Abstract
Path planning is crucial for characterizing the driving ability of autonomous vehicles. The ant colony algorithm is a heuristic searching algorithm that simulates ant foraging. When used for the path planning of autonomous vehicles, this algorithm may suffer from slow convergence speed and unsmooth corners, and the solutions may fall into local extremes. An improved ant colony algorithm was proposed herein for reducing the risk of collision and improving the quality and efficiency of path planning. The heuristic function 
and pheromone update rules of the traditional ant colony algorithm were modified. 
was calculated based on the distance and angle from the current node to the target node. A pheromone regulatory factor C was introduced; its value decreased when the path length was greater than the average path length and increased otherwise. Simulation of the improved algorithm on 20 × 20 and 30 × 30 grid maps revealed that the path length and number of iterations decreased by an average of 9.8% and 64.3%, respectively, compared with those of the traditional ant colony algorithms. Experimental results showed that autonomous vehicles could move from the starting point to the end point along the simulated paths using the proposed algorithm. The findings of this study are of considerable significance for autonomous vehicles.
Subject terms: Computer science, Information technology
Introduction
Autonomous driving technology has garnered considerable research attention with the rapid development of artificial intelligence and computer information processing1–5. Electrification, intelligence, networking, and sharing have collectively promoted the development of the automotive industry. Path planning is crucial for characterizing the autonomous driving capability of vehicles targeted toward finding the shortest path, ensuring the highest safety level, and consuming the least time in traffic environments6,7. Intelligent algorithms were used to plan a collision-free path that connected the current and target goal positions of vehicles.
The current autonomous path planning techniques are based on traditional and intelligent algorithms. Traditional algorithms mainly include the A* algorithm8, Dijkstra algorithm9, and artificial potential field (APF) algorithm10. The A* algorithm uses heuristics to find the shortest path. The Dijkstra algorithm is a greedy algorithm and can plan a globally optimal path; however, it has some redundant inflection points and yields low efficiency. The paths planned by the APF algorithm are smooth with low computational complexity; however, some fall into the middle equilibrium point and local minimum value. Genetic algorithm11, simulated annealing algorithm12, ant colony algorithm13–16 are a few main intelligent algorithms. The genetic algorithm has the characteristics of population search but has some drawbacks, such as rough paths and ineffective crossing between the same chromosomes. The simulated annealing algorithm can find the global shortest path based on probability calculation. However, its performance is considerably impacted by initial parameter setting and slow path calculation speed. The ant colony algorithm is a heuristic searching algorithm that simulates ant foraging. It is highly robust and offers distributed computing during path planning; however, it has a slow convergence speed, and the solution can fall into local extremum points17,18. Therefore, several researchers delivered a lot of improvement measures to solve these problems. Chao Liu et al. proposed a novel variant of ant colony optimization that contains four improved mechanisms including adaptive pheromone concentration setting, heuristic mechanism with directional judgment, improved pseudo-random transfer strategy, and dynamic adjustment of the pheromone evaporation rate19. By introducing these three mechanisms into the combination of ant colony optimization and artificial bee colony, a new hybrid approach for solving path planning problems is proposed, called the improved ant colony optimization-artificial bee colony algorithm (IACO-IABC) algorithm20.
The ant colony algorithm exhibited issues such as slow convergence speed, approaching the boundary of obstacles, and unsmooth corners when used for the path planning of autonomous vehicles. Even though the ant colony optimization and its variants can produce satisfactory solutions in solving path planning problems, with regard to effectiveness and efficiency, there is still space for further improvement of ant colony optimization performance. An improved ant colony algorithm was proposed to address these issues and meet the dynamic constraints of vehicles, thereby reducing collision risks and improving the quality and efficiency of path planning. The heuristic function 
and pheromone update rules of the traditional ant colony algorithm were modified. 
was calculated based on the distance and angle from the current node to the target node. A pheromone regulatory factor C was introduced. Comparing with the traditional ant colony algorithms, the distance of the least path and iterations are significantly reduced. The main contributions are summarized as follows:
The heuristic function
was calculated based on the distance and angle from the current node to the target node. It makes path search more directional.A pheromone regulatory factor C was introduced; its value decreased when the path length was greater than the average path length and increased otherwise. It accelerated the convergence speed.
The remaining structure of this study is as follows: Section “Principles” provides a detailed principles introduction to the traditional and improved ant colony algorithms. Section “Simulation Analysis” presents simulation experiments on 20 × 20 and 30 × 20 grid maps. The experimental setups are presented in Section “Experiments”. The comparison experiments with A*, Dijkstra’s, RRT and other improved ant colony algorithms are discussed in Sect. “Discussion”. Finally, Sect. “Conclusions” provides the conclusion of the study.
Principles
Principles of the traditional ant colony algorithm
Inspired by the path-finding behavior of ants during foraging, Marco Dorigo proposed the ant colony algorithm in 1991. Ants depart from the ant nest randomly in all directions for foraging while releasing pheromones. They move forward along the routes that have high pheromone concentration. If some ants find a shorter path, the ant colony will gradually gather therein because the pheromone concentration is inversely proportional to the path length. The ant colony algorithm similarly finds an optimal path via continuous iteration and updating.
In traditional ant colony algorithms, the path searching direction is determined by the pheromone concentration and heuristic function. The concentration of a constant pheromone amount decreases as the path length increases. The transfer probability of the k-th ant from node (gride) i to node (gride) j is demonstrated as Eq. (1):
 |
1 |
where 
is the pheromone concentration. The higher the 
, the greater the probability of ants choosing that path. 
is the information inspiration factor. When the ant chooses the j-th node, 
characterizes the degree of pheromones. 
is a heuristic function that represents the expected level from node i to node j of ants. 
is the expected heuristic factor that represents the importance of heuristic information during ant path selection. 
is the set of nodes that the ant k does not access in the t-th iteration. As the amount of ant access nodes increases, the number of elements in the set 
decreases until it is empty; this indicates that the ants have accessed all nodes. The heuristic function 
can be determined using Eq. (2):
 |
2 |
where 
is the Euclidean distance from node i to node j, as shown in Eq. (3):
 |
3 |
The coordinates of node i and j can be expressed as 
and 
in Eq. (3). Pheromones are volatile, and their concentration gradually approaches zero over time. Assuming 
expresses the volatilization factor of pheromone concentration, the pheromone concentration is updated after the path search has been completed. In other words, the portion of pheromone volatilization is subtracted, and the new pheromone concentration resulting from the ants walking through this path is added. The updated rules are expressed as follows:
 |
4 |
 |
5 |
 |
6 |
In Eq. (4), 
denotes the increase in pheromone concentration in this iteration on the path from node i to node j. In Eq. (5), m denotes the number of ants. In Eq. (6), Q is the total amount of pheromones released by ant k in one iteration and 
is the total path length that ant k traverses along all the nodes in one iteration.
Improved ant colony algorithm
In the traditional ant colony algorithm, the heuristic function 
is inversely proportional to the distance between two nodes (Eqs. (1) and (2)). The larger the distance, the smaller the heuristic function and the lower the heuristic effect on probability. This function cannot inspire and guide the ants to follow the optimal path to reach the endpoint. The heuristic function was improved as follows:
 |
7 |
where 
represents the horizontal and vertical components from the current node to the target node. 
is a constant with a value of 
, 
denote the L2 norm. P is distance factor expressed as follows:
 |
8 |
where 
and 
represent the coordinates of the current node and target node. S is the starting point. dw and aw are the distance and angle weights, respectively. Angle is constituted by the vector pointing from the current node to the target node as well as the horizontal vector, 
. By improving the heuristic function, angle guides ants to choose a better direction of movement during path finding. The pheromone regulatory factor C is introduced to increase the convergence speed of the traditional ant colony algorithm. This factor is regulated based on the average path length. Different strategies are adopted for different ants in the algorithm. When the path length is higher than the average path length, C is decreased; else, the pheromone concentration is increased. Generally, the range of C is (0, 1] in inferior paths. From 0.9 to 0.1, C decreases by 0.1 each time. By observing the simulation results, select the optimal value of C from {0.1, 0.2, 0.3… 0.9}. Similarly, in high-quality paths, the range of C is [5, 100]. C increases by 5 each time. By observing the simulation results, select the optimal value of C from {5, 10, 15…100}. Consequently, the path-finding speed is improved. The updating rules of pheromones were modified as follows. The average path length was calculated, and the pheromone concentration was regulated based on the path length. Thus, Eq. (6) changes to Eq. (9):
 |
9 |
Simulation analysis
Simulation on a 20 × 20 grid map
The traditional and improved ant colony algorithms were simulated. An environment map was generated for autonomous vehicles using the grid method. Then, the real environment was mapped to the binary visual grid map, where every grid represented a region. Whether the autonomous vehicle can pass through the region was determined based on the grid value. “1” represents the region with obstacles, i.e., an impassable route. “0” represents a passable route. The i-th grid can be expressed as 
, and the grid environment map is shown in Fig. 1.
Fig. 1.
Grid map of the environment.
As shown in Fig. 1, grid (1,1) is the starting point of path planning and grid (20,20) is the destination. Figure 2a,b show the simulation paths of traditional and improved ant colony algorithms, respectively. The path length of the improved ant colony algorithm is smaller than that of the traditional ant colony algorithm. Figure 2c shows the convergence curves; the curve performance enhanced for the improved ant colony algorithm.
Fig. 2.
Simulation results of path planning: simulation path of (a) traditional ant colony algorithm and (b) improved ant colony algorithm. (c) Convergence curves.
Table 1 compares the performance of both algorithms. The traditional colony algorithm outputted the shortest path length of 36.1 m in 56 iterations and 8.6 s, with 13 inflection points and 630°total rotation angle values. The improved colony algorithm outputted the shortest path length of 29.8 m in 8 iterations and 4.8 s, with 5 inflection points and 225°total rotation angle values. Due to introduction of the angle information and its weight in Eq. (7), path planning is more directional; and the pheromone regulatory factor C increases the convergence speed. The improved ant colony algorithm exhibited better performance than the traditional ant colony algorithm.
Table 1.
Comparison of the performances of traditional and improved ant colony algorithms.
| Algorithm | Items | ||||
|---|---|---|---|---|---|
| Distance of the least path (m) | Iterations | Time (s) | Inflection point | Total rotation angle values (°) | |
| Traditional ant colony algorithm | 36.1 | 56 | 8.6 | 13 | 630 |
| Improved ant colony algorithm | 29.8 | 8 | 4.8 | 5 | 225 |
Figure 2b shows that the simulation path is near the obstacle edges, posing a collision risk. To reduce the probability of collision, obstacles are expanded by placing virtual obstacles around real obstacles. The corresponding grid map is shown in Fig. 3, wherein the red obstacles represent virtual obstacles.
Fig. 3.
Grid map obtained after the expansion of obstacles.
Figure 4 shows the path planning routes of traditional and improved ant colony algorithms. Table 2 shows their corresponding performances, which indicates that their path planning results are nearly the same. The iterations, computing time, number of inflection points and total rotation angle values decreased from 42 times, 12.4 s, 11 and 495°to 20 times, 7.9 s, 5 and 225°, respectively.
Fig. 4.
Path planning after placing virtual obstacles. (a) Traditional ant colony algorithm, (b) improved ant colony algorithm, and (c) convergence curves.
Table 2.
Performance comparison of traditional and improved ant colony algorithms after the expansion of obstacles.
| Algorithm | Items | ||||
|---|---|---|---|---|---|
| Distance of the least path (m) | Iterations | Time (s) | Inflection point | Total rotation angle values (°) | |
| Traditional ant colony algorithm | 32.7 | 42 | 12.4 | 11 | 495 |
| Improved ant colony algorithm | 30.9 | 20 | 7.9 | 5 | 225 |
Simulation on a 30 × 30 grid map
Figure 5 shows the results of the simulation performed on a 30 × 30 grid map. Figure 5a,b show the paths planned by the traditional and improved ant colony algorithms, respectively. The path length and iterations of the improved ant colony algorithm were considerably smaller than those of the traditional ant colony algorithm. Figure 5c shows the convergence curves, indicating that the improved anti-colony algorithm performed better than the traditional one on a 30 × 30 grid map. Table 3 compares the performances of the two algorithms. The path length, iterations, computing time, number of inflection points and total rotation angle values of the traditional ant colony algorithm are 48.9 m, 70 times, 28.8 s, 27, and 1665°respectively, whereas those of the improved ant colony algorithm decreased to 44.8 m, 18 times, 15.6 s, 8 and 360°, respectively.
Fig. 5.
Simulation results of path planning on a 30 × 30 grid map. (a) Traditional ant colony algorithms (b) Improved ant colony algorithms (c) Convergence curves.
Table 3.
Comparison of traditional and improved ant colony algorithm performance on a 30 × 30 grid map.
| Algorithm | Items | ||||
|---|---|---|---|---|---|
| Distance of the least path (m) | Iterations | Time (s) | Inflection points | Total rotation angle values (°) | |
| Traditional ant colony algorithm | 48.9 | 70 | 28.8 | 27 | 1665 |
| Improved ant colony algorithm | 44.8 | 18 | 15.6 | 8 | 360 |
Figure 5a shows the expanded grid map with the disconnected path. The path planning performance of both algorithms was compared after the 30 × 30 grid map was expanded. Figure 6a,b show the path planning performances of traditional and improved ant colony algorithms, respectively. Figure 6c shows the convergence curves of traditional and improved ant colony algorithms, and Table 4 shows their performance comparison. The path lengths, iterations, computing time, number of inflection points and total rotation angle values decreased from 52.4 m, 47 times, 33.5 s, 22 and 1395°, respectively, for the traditional ant colony algorithm to 48.3 m, 26 times, 26.2 s, 9, and 405°, respectively, for the improved ant colony algorithm. These findings indicated that the improved algorithm exhibited considerably better path-finding performance.
Fig. 6.
Simulation results of path planning on the expanded 30 × 30 grid map. (a) Traditional ant colony algorithm (b) Improved ant colony algorithm (c) Convergence curves.
Table 4.
Comparison of traditional and improved ant colony algorithm performance on the expanded 30 × 30 grid map.
| Algorithm | Items | ||||
|---|---|---|---|---|---|
| Distance of the least path (m) | Iterations | Time (s) | Inflection points | Total rotation angle values (°) | |
| Traditional ant colony algorithm | 52.4 | 47 | 33.5 | 22 | 1395 |
| Improved ant colony algorithm | 48.3 | 26 | 26.2 | 9 | 405 |
Experiments
Experimental vehicles
Figure 7 shows the autonomous vehicles used during the experiments, which comprise a wire-controlled chassis module, environment sensing module, and navigation module. Table 5 details their specific configuration.
Fig. 7.
Autonomous vehicles used in experiments.
Table 5.
Specific configurations of the autonomous vehicles used in experiments.
| No | Module | Type | Attributes |
|---|---|---|---|
| 1 | Wire-controlled chassis | AGILEX, Scout Mini | Four wheels and independent suspension |
| 2 | On-board computer | NUC11 | 11th generation Core i7, 4 cores and 8 threads |
| 3 | Lidar | Robo Sense | Wavelength: 905 nm; Frequency:10/20 Hz |
| 4 | Camera | Microsoft, Kinect DK | Depth camera: 1 million pixels; RGB camera: 12 million pixels |
| 5 | Millimeter wave radar | Delphi, ESR | 77 GHz |
| 6 | Inertial navigation system | Wit Motion, HWT9053 | Accuracy: 0.001°; 9 axes |
| 7 | GPS | ComNav, M100 | Accuracy: Horizontal: ± (10 + 1 × 10−6 XD) mm; Vertical: ± (20 + 1 × 10−6 XD) mm |
| 8 | Screen | 14 inches | Proportion: 16:9; Resolution: 1920 × 1080 |
The wire-controlled chassis was a Scout Mini obtained from Agilex Robotics Co., Ltd.; it contained features such as lights, in-wheel motors, four-wheel differential drive, independent suspension, and in situ differential rotation. Intel NUC11 was used as the onboard computer, with an 11th-generation Core i7 CPU comprising four cores and eight threads. Lidar was obtained from RoboSense with 32 lines, a wavelength of 905 nm, and a frequency of 10/20 Hz. Kinect DK from Microsoft was used as the depth camera with 1 million pixels, and the RGB camera had 12 million pixels. Delphi ESR medium-range millimeter wave radar with a frequency of 77 Hz was used as the millimeter wave radar. The 9-axis HWT9053 was used as the navigation system, with a measurement accuracy of 0.001°. M100 from ComNav Technology Ltd was used as the GPS, with a horizontal accuracy of ± (10 + 1 × 10−6 XD) mm and vertical accuracy of ± (20 + 1 × 10−6 XD) mm. The screen was 14-inch wide, with a 16:9 proportion and a 1920 × 1080 resolution.
Maps used for experiments
Figure 8 shows the real map built based on the grid map shown in Fig. 1, and cone barrels were used as the obstacles. Based on the obstacle model shown in Fig. 1, 37 cone barrels were placed such that a 20 × 20 real grid map was created (Fig. 8a). As shown in Fig. 8a, the autonomous vehicle moves from grid (1,1) to grid (20,20). Results showed that the autonomous vehicles could run from the start point to the end point following the path shown in Fig. 4b using the improved ant colony algorithm.
Fig. 8.
Experimental scene maps. (a) 20 × 20 scene map; (b) Map acquired by Lidar.
Discussion
Comparison of improved ant colony algorithms
Several researchers have improved the traditional ant colony algorithms. For instance, Yong Tao proposed a pheromone update rule using fuzzy control, in which the values of pheromone and expectation heuristic factors were changed to adjust the evaporation rate of pheromone in stages21, which is called fuzzy control ant colony algorithms (FCACA).
Khaled Akka introduced an improved ant colony algorithm that used a stimulating probability to help the ant to select the next grid. They employed new heuristic information based on the principle of unlimited step length to expand the vision field and enhance the visibility accuracy22 which is called stimulating probability ant colony algorithms (SPACA). The performance of the improved ant colony algorithm proposed herein was compared with those of previously reported improved ant colony algorithms21,22. Figure 9a–c show the paths planned by traditional ant colony algorithms and FCACA proposed previously22 and herein, respectively. Figure 9d shows their corresponding convergence curves, and Table 6 compares their performances. The shortest paths length found by both the improved ant colony algorithms22 were the same and smaller than those found by the traditional ant colony algorithm. The iterations, inflection points and total rotation angle values of the proposed improved ant colony algorithm were lower than the other two algorithms.
Fig. 9.
Experimental results of all compared algorithms. (a) Path planned by the traditional colony algorithm; (b) Path planned by the improved ant colony algorithm [21]; (c) Path planned by the proposed improved algorithm; (d) Convergence curves of different algorithms.
Table 6.
Comparison of the performances of traditional and improved ant colony algorithms on a 20 × 20 grid map.
| Algorithm | Items | |||
|---|---|---|---|---|
| Distance of the least path (m) | Iterations | Inflection points | Total rotation angle values (°) | |
| Traditional ant colony algorithm | 30.6 | 29 | 12 | 585 |
| Reference [20] | 29.8 | 25 | 10 | 450 |
| Improved ant colony algorithm | 29.8 | 16 | 9 | 405 |
Figure 10a–c show the path planned by the traditional ant colony algorithm and those by the SPACA proposed previously22 and herein, respectively. Figure 10d shows the convergence curves of different algorithms, and Table 7 shows their corresponding performances.
Fig. 10.
Experimental results that compared with reference [22]; (a) Path planned by the traditional colony algorithm; (b) Path planned by the improved ant colony algorithm [22]; (c) Path planned by the proposed improved algorithm; (d) Convergence curves of different algorithms.
Table 7.
Comparison of traditional and improved ant colony algorithm performances on a 30 × 30 grid map.
| Algorithm | Items | |||
|---|---|---|---|---|
| Distance of the least path (m) | Iterations | Inflection points | Total rotation angle values (°) | |
| Traditional ant colony algorithm | 50.0 | 65 | 22 | 990 |
| Reference [21] | 43.7 | 25 | 18 | 810 |
| Improved ant colony algorithm | 44.3 | 13 | 18 | 810 |
The least path of the proposed improved ant colony algorithm was longer than SPACA22 but shorter than the traditional ant colony algorithm. The iterations of the proposed algorithm were lesser than the SPACA22. Their inflection points and total rotation angle values were the same.
The performance of the improved ant colony algorithm proposed herein was compared with IACO-IABC20. The experimental results are shown in Fig. 11. The path length, number of inflection points and total rotation angle values of the improved ant colony algorithm are 28.6 m, 7 and 315°respectively, whereas those of the IACO-IABC are 31.3 m, 3 and 135°, respectively. The path length of improved ant colony algorithm is better than IACO-IABC.
Fig. 11.
Compared with IACO-IABC.
Comparisons with A*, Dijkstra’s, and RRT
Comparison experiments with A*, Dijkstra’s, and RRT algorithms on 20 × 20 grid map and 30 × 30 grid map have been conducted respectively, and the results is shown in Fig. 12. In Fig. 12a, the improved colony algorithm outputted the shortest path length of 29.2 m, with 5 inflection points and 90°rotation angle values. Its’ number of inflection points and total rotation angle are smaller than the other three methods. The detailed comparison results are shown in Table 8.
Fig. 12.
Comparison experiments with other widely used algorithms. (a) Path planned on a 20 × 20 grid map; (b) Path planned on a 30 × 30 grid map.
Table 8.
Comparison with A*, Dijkstra’s, RRT on a 20 × 20 grid map.
| Algorithm | Items | ||
|---|---|---|---|
| Distance of the least path (m) | Inflection points | Total rotation angle values (°) | |
| Improved ant colony algorithm | 29.2 | 5 | 90 |
| A* | 29.2 | 10 | 495 |
| Dijkstra’s | 29.2 | 9 | 360 |
| RRT | 34.1 | 15 | 1170 |
In Fig. 12b, the improved colony algorithm outputted the shortest path length of 46.5 m, with 9 inflection points and 360°rotation angle values. Similarly, its’ number of inflection points and total rotation angle are smaller than the other three methods. The detailed comparison results are shown in Table 9.
Table 9.
Comparison with A*, Dijkstra’s, RRT on a 30 × 30 grid map.
| Algorithm | Items | ||
|---|---|---|---|
| Distance of the least path (m) | Inflection points | Total rotation angle values (°) | |
| Improved ant colony algorithm | 46.5 | 9 | 360 |
| A* | 45.1 | 12 | 585 |
| Dijkstra’s | 45.1 | 12 | 585 |
| RRT | 54.5 | 28 | 1620 |
Sensitivity of weight parameters
Sensitivity Experiments of weight parameters that dw and aw in Eq. (7) are conducted on 20*20 grid map which is shown in Fig. 12a, and the results are shown in Fig. 13. In Fig. 13a, the range of aw is 0.1–1. Path length is not affected by changes of aw values. When aw is 0.3 and 0.4, the quantity of inflection points is 27. In Fig. 13b, the range of dw is 0.1–1. Path length decreases with dw increasing, and finally equals 29.2. Overall, the inflection points quantity decreases with dw increasing.
Fig. 13.
Sensitivity experiments of weight parameters (a) Sensitivity of path length and Inflection points numbers to aw (b) Sensitivity of path length and Inflection points numbers to dw.
Path smoothing
The simulated paths contained several turning points with discontinuous derivatives and could not meet the dynamic constraints of vehicles. Cubic spline interpolation was applied to the inflection points on the path to generate a smooth curve and address the aforementioned issue. Figure 14a,b show the paths planned on 20 × 20 and 30 × 30 grid maps after smoothing, respectively. No inflection points were observed on the paths after smoothing.
Fig. 14.
Path planning after smoothing. (a) Path planned after smoothing on a 20 × 20 grid map; (b) Path planned after smoothing on a 30 × 30 grid map.
The improved ant colony algorithm in this paper could reduce the distance of the least path and iterations significantly. However, its ability to handle dynamic changes, real-time path re-planning lacks theoretical analysis and experimental scenario verification. It is very important that how the improved ant colony algorithm handles dynamic changes, real-time path re-planning, or its efficiency in continuously evolving road scenarios. Due to limitations in paper length and revision time, it will be research in depth in next paper.
Real-time path re-planning
The real-time path re-planning is showed in Fig. 15. Figure 15a shows the initial planned path in original path. Figure 15b shows that a red grid cells become occupied mid-run, the path is replanned. It indicates that the improved ant algorithm could handle dynamic changes.
Fig. 15.
Real-time path re-planning. (a) The initial planned path; (b) The re-plans path after the maps chang.
Conclusions
Herein, an improved ant colony algorithm was proposed to reduce the collision risk and improve the quality and efficiency of path planning. The heuristic function 
was calculated based on the distance and angle from the current node to the target node. A pheromone regulatory factor C was introduced; its value decreased when the path length was greater than the average overall path length and increased otherwise. The simulation results of the proposed improved ant colony algorithm on 20 × 20 and 30 × 30 grid maps showed that its path length and number of iterations decreased by an average of 9.8% and 64.3%, respectively, compared with the traditional ant colony algorithm. Its number of iterations and inflection points decreased by 36% and 10% compared with the FCACA, respectively. Its number of iterations also decreased by 48% compared with that of SPACA. To meet the dynamic constraints of autonomous vehicles and generate a smooth curve, cubic spline interpolation was applied to the inflection points on the path. Experimental results revealed that autonomous vehicles could move from the starting point to the end point along the simulated paths using the proposed algorithm.
Acknowledgements
This work was supported by Scientific research projects of Tianjin Municipal Education Commission under Grant 2020KJ086 and Tianjin Enterprise Science and Technology Commissioner Project under Grant under Grant 25YDTPJC00820.
Author contributions
G. W., Z. G. and H. Z. proposed the improved ant colony algorithm; R. L. and R. D. conduct experiments; G. W. analyzed the results. All authors reviewed the manuscript.
Data availability
All data generated or analyzed during this study are included in this published article.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Guoqiang Wen, Email: wgqdiamond@126.com.
Zhiwei Guan, Email: zhiwguan@163.com.
Hongxia Zhang, Email: hxzhang@tju.edu.cn.
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Data Availability Statement
All data generated or analyzed during this study are included in this published article.