3.1. The ACO algorithm before improvement

ACO algorithm is a typical representative of meta-heuristic algorithms, inspired by the collaborative behaviour of ant colonies foraging in nature [Reference Dorigo, Maniezzo and Colorni39]. Research has found that when ants search for food, they release pheromones along the way, forming chemical traces that can be perceived by other ants. The paths passed multiple times become more attractive due to the accumulation of pheromones, constituting a positive feedback effect. The natural volatilisation of pheromones maintains a dynamic balance, enabling the ant colony to strike a balance between exploring new routes and optimising existing ones. ACO algorithm simulates this process by constructing solutions through artificial ants and updating pheromones in iterations to optimise the objective function.

In the ACO algorithm, each ant selects the next path based on the pheromone concentration and heuristic information. The probability that ant k chooses to go to node j at node i is as follows:

(2) \begin{equation} P_{ij}^{k}\left(t\right)=\left\{\begin{array}{c@{\quad}c} \dfrac{\left[\tau _{ij}\left(t\right)\right]^{\alpha }\cdot \left[\eta _{ij}\left(t\right)\right]^{\beta }}{\sum\limits_{j\in \text{allowed}_{k}}\left[\tau _{ij}\left(t\right)\right]^{\alpha }\cdot \left[\eta _{ij}\left(t\right)\right]^{\beta }}, & \text{if }j\in \text{allowed}_{k}\\[10pt] 0, & \text{otherwise} \end{array}\right. \end{equation}

where $\alpha$ serves as the pheromone heuristic factor and is used to adjust the weight of pheromone concentration in the decision-making process; $\beta$ , as an expectation heuristic factor, influences the expectation of ants in choosing paths; $\eta _{ij}(t)$ represents the heuristic information from the ith grid to the jth grid at time t, which is calculated by Eq. (3); and $\textit{allowed}_{k}$ includes all the subsequent nodes that ant k can reach during its path exploration process:

(3) \begin{equation} \eta _{ij}\left(t\right)=\frac{1}{d_{ij}} \end{equation}

When ants complete a round of search iterations, the update of pheromones is indispensable. At the next moment, the pheromone concentration on the path will be recalculated and adjusted according to a specific update equation. The specific update equation is as follows:

(4) \begin{equation} \tau _{ij}\left(t+1\right)=\left(1-\rho \right)\tau _{ij}\left(t\right)+\Delta \tau _{ij}\left(t\right) \end{equation}

(5) \begin{equation} {\unicode{x0394}} \tau _{ij}\left(t\right)=\sum _{k=1}^{M}{\unicode{x0394}} \tau _{ij}^{k}\left(t\right) \end{equation}

(6) \begin{equation} {\unicode{x0394}} \tau _{ij}^{k}=\left\{\begin{array}{l} \dfrac{1}{L_{k}}\\ 0 \end{array}\right. \end{equation}

where $\rho$ is the pheromone volatilisation factor, $L_{k}$ is used to represent the length of the path walked by the kth ant, and ${\unicode{x0394}} \tau _{ij}(t)$ represents the total release of pheromones on the path from the ith grid to the jth grid.

3.2. Bresenham algorithm

The Bresenham algorithm is a classic computer graphics algorithm used to draw straight lines on discrete grids, such as pixel grids. It was proposed by Jack E. Bresenham in 1962 and has been widely used due to its high efficiency and simplicity. The prominent feature of this algorithm is that it uses integer operations and avoids floating-point operations, making it highly suitable for computing environments with limited resources.

The core of the Bresenham algorithm is to select the pixel closest to the true line by controlling the error. Given a starting point $(x_{0,}y_{0})$ and an endpoint $(x_{n,}y_{n})$ , the algorithm decides at each step whether to move horizontally or diagonally to approximate the line path. As shown in Figure 2.

Assume a starting point $(x_{0,}y_{0})$ to an endpoint $(x_{n,}y_{n})$ , where $x_{0}\lt x_{n}$ , $y_{0}\lt y_{n}$ , and the slope is calculated by Eqs. (7), (8), and (9):

(7) \begin{equation} m=\frac{\Delta y}{\Delta x}\leq 1 \end{equation}

(8) \begin{equation} \Delta x=x_{n}-x_{0} \end{equation}

(9) \begin{equation} \Delta y=y_{n}-y_{0} \end{equation}

First, the error term e is initialised, with the calculation equation as shown below:

(10) \begin{equation} e=2\Delta y-\Delta x \end{equation}

Next, the current pixel $(x,y)$ is drawn. If $e\lt 0$ , node $(x+1,y)$ is selected and the error term e is updated using Eq. (11); if $e\gt 0$ , node $(x+1,y+1)$ is selected and the error term e is updated using Eq. (12):

(11) \begin{equation} e=e+2\Delta y \end{equation}

(12) \begin{equation} e=e+2\left(\Delta y-\Delta x\right) \end{equation}

Finally, this iteration process is repeated until $x=x_{n}$ . In this way, the Bresenham algorithm ensures that the generated pixel path visually approximates the true line while maintaining efficiency.

4.1. Bresenham pheromone enhancement strategy

In the traditional ACO algorithm, the pheromone matrix is typically initialised uniformly, which may lead to low search efficiency in complex environments. To improve search efficiency, the Bresenham algorithm can be used to generate a path from the starting point to the endpoint that closely approximates a straight line, and the pheromone concentration can be enhanced on the edges along this path. This way, ants will prioritise exploring along the approximate straight-line direction in the early stages of the search, reducing ineffective searches and accelerating the algorithm’s convergence speed.

Figure 2.

Bresenham straight line.

First, use the Bresenham algorithm to generate a raster path between the starting point and the endpoint that closely approximates a straight line. Then, enhance the pheromone concentration of this path according to Eq. (13), as shown in Figure 3.

Figure 3.

Bresenham straight line.

(13) \begin{equation} \tau _{ij}=\tau _{0}\cdot D \end{equation}

where $\tau _{0}$ is the initial pheromone value of the grid cell and $D$ is the enhancement factor, typically $D\gt 1$ , used to increase the pheromone concentration on the Bresenham straight-line path.

Initialising pheromones using the Bresenham algorithm effectively guides ants to select paths close to a straight line in the early stages of the search. This method significantly improves the search efficiency and convergence speed of the ACO algorithm, particularly in the field of path planning. By combining the simplicity and efficiency of the Bresenham algorithm with the global optimisation capabilities of ACO, this strategy demonstrates strong potential in practical applications.

4.2. Introduction of the direction continuity factor

The traditional ACO algorithm guides ants to find optimal paths using pheromones and heuristic information. However, in complex environments, the generated paths may contain numerous sharp turns or unsmooth trajectories, particularly in applications like robot path planning, which increases the difficulty and cost of practical execution.

To improve path smoothness, this study introduces a direction continuity factor to encourage ants to prioritise paths consistent with the previous movement direction. This approach quantifies directional consistency and adjusts the state transition probability, resulting in paths that better meet continuity and smoothness requirements, reducing abrupt turns.

The core idea of the direction continuity factor is to leverage the relationship between the ant’s current movement direction and the direction of candidate path nodes, increasing the probability of selecting paths aligned with the previous direction. The specific implementation is based on the following principle: if the ant’s previous direction is close to the next direction (i.e., the angle between them is small), the path is considered more optimal. Directional similarity is calculated as the ratio of the vector dot product to the vector magnitudes, which is then converted into a probability weight. The improved state transition probability equation is as follows:

(14) \begin{equation} P\left(i,j\right)=\frac{\left[\tau \left(i,j\right)\right]^{\alpha }\cdot \left[\eta \left(\,j\right)\right]^{\beta }\cdot \left[s_{ij}\right]^{\omega }}{\sum\limits_{k\in \text{allowed}}\left[\tau \left(i,k\right)\right]^{\alpha }\cdot \left[\eta \left(k\right)\right]^{\beta }\cdot \left[s_{ik}\right]^{\omega }} \end{equation}

where $s_{ij}$ is the direction continuity factor, calculated by Eq. (15) and $\omega$ is the direction continuity weight parameter, used to adjust the strength of the direction continuity factor’s influence.

(15) \begin{equation} s_{ij}=\exp \left(\frac{\cos \theta }{2}\right) \end{equation}

where $cos\theta$ is the direction cosine, representing directional similarity using the cosine of the angle between vectors, calculated by Eqs. (16), (17), and (18).

(16) \begin{equation} \cos \theta =\frac{\mathbf{v}_{prev}\cdot \mathbf{v}_{\textit{current}}}{|\mathbf{v}_{prev}|\cdot |\mathbf{v}_{\textit{current}}|} \end{equation}

where $v_{prev}$ is the previous step direction vector, calculated by Eq. (17), $v_{\textit{current}}$ is the current step direction vector, calculated by Eq. (18), $v_{prev}\cdot v_{\textit{current}}$ is the vector dot product, and $|v_{prev}| \cdot | v_{\textit{current}}| $ is the product of the vector magnitudes.

(17) \begin{equation} \mathbf{v}_{prev}=\left(x_{\textit{current}}-x_{prev},y_{\textit{current}}-y_{prev}\right) \end{equation}

(18) \begin{equation} \mathbf{v}_{\textit{current}}=\left(x_{next}-x_{\textit{current}},y_{next}-y_{\textit{current}}\right) \end{equation}

where $x_{prev}$ and $v_{prev}$ are the x-coordinate and y-coordinate of the node preceding the current node; $x_{next}$ and $v_{next}$ are the x-coordinate and y-coordinate of the next selectable node from the current node; and $x_{\textit{current}}$ and $v_{\textit{current}}$ are the x-coordinate and y-coordinate of the current node.

By introducing a direction continuity factor to quantify the consistency of movement direction, the state transition probability of the traditional ACO algorithm is modified, significantly improving path smoothness. This method effectively reduces unnecessary turns in complex environments, making the generated paths better aligned with practical requirements.

4.3. Pheromone update affected by path curvature

The traditional ACO algorithm is modelled on the foraging behaviour of ant colonies, guiding the search process through a dynamic pheromone updating mechanism. Specifically, ants deposit pheromones along the paths they traverse, and the concentration of pheromones increases on shorter or more frequently chosen paths. This, in turn, attracts more ants to these paths, forming a positive feedback loop that iteratively optimises the solution. The algorithm typically uses path length or cost as a single optimisation objective, gradually approaching the global optimum through repeated iterations. However, in practical applications such as robot path planning or vehicle navigation, focusing solely on path length may result in routes with numerous sharp turns or high-curvature segments. These features not only increase execution difficulty but can also lead to greater energy consumption and reduced system stability, posing potential safety risks. Therefore, it is necessary to incorporate geometric constraints such as path smoothness or curvature into the optimisation objective framework.

The pheromone update influenced by path curvature is an improvement to the traditional ACO algorithm. By introducing the factor of path curvature into the pheromone update process, ants are encouraged to choose paths with lower curvature (i.e., smoother paths), thereby enhancing the quality and practicality of the resulting paths. The underlying principle is as follows: the curvature is quantified by calculating the angle formed by each set of three consecutive points along the path. The smaller the curvature, the smoother the path. More pheromone is deposited on paths with lower curvature to guide subsequent ants to preferentially select these paths. The improved pheromone update equation is shown in Eq. (19):

(19) \begin{equation} {\unicode{x0394}} \tau _{ij}^{k}=\frac{1}{L_{k}}\cdot \left(1+\frac{C}{2}\right) \end{equation}

where the path curvature influences the value of C; the smaller the curvature and the smoother the path, the larger the value of C, which is calculated according to Eq. (20).

For three consecutive points $P_{i-1}$ , $P_{i}$ , and $P_{i+1}$ on a path, the vectors $v_{1}=P_{i}-P_{i-1}$ and $v_{2}=P_{i+1}-P_{i}$ are calculated, and the sum of the absolute values of the cosine of the angles between these vectors is then computed:

(20) \begin{equation} C=\sum _{i=2}^{n-1}\left| \frac{v_{1}\cdot v_{2}}{|v_{1}|\cdot |v_{2}|}\right| \end{equation}

By introducing a path curvature factor, the pheromone update becomes associated with the curvature of the path. The straighter the path, the higher the curvature value (C-value), and consequently, the greater the pheromone increment. This mechanism improves upon the traditional pheromone updating method by guiding the algorithm to favour paths with lower curvature and smoother trajectories. As a result, it significantly enhances the practicality and adaptability of the generated paths in complex scenarios such as robot navigation.

4.4. Path smoothing strategy based on the spring model

The core objective of robot path planning is to find a path that not only avoids obstacles but also maintains smoothness and efficiency. However, this task becomes particularly challenging in environments with dense obstacles. In such scenarios, the path planning algorithm must navigate around obstacles while minimising the overall path length and curvature, all the while ensuring the path remains feasible. For example, in an environment filled with obstacles, the robot may need to make frequent turns or detours, which can increase the path length and introduce sharp turns or unnecessary deviations. Traditional path planning algorithms often struggle to balance obstacle avoidance, smoothness, and efficiency simultaneously, resulting in suboptimal paths.

To address the aforementioned limitations, many researchers in recent years have proposed improvements to path planning algorithms by incorporating various path smoothing techniques. Poy et al. introduced an enhanced particle swarm optimisation (EPSO) algorithm, which combines a novel path planning scheme with a cubic Bézier curve-based trajectory smoothing method. This approach reduces the tortuosity of the generated path and avoids frequent turning [Reference Poy, Loke, Darmaraju, Goh, Kwan, Liu and Ng40]. Mohaghegh et al. proposed a smoothing particle swarm optimisation with improved potential field (SPSO-IPF), which integrates PSO with a potential field (PF) method to produce smoother and safer paths [Reference Mohaghegh, Jafarpourdavatgar and Saeedinia41]. Cao et al. developed a multi-objective path smoothing algorithm based on node adjustment and ramp smoothing. Their method filters and moves nodes in the original path using a path point adjustment algorithm. Then, B-spline curves are used to smooth discontinuities at turns, and control points are adjusted to ensure obstacle avoidance while satisfying the maximum curvature constraint [Reference Cao, Yi, Zhang and Qiu42].

Existing path-smoothing methods proposed by scholars have achieved significant results in improving the smoothness of paths generated by meta-heuristic algorithms, reducing path inflection points, and avoiding frequent turns. However, these methods still suffer from limitations in local optimisation. To address the shortcomings of traditional path smoothing strategies in global smoothness control and redundant node elimination, this study innovatively introduces the spring model into the field of path planning, proposing a path optimisation method based on dynamic principles. The core idea of the spring model is to treat nodes on the path as mass points connected by virtual springs between adjacent nodes. By simulating the stretching and compressive forces of the springs, the node positions are dynamically adjusted, ultimately achieving continuous path optimisation and autonomous elimination of redundant nodes. The specific operations are as follows.

For each node on the path (except the fixed start and endpoints), the model calculates the resultant force acting on it.

First, the spring force between node i and its previous node i-1, as well as the spring force between node i and its next node i + 1, is calculated as follows:

(21) \begin{equation} \overset{\rightarrow }{F}_{1}=\left(\| x_{i}-x_{i-1}\| -\frac{1}{2}\right)\cdot \frac{x_{i-1}-x_{i}}{\| x_{i-1}-x_{i}\| } \end{equation}

(22) \begin{equation} \overset{\rightarrow }{F}_{2}=\left(\| x_{i+1}-x_{i}\| -\frac{1}{2}\right)\cdot \frac{x_{i+1}-x_{i}}{\| x_{i+1}-x_{i}\| } \end{equation}

where $x_{i}$ represents the coordinates of the current node, $x_{i-1}$ represents the coordinates of the previous node of node i, and $x_{i+1}$ represents the coordinates of the next node of node i.

Then, the smoothing force and the direct connection force are calculated as follows:

(23) \begin{equation} \overset{\rightarrow }{F}_{3}=\frac{1}{2}\left(x_{i+1}-2x_{i}+x_{i-1}\right) \end{equation}

(24) \begin{equation} \overset{\rightarrow }{F}_{4}=\frac{1}{2}\left(\frac{x_{i+1}+x_{i-1}}{2}-x_{i}\right) \end{equation}

Finally, the sum of all the forces is calculated as follows:

(25) \begin{equation} \overset{\rightarrow }{F}_{\textit{total}}=\overset{\rightarrow }{F}_{1}+\overset{\rightarrow }{F}_{2}+\overset{\rightarrow }{F}_{3}+\overset{\rightarrow }{F}_{4} \end{equation}

The calculated resultant force acting on node i is substituted into the position update Eq. (26), and the new coordinates of the node are obtained:

(26) \begin{equation} x_{i}^{new}=x_{i}+\frac{1}{20}\cdot \overset{\rightarrow }{F}_{\textit{total}} \end{equation}

The spring force and direct connection force encourage nodes to move toward a straighter path, significantly reducing path length, while the smoothing force reduces sharp corners in the path. Through the above formula calculations and multiple iterations, the nodes on the path are gradually adjusted, ultimately resulting in a shorter and smoother path, generating a more natural trajectory. As shown in Figure 4, the blue path represents the original path, and the red path represents the smoothed path.

Figure 4.

Path optimisation graph based on spring model.

The path smoothing strategy based on the spring model optimises the paths generated by the ant colony algorithm through the synergistic effects of spring forces, smoothing forces, and direct connection forces. The optimisation process integrates iterative updates and collision detection, ensuring that the path is both shortened and smoothed while avoiding obstacles. This method is highly practical in robot path planning, effectively enhancing path quality.

4.5. Steps and processes of the BCD-ACO algorithm

The enhanced algorithm begins its process by leveraging the Bresenham algorithm to establish initial pheromone trails strategically laid out to steer ants along a search trajectory that closely resembles a straight line. This initialisation phase ensures that the ants‘ early exploration efforts are directed efficiently toward their intended destination, setting a solid foundation for subsequent refinements. Following this, the algorithm focuses on improving the smoothness of the path through a series of repeated search iterations, integrating a direction continuity factor, a mechanism designed to maintain consistent movement directions. This factor reduces sudden shifts or jagged transitions in the ants‘ paths, resulting in a more cohesive and fluid route.

Next, the algorithm adjusts the pheromone concentrations dynamically, considering the curvature of the paths that emerge during the ants‘ exploration. By favouring paths with gentler curves and fewer sharp bends, this pheromone update process encourages the selection of smoother trajectories, enhancing the overall quality and navigability of the routes chosen by the ants. Finally, the algorithm employs a sophisticated path-smoothing strategy rooted in a spring model, which simulates elastic forces to refine the path further. This approach minimises irregularities and sharp turns, optimising the route for efficiency and safety while validating its practicality. Through this multi-step methodology, the algorithm ultimately produces a highly refined path that is not only efficient in terms of exploration but also smooth and secure for traversal. The flow chart of the BCD-ACO algorithm is shown in Figure 5. The pseudocode of the BCD-ACO algorithm is shown in Table I.

Figure 5.

Flowchart of the BCD-ACO algorithm.

5.1. Parameter tuning

In the BCD-ACO algorithm, parameter tuning is crucial for performance optimisation. By optimising parameter settings, the algorithm can achieve a dynamic balance between exploration and exploitation, which helps enhance solution quality and search efficiency. Moreover, carefully tuned parameters significantly improve the algorithm’s stability and reliability across multiple runs, reducing performance fluctuations caused by randomness. This enables the BCD-ACO algorithm to exhibit greater adaptability and robustness when addressing complex optimisation problems, providing strong support for efficient problem-solving in real-world applications. This study uses the grid environment from ref. [Reference Cui, Wu, Huang, Xu, Liu and Xiao36] (as shown in Figure 6). A controlled variable method is applied to tune the parameters $\alpha$ , $\beta$ , and $\omega$ sequentially. For each parameter value, 10 experiments are conducted to analyse its impact on key performance metrics, including the shortest path length ( $PL_{min}$ ), longest path length ( $PL_{max}$ ), average path length ( $PL_{mean}$ ), standard deviation of path length ( $PL_{std}$ ), best iteration count ( $I_{min}$ ), worst iteration count ( $I_{max}$ ), average iteration count ( $I_{mean}$ ), and standard deviation of iteration count ( $I_{std}$ ).

Figure 6.

Tuning environment diagra.

5.1.1. α parameter tuning

In the ACO algorithm, the parameter α functions as the pheromone influence coefficient, directly governing how strongly the existing pheromone trails steer each ant’s decision-making when selecting the next node. By increasing α, one amplifies the weight of pheromone concentration in the probabilistic transition rule, thereby encouraging ants to favour routes that have been heavily traversed in previous iterations. Conversely, a lower α reduces this bias, allowing heuristic information, such as distance or cost, to play a more prominent role. Since α fundamentally balances exploration and exploitation, its proper adjustment is essential for achieving rapid convergence without premature stagnation. For this reason, our tuning methodology is anchored first and foremost on carefully selecting the optimal value of α before proceeding to refine other algorithmic parameters. The tested range for $\alpha$ is {1,2,3,4,5}, while the other parameters are fixed at $K=100$ , $M=50$ , $\beta =10$ , $\rho =0.3$ , $\omega =5$ , and $D=2$ . The experimental results are shown in Figure 7.

Figure 7.

The influence of different parameter α values on various indicators.

Figure 7 illustrates the relationship between the α parameter and the variability of the algorithm’s convergence behaviour. As the plot indicates, setting α to 4 yields the smallest standard deviation in the number of iterations required for convergence, suggesting a slightly more consistent performance across repeated runs. However, this reduction in variability is very slight when compared to the results obtained with α = 3, and the difference is practically negligible in the context of our simulation experiments. More importantly, when α is set to 3, all other key performance metrics, including average convergence speed, total path length, and computational time, reach their minimum values. This combination of fast convergence, smooth path trajectories, and low-resource consumption demonstrates that α = 3 provides the best overall balance of performance characteristics. Consequently, we adopt α = 3 for the BCD-ACO algorithm to maximise search efficiency and ensure robust, high-quality path planning results.

5.1.2. β parameter tuning

In the ACO algorithm, the parameter β acts as the heuristic information influence coefficient, determining the weight given to the inverse distance (or other heuristic measure) when an ant selects its next node. Increasing β biases the algorithm toward shorter paths, while decreasing β strengthens the role of pheromone trails, promoting broader exploration. Therefore, once the optimal value of α has been established, we immediately tune β to further balance exploration and exploitation. The range of values tested for $\beta$ is {8, 10, 12, 14, 16, 18, 20, 22}, with all other parameters fixed at: $K=100$ , $M=50$ , $\alpha =3$ , $\rho =0.3$ , $\omega =5$ , and $D=2$ . The experimental results are shown in Figure 8.

Figure 8.

The influence of different parameter $\beta$ values on various indicators.

As shown in Figure 8, the minimum path length ( $PL_{min}$ ) consistently reaches the lowest value of 41.46 across all tested values of parameter $\beta$ . However, when $\beta$ is set to 20, all performance metrics reach their lowest levels, indicating that the algorithm achieves the best overall performance under this condition. Therefore, the value of the parameter $\beta$ is set to 20 to enhance the performance of the BCD-ACO algorithm, with the goal of reducing the energy consumption of the robot.

5.1.3. ω parameter tuning

Parameter $\omega$ , which serves as the weight of the directional continuity factor, determines the strength of its influence. Tuning this parameter plays an important role in improving the smoothness of the paths generated by the algorithm. Therefore, it is necessary to optimise the value of $\omega$ .

The tested range for $\omega$ is {1, 2, 3, 4, 5}, with the other parameters fixed at: $K=100$ , $M=50$ , $\alpha =3$ , $\rho =0.3$ , $\beta =20$ , and $D=2$ . The experimental results are shown in Figure 9.

Figure 9.

The influence of different parameter $\omega$ values on various indicators.

As shown in Figure 9, the minimum path length ( $PL_{min}$ ) consistently reaches the lowest value of 41.46, and the maximum path length ( $PL_{max}$ ) remains constant at 41.48 across all tested values of parameter $\omega$ . When $\omega$ is set to 3, the standard deviation of path length ( $PL_{std}$ ) reaches its minimum; however, since $PL_{std}$ remains within a narrow range of 0.23 to 0.27 regardless of the $\omega$ value, the impact of $\omega$ on this metric can be considered negligible. When ω is set to 5, all performance metrics except $PL_{std}$ reach their lowest values, indicating that ω = 5 offers the best overall performance. Therefore, the value of $\omega$ is set to 5 to enhance the smoothness of the paths generated by the BCD-ACO algorithm.

5.2. Gradient experiment

In order to more precisely and quantitatively assess the individual contributions of each improvement strategy to the algorithm’s performance, this paper designs and conducts a series of systematic gradient experiments in two different grid environments. The experiments employ a stepwise introduction approach, incrementally adding the various improvement strategies proposed here to the classical ACO algorithm. Using this progressive method, we analyse and compare both the impact of each improvement strategy on algorithm performance and the synergistic effects arising from their combinations. The experimental results are shown in Figures 10 and 11. Here, ACO-1 denotes the classical ACO algorithm augmented with the Bresenham-based pheromone enhancement strategy; ACO-2 builds on ACO-1 by introducing a directional continuity factor; ACO-3 further extends ACO-2 by incorporating a path curvature-influenced pheromone update; and, finally, BCD-ACO adds to ACO-3 a spring model-based path smoothing strategy.

Table II.

Final parameter value table.

Figure 10.

Gradient experiment 1 comparison results.

Figure 11.

Gradient experiment 2 comparison results.

As shown by the results in Figure 10, the traditional ACO algorithm exhibits poor convergence, requiring approximately 60 iterations to converge. This indicates relatively low search efficiency and limited path optimisation capability in complex environments. In contrast, the ACO-1 algorithm, which introduces the Bresenham pheromone enhancement strategy based on the traditional ACO, significantly improves convergence speed, converging in only three iterations and greatly increasing convergence efficiency. However, the paths generated by ACO-1 contain many sharp turns and lack smoothness, indicating that while this strategy improves search efficiency, it does not adequately account for the geometric characteristics of the path.

Building upon the ACO-1 algorithm, the introduction of the directional continuity factor in the ACO-2 algorithm effectively reduces the number of sharp turns in the path and improves its smoothness. This improvement demonstrates that the directional continuity factor can guide ants to preferentially select smoother paths during the search process, thereby enhancing the overall path quality.

Building upon the ACO-2 algorithm, the ACO-3 algorithm introduces pheromone updating influenced by path curvature, which further improves the convergence speed, achieving convergence in just one iteration. This result indicates that the pheromone update strategy based on path curvature dynamically adjusts the pheromone distribution, significantly enhancing the algorithm’s global search capability and convergence efficiency.

Finally, based on ACO-3, the BCD-ACO algorithm further reduces path length and enhances overall path quality by incorporating a spring model-based path smoothing strategy.

The experimental results show that the Bresenham-based pheromone enhancement strategy endows the algorithm with highly efficient search capability, the directional continuity factor improves path smoothness, the curvature-influenced pheromone update further accelerates convergence, and the spring model-based path smoothing strategy achieves final optimisation in both path length and quality. The incremental introduction of each strategy demonstrates clear cumulative effects and synergy. By integrating all these improvements, the BCD-ACO algorithm excels in convergence speed, path smoothness, and overall path length.

To further validate the coordination and performance contributions of the improvement strategies in more complex environments, this study conducted a second gradient experiment in a 30 × 30 grid environment. Compared to the first experiment’s 20 × 20 grid, the new environment is larger and more complex, posing stricter challenges to the algorithm’s search efficiency and path optimisation capability.

The traditional ACO algorithm performs poorly in the 30 × 30 grid environment, struggling to converge and generating paths that are relatively long with numerous sharp turns. This reflects its limitations in complex environments, indicating that the traditional ACO algorithm lacks sufficient search capability and path optimisation ability when dealing with large-scale, complex grids, making it difficult to meet the demands of efficient path planning.

The ACO-1 algorithm, which incorporates the Bresenham pheromone enhancement strategy, significantly improves the algorithm’s performance. The convergence speed is greatly increased, requiring only 30 iterations to converge, and the planned path length is noticeably shorter compared to the traditional ACO algorithm. This improvement demonstrates that the Bresenham pheromone enhancement strategy effectively optimises pheromone distribution, thereby enhancing the algorithm’s search efficiency and path quality in complex environments.

Building upon the ACO-1 algorithm, the introduction of the directional continuity factor in the ACO-2 algorithm further enhances performance, with convergence accelerating to just four iterations. The addition of the directional continuity factor effectively reduces sharp turns in the path and improves its smoothness, indicating that this strategy guides ants to choose better path directions in complex environments, thereby balancing convergence speed and path quality.

Based on the ACO-2 algorithm, the ACO-3 algorithm incorporates pheromone updating influenced by path curvature, demonstrating even more outstanding performance by converging to the optimal solution in just one iteration. This result shows that the pheromone update strategy influenced by path curvature dynamically adjusts pheromone distribution, further strengthening the algorithm’s global search capability and convergence efficiency, enabling it to quickly adapt to the challenges of complex environments.

Finally, based on the ACO-3 algorithm, the BCD-ACO algorithm, which incorporates a spring model-based path smoothing strategy, further compresses the path length, significantly enhances overall path quality, and demonstrates its unique advantages in path optimisation.

Overall, the second series of stepwise experiments confirms the effectiveness and synergy of each improvement strategy in larger, more complex grid environments. The Bresenham-based pheromone enhancement lays the groundwork for efficient search; the directional continuity factor improves path smoothness; the curvature-influenced pheromone update accelerates convergence; and the spring model-based path smoothing strategy further shortens the path length. As the final optimised version, the BCD-ACO algorithm integrates all of these enhancements and, in terms of convergence speed, path smoothness, and path length, significantly outperforms the traditional ACO algorithm.

5.3. Comparative experiment

To comprehensively validate the superior performance of the proposed BCD-ACO algorithm in path planning tasks, this study designs a series of comparative experiments that systematically contrast BCD-ACO with multiple existing path planning algorithms. The experiments include, as ant-colony-based benchmarks, the IAACO algorithm from ref. [Reference Miao, Chen, Yan and Wu31], the PS-ACO algorithm from ref. [Reference Shuang, Chen and Li32], the AIACSE algorithm from ref. [Reference Zhang, Pu and Si33], the IACO algorithm from ref. [Reference Zhao, Cheng and Hao34], the MRCACO algorithm from ref. [Reference Zhang, You, Liu and Pan35], the MsAACO algorithm from ref. [Reference Cui, Wu, Huang, Xu, Liu and Xiao36], and the classical ACO algorithm, in order to assess BCD-ACO’s improvements in convergence speed, path quality, and adaptability to complex environments. Additionally, to further verify BCD-ACO’s competitiveness among a broader set of path planning methods, the study incorporates Q-learning, breadth-first search (BFS), depth-first search (DFS), the A^* algorithm, and Dijkstra’s algorithm as baseline comparisons. These algorithms span heuristic search, reinforcement learning, and graph-search paradigms, representing diverse technical approaches within the path planning field. All tests are conducted on the grid maps used in ref. [Reference Cui, Wu, Huang, Xu, Liu and Xiao36]; performance data for IAACO, PS-ACO, AIACSE, MRCACO, IACO, and MsAACO are cited directly from their original publications to ensure fairness and consistency. Each algorithm is evaluated under uniform conditions with 20 repeated runs per environment, comparing metrics such as path length, convergence speed, and number of turning points. By benchmarking across grid environments of varying scale and complexity, the experiments aim to elucidate the performance advantages of BCD-ACO and reveal its potential value in convergence efficiency, path length reduction, smoothness, and environmental adaptability.

Figure 12.

Comparative experiment 1 path comparison diagram.

Table III.

Results of comparative experiment 1.

Figure 13.

Comparison of convergence curves of improved ant colony optimisation algorithms in comparative experiment 1.

5.3.2. Comparative experiment 2

Comparative experiment 2 aims to systematically evaluate the performance of the BCD-ACO algorithm in a 30 × 30 grid environment and comprehensively compare it with several mainstream algorithms. Specifically, this experiment selects the PS-ACO algorithm proposed in ref. [Reference Shuang, Chen and Li32], the AIACSE algorithm from ref. [Reference Zhang, Pu and Si33], the MRCACO algorithm from ref. [Reference Zhang, You, Liu and Pan35], the MsAACO algorithm from ref. [Reference Cui, Wu, Huang, Xu, Liu and Xiao36], as well as traditional algorithms such as BFS, A^*, and Dijkstra as comparison objects. These algorithms are representative and widely used in the field of path planning, which effectively demonstrates the superiority and adaptability of the BCD-ACO algorithm in complex environments.

During the experimental phase, to ensure fairness and reproducibility, all algorithms were subjected to multiple independent tests in the same 30 × 30 grid environment. By systematically collecting and analysing the resulting data, we conducted a comprehensive evaluation of each algorithm’s performance across three key dimensions: path length, number of iterations, and optimisation capability. The aggregated results, presented as convergence curves in Figure 14 and numerical comparisons in Table IV, demonstrate the significant advantages of the BCD-ACO algorithm in terms of path planning quality, computational efficiency, and global search performance when compared to the other algorithms.

Figure 14.

Comparative experiment 2 path comparison diagram.

The experimental results from Figure 14 and Table IV clearly demonstrate that the proposed BCD-ACO algorithm exhibits significant advantages across multiple key metrics in path planning, outperforming the other compared algorithms. Specifically, the paths generated by BCD-ACO show superior smoothness, featuring more continuous and fluent curves that effectively reduce sharp turns and unnecessary fluctuations. This improvement enhances the stability and safety of the robot during movement.

Furthermore, the experimental data indicate that BCD-ACO produces shorter path lengths, optimising movement efficiency while reducing energy consumption and time costs. Additionally, the number of turns in the paths is significantly decreased, resulting in a more streamlined path structure. This not only further improves path smoothness but also minimises mechanical wear and control complexity caused by frequent turns during robot navigation.

Table IV.

Results of comparative experiment 2.

Figure 15.

Comparison of convergence curves of improved ant colony optimisation algorithms in comparative experiment 3.

5.3.3. Comparative experiment 3

Comparative Experiment 3 comprehensively compares BCD-ACO against several existing algorithms in a 20 × 20 grid environment. The algorithms compared are the IACO algorithm from ref. [Reference Zhao, Cheng and Hao34], the MsAACO algorithm from ref. [Reference Cui, Wu, Huang, Xu, Liu and Xiao36], the classical ACO algorithm, the A^* algorithm, and Dijkstra’s algorithm. The experimental results are shown in Figures 15 and 16 and Table V.

Table V.

Results of comparative experiment 3.

Figure 16.

Comparative experiment 3 path comparison diagram.

Based on the experimental data from Figures 15, 16, and Table V, the proposed BCD-ACO algorithm demonstrates significant advantages across multiple core metrics in path planning, outperforming the other compared algorithms.

First, it exhibits excellent path smoothness, generating smooth and continuous paths that reduce abrupt turns and fluctuations, thereby enhancing the robot’s movement stability and lowering potential collision risks. Second, the BCD-ACO algorithm plans shorter paths, optimising movement efficiency, which helps save energy and time, particularly beneficial in resource-constrained scenarios. Additionally, the number of turns in the paths is significantly reduced, resulting in a more streamlined path structure. This not only improves smoothness further but also decreases mechanical wear and control complexity caused by frequent turning.

These combined advantages make the BCD-ACO algorithm highly practical for path planning in fields such as robot navigation, logistics optimisation, and autonomous driving, providing robust support for efficient and safe solutions.