Research on path planning of mobile robots based on improved A* algorithm

Grid-based environment modeling

Environment modeling is a critical aspect in the study of path planning for mobile robots. An appropriate environmental map model can significantly enhance the search efficiency of algorithms. Commonly used modeling techniques include grid-based methods (Weikuan & Baoping, 2019), topological methods (Chong et al., 2006), and visibility graph methods (Xiaodan, 2021). Given the advantages of visualization and interpretability offered by the grid-based method, along with its ease of data manipulation during algorithm simulation and validation, this article adopts the grid-based method for environmental modeling.

To facilitate the effective positioning of the mobile robot, the path planning nodes are constrained to the centers of the grid cells. The robot is considered as a mobile point mass, with its position determined by the grid cells on the map. This approach effectively reduces the complexity of the path. Figure 1 illustrates a grid example of size 10 × 10, wherein the red grid cell represents the starting position of the mobile robot, the blue grid cell denotes the target position, the black grid cells indicate obstacles, and the white grid cells signify passable areas.

Once the grid model is established, each grid cell must undergo an encoding process. Each grid cell can have one of two states: vacant or occupied. When the encoding of a grid cell is set to 1 (i.e., when the color of the cell is black), it signifies that the grid is in an occupied state, indicating the presence of an obstacle. Conversely, when the encoding is set to 0 (i.e., when the cell is white), it denotes that the grid is vacant, allowing the robot to pass through without obstruction.

Traditional A* algorithm

The traditional A* algorithm is a heuristic approach that combines the advantages of both Dijkstra’s algorithm and the Breadth-First Search (BFS) algorithm, effectively addressing the pathfinding problem. Once the initial point is determined, the algorithm employs a heuristic search method. Specifically, it calculates the cost associated with each reachable node, which is the sum of the cost incurred to reach that node and the estimated cost from that node to the target point. The node with the lowest total cost is then selected as the next node to explore, and the search continues until the target point is reached.

The heuristic function of the A* algorithm is defined as Eq. (1):

(1) $$f\left(n\right)=g\left(n\right)+h\left(n\right).$$

In the equation, $n$ represents the currently explored node, $f(n)$ denotes the total cost, $g(n)$ represents the cost incurred from the start node to the current node, and $h(n)$ indicates the estimated cost from the current node to the target point.

Common methods for calculating the estimated cost $h(n)$ include the Manhattan distance, Euclidean distance, and Chebyshev distance, which are expressed mathematically as Eqs. (2)–(4):

(2) $$h\left(n\right)=\left|x_2-x_1\right|+\left|y_2-y_1\right|$$ (3) $$\begin{array}{c}h\left(n\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\end{array}$$ (4) $$h\left(n\right)=max\left(\left|x_2-x_1\right|,\left|y_2-y_1\right|\right).$$

The choice of heuristic function has a significant impact on the performance of the A* algorithm, making the selection of the estimated cost $h(n)$ critically important. The Euclidean distance effectively approximates the actual distance from the current node to the target node, guiding the algorithm in the direction of the target. Furthermore, utilizing the Euclidean distance allows the paths generated by the algorithm to more closely align with practical application scenarios. Therefore, the improved A* algorithm proposed in this article employs the Euclidean distance function.

Neighborhood improvement

Neighborhood expansion involves the outward extension of nodes from the current node in various directions. The traditional A* algorithm commonly employs an eight-directional neighborhood expansion method, wherein nodes are expanded in eight different directions from the current node, with each direction separated by an angle of 45°. The schematic representation of this principle is illustrated in Fig. 2A, and the set of coordinates can be expressed as Eq. (5):

(5) $$T_n=\{\left(x_n\pm 1,y_n\right),\left(x_n,y_n\pm 1\right),\left(x_n\pm 1,y_n\pm 1\right)\}.$$

In the equation, $x_n$ represents the x-coordinate of the current search node $n$, and $y_n$ represents the y-coordinate of the current search node $n$.

Due to the eight-directional search method, the algorithm only expands to eight neighboring nodes at a time. In environments with few obstacles, this can lead to low search efficiency and an excessive number of expanded nodes. To address this issue, we propose a twenty-four directional search method. This method extends the search range from one layer of nodes surrounding the current node to two layers, allowing the algorithm to search towards 24 different nodes simultaneously. The schematic representation of this method is illustrated in Fig. 2B, and the set of coordinates can be expressed as Eq. (6):

(6) $$T_n=\left\{\begin{array}{c}\left(x_n\pm 1,y_n\right),\left(x_n,y_n\pm 1\right),\left(x_n\pm 1,y_n\pm 1\right),\left(x_n\pm 2,y_n\right),\\ \left(x_n,y_n\pm 2\right),\left(x_n\pm 2,y_n\pm 2\right),\left(x_n\pm 2,y_n\pm 1\right),\left(x_n\pm 1,y_n\pm 2\right)\end{array}\right\}.$$

While the twenty-four directional search method allows for the exploration of multiple nodes simultaneously (Baoxia, Miaochi & Yong, 2018; Zhimin et al., 2023), its search efficiency can significantly diminish when there are obstacles present within the first layer of neighboring nodes. When the algorithm extends to the second layer of nodes, it is necessary to first assess whether any of the first-layer nodes it traverses contain obstacles. Moreover, the multi-point search approach can result in the algorithm exploring an excessive number of useless nodes, which further reduces its overall efficiency.

In response to the aforementioned issues, this article proposes a novel sixteen-directional extension method. The schematic representation of this method is illustrated in Fig. 3, and the set of coordinates can be expressed as Eq. (7):

(7) $$T_n=\left\{\begin{array}{c}\left(x_n\pm 2,y_n\right),\left(x_n,y_n\pm 2\right),\left(x_n\pm 2,y_n\pm 2\right),\\ \left(x_n\pm 2,y_n\pm 1\right),\left(x_n\pm 1,y_n\pm 2\right)\end{array}\right\}.$$

The new sixteen-directional extension method is based on traditional eight-directional and twenty-four-directional extension methods. By removing the eight neighboring nodes surrounding the first layer of nodes in the twenty-four directional framework, the newly formed sixteen-directional neighborhood is established. When there are no obstacles in the eight neighboring nodes of the current node, the algorithm can directly explore the second layer of nodes within the new sixteen-directional neighborhood.

The specific search process of the algorithm is founded on eight-directional exploration, wherein the presence of obstacles in the eight neighboring nodes of the current node is evaluated. If obstacles are found, the algorithm continues to employ the eight-directional extension method. Conversely, if no obstacles exist within the current node’s eight neighboring nodes, the new sixteen-directional extension method is utilized for further exploration.

This hybrid search method determines the use of either the eight-directional extension or the new sixteen-directional extension based on the presence of obstacles in the eight neighboring nodes of the current node. This approach enables effective cross-node exploration, significantly enhancing the search efficiency of the algorithm while reducing the expansion of unnecessary nodes. Figure 4 illustrates a scenario where obstacles are present within the eight-directional neighborhood, with the black grid representing obstacles.

Node optimization

The improved hybrid search method utilizing the eight-directional and new sixteen-directional neighborhoods generates search nodes as illustrated in Fig. 5. During the exploration process from the current node to the target node, both the eight-directional and new sixteen-directional searches may yield invalid nodes. The existence of these invalid nodes results in a waste of resources during the algorithm’s search. To enhance the search efficiency of the algorithm, this article proposes the removal of unnecessary nodes that do not impact the search results.

Assuming the angle between the line connecting the current node and the target node and the positive direction of the x-axis is denoted as α, invalid nodes can be eliminated based on the range of angle α. The specific implementation method is as follows:

When employing the eight-directional search method, as shown in Table 1, three unnecessary nodes are removed based on the range of angle α, while five valid nodes are retained.

When utilizing the new sixteen-directional search method, as illustrated in Table 2, nine unnecessary nodes are eliminated based on the range of angle α, while seven valid nodes are retained.

Bidirectional search

Although the traditional A* algorithm has significantly improved its search speed compared to other heuristic algorithms, it still exhibits relatively slow performance, particularly in large-scale map environments, which leads to a noticeable decrease in search efficiency. To address this issue, this article employs a bidirectional search strategy aimed at reducing the number of nodes traversed by the algorithm and enhancing its search speed. The bidirectional search is divided into forward search and backward search. After determining the coordinates of the start and end points, the forward search proceeds from the start point toward the end point, while the backward search goes from the end point back to the start point. The current node of the algorithm is defined as the parent node (Pnode), and the neighboring search nodes are defined as child nodes (Snode). The specific search process of the algorithm is outlined as follows:

Step 1: Initialize the forward search’s Openlist1 and Closelist1, adding the start point to Closelist1 and defining it as Pnode1. Also, initialize the backward search’s Openlist2 and Closelist2, adding the end point to Closelist2 and defining it as Pnode2.

Step 2: Define the neighboring nodes surrounding Pnode1 as SPnode1 and add them to Openlist1. Compute the total cost values of all nodes in Openlist1 using the cost calculation function. Select the node with the minimal cost value and define it as SPnode2, then remove it from Openlist1 and add it to Closelist1.

Step 3: Define the neighboring nodes surrounding Pnode2 as SRnode1 and add them to Openlist2. Similarly, compute the total cost values of all nodes in Openlist2 using the cost calculation function. Select the node with the minimal cost value and define it as SRnode2, then remove it from Openlist2 and add it to Closelist2.

Step 4: Check whether SPnode2 exists in Openlist2 and whether SRnode2 exists in Closelist1. If either exists, it indicates that the forward and backward searches have intersected at a point, and the algorithm concludes. If neither exists, redefine SPnode2 as the new parent node Pnode1 and SRnode2 as the new parent node Pnode2. Steps 2 and 3 are then repeated until an intersection point is found.

Dynamic evaluation function

During bidirectional search, when the start and end points are situated at a considerable distance from one another, there is a possibility that the forward search and backward search may either intersect along a convoluted path or fail to intersect altogether. This situation can lead to a reduction in both the efficiency and accuracy of the algorithm. To address this issue, this article employs a dynamic weight coefficient $w(n)$ to adjust the heuristic function, dynamically balancing the ratio between $g(n)$ and $h(n)$, while controlling the search speed and range of the algorithm.

With the introduction of the dynamic weight coefficient $w(n)$, the total cost calculation for the A* algorithm can be expressed as Eq. (8):

(8) $$f\left(n\right)=g\left(n\right)+w\left(n\right)\cdot h\left(n\right).$$

In the case of forward search, the dynamic weight coefficient is represented as Eq. (9):

(9) $$\begin{array}{c}w\left(n\right)=e^{\left|x_t-x_g\right|+\left|y_t-y_g\right|}\cdot \left(l{n}^{{\displaystyle \frac{\left|x_t-x_g\right|+\left|y_t-y_g\right|}{\left|x_s-x_g\right|+\left|y_s-y_g\right|}}}+1\right).\end{array}$$

In the case of backward search, the dynamic weight coefficient is represented as Eq. (10):

(10) $$\begin{array}{c}w\left(n\right)=e^{\left|x_u-x_g\right|+\left|y_u-y_g\right|}\cdot \left(l{n}^{{\displaystyle \frac{\left|x_u-x_g\right|+\left|y_u-y_g\right|}{\left|x_s-x_g\right|+\left|y_s-y_g\right|}}}+1\right).\end{array}$$

Here, $(x_s,y_s)$ are the coordinates of the start point, $(x_g,y_g)$ are the coordinates of the end point, $(x_t,y_t)$ are the current coordinates in the forward search, and $(x_u,y_u)$ are the current coordinates in the backward search.

The dynamic weight coefficient combines exponential and logarithmic functions. When the current node is relatively far from the target endpoint, the search speed is accelerated. Conversely, when the current node is closer to the target endpoint, the search speed is reduced to enhance search accuracy and minimize exploration of ineffective areas. This approach improves search efficiency and ensures that the forward and backward searches converge at an intermediate node.

Redundant waypoint deletion strategy

Even after the A* algorithm has planned the optimal path, a number of redundant waypoints may still exist. To address this issue, this article proposes a waypoint deletion strategy that effectively removes unnecessary waypoints by assessing whether obstacles exist between nodes. As illustrated in Fig. 6, the specific implementation method is as follows:

Step 1: Starting from the endpoint P10, connect it to the adjacent waypoint P8 and check for the presence of obstacles along the connecting line. If no obstacles are present, the waypoint P9 can be deleted.

Step 2: Connect P10 to P7 and examine whether there are obstacles along this line. If obstacles are detected, waypoint P8 is designated as a non-deletable waypoint. If no obstacles are found, waypoint P8 can be removed.

Step 3: Use P8 as the new starting waypoint and repeat Steps 1 and 2 until reaching the starting point P1. All non-deletable waypoints are then reconnected to form the new optimal path.

Overall process of the improved algorithm

In summary, the overall flowchart of the improved A* algorithm is illustrated in Fig. 7.

In order to help readers understand the process of the algorithm more intuitively, we use pseudocode to describe the process of the entire algorithm, and the specific algorithm pseudocode is as follows:

Experimental environment

To evaluate the performance of the proposed algorithm, we constructed a simulation environment and conducted simulation experiments. All experiments were executed on a computer equipped with an Intel Core i9 processor, 128 GB of RAM, and two NVIDIA RTX 4090 GPUs with 24 GB of video memory each, utilizing the Python programming language for implementation.

Comparison experiment of improved A* algorithm 1

First, to demonstrate the advantages of the improved neighborhood search mechanism proposed in this article, comparative experiments were conducted against the four-neighborhood and eight-neighborhood A* algorithms. Two different environmental maps were created randomly: a 30 × 30 grid and a 50 × 50 grid. In the 30 × 30 map, the starting point and endpoint were set at (1, 1) and (30, 30), respectively, while in the 50 × 50 map, the starting point and endpoint were set at (1, 1) and (50, 50), respectively. In this experiment, only the improved algorithm in “Neighborhood Improvement” and “ Node Optimization” is verified, which is called “Improved A* Algorithm 1”.

The simulation results are shown in Figs. 8 and 9, where the black grids represent obstacles, the gray grids represent expanded nodes, the green grids represent path nodes, and the yellow lines indicate the planned path. Tables 3 and 4 present the algorithm’s performance parameters under different map environments.

By comparing Figs. 8A and 8C, it can be observed that the Improved A* Algorithm 1 demonstrates significant enhancements in the neighborhood expansion of nodes compared to the four-neighborhood A* algorithm in different map environments. Concurrently, the algorithm’s search efficiency has markedly increased. Furthermore, by comparing Figs. 9B and 8C, it is evident that the Improved A* Algorithm 1 optimizes neighborhood expansion nodes in relation to the eight-neighborhood A* algorithm across different map environments, along with an improvement in search efficiency.

Additionally, as shown in Tables 3 and 4, the performance indicators of the proposed Improved A* Algorithm 1 exhibit enhancements relative to the four-neighborhood A* algorithm, with the number of expanded nodes reduced by 69.08% and 78.86%, respectively. In comparison with the eight-neighborhood A* algorithm, while the path lengths remain relatively unchanged, the number of expanded nodes was decreased by 14.03% and 11.65%, respectively. The significant reduction in the number of expanded nodes effectively validates the efficacy of the neighborhood improvement strategy proposed in this article.

Comparative experiment of the improved A* Algorithm 2

To validate the effectiveness of the proposed improved A* algorithm (All improvement methods in “Problem Description”, referred to as “Improved A* Algorithm 2”), comparative experiments were conducted between the Improved A* Algorithm 2 and both the four-neighborhood A* algorithm and the eight-neighborhood A* algorithm in 64 × 64 and 128 × 128 grid map environments. The simulation environment was constructed as a two-dimensional grid map, where black grids represent obstacles, gray grids indicate forward expansion nodes, orange grids signify backward expansion nodes, green grids denote path nodes, and yellow lines illustrate the planned path. In the 64 × 64 map, the starting point and the endpoint are set at coordinates (1, 1) and (64, 64) respectively; whereas, in the 128 × 128 map environment, the starting point and endpoint are set at (1, 1) and (128, 128).

The simulation results are presented in Figs. 10 and 11, with the data comparisons delineated in Tables 5 and 6 below.

By comparing the experimental results presented in Figs. 10 and 11, it can be observed that the proposed Improved A* Algorithm 2 significantly reduces the number of expanded nodes and path points, shortens the path length, and enhances efficiency in various map environments compared to the four-neighborhood A* algorithm and the eight-neighborhood A* algorithm. As shown in Table 5, in the 64 × 64 grid map environment, the Improved A* Algorithm 2 reduces the number of expanded nodes by 89.98% and 18.5% compared to the four-neighborhood and eight-neighborhood A* algorithms, respectively. The path length is shortened by 26.84% and 5.25%, respectively, while the number of path points decreases by 77.16% and 62.82%, and the search time is reduced by 99.4% and 99.07%. Table 6 indicates that in the 128 × 128 grid map environment, the Improved A* Algorithm 2 reduces the number of expanded nodes by 89.75% and 12.79% compared to the four-neighborhood and eight-neighborhood A* algorithms, respectively. The path length is shortened by 28.13% and 3.1%, while the number of path points decreases by 79.21% and 62.93%, and the search time is reduced by 99.22% and 98.02%. These results demonstrate that the performance indicators of the Improved A* Algorithm 2 surpass those of both the four-neighborhood and eight-neighborhood A* algorithms in all aspects, significantly decreasing the number of expanded nodes and path points, thereby greatly enhancing the search efficiency of the algorithm while also shortening the path length. This further validates the effectiveness and superiority of the Improved A* Algorithm 2.