# Improved Hybrid A* Algorithm Based on Lemming Optimization for Path Planning of Autonomous Vehicles
## 2 Algorithm Overview
### Lemming Optimization Algorithm
#### 2.2.2 Behavioral Patterns  
Lemmings exhibit four key behaviors. They migrate across long distances, dig complex burrow systems, constantly search for food sources, and use various methods to escape predators. (1) Long-distance migration  
Lemmings explore their surroundings based on their current locations and the positions of random individuals within their populations, seeking habitats that are abundant in food sources to secure better living conditions and resources. Notably, factors such as the ecological conditions influence their migration, in which both the direction and distance of their movement are not fixed. This behavior can be modeled using the following Equation (3):  
\[\overset{\rightarrow}{Z}_{i}(t+1)=\overset{\rightarrow}{Z}_{best}(t)+F \times\overset{\rightarrow}{BM}\times\left(\overset{\rightarrow}{R}\times \left(\overset{\rightarrow}{Z}_{best}(t)-\overset{\rightarrow}{Z}_{i}(t) \right)+\left(1-\overset{\rightarrow}{R}\right)\times\left(\overset{ \rightarrow}{Z}_{i}(t)-\overset{\rightarrow}{Z}_{a}(t)\right)\right) \tag{3}\]  
where \(\overset{\rightarrow}{Z}_{i}(t+1)\) denote the \(ith\) position of the search agent during iteration \((t+1)\),while \(\overset{\rightarrow}{Z}_{best}(t)\) indicates the current global optimum solution. The direction control parameter \(F\), calculated as in Equation (5), facilitates exploration by periodically altering the search trajectories to escape local optima. The stochastic component \(\overset{\rightarrow}{BM}\) implements Brownian motion with step sizes derived from a normal distribution--see Equation (4)--enabling dynamic space exploration. Additionally, \(\overset{\rightarrow}{R}\) represents a uniformly distributed random vector [\(-1\),\(1\)] (where d is the problem dimensionality), generated via Equation (6), which enhances the solution space coverage. \(\overset{\rightarrow}{Z}_{i}(t)\) denotes the position vector of the \(ith\) search agent, while \(\overset{\rightarrow}{Z}_{a}(t)\) represents a randomly chosen individual from the population. The integer parameter a is between 1 and \(N\). governs the directional movement between the current optimal solution and randomly selected individuals, mathematically characterizing the inter-agent interactions during migration. The two-dimensional mathematical representation of this long-distance migration mechanism is depicted in Figure 3.  
\[f_{BM}(x;0,1)=\frac{1}{\sqrt{2\pi}}\times\exp\left(-\frac{x^{2}}{2}\right) \tag{4}\]  
\[F= \begin{cases}1&if[2\times rand+1]=1\\ -1&if[2\times rand+1]=2\end{cases}  \tag{5}\]  
\[\overset{\rightarrow}{R}=2\times rand(1,Dim)-1 \tag{6}\]  
(2) Digging holes  
The second behavior of lemmings is the digging of burrows in their habitats, forming complex tunnels that provide safe shelter and storage for food. Lemmings will randomly dig new burrows based on the current location of the burrow and the locationsof random individuals in the population. This design assists them in quickly escaping the threats of predators and searching for food more efficiently. The burrowing behavior is mathematically modeled as in Equation (7), with Figure 4 providing a simplified schematic representation. \[\overset{\rightarrow}{Z}_{i}(t+1)=\overset{\rightarrow}{Z}_{i}(t)+F\times L \times\left(\overset{\rightarrow}{Z}_{best}(t)-\overset{\rightarrow}{Z}_{b}(t)\right) \tag{7}\]  
where \(L\) represents an iteration-dependent stochastic parameter, while \(\overset{\rightarrow}{Z}_{b}\) denotes a randomly sampled population member. The discrete control parameter b is between 1 and \(N\). \(L\) and \(\overset{\rightarrow}{Z}_{b}\) represent the interaction between the solution vectors during the adjustment process.