# Improved Trimming Ant Colony Optimization Algorithm for Mobile Robot Path Planning
## 3 Improved Trimming Ant Colony Optimization (ITACO) Algorithm  
### Improved State Transition Probability Formula  
The traditional ant colony algorithm possesses several advantages, including local obstacle avoidance, strong robustness, positive feedback, and parallelism. However, it also suffers from certain limitations, such as slow convergence speed and a tendency to fall into local optima [34; 35; 36]. To overcome these limitations, this paper introduces a weighting factor \(\partial(\partial\in(0,1))\) into the state transition probability formula. The enhanced state transition formula is presented in Equations (5) and (6). \[P_{ij}^{k}(t)= \begin{cases}\partial\cdot\frac{[\tau_{ij}(t)]^{\kappa}\cdot[ \eta_{ij}(t)]^{\beta}}{\sum_{k\in[N-taba_{k}]}[\tau_{ij}(t)]^{\kappa}\cdot[ \eta_{ij}(t)]^{\beta}}&,\,j\in\{N-tabu_{k}\}\\ 0&,\,\,other\end{cases}  \tag{5}\]  
\[\partial=\sin\biggl{(}\frac{\pi t}{2T}\biggr{)} \tag{6}\]  
In the early stages of the algorithm, ants exhibit strong exploratory behavior, as the algorithm has not yet determined which path or solution is better. At this stage, the value of \(\partial\) is relatively small, allowing the heuristic function to exert a greater influence on the selection probability. This encourages ants to extensively explore a wide range of paths or solution spaces, increasing the likelihood of discovering high-quality solutions and preventing premature convergence to local optima. In the later stages of the algorithm, ants have accumulated pheromone information about good paths through earlier exploration. The value of \(\partial\) increases, enhancing the influence of pheromone concentration on the selection probability. As a result, the algorithm becomes more inclined to exploit the high-quality paths already identified, which accelerates convergence to the global optimum or a near-optimal solution. This dynamic weighting mechanism also enables the algorithm to handle other optimization problems, such as coverage maximization tasks, by changing path minimization goal to coverage maximization goal. For applications requiring area coverage (e.g., cleaning or surveillance), the heuristic component \(\eta_{ij}\) can be modified as the inverse of the visitation frequency to prioritize unvisited regions. Combined with our pruning strategy (Section 3.3), this adaptation can mimic the memory effect of chaos-based methods like P-RBMCA [37], while retaining convergence guarantees of ITACO. Additionally, this paper employs an artificial potential field as the heuristic function to replace the traditional heuristic function, thereby further enhancing the algorithm's convergence speed and optimization capability. The artificial potential field operates on the principle that a virtual potential field exists within the grid map [38]. In a virtual potential field, the potential energy to which the robot is subjected is proportional to the distance between the robot and the target point. Under the influence of the attractive field, the robot is continuously drawn toward the target point, enabling the planning of collision-free paths. The heuristic function incorporating the attractive potential field is defined in Equation (7), and the potential field attraction formula is given in Equation (8), as follows:  
\[\eta_{ij}=\left(\frac{1}{2}\right)^{fa}\cdot\frac{1}{\left(\left(x_{i}-x_{g} \right)^{2}+\left(y_{i}-y_{g}\right)^{2}\right)^{0.5}} \tag{7}\]  
\[fa=\sigma\cdot\left(x_{i}-x_{g}\right)^{2}+\left(y_{i}-y_{g}\right)^{2} \tag{8}\]  
where \(\sigma\) represents a potential field constant greater than 0, which is used to adjust the intensity of the potential field; \(\left(x_{i},y_{i}\right)\) represents the current coordinates of the robot; and \(\left(x_{g},y_{g}\right)\) denotes the coordinates of the target point. A larger \(\sigma\) results in a stronger attractive force of the potential field, promoting a broader search range for the algorithm, whereas a smaller \(\sigma\) leads to a weaker attractive force, enabling a local optimization of the path.