## 3 Path Planning for Mobile Robots  
### Global Path Planning Based on Improved Ant Colony Algorithm  
This paper proposes two improvements for traditional algorithms in terms of slow convergence speed and local optimization problems. (1) Use an improved heuristic function to provide an initial selection of the ant colony to guide the direction. (2) Use a hybrid strategy to update the pheromone. 1. Initial state: In the initial state, the probability of the ants choosing a feasible path is equal, and the ant will determine the direction of the next node according to the pheromone content of the state transition point on the selection path. Setting the state transition probability of ant \(b\) from point \(I\) to point \(g\) on the grid graph as \(p_{ig}^{(b)}(t)\), the pheromone content \(T_{ig}(t)\) from point \(i\) to point \(g\), heuristic function \(N_{ig}(t)\), pheromone heuristic factor \(a\), and expectation the heuristic factor \(w\) are determined. Then the state transition probability formula is as follows: \[p_{ig}^{(\mathrm{b})}(t)=\left\{ \begin{array}{l}T_{ig}^{a}\cdot N_{ig}^{w}(t )/\sum\limits_{g\in\mathit{all}(i)}T_{ig}^{a}\cdot N_{ig}^{w}(t)\\ 0,\ otherwise\end{array} \right.\] (11) where _all(i)_ represents the set of transfer nodes that ant \(b\) can select at node \(i\), and \(N_{ig}(t)\) is inversely proportional to the distance \(l_{ig}\) between nodes \(i\) to \(g\). 2. Improved heuristic function: On the basis of the square of the sum of the distance from the current node to the next node and the distance from the next node to the target in [23], the heuristic function is introduced to speed up the search speed of the ant colony, which is \[N_{ig}(t)=1/\left(\sigma\cdot l_{ig}+(1-\sigma)\cdot l_{ig}\right)^{2}\] (12) where \(\sigma\in[0,1]\), determined by the real-time environment; \(l_{ig}\) represents the distance between the current node and the next node; and \(l_{ig}\) represents the distance between the current node and the target point, and increases the ant colony's selection and guidance effect on the next target point, which helps to reduce the search time of the ant colony. 3. Pheromone update: In order to speed up the search time of ants, to reduce the probability of choosing the route that has been taken, and to avoid the pheromone concentration being too low or accumulating after volatilization, it is necessary to determine the maximum value \(T_{max}\) and the minimum value \(T_{min}\) of the pheromone concentration between two points. In this paper, the maximum and minimum values of pheromone were determined according to the volatilization degree and optimal path of the pheromone, which is \[\left\{ \begin{array}{l}T_{max}=[1/2(1-\lambda)]\times[1/D_{s}]+1/D_{s}\\ T_{min}=T_{max}/c\end{array} \right.\] (13) where \(D_{s}\) is the optimal path length after a certain iteration, and \(c\) is the number of cycles each time. After the ants finish each cycle, the pheromone information on the path changes. In order to obtain the optimal path, the pheromone on the entire path needs to be adjusted. The update conditions are as shown in Equation (14). At the same time, in order to prevent the ant colony from using the pheromone on the local optimal path for path planning, the pheromone concentration increment of the b-th ant between the two points adopts the update strategy shown in (16) to ensure that each ant can find a feasible path. \[T_{ig}(t+1)=(1-\lambda)\cdot T_{ig}(t)+\Delta T_{ig}(t) \tag{14}\]  
\[\Delta T_{ig}(t)=\sum_{b=1}^{m}\Delta T_{ig}^{b}(t) \tag{15}\]  
\[\Delta T_{ig}^{b}=\left\{ \begin{array}{ll}T_{\min}&\Delta T_{ig}^{b}+(1- \lambda)T_{ig}^{b-1}{<T_{\min}}\\ 1/r_{s}&T_{\min}<\Delta T_{ig}^{b}+(1-\lambda)T_{ig}^{b-1}{<T_{max}}\\ T_{max}&\Delta T_{ig}^{b}+(1-\lambda)T_{ig}^{b-1}{>T_{max}}\end{array} \right.