A Path-Planning Approach for an Unmanned Vehicle in an Off-Road Environment Based on an Improved A* Algorithm

Abstract

Path planning for an unmanned vehicle in an off-road uncertain environment is important for navigation safety and efficiency. Regarding this, a global improved A* algorithm is presented. Firstly, based on remote sensing images, the artificial potential field method is used to describe the distribution of risk in the uncertain environment, and all types of ground conditions are converted into travel time costs. Additionally, the improvements of the A* algorithm include a multi-directional node search algorithm, and a new line-of-sight algorithm is designed which can search sub-nodes more accurately, while the risk factor and the passing-time cost factor are added to the cost function. Finally, three kinds of paths can be calculated, including the shortest path, the path of less risk, and the path of less time-cost. The results of the simulation show that the improved A* algorithm is suitable for the path planning of unmanned vehicles in a complex and uncertain environment. The effectiveness of the algorithm is verified by the comparison between the simulation and the actual condition verification.

1. Introduction

In much off-road environmental work, unmanned vehicles play important roles, including rapid transportation, emergency rescue, and environmental reconnaissance [1,2,3]. The driving conditions of the unmanned vehicle in the off-road uncertain environment are different from those in the structured road condition, and there are many kinds of complex terrain in the off-road environment. At the same time, there are many kinds of risks in the uncertain environment, such as hidden minefields, vehicles at risk of explosion, and combustion points at risk of explosion. Therefore, when the path-planning algorithm of an unmanned vehicle in an off-road uncertain environment is studied, it should meet the requirements of avoiding collision with obstacles in the general path-planning algorithm, and also, the ground condition situation and risk situation in the off-road uncertain environment should be analyzed [4,5,6,7]. For these reasons, planning a passable path for unmanned vehicles to a destination is a challenging task.

The premise of path planning is to create a map which can accurately describe the condition and adapt to the path-planning algorithm. Traditional path-planning maps include raster maps, topological maps, and vector maps [8,9]. The distinguishing feature of a topological diagram is the intuitive expression of the position relationships between various points. It is generally used for route planning under the conditions of a structured highway network. [10,11]. A raster map uses a grid as the smallest unit to describe the condition, and the construction method is simple, which can reflect the obstacle condition in the map. Raster maps are suitable for most node search algorithms, such as the A* algorithm, particle swarm algorithm, and Q-learning algorithm [12,13,14,15]. Vector maps encode and define the attributes of various contents on the map, and determine the categories, levels, and characteristics of map elements. The contents on the map can then be represented by their code, attribute description, and corresponding coordinate position. Vector maps are characterized by the ability to reflect the characteristics of different areas [16,17]. When constructing the path-planning map of an off-road environment, on the one hand, the obstacles should be accurately described, and on the other hand, the characteristics of the off-road uncertain environment should be reflected, and finally, the complexity of the calculation should be ensured within the scope of practical application. Combining the advantages of a raster map with those of a vector map, the raster map method can be used to describe the position of obstacles, and the vector map method can be used to describe the ground conditions and risk situation in the environment so that the obstacle information and environment information can be added to the path-planning map [18].

In terms of path-planning algorithms, many researchers have improved many classical algorithms, including the heuristic search A* algorithm, rapidly-exploring random trees (RRT) algorithm, bionic ant colony algorithm, and machine learning Q-learning algorithm [19,20,21,22]. The principle of the RRT algorithm is to start from the starting point, adopt a tree-shaped branch structure, randomly sample the map, find the point closest to the sampling point and an accessible connection in the path tree, connect the point with the sampling point, and add the sampling point to the path tree until the area near the end point is explored [23]. On the basis of the RRT algorithm, the steps of re-selecting the parent node and rerouting are added to define the RRT* algorithm, which can determine a shorter distance path [24,25]. The ant colony algorithm is a heuristic algorithm that simulates the activities of ants in nature. The path node traversed by the ant colony generates a random information concentration which is determined by the ant’s state transition rule in the subsequent selection of the next path node and leaves the pheromone concentration in the next path node. As the number of times a better node is selected increases, the pheromone concentration of that node also increases to attract more ants, and the nearest path is generated after multiple searches for the target point [26,27,28]. Q-learning is a kind of reinforcement learning algorithm, which is a model-free algorithm based on Q value iteration. Q (Si, Ai) is the estimate of the return from taking action Ai in the state Si at that moment. After exploring the state Si and all possible actions, A(Si), at a given moment, an optimal action can be selected to enter the next state, Si + 1, according to the feedback and return information of the environment; this is repeated until the end or artificial termination, and a passable path is obtained [29,30,31]. The application environment of the above path-planning algorithms is the traditional obstacle environment. If it needs to be applied in the off-road environment, these algorithms should be improved.

The heuristic search A* algorithm uses the cost function to obtain the heuristic in-formation related to the target point and guides the search direction. The A* algorithm can find the optimal path while maintaining high efficiency, and the design of heuristic functions is flexible [32,33,34,35]. Different heuristic functions will affect the performance of the algorithm. The A* algorithm has been widely used in various fields such as drones, autonomous vehicles, and mobile robots. However, its application in the field of navigation path planning is less developed. Although the traditional A* algorithm can calculate the shortest path, it is limited by the fixed search direction and the fixed search-step size, and the obtained path may lack smoothness. Only the distance factor can be calculated, while other factors cannot be calculated [36,37,38]. Xie et al. used flexible search angle and variable search-step size, and then represented the distribution of obstacles in the form of an artificial potential field distribution, added the distance to the obstacles to the heuristic function of the A* algorithm, which could make the ship’s sea-path planning away from obstacles, and finally obtained a safe sea ship A* path-planning algorithm [39]. Yuan et al., designed a 3D environment octree map model combined with the line-of-sight algorithm to transform the search-step size in the 3D environment. In addition, the heuristic function of the A* algorithm added the correlation coefficient of the distance between the search node and the starting point and the end point, thus improving the efficiency of long-distance path planning for UAVs and ensuring the accuracy of short-distance path-planning [40]. Zhang et al. used the six-direction search method, and then added the equipment driving capacity limit and the risk in the environment into the cost function, and further optimized the path-planning algorithm in the environment with risk factors. [41]. In planning the movement path of a vehicle, the basic parameters of the movement path, such as speed, accuracy of the movement path, and the type of trajectory itself, play an important role [42].

On the basis of previous research, a new A* algorithm is proposed in this paper. According to the actual situation of the environment, the distribution map of the risk is established. The distribution map of the passing-time cost is established according to the ground conditions. The search method and cost function of the traditional A* algorithm are improved. A path-planning simulation experiment is conducted, and it is determined that the improved A* algorithm can calculate a path with less risk and a path with less passing-time cost in the off-road uncertain environment. The main contributions are summarized as follows:

(1)

This paper designed a grid map with a pixel as the cell. An artificial potential field model was used to describe the distribution of risk in the environment, as presented. In addition, according to the ground conditions and the performance of the vehicle, the passing-time cost of the vehicle was analyzed, and a passing-time cost distribution map was established.

(2)

A multi-direction search method was used to select the child nodes. In addition, combining the distribution of risk and the distribution of time-cost, a new calculation method of cost function was designed.

(3)

Path-planning simulation experiments validated that the improved A* algorithm proposed in this paper can determine the path with the shortest length, the path with the least risk, and the path associated with the least amount of time.

2. Identification of Environment

2.1. Set Pixel Points to Grid Points

The grid map is a map composed of many grids, each of which can represent the traffic conditions of a corresponding area. The function of the grid diagram is to discretize the actual traffic conditions. After the actual conditions were divided into grids, there were three mesh states: no obstacles in the grid at all, obstacles in the local grid, and obstacles in the entire area composed of the grids [39]. Therefore, the size of the grid is the most important indicator of map rasterization; the smaller the size of the grid and the finer the grid, the more accurate the description of the map, as shown in Figure 1. The red curve is the shape of the obstacle, and the gray grid is the grid occupied by the obstacle.

Based on remote sensing images, this research effort first used traditional image processing methods and deep learning methods to classify and recognize remote sensing images, and then segmented different types of regions in remote sensing images. The obstacle areas existing in the off-road environment, such as buildings, bodies of water, and trees, were obtained, as well as the passable areas, such as hardened roads, muddy roads, smooth surfaces, rough surfaces, grasslands, and farmlands. Using the pixel of the image as the grid of the raster map, the obstacle area in the map could be accurately expressed. Finally, an obstacle map was constructed based on the segmentation results of remote sensing images, and was represented by binarized images, with white representing a passable area and black representing an obstacle area. The result is shown in Figure 2.

2.2. Identification of Risk Areas

In the off-road uncertain environment, in addition to obstacles and ground conditions, other potentially dangerous risk conditions need to be analyzed. These potential risks include potentially explosive areas, mined areas, burning fires, hidden traps, and other types of attack risks. The typical potentially-dangerous-risk distribution model is the radiation model from the center to the periphery. The center of the circle is the center risk-point, and the risk decreases as the distance from the center of the circle increases. Using ranges of different sizes to represent risk areas can affect the results in path planning. Therefore, the use of a fixed size range indicates that the scope of the risk area is not suitable for path planning in an uncertain environment, as shown in Figure 3. The yellow areas represent the high risk areas, the green areas represent the medium risk areas, and the blue areas represent low risk areas.

According to the risk distribution model, the risks are evenly distributed in all directions, so the APF model (artificial potential field) can be used to represent the probability that the vehicle will encounter a risk inside the uncertain environment. There are many kinds of potential field functions in an artificial potential field. In this paper, the two-dimensional Gaussian distribution function is used as the risk potential field distribution function [43,44,45]. The general function expression is expressed as

$$p(x,y)=\frac{T{H}_0}{2\mathrm{\pi }{\left|{\Sigma }_0\right|}^{1/2}}\exp \left[-\frac{1}{2}{\left(X-{\mu }_0\right)}^{\mathrm{T}}{\Sigma }_0^{-1}\left(X-{\mu }_0\right)\right];X=\left(\begin{array}{l}x\\ y\end{array}\right);{\mu }_0=\left(\begin{array}{l}{x}_0\\ y_0\end{array}\right);{\Sigma }_0=\left(\begin{array}{cc}{\sigma }_{x0}^2& 0\\ 0& {\sigma }_{y0}^2\end{array}\right)$$

(1)

where ${\mu }_0$ is the center point of the risk and $T{H}_0$ is the risk parameter. These are set to different values according to different types of risk. Generally, risk distribution has the same probability distribution in each direction. Therefore, the covariance ${\Sigma }_0$ is the multiplication of positive real numbers with the identity matrix. The probability correlation coefficient ${\sigma }_{xy}$ in the X and Y directions is 0. This model has low requirements as to calculation difficulty and data storage.

When there are many risk-points in the environment, two-dimensional Gaussian mixture probability distribution is used to represent the mathematical model of risk. The function expression is

$$p(x,y)={\displaystyle \sum _{k=0}^k\frac{T{H}_k}{2\mathrm{\pi }{\left|{\Sigma }_k\right|}^{1/2}}\exp \left[-\frac{1}{2}{\left(X-{\mu }_k\right)}^{\mathrm{T}}{\Sigma }_k^{-1}\left(X-{\mu }_k\right)\right]};{\mu }_k=\left(\begin{array}{l}{x}_k\\ y_k\end{array}\right);{\Sigma }_k=\left(\begin{array}{cc}{\sigma }_{xk}^2& 0\\ 0& {\sigma }_{yk}^2\end{array}\right)$$

(2)

where k is the number of risk-points.

It is assumed that there are one or two potential explosion points in the environment, and the potential location of the center coordinate of the explosion point is the center point of the probability distribution. The probability distribution of a single risk-point is shown in Figure 4a, and the probability distribution of two risk-points is shown in Figure 4b. The color yellow to blue in the graph indicates the value of the distribution function from large to small.

In order to obtain a risk probability map suitable for path planning, the two-dimensional Gaussian distribution is discretized. Firstly, combining the pixel size of the remote sensing image, a matrix consistent with the image pixel size is constructed. Second, the probability distribution value at each pixel coordinate point is calculated, and this value is filled into the corresponding position of the matrix to obtain a risk probability matrix with the same size as the image pixel. Finally, the values in the matrix are normalized to obtain the explosive risk distribution map for path planning, as shown in Figure 5. The color yellow to blue in the graph indicates the value of the distribution function from large to small. The function values in each grid are magnified in Figure 5.

The risk probability map is used to represent the risk distribution in the uncertain environment. This method is suitable for expressing the characteristics of uncertainty and complexity of the off-road environment and can reduce the complexity of the path-planning space.

2.3. Identification of Ground Conditions

In the off-road environment condition, as for unmanned vehicles with a certain off-road capability, they can travel under a variety of ground conditions. There many types of ground conditions, including hardened roads, hardened ground, gravel ground, rough ground, grass, muddy ground, farmland, etc. The ground condition is related to the driving speed of the vehicle; the better the ground passing condition, the higher the driving speed of the vehicle, and the shorter the passing-time cost of the vehicle. The worse the ground traffic conditions, the lower the vehicle speed, the longer the driving time, and the greater the cost of vehicle traffic time. By analyzing the ground characteristics of these passable areas and combining the determinations of field-driving capability of unmanned vehicles, this study quantifies the passing-time cost of different types of areas, setting the passing-time cost for all types of ground conditions. In order to show the distinction clearly, different colors are set to represent different ground conditions, as shown in

Table 1. The features and parameters of different ground conditions.

Ground Type	Picture	Feature	Time Cost	Display Color
Hardened road		High speed	0.1
Hardened ground		Relatively high speed	0.2
Gravel ground		Medium speed	0.3
Rough ground		Medium speed	0.4
Grass		Slow	0.6
Muddy ground		Relatively slow	0.7
Farmland		Hyperslow	0.8

.

All of the kinds of ground conditions in the map are segmented, and then the distribution of the passing-time cost of each point in the map is expressed by a pixel as the unit, and a passing-time cost matrix with the same size as the pixel map is obtained. This matrix is the time–cost map for path planning, as shown in Figure 6. The colors represent the same generational value as Table 1.

3. Improved A* Algorithm

3.1. Principle of the A* Algorithm

An A* algorithm is a classic heuristic search algorithm based on Dijkstra’s algorithm and combined with heuristic rules. By calculating the distance attribute of the node, an A* algorithm can ensure that the obtained path is the shortest path, and at the same time it can ensure a quick computing speed. The core calculation expression of an A* algorithm is

$$f(n)=g(n)+h(n)$$

(3)

where g(n) is the cost value from the starting point to the current point, representing the sum of the costs of the path traveled; h(n) is the estimated cost value of the current point to the end point; f(n) is the total cost value from the starting point to the target point, and the value is the sum of g(n) and h(n). The operation method of the A* algorithm is to start the search from the starting grid, calculate the g(n), h(n), and f(n) of the neighboring grids successively, and then select the grid with the smallest f(n) value as the new search starting point to continue to expand around; the algorithm ends when the search reaches the target grid. The core of the A* algorithm is h(n), also known as the heuristic function, which is able to guide the path towards the target point continuously. The diagram of the A* algorithm is shown in Figure 7. The green grid is the starting point, the red grid is the end point, the yellow grid is the path, the white grid is the optional area, and the black grid is the obstacle area.

3.2. Multi-Direction Search Method

The search direction of the traditional A* algorithm is generally four or eight directions, and the search points are adjacent grids, as shown in Figure 8a,b. In this paper, a multi-direction search method is proposed. Firstly, a reasonable search radius size is set, and then a circle search is conducted, with the parent node as the center and the search radius size as the radius. This search method can break through the limitation of angle and complete a multi-direction search. In addition, the steering angle limit of the vehicle can be combined with the search angle. The multi-direction search method is shown in Figure 8c. The red grid is the search starting point, the green grid is the search reachable point, and the arrow direction is the path direction.

Different from the search method of adjacent grids, the multi-direction search method needs to determine whether the point passed by the search radius is an obstacle. A new line-of-sight algorithm is designed in this paper. When the unmanned vehicle is driving in the off-road environment conditions, it has a certain width and volume, but the planned path is a straight line in the ideal state without any calculation of the width. In order to solve this problem, the line-of-sight algorithm designed in this paper judges the distance between the pixel and the line of the path. If the distance between the pixel and the line of the path is less than half the width of the vehicle (W), it is determined that the pixel is the one the vehicle passes through. Then it is determined whether there are obstacles in the pixels contained in the path. The method is shown in Figure 9. The red grid is the search starting point, the green grid is the search reachable point, the blue grid represents the grid through which the vehicle travels, and the arrow direction is the path direction.

The flow chart of the multi-direction search method is shown in Figure 10.

3.3. Set Cost Function

The core step of the traditional A* algorithm is to select a point in the constantly updated set of child nodes so that the sum of the moving-cost from the starting point to the selected node to the destination point is minimized. In the process of calculating the cost, the value of the cost function changes with the change of the search location information. According to the multi-directional search method designed, as described in Section 3.2, $g_{\mathrm{short}}(n)$ in the A* algorithm represents the cost from the starting point to the search point, which can be expressed as

$$g_{\mathrm{short}}(n)=nR$$

(4)

where n is the number of search steps from the starting point to the search point, and R is the search radius. The search step can represent the distance cost, and it can alsop be used to represent the risk in the environment, as well as the time-cost. In this paper, the risk factor and the time-cost are added to the cost function g(n).

Consistent with the risk map established in Section 2.2, the calculation method of risk value in the search step is as follows: the multi-direction search algorithm in this paper (Section 3.2) is used to determine the pixels passed by the path from the starting point to the search point, and then the corresponding values of these pixels in the risk map are summed. The total risk value of the nth node can be expressed as

$$g_{\mathrm{risk}}(n)=g_{\mathrm{risk}}(n-1)+{\displaystyle \sum _{i=1}^R{P}_{\mathrm{risk}}(i)}$$

(5)

where $g_{\mathrm{risk}}(n-1)$ is the sum of the risk value from the starting point to the parent node; R is the search radius; and $P_{\mathrm{risk}}(i)$ is the risk value of each pixel in the search radius.

According to the passing-time cost map established in Section 2.3, on this basis, the passing-time cost as to path planning is analyzed. The calculation method of time-cost in the search step is similar to the calculation method of risk value above. Firstly, the pixels passed by the path from the starting point to the search point are determined, and then the corresponding values of these pixels in the time-cost map are summed. The total time-cost of the nth node can be expressed as

$$g_{\mathrm{time}}(n)=g_{\mathrm{time}}(n-1)+{\displaystyle \sum _{i=1}^R{P}_{\mathrm{time}}(i)}$$

(6)

where $g_{\mathrm{time}}(n-1)$ is the sum of the time-cost from the starting point to the parent node; R is the search radius; and $P_{time}(i)$ is the time-cost of each pixel in the search radius.

Therefore, g(n) can be selected in three ways, namely, the representing distance cost, the risk value and the time-cost, respectively. The g(n) function is expressed as

$$g(n)=\left\{\begin{array}{l}{g}_{\mathrm{short}}(n)\\ g_{\mathrm{risk}}(n)\\ g_{\mathrm{time}}(n)\end{array}\right.$$

(7)

In addition, the heuristic function h(n) in the A* algorithm plays a role in guiding the search path to the end point. In this study, the traditional Euclidean distance is used as the heuristic function to ensure the search integrity of the A* algorithm.

$$h(n)=\sqrt{{(n.x-goal.x)}^2+{(n.y-goal.y)}^2}$$

(8)

According to different task requirements, the cost function of the improved A* algorithm can be divided into several categories.

When executing the task that requires the shortest path, the cost function is as follows:

$$f_{\mathrm{short}}(n)=g_{\mathrm{short}}(n)+h(n)$$

(9)

When executing a task that requires high security, select the path scheme with less risk value. The cost function is as follows:

$$f_{\mathrm{risk}}(n)=g_{\mathrm{risk}}(n)+h(n)$$

(10)

When executing the task that requires timeliness, select the path that consumes less time-cost. The cost function is as follows:

$$f_{\mathrm{time}}(n)=g_{\mathrm{time}}(n)+h(n)$$

(11)

The main steps of the improved A* algorithm in this paper are as follows:

Step 1: The remote sensing map is processed to distinguish the passable area and the obstacle area in the map, and the obstacle map for path planning is obtained.

Step 2: According to the actual situation of an off-road uncertain environment, the distribution model of risk-points is established, and a risk distribution map is established.

Step 3: According to the type of ground conditions, the time-cost is computed and the distribution map of time-cost is obtained.

Step 4: The search radius size and search angle are set in the multi-direction search method, and the starting point and ending point are set. The closed list and the open list are set, and the starting point is added to the open list.

Step 5: It is determined whether the open list is empty. If not, Step 6 is commenced. If the open list is empty, the algorithm stops running.

Step 6: According to different task requirements, the different types of cost functions f(n) of the nodes in the open list are calculated, and then the node with the smallest value of f(n) is selected and put into the closed list.

Step 7: The multi-direction search method is used to search for the nodes reachable by the parent node, and the qualified nodes are stored in the open list, and the cost functions f(n) of these nodes are calculated.

Step 8: It is determined whether the end point is reachable within the search range. If the end point is reachable, Step 9 commences. If the end point is unreachable, Step 5 commences.

Step9: The nodes in the closed list are saved and connected to determine the final planned path.

The main program flow chart of the improved A* algorithm is shown in Figure 11.

In order to explain the algorithm more clearly, the pseudo-code of the algorithm is included below. The algorithm’s main function, as shown in the pseudocode, is given in Algorithm 1. The Helper Function is divided into four parts. The Helper Function1, FunctionMin, calculates the cost function value of the node and selects the node with the lowest cost function value, corresponding to line 8 in the main function. This function is a multi-mode function, one which corresponds to three calculation modes of cost function in Section 3.3. Different calculation cases can be selected according to the task requirements in order to obtain corresponding calculation results of the cost function. The Helper Function2, FunctionSearch, searches for children around the parent node and then obtains a set of children that satisfy the condition roundlist(i), corresponding to line 13 in the main function. This function uses the multi-direction search method in Section 3.2 to determine the node. The Helper Function3, FunctionReelect, first selects the node with the same coordinates in the child node set roundlist(i) and the open list, then compares the cost function value of the parent node of the two nodes and sets the parent node with the smaller cost function value as the new parent node, corresponding to line 19 in the main function. The Helper Function4, Functionlineofsight, is the function that FunctionSearch needs, which determines whether there are obstacles inside the step size, corresponding to line 9 in the FunctionSearch. This function uses the line-of-sight algorithm in Section 3.2.

There are also some simple Helper Functions: length is the number of points in the computed set; min selects the smallest point in the set; int is the part of an integer that takes a number; abs is to calculate the absolute value; det is to calculate the determinant of the matrix; norm is to calculate the distance between two points; and atan is to calculate the inverse tangent angle value.

The information contained in each node includes the following: _row represents the horizontal coordinate of the point, _col represents the vertical coordinate of the point, _directedangle represents the orientation angle of the point, _f represents the f(n) of the point, _g represents the g(n)t of the point, and _h represents the h(n) of the point.

Algorithm 1 Main Function

1: function Improved A* Function ()

2: Map(obstacle); Map(risk); Map(time cost); Steplength; Case(j)j ∈ {1,2,3};

3: start_point; end_point;

4:  openlist() = Ø; closedlist() = Ø;

5:  openlist(1) = start;

6:  while openlist() ≠ Ø;

8:     Fmin= FunctionMin (openlist());

9:     if Fmin_h < L

10:         return ‘get path’

11:     end if

12:        closedlist = [closedlist: Fmin]

13:      roundlist() = FunctionSearch(Fmin,Case(j));

14:      for i = 1: length(roundlist)

15:         if roundlist(i) ∉ closedlist()

16:        if roundlist(i) ∉ openlist()

17:          openlist = [openlist: roundlist(i)];

18:        else

19:           FunctionReelect(roundlist(i))

20:        end if

21:       end if

22:     end for

23:   end while

24: end function

FunctionMin

1: function FunctionMin ()

2:  openlist(); Case(j) j ∈ {1,2,3};

3:   for i = 1: length(openlist)

4:        switch Case(j)

5:       case Case(1)

6:         openlist(i)_f = openlist(i)_gshort + openlist(i)_h

7:            Fmin = min(openlist(i)_f)

8:       case Case(2)

9:         openlist(i)_f = openlist(i)_grisk + openlist(i)_h

10:            Fmin = min(openlist(i)_f)

11:       case Case(3)

12:          openlist(i)_f = openlist(i)_gtime + openlist(i)_h

13:             Fmin = min(openlist(i)_f)

14:        end switch

15:       end for

16:  end function

FunctionSearch

1: function FunctionSearch ()

2:  Fmin_coord_row; Fmin_coord_col; Fmin_ directedangle

3:  Steplength; Case(j); Angleelement;

4:      roundlist() = Ø

5:     for     k = 1: int(2pi/Angleelement)

6:     roundlist(k)_coord_row = int(Fmin_coord_row+                Steplength×sin(Fmin_ directedangle + k×Angleelement))

7:     roundlist(k)_coord_col = int(Fmin_coord_col+                Steplength×cos(Fmin_ directedangle + k×Angleelement))

8:    roundlist(k)_directedangle = atan((roundlist(k)_coord_row-Fmin_coord_row)/(round        list(k)_coord_col-Fmin_coord_col))

9:           if Functionlineofsight (roundlist1(k)) = Y

10:         roundlist() = roundlist1(k)

11:           end if

12:      end for

13:       return ‘roundlist()’

14:   end function

FunctionReelect

1: function FunctionReelect ()

2:       Fmin; roundlist(i)_parent; Case(j) j ∈ {1,2,3};

3:        If Fmin_f < roundlist(i)_parent_f

4:         roundlist(i)_parent = Fmin

5:         End if

6: End function

Functionlineofsight

1: function Functionlineofsight ()

2: Fmin; roundlist(i)_row; roundlist(i)_col;

3:           numx = roundlist(i)_row-Fmin_row

4:         numy = roundlist(i)_col-Fmin_col

5:          For m = 1: numx

6:                for n = 1: numy

7:              p = (Fmin_row + m, Fmin_col + n)

8:     dis = abs(det([Fmin − roundlist(i),p-roundlist(i)]))/norm(Fmin − roundlist(i));

9:              if dis < 1

10:                If Map(p) = 0

11:                return ‘N’

12:                else

13:                 return ‘Y’

14:                End if

15:             End if

16:           End for

17:         End for

18:  End function

4. Simulation Results and Analysis

To verify the effectiveness of the proposed improved A* algorithm, Windows 11 is used as the platform and Matlab2022 is used to simulate the programming condition. The emulation hardware platform is an Intel Core (TM) i7-8700 CPU @GHz processor with a clock speed of 3.20 GHz and 16 GB of RAM. In order to further explain the function of the algorithm, we choose the improved RRT* algorithm and the traditional eight-direction A* algorithm as the comparison object. Under the premise of setting the same starting point, target point, and map, the simulation data results of the three algorithms are compared.

4.1. Design of a Map

The map pixel size is 512 × 512, corresponding to the actual length of 712.5 m × 712.5 m, with each pixel corresponding to a length of 1.39 m. As shown in Figure 12, the white area is the passable area, and the black area is the obstacle area. The starting point coordinates are (30, 30) and the target point coordinates are (500, 500).

4.2. Path Comparison

Firstly, the traditional eight-direction A* algorithm is used to calculate the path, as shown in Figure 13a. Secondly, the RRT* algorithm is used to calculate the path, as shown in Figure 13b. Finally, the improved A* algorithm described in this paper is used for path planning; the model is set to the shortest path, and the search radius is set to 30. The results of the simulation are shown in Figure 13c. The comparison of the calculated results of the three algorithms is shown in

Table 2. The parameters of the different algorithms.

Algorithm	Computation Time/ms	Number of Nodes	Length/m
A*	166 ms	25	757.07 m
RRT*	273 ms	15	737.23 m
Improved A* (shortest)	226 ms	24	729.64 m

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Compared with the results of the simulation of the three algorithms, it can be determined from the table data that the improved A* algorithm in this paper provides more flexibility in selecting nodes than do the traditional eight-direction A* algorithm and the RRT* algorithm, the obtained path is smoother, and the total length of the path is shorter. The shortest path length obtained by the A* improved algorithm is 27.4 m less than that of the traditional A* algorithm and 7.56 m less than that of the RRT* algorithm.

4.3. Less-Risk Path Simulation

In order to verify that the proposed algorithm has better performance properties in the environment with risk-points, the possible danger points are set in the path-planning environment. On the basis of the obstacle map, the two coordinates of central point of the risk are set as (100, 100) and (400, 400). According to the environment risk situation, the risk parameter value of the risk-points TH₀ and TH₁ are set as 0.98, and the two distribution values of the Gaussian function are set as 6000. The risk range function is expressed as

$$\begin{array}{l}p(x,y)=\frac{T{H}_0}{2\mathrm{\pi }{\left|{\Sigma }_0\right|}^{1/2}}\exp \left[-\frac{1}{2}{\left(X-{\mu }_0\right)}^{\mathrm{T}}{\Sigma }_0^{-1}\left(X-{\mu }_0\right)\right]+\frac{T{H}_1}{2\mathrm{\pi }{\left|{\Sigma }_1\right|}^{1/2}}\exp \left[-\frac{1}{2}{\left(X-{\mu }_1\right)}^{\mathrm{T}}{\Sigma }_1^{-1}\left(X-{\mu }_1\right)\right]\\ X=\left(\begin{array}{l}x\\ y\end{array}\right);{\mu }_0=\left(\begin{array}{l}100\\ 100\end{array}\right);{\Sigma }_0=\left(\begin{array}{cc}6000& 0\\ 0& 6000\end{array}\right);{\mu }_1=\left(\begin{array}{l}400\\ 400\end{array}\right);{\Sigma }_1=\left(\begin{array}{cc}6000& 0\\ 0& 6000\end{array}\right);T{H}_0=0.98;T{H}_1=0.98\end{array}$$

(12)

The distribution of risk is shown in Figure 14a.

The improved A* algorithm in this paper is used for path planning; the mode is set as the less-risk coefficient, and the search radius is set as 30. The results of the simulation are shown in Figure 14b. The results of the algorithm comparison are shown in

Table 3. The parameters of the different algorithms.

Algorithm	Computation Time/ms	Length/m	The Risk Factor of Path
A*	166 ms	757.07 m	306.28
RRT*	273 ms	737.23 m	261.20
Improved A* (less risk)	426 ms	812 m	201.83

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It can be seen from the results of the simulation that the improved A* algorithm can determine a complete path with less risk. Although the length of the path calculated by the improved A* algorithm in this paper is increased, the risk coefficient of the whole comparison path is lower. Compared with the traditional A* algorithm, the risk value of the proposed improved A* algorithm is reduced by 34.1%, and the risk value is reduced by 22.7%, compared with the RRT* algorithm.

4.4. Less Time-Cost Path Simulation

In order to verify that the proposed algorithm has better performance properties in different ground conditions, different ground conditions are set in the path-planning environment. On the basis of the obstacle map, different ground conditions are set in different areas. The distribution of ground conditions is shown in Figure 15a. The improved A* algorithm in this paper is used for path planning, the mode is set as less time-cost, and the search radius is set as 30. The results of the simulation are shown in Figure 15b. The results of the algorithm comparison are shown in

Table 4. The parameters of the different algorithms.

Algorithm	Computation Time/ms	Length/m	Time-Cost of Path
A*	166 ms	757.07 m	306.28
RRT*	273 ms	737.23 m	261.20
Improved A* (less time)	383 ms	847.07 m	180.62

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It can be seen from the results of the simulation that the improved A* algorithm in this paper can determine a path with less time-cost. Because the area through which the path passes is mostly the area with less time-cost, although the path length is longer than the result of other algorithms, the time-cost is less. Compared with the traditional A* algorithm, the time consumption of the proposed algorithm is reduced by 52.5%, and the time-consuming of the proposed algorithm is reduced by 51.3%, compared with the RRT* algorithm.

5. Actual Condition Verification

5.1. Experimental Settings

In order to evaluate the algorithm proposed in this paper, we design a vehicle experiment. First, select an area as the uncertain off-road environmental. The selected map size is 960 m × 960 m, the pixel size of the remote sensing image is 512 × 512, and the actual length represented by each pixel is 1.875 m. The starting point of the experiment is (96, 13) and the end point is (483, 333). The experimental map is shown in Figure 16.

In this paper, a modified electric drive off-road test vehicle with automatic navigation function is used as the experimental platform, as shown in Figure 17. This test vehicle has remote control and positioning and navigation functions. It is equipped with an automatic steering system, remote control system, millimeter wave radar, visual measurement system, remote information transmission system, monitoring system, and other information and control systems. The specific parameters of the test vehicle are shown in

Table 5. The parameters of the test vehicle.

Model	Value	Model	Value
Length	5316 mm	Wheel radius	478.3 mm
Width	2144 mm	Left and right wheel-base	1873 mm
Height	2182 mm	Ground clearance	462.3 mm
Wheelbase	3466 mm	Maximum Traction	96 kN
Weight	2169 kg	PTO power	323 kW

.

5.2. Path Planning

Firstly, the obstacle area in the map is marked to determine the obstacle map. The improved A* algorithm described in this study was used to calculate the shortest path; the mode is set to the shortest path, and the search radius is set to 10. The result for the path is shown in Figure 18. The parameters for the path are shown as Path 1 in

Table 6. The parameters of the three paths.

Path	Length/m	Risk Value of Path	Time-Cost of Path
Path1	510.4 m	200.29	414.12
Path2	519.49 m	99.9	402.63
Path3	690.7 m	225.89	195.13

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Second, the coordinates of the central point of the risk are set as (256, 180). According to the results of the simulation of the environment risk, the value of $T{H}_0$ is set as 0.98, and the two distribution values of the Gaussian function are set as 6000. The risk range function is expressed as

$$p(x,y)=\frac{T{H}_0}{2\mathrm{\pi }{\left|{\Sigma }_0\right|}^{1/2}}\exp \left[-\frac{1}{2}{\left(X-{\mu }_0\right)}^{\mathrm{T}}{\displaystyle \sum _0^{-1}\left(X-{\mu }_0\right)}\right];X=\left(\begin{array}{l}x\\ y\end{array}\right);{\mu }_0=\left(\begin{array}{l}256\\ 180\end{array}\right);{\Sigma }_0=\left(\begin{array}{cc}6000& 0\\ 0& 6000\end{array}\right)$$

(13)

The improved A* algorithm in this paper is used for path planning; the mode is set as less risk, and the search radius is set as 10. The result for the path is shown in Figure 19. The parameters for the path are shown as Path 2 in Table 6.

Thirdly, according to different ground conditions, the distribution map of the passing-time cost is divided according to the classification of the passing-time cost in Section 3.3 and marked with the specified color, and the time-cost map is finally obtained. The improved A* algorithm in this paper is used for path planning; the mode is set as less time-cost, and the search radius is set as 10. The result for the path is shown in Figure 20. The parameters for the path are shown as Path 3 in Table 6.

The path length, the risk value and the passing-time cost of the three results of the simulation were analyzed; the comparison results are shown in Table 6.

5.3. Analysis of Algorithm Optimization Effect

According to the visualization results and path data, the improved A* algorithm can plan three kinds of paths for an unmanned autonomous vehicle, namely, the shortest path in length, the less-risk path, and the less time-cost path. From the experimental data, it can be seen that the value of the distance for the shortest path is 510.4 m. The risk value of the path with less risk is reduced by 50.12%, compared to the path without risk factors. The time-cost of the path with less time-cost is reduced by 52.88%, compared to the path without time-cost factors.

6. Conclusions

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The improved algorithm can complete the path planning for the path with the less risk in a risky environment.

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The improved algorithm can obtain the path-planning scheme involving the less time in a complex ground environment.

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The improved algorithm can obtain a smoother short path.

Determining a reasonable trajectory within an off-road uncertain environment is important for the safety and efficiency of an unmanned vehicle in the execution of its missions. Based on environmental remote sensing images, pixels are used as grid units of raster maps to improve the resolution of raster maps and mark obstacles in maps more accurately. The APF (artificial potential field) method is introduced to describe the potential risk in the environment. The Gaussian probability distribution function is used as the distribution function of the artificial potential field, and it is discretized to determine a risk map consistent with the pixel matrix size. This paper quantifies the time-cost of vehicles under different ground conditions in an off-road environment, divides the map into regions according to the different time-costs, and obtains the time-cost map with an identical pixel matrix size. We propose an improved A* algorithm which shows good flexibility for path planning. The multi-direction search method can search nodes efficiently in the raster map with a pixel as the unit and solves the problem of a path not being smooth due to a fixed search direction. An improved line-of-sight algorithm is proposed, one which includes a vehicle-width factor. Using the improved line-of-sight algorithm to judge the obstacles on the path, the vehicle can keep a safe distance from the obstacles. The improved A* algorithm uses three models of cost function, namely, shortest path, less risk, and less time-cost.

Compared with the traditional eight-direction A* algorithm and RRT* algorithm, the improved A* algorithm described in this paper obtains a shorter and smoother path length, which accords with the actual driving conditions of vehicles. In addition, the improved A* algorithm in this paper can calculate three path costs of nodes according to task requirements, and obtain the shortest path, a less risk path, and a path with less time consumption, respectively. Therefore, the improved A* algorithm can quickly, smoothly, and safely plan three paths in battlefield conditions. The results of the simulation indicate that the proposed approach is a useful tool for the path planning of unmanned vehicles navigating within off-road environments.

The algorithm in this paper is a global path planning mechanism. At the same time, unmanned vehicles also face suddenly arising obstacles in carrying out their tasks. It is sometimes necessary to make quick obstacle-avoidance path planning. Therefore, the next step is to study the local path-planning methods of unmanned vehicles under dynamic conditions.