

[0098] To simulate the hierarchical system of grey wolves while simplifying the algorithm, it is assumed that there is one alpha wolf, one beta wolf, and one delta wolf as the leaders of the grey wolves. The optimal solution is regarded as the alpha wolf, and the second and third best solutions are regarded as the beta wolf and the delta wolf respectively. The remaining candidate solutions are assumed to be omega wolves. In the grey wolf optimization algorithm, the hunting behavior is mainly guided by the alpha, beta, and delta wolves, and the omega wolves follow these three wolves.

[0099] Define the behavior of grey wolves surrounding their prey as follows:

[0100] D i =|C i ·X i (t)-X(t)|

[0101] In the above formula, D i represents the distance between the alpha, beta, and delta wolves and the omega wolf;

[0102] X i (t + 1) = X i (t)-A i ·D i ,

[0103] The above formula represents the position update formula for the omega wolf following the alpha, beta, and delta wolves respectively;

[0104]

[0105] The above formula represents the final position formula of the omega wolf;

[0106] i = 1, 2, 3; X 1 (t), X 2 (t), X 3 (t) are the current positions of the alpha, beta, and delta wolves respectively, and X 1 (t + 1), X 2 (t + 1), X 3 (t + 1) are the positions generated by the omega wolf following the alpha, beta, and delta wolves respectively. D 1 ,D 2 ,D 3 are the distances between the alpha, beta, and delta wolves and the omega wolf respectively, and A 1 ,A 2 ,A 3 and C 1 ,C 2 ,C 3 is a randomly generated vector, and the calculation formula is A i= 2a·rand() - a, C i = 2·rand(), where a is the convergence factor, which decreases linearly with the number of iterations, and rand() is a random number between 0 and 1.

[0107] After obtaining the global path L by the Grey Wolf Optimization Algorithm, extract the inflection points of the path and the coordinates of all vertices of the obstacles.

[0108] Step 3, the influence of the thin plate spline interpolation algorithm on points on the plane is as Figure 4 shown. Apply the improved thin plate spline interpolation method to the global path L for spatial deformation, perform secondary path modification, connect the shortest straight line between the start point and the end point, and regard it as a deformable metal thin plate independent of the working environment. Calculate the coefficients of spatial deformation using the information of each point on the global path and apply them to the metal thin plate. The deformation of the thin plate causes the deformation of the path, and the deformation process is as Figure 5 shown. In complex cases, the information of each point refers to the start point and end point of the global path, the inflection points of the global path, the vertices of the obstacles, and the boundary points of the map.

[0109] Specifically, regard the global path and the straight line between the start point and the end point as deformable arbitrarily. Regard the non-smooth inflection points of the global path finally obtained in Step 2 as the task points that the optimal path must pass through, that is, the target points (x t , y t ). According to the segmentation ratio of the target points in the original global path, determine the control points (x c , y c ) on the connection line d between the start point and the end point. The control points correspond to the target points one by one. Apply the spatial deformation algorithm based on the TPS algorithm to deform the space where the thin plate is located.

[0110] According to the TPS algorithm, when the bending energy applied during the deformation process is the smallest, the deformation function of any point is

[0111]

[0112] where z i is the control point, n is the number of control points; z is any point to be deformed; U is the radial basis function of ||x - x i ||. It represents the influence of other control points on any point during deformation.

[0113] The expression of the deformation coefficient M is as follows:

[0114] M = E -1 Y

[0115]

[0116] Among them, Y is a matrix composed of target point coordinates and 0 padding; E mainly consists of two parts. One part is the matrix P composed of control point coordinates and 1 padding, and the other part is the radial basis K composed of the Euclidean distances between control points. Due to the special nature of the path, no deformation occurs at the starting point and the ending point.

[0117] In the deformation of the flight path, obstacle avoidance needs to be considered. Therefore, the thin plate spline interpolation algorithm is optimized and improved. When actually modifying the path deformation, the Y matrix and the P matrix are dimensionally increased, and the constraints of the obstacle vertices are added.

[0118] In addition, the constraints of the working environment boundary should also be considered. The minimum bending energy under multiple constraints can achieve deformation. At this time, after the improved spatial deformation algorithm, the specific components of the Y matrix and the E matrix are as follows.

[0119]

[0120]

[0121] Among them: (x o , y o ) are the coordinates of each vertex of the obstacle. If the obstacle has four vertices in the map grid, they are respectively represented as (x o1 , y o1 ), (x o2 , y o2 ), (x o3 , y o3 ) and (x o4 , y o4 ), (x p , y p ) are the boundary points that make up the map model. The map model has four boundary points, which are respectively represented as (x p1 , y p1 ), (x p2 , y p2 ), (x p3 , y p3 ), (x p4 , y p4 ).

[0122] Thus, the optimal path L' after spatial deformation can be obtained.

[0123]

[0124] According to the above steps, set the risk calculation parameters, such as safety index, shielding coefficient, air density, UAV mass, etc., calculate the risk cost information of each grid and expand the obstacle coordinates; use the hybrid path planning method based on spatial deformation for planning. First, use the grey wolf algorithm for global planning, and the generated path is as Figure 7As shown, the length is 30.9706, the risk cost is 13998, and there are many uneven inflection points. Therefore, based on the improved thin plate spline interpolation for spatial deformation, the path is modified twice, the deformation coefficient M is calculated, and the path is spatially deformed to generate an optimal path as Figure 8 shown, with a length of 30.8811 and a risk cost of 14030. The curvature curve of the optimal path is as Figure 9 shown, all < 1, which is a reasonable flyable path. The path planned by the method proposed in the present invention is compared with the path planned by the traditional grey wolf algorithm without considering the safety margin. The path length of the traditional grey wolf algorithm is 29.2132, and the risk cost is 17276. At the cost of losing 5.7% of the path length, this method plans a path that is safer and more in line with the flight dynamics of the UAV. It is more conducive to flight in complex urban situations.

[0125] The above are only the preferred embodiments of the present invention, and do not impose any form of limitation on the present invention. Those skilled in the art can make various equivalent changes and improvements based on the above embodiments. Any equivalent changes and modifications made within the scope of the claims shall fall within the protection scope of the present invention.

Claims

1. Unmanned Aerial Vehicle (UAV) path planning method based on spatial deformation, characterized in that, it specifically includes the following steps: Step 1: Establish a map model, expand the safety margin of obstacles, and add risk cost information to the map model. The specific steps of Step 1 are as follows: Step 1.1, establish an environmental map model, rasterize the environmental map, and the formula for expanding the safety margin of obstacles can be expressed as: where p represents the actual length of the obstacle, q represents the actual width of the obstacle, d represents the actual distance represented by the side length of a unit grid, g(a,b) represents whether this area is traversable, taking 1 as non-traversable and 0 as traversable; x and y represent the coordinates in the grid map; Step 1.

2. During actual flight, the risks brought by the UAV need to be considered, so a risk assessment matrix R is established (n×m) , and a risk factor value R is assigned to each grid in the grid map ij information: where n is the number of rows, m is the number of columns, and R ij represents the risk factor value after quantization of the area corresponding to the grid in the i-th row and j-th column of the matrix. For risk estimation, 1 ≤ i ≤ n and 1 ≤ j ≤ m; The calculation formula for the risk factor value R is as follows: R = fMP where f is the safety index of the UAV, representing its crash probability, M is the number of people affected after a crash, which depends on the population density ρ and the ground impact area S, and P is the casualty of the affected population: where E is the kinetic energy generated when the UAV collides, d is the environmental shielding coefficient, p is the degree of casualty, e is the impact energy required for the mortality rate to reach 50% when d = 0.5, and β is the impact energy required for the mortality rate to reach 100% without shielding; Among them, v max is the maximum running speed, v t is the real-time speed of the drone's fall, v is the real-time speed of the drone, ρ 空 is the local actual air density, m 0 is the mass of the drone, S is the ground impact area, R t is the drag coefficient of the drone; Therefore, the risk cost generated by the UAV flight is as follows: C = R 总 × 10 4 / R 标 wherein, R 标 indicates that the operation of the drone meets appropriate safety standards, and R 总 represents the total sum of the risk cost values accumulated during the actual operation of the drone; Step 1.1 also includes removing the grid of the safe area that is not adjacent to the obstacle to obtain the obstacle grid with a final safety distance; Step 2: Use the Grey Wolf Algorithm to plan the global path L. The specific steps of Step 2 are as follows: Step 2.1, initialize the size of the wolf pack population and the positions of each grey wolf, and set the maximum number of iterations; Step 2.2, update the current positions of each grey wolf and calculate the fitness of all grey wolves; Step 2.3, select new α, β, and δ wolves; Step 2.4, update the number of iterations, and end if it exceeds; Step 2.5, the optimal solution output is the position of the α wolf, representing the global path finally obtained by the algorithm; Step 3, perform spatial deformation on the global path L using the improved thin plate spline interpolation method to obtain the optimal path L'; the specific steps of Step 3 are as follows: Step 3.1, regard the straight line d between the starting point and the ending point of the global path L as a thin plate that can be arbitrarily deformed; Step 3.2, calculate the inflection points in the global path L, regarded as the task points that the optimal path L’ must pass through, which are the deformed target points (x t , y t ); Step 3.2, according to the segmentation ratio of the target point (x t , y t ) in the global path L, determine the control point (x c , y c ) on the connection line d between the start point and the end point, and the control points correspond to the target points one by one; Step 3.3, use the spatial deformation algorithm based on the TPS algorithm to deform the space where the thin plate is located; According to the TPS algorithm, when the bending energy applied during the deformation process is the smallest, the deformation function of any corresponding point is: Among them, z i represents two-dimensional control points (x c , y c ), where n is the number of control points; z is an arbitrary point to be deformed; U is the radial basis function of ||z - z i ||, representing the influence of other control points on an arbitrary point during deformation; The expression of the deformation coefficient M is as follows: M = E -1 Y Among them, Y is a matrix composed of the target point coordinates (x t , y t ) and 0 padding; E consists of two parts. One part is a matrix P composed of the control point coordinates (x t , y t ) and 1 padding, and the other part is a radial basis K composed of the Euclidean distances between the control points; Optimize and improve the thin plate spline interpolation algorithm. When actually modifying the path deformation, raise the dimensions of the Y matrix and the P matrix and add the constraints of the obstacle vertices; In addition, the constraints of the working environment boundary are also considered. After the improved spatial deformation algorithm, the specific components of the Y matrix and the E matrix are as follows: Among them, (x o , y o ) are the coordinates of each vertex of the obstacle. If the obstacle has four vertices in the map grid, they are respectively represented as (x o1 , y o1 ), (x o2 , y o2 ), (x o3 , y o3 ) and (x o4 , y o4 ). (x p , y p ) are the boundary points that make up the map model. There are four boundary points of the map model, which are respectively represented as: (x p1 , y p1 ), (x p2 , y p2 ), (x p3 , y p3 ), (x p4 , y p4 ) Thus, the optimal path L' after spatial deformation can be obtained:

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