Research on Robot Path Planning Based on Improved A* Algorithm

doi:10.21203/rs.3.rs-4723600/v1

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Research on Robot Path Planning Based on Improved A* Algorithm

https://doi.org/10.21203/rs.3.rs-4723600/v1

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With the advent of the industrial 4.0 era of intelligent manufacturing, industrial production is becoming more and more unmanned, such as drones, self-driving cars, industrial robotic arms, unmanned aerial vehicles, etc. This paper proposes an improved A* path planning algorithm, which achieves fast and accurate path planning to find the target location. Aiming at the problems of blind expansion, large number of sampling nodes, and slow speed of the traditional A* algorithm, this paper proposes an improved A* path planning algorithm, which retains the heuristic function and uses a fixed sampling function to find the fixed sampling points, and makes the starting point and the target point simultaneously perform path planning towards the sampling points. Thus, the problems of a large number of sampling nodes and slow speed are solved. The simulation results show that compared with the traditional A* algorithm, the number of sampling nodes is reduced by 65.30%, and compared with the A* algorithm with heuristic function, the number of sampling points is reduced by 55.35%. At the same time, the optimal path is retained. It meets the requirements of path planning.

Robot

Improved

A* algorithm

Path planning

Robot autonomous navigation typically involves perception, localization, path planning, and control. As a critical technology in robot navigation, path planning has become a primary research focus. It involves finding a feasible route from the initial position to the target position in the configuration space using a planning algorithm, avoiding any obstacles[1]. This technology is extensively used in industrial robots, bionic robots, drones, self-driving cars, and other fields. Effective path planning algorithms can devise optimal routes, enhance operational efficiency, and swiftly and safely locate target destinations.

In recent years, numerous studies on path planning algorithms have been conducted by scholars globally[2, 3]. These include A*[4, 5], D*[6], ant colony algorithm[7], genetic algorithm (GA)[8], rapidly exploring random tree algorithm (RRT)[9], and artificial potential field method[10]. The ant colony algorithm, inspired by the positive feedback mechanism of ants searching for food, is highly efficient but has a slow convergence rate at the search's outset[11]. The genetic algorithm is notable for its parallelism and global search capabilities but often falls into local optima, with genetic factors being challenging to determine[12]. Rapidly exploring random tree algorithms offer probabilistic completeness, scalability, and fast detection speeds, making them popular in robot path planning. However, traditional RRT algorithms, while powerful, suffer from low search efficiency due to random sampling, leading to numerous invalid samples and slow convergence; this randomness can cause significant deviations in the generated path[13, 14]. The artificial potential field method, though simple and computationally light, struggles with unreachable targets and local minima. The A* algorithm, relying on a heuristic function, excels in finding optimal paths but requires an appropriate heuristic function and is computationally intensive in large-scale environments[15, 16].

To address these shortcomings, various enhanced A* algorithms have been proposed. For example, Li et al.[17] introduced a bidirectional A* algorithm that searches simultaneously from the start and end points, improving search efficiency. Likhachev et al.[18] proposed the ARA* (Anytime Repairing A*) algorithm, which quickly generates an initial path and continuously refines it within a given timeframe. Lai et al.[19] developed a path-planning algorithm combining an improved A* algorithm with a segmented second-order Bézier curve, adjusting the traditional 8-neighborhood search strategy to 5-neighborhood, and introducing a dynamic weight factor for the heuristic function. Zelin Wang et al.[20] incorporated the artificial potential field (APF) factor into the A* algorithm, generating safe and collision-free paths and optimizing the number of turns. Sun et al.[21] combined the A* algorithm with the dynamic window method, addressing collision and path failure issues and removing redundant points. Zhen et al.[22] proposed a hybrid of A* and adaptive DWA, adding safety distance judgment to avoid diagonal obstacle crossings and smoothing the path to align with a mobile robot's movement curve.

Despite these advancements, existing algorithms still face issues like extensive node traversal and slow running speeds. This paper proposes an enhanced A* algorithm that employs a fixed sampling function to identify a midpoint between the start and target points, concurrently sampling from both ends to this fixed point while maintaining the heuristic function, thereby improving the A* algorithm's efficiency.

The remainder of this paper is organized as follows: The next section introduces the A* algorithm with a heuristic function. Section 3 details the proposed improvements to the robot path planning algorithm, including the sampling function and search method optimization. The paper concludes with a summary of findings.

A* algorithm was published by Hart eta[23]. in 1968. It is a common path-finding and graph traversal algorithm, that combines the advantages of the Dijkstra algorithm and best-first search. It uses a heuristic function to guide the search direction and improves efficiency and accuracy. The core of the A* algorithm is the evaluation function f (n), which represents the estimated cost of the optimal path from the start point to the endpoint through node n. The evaluation function consists of two parts, one is g (n), which represents the actual cost from the start point to node n; the other is h (n), which represents the expected cost from node n to the end point, that is, the heuristic function. The choice of heuristic function has a great impact on the performance and accuracy of the A* algorithm. If the heuristic function h (n) is always 0, then the A* algorithm degenerates into the Dijkstra algorithm; if the heuristic function h (n) is always less than or equal to the actual cost from node n to the endpoint, then A* algorithm can guarantee to find the shortest path; if the heuristic function h (n) is exactly equal to the actual cost from node n to the endpoint, then A* algorithm will find the best path and be very fast; if the value of heuristic function h (n) is larger than the actual cost from node n to the endpoint, then A* algorithm cannot guarantee to find the shortest path, but it will be very fast.

ts search weight function is:

$\:\text{f}\left(\text{n}\right)\:=\:\text{g}\left(\text{n}\right)\:+\:\text{h}\left(\text{n}\right)$ (Eq. 1)

where g(n) is the cost of the node from the start point, and h(n) is the estimated cost of the node n to the endpoint, that is, the heuristic function.

Common heuristic functions include Manhattan distance, diagonal distance, and Euclidean distance. Their calculation methods are as follows:

Manhattan distance:

Diagonal distance:

$\:\text{h}\left(\text{n}\right)\:=\:\text{D}\ast\:\left(\right|\text{n}.\text{x}\:-\:\text{g}\text{o}\text{a}\text{l}.\text{x}|\:+\:|\text{n}.\text{y}\:-\:\text{g}\text{o}\text{a}\text{l}.\left|\right)+（\text{D}2\:-\:2\ast\:\text{D}）\ast\:\:\text{m}\text{i}\text{n}\left(\right|\text{n}.\text{x}\:-\:\text{g}\text{o}\text{a}\text{l}.\text{x}|\:+\:|\text{n}.\text{y}\:-\:\text{g}\text{o}\text{a}\text{l}.\left|\right)$ (Eq. 3)

Euclidean distance:

$\:\text{D}\ast\:\sqrt{(\text{n}.\text{x}\:-\:\text{g}\text{o}\text{a}\text{l}.\text{x})²+\:（\text{n}.\text{y}\:-\:\text{g}\text{o}\text{a}\text{l}.\text{y}）²}$ (Eq. 4)

where D is the moving cost between adjacent nodes, and D2 is the moving cost between diagonally adjacent nodes.

The flowchart of the A* algorithm is shown in Fig. 1, and the specific steps are as follows:

Step 1: Initialize the open list and the closed list, and add the starting point as the root node to the open list.

Step 2: If the open list is not empty, select the node n with the smallest evaluation function f (n) from the open list as the current node: if node n is the end point, then: trace back the parent nodes from the end point to the starting point; return the found result path, and end the algorithm; if node n is not the end point, then: delete node n from the open list and add it to the closed list; traverse all the neighboring nodes m of node n : if the neighboring node m is in the closed list or not passable, then: skip it and select the next neighboring node; if the neighboring node m is not in the open list, then: set node n as the parent node of node m; calculate the evaluation function f (m), g (m) and h (m) of node m, where g (m) is the actual cost from the starting point to node m, h (m) is the estimated cost from node m to the end point, f (m) = g (m) + h (m) ; add node m to the open list; if the neighboring node m is already in the open list, then: check whether there exists a better path through node n to reach node m, that is, compare g (n) + cost (n, m) and g (m), where cost (n, m) is the moving cost from node n to node m ; if g (n) + cost (n, m) is smaller, then: set node n as the parent node of node m; update the evaluation function f (m) and g (m) of node m .

Although the heuristic A* algorithm improves some problems of traditional A* algorithm such as traversing many nodes and slow speed, its efficiency is still not high. If the configuration space is too large and there are many obstacles in between, the number of nodes traversed by it will still be very large, and its speed will not be much improved compared with the traditional A* algorithm.

Improved A* algorithm

Fixed sampling function

Li eta[17]. proposed an improved bidirectional A* (Bidirectional A*) algorithm based on the A* algorithm, which introduced a bidirectional alternating search strategy. Although this algorithm improved the search speed, it still needed to traverse a large number of nodes in the expansion process due to the lack of random sampling. Xin eta[24]. proposed a strategy of sampling a fixed point in an improved RRT algorithm, which specified the exploration direction and improved the path search efficiency.

There are two cases for determining the sampling point Qfix. If the midpoint of the line between Qstart(xs, ys) and Qgoal(xg, yg) does not collide with obstacles, use formula (Eq. 5) to calculate this midpoint Qcenter(xc, yc), and use it as the sampling fixed point.

$$\:\:\text{x}\text{c}\:=\:(\text{x}\text{s}\:+\:\text{x}\text{g})\:/\:\:2\:$$

$\:\text{y}\text{c}\:=\:(\text{y}\text{s}\:+\:\text{y}\text{g})\:\:/\:\:2$ (Eq. 5)

When the midpoint collides with the obstacle, use the fixed sampling function to determine the fixed point. When Qcenter is in the obstacle area, first draw a circle with Qcenter as the center and R as the radius, initialize the radius Rs as 1, traverse from 0° to 360° every 10° on the circle, calculate the coordinates on the circle, if the node is safe, use this node as the sampling point. If no sampling point is found, increase the radius of the circle and continue to find a safe node. Its expression is:

The formula for the generated circle is:

$\:(\text{x}\:-\:\text{x}\text{c})\:+\:(\text{y}\:-\:\text{y}\text{c})\:=\:\text{r}²$ (Eq. 6)

The formula for the sampling point is:

$\:(\text{x}\text{n}\:,\:\text{y}\text{n})\:=\:(\text{x}\text{c}\:+\:\text{r}\text{c}\text{o}\text{s}\text{n}\:\times\:\:10^\circ\:，\text{y}\text{c}\:+\:\text{r}\text{s}\text{i}\text{n}\text{n}\:\times\:\:10^\circ\:),\text{n}\:=\:\text{1,2},\text{3,4}...35$ (Eq. 7)

The idea of the Sampling algorithm is shown in Algorithm 1.

Algorithm 1 Sampling

Input: Map, k;

Output: x, y;

1 : while True do

2 : x ← Random(min_x, max_x);

3 : y ← Random(min_y, max_y);

4 : if k % 10 = = 0 then

5 : x ← gx;

6 : y ← gy;

7 : if verify_node(x, y) then

8 : return x, y;

Improvement of search method

A* algorithm has a heuristic function, which explores by estimating the cost from the starting point to the target point, but the results are not satisfactory through experimental verification. Therefore, this section recommends how to improve the search method. Similar to the basic A* algorithm, it finds a midpoint Qc from the line between the starting point Qs and the target point Qg, and uses a sampling function to find a fixed point Qf, and then explores from the starting point and target point to the fixed point at the same time. The improved A* algorithm is shown in Fig. 3.

The steps of the A* algorithm using the fixed samplingfunction are:

Step1, The midpoint Qc of the connection between the initial point Qs and the target point Q is calculated by Eq. (4), and it is determined whether this point is located in the barrier area and performs 3 if this point is not in the obstacle area, otherwise perform Step2.

Step 2. Use the sampling function to determine whether the sampling point is in the obstacle or not. If not, determine the sampling point. If it is in the obstacle, enlarge the radius of the search circle and resample until a fixed sampling point Qf is determined.

Step 3. The starting point Qs and target point Qg explore towards the fixed sampling point at the same time. When in some special obstacle environments, if starting point Qs or target point Qg can directly point to target or when starting point Qs and target point Qg traverse to the same node during the exploration process, discard fixed sampling point Qf and directly generate path. Otherwise, execute Step 4.

Step 4. Starting point Qs and target point Qg find fixed sampling point Qf, connect two paths, generate a final path, and the algorithm ends.

The complete idea of the Improved A* algorithm is shown inAlgorithm 2.

Algorithm 2 Improved A*

Input: Map, sx, sy, gx, gy, xp, yp;

Output: path1, path2;

1 : start_node ← new Node(calc_xy_index(sx, min_x), calc_xy_index(sy, min_y), 0.0,−1);

2 : goal_node ← new Node(calc_xy_index(gx, min_x), calc_xy_index(gy, min_y), 0.0,−1);

3 : state_node ← new Node(calc_xy_index(xp, min_x), calc_xy_index(yp, min_y), 0.0,−1);

4: open_set ← new dict();

5: closed_set ← new dict();

6: open_set1 ← new dict();

7: closed_set1 ← new dict();

8: open_set[calc_grid_index(start_node)] ← start_node;

9: open_set1[calc_grid_index(goal_node)] ← goal_node;

10: while True do

11: if open_set is empty then

12: c_id ← min(open_set, key = lambda o: open_set[o].cost + calc_heuristic(state_node, open_set[o]));

13: current ← open_set[c_id];

14: if current.x = = state_node.x and current.y = = state_node.y then

15: path1 ← calc_final_path(state_node, closed_set);

16: if open_set1 is empty then

17: c_id1 ← min(open_set1, key = lambda o: open_set1[o].cost + calc_heuristic(state_node, open_set1[o]));

18: current1 ← open_set1[c_id1];

19: if current1.x = = state_node.x and current1.y = = state_node.y then

20: path2 ← calc_final_path(state_node, closed_set1);

21: return path1, path2;

Simulation environment and parameter setting

This paper proposes a grid environment model of a robot working scenario with a 70×70 plane area. The environment model is based on Python built-in Pycharm, and the path planning algorithm runs in Pycharm as well. The robot radius is 1, the grid map resolution is 2.0, the starting point coordinates are (-5, -5), and the target point coordinates are (50, 50). Two long obstacles are generated at x coordinates 20 and 40 respectively. Experiments are carried out for the traditional A* algorithm, the heuristic A* algorithm, and the improved A* algorithm respectively, and the parameters required by the algorithms are the same. The number of traversed nodes, running time, and path length of the above algorithms are compared and analyzed.

Simulation results and analysis

When performing the above algorithm experiments on the robot grid environment model, the starting point coordinates are the green dot in the figure, the blue cross is the target point position, and the black cross is the fixed sampling point position. The two-dimensional coordinates (X, Y) of these three points can be obtained from the figure.

Figure 4 shows the path planning result of the traditional A* algorithm. It can be seen that the traditional A* algorithm traverses a large number of nodes, almost occupying the entire map, but finally finds a path successfully.

Figure 5shows the path planning result of the heuristic A* algorithm. It can be seen that compared with the traditional A* algorithm, the heuristic A* algorithm traverses fewer nodes, and also finds a path, improving the running efficiency of the robot, but there are a lot of twists and turns on the path near the target point.

Figure 6 shows the path planning res mproved A* algorithm. The improved algorithm finds a fixed point in the middle area of the starting point and target point, and uses the position of the fixed point as the expansion direction of the two points, reducing the path length and node number. Compared with the previous two algorithms, the number of traversed nodes is greatly reduced, and the search efficiency is improved.

Repetitive experiment analysis

Three algorithms were tested 50 times in the robot grid environment model, and the results are summarized in Table 1. The improved algorithm demonstrates a significantly lower average number of traversed nodes, with a count of 2247, outperforming the traditional A* algorithm and the heuristic A* algorithm by 65.30% and 55.35%, respectively. Figure 7 illustrates that the improved algorithm effectively leverages the benefits of the heuristic A* algorithm, substantially reducing the number of traversed nodes.

From Table 1 and Fig. 8, the running times of the three algorithms are compared. The improved algorithm boasts an average running time of just 0.83 seconds, which is 64.83% and 57.43% faster than the traditional A algorithm and the heuristic A algorithm, respectively. This indicates that the running time of the A algorithm is heavily influenced by the number of traversed nodes. The improved A* algorithm, by exploring from both the starting point and the target point, further reduces the running time, resulting in a significantly faster performance compared to the other two algorithms.

Path length is a critical metric for evaluating the quality of the path, as it determines the distance the robot can control. The average path length for the improved algorithm, as presented in Table 1, is 243.87 cm, which is shorter than the traditional A algorithm and the heuristic A algorithm by 0.78 cm and 0.09 cm, respectively. Figure 9 shows that the path lengths of the three algorithms remain relatively stable across the experiments, with minimal differences among them.

From the perspective of the fixed sampling strategy, Table 1 reveals that the improved algorithm, utilizing a fixed sampling function, results in fewer traversed nodes and shorter path connection times, thereby enhancing the path finding speed. The 50 experiments demonstrate that the improved algorithm performs exceptionally well in terms of the number of traversed nodes, running time, and path length.

Table 1

.Experimental data of three algorithms
algorithm	Traditional A* algorithm	heuristic A* algorithm	improved A* algorithm
Number of nodes(dot)	4685	3144	2247
Running time(s)	2.36	1.95	0.83
Path length(cm)	244.65	243.96	243.87

With the advent of Industry 4.0, robots have been applied to many fields of life and production. Aiming at the problem of low efficiency of robot path planning algorithm, this paper carried out the following research: Based on the A* algorithm, a fixed sampling function was designed to find a fixed point in the middle position and search from the starting point and target point to the fixed sampling point at the same time. Finally, compared with the unimproved A* algorithm and the heuristic A* algorithm, the improved A* algorithm has obvious improvement in traversed node number and running time. It meets the requirements of robot path planning.

The limitations and prospects of this paper are as follows: In some special environments, the fixed sampling point may lose its effect, and then the algorithm will degenerate into a bidirectional A* algorithm. This method can be improved in the future to keep the exploration direction correct. It can meet the needs of most tasks in normal situations. In further research, we will consider limiting the exploration direction of the A* algorithm, so that it always explores towards the target point direction.

Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.

Conflicts of Interest:

The authors declare no conflict of interest.

Author Contribution

All authors reviewed the manuscript

Acknowledgments

Research on Shunting Operations Remote Control Technology Based on STP System and Video Monitoring, 2023 Key Scientific Research Projects of CARS, Project No. 2023YJ117

Data availability statements:

Data sets generated during the current study are available from the corresponding author on reasonable request.

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No competing interests reported.

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Research on Robot Path Planning Based on Improved A* Algorithm

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Abstract

Figures

Introduction

Heuristic A* algorithm

Improved A* algorithm

Fixed sampling function

Improvement of search method

Results and analysis

Simulation environment and parameter setting

Simulation results and analysis

Repetitive experiment analysis

Conclusion

Declarations

Conflicts of Interest:

Author Contribution

Acknowledgments

Data availability statements:

References

Additional Declarations

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