# A Hybrid Bald Eagle Search Algorithm for Time Difference of Arrival Localization
## 3 Proposed Localization Algorithm
### Hybrid Bald Eagle Search Optimization Algorithm  
#### 3.2.1 Bald Eagle Search Optimization Algorithm  
The bald eagle search algorithm [30] is a new metaheuristic algorithm inspired by the predatory behaviors of the bald eagle. Bald eagles use multiple attack methods to assess the energy cost of a hunting action, the energy contained in the prey, and the successrate in all kinds of habitats. They follow an intelligent strategy of selecting the correct area with prey, searching the selected area, and capturing the prey at the right time. Wise hunting strategies are also used to take advantage of wind speeds and stormy air. Bald eagle hunting behaviors involve selecting the right search area, searching the selected area, and swooping in on prey at the right time. A bald eagle usually chooses a search area that is spatially closer to the previous search area, and each of its movements is determined by the previous movements. The basic process of BES algorithm can be divided into three stages: choosing a seek space, seeking the space for prey, and diving for prey.  
Step 1: Selecting a search space: The bald eagle randomly searches a large area for prey, deciding whether the area is suitable for hunting in accordance with the prey concentrative degree and its own experience. The equation for updating a bald eagle's location is:  
\[P_{i,new}=P_{best}+\alpha\times r(P_{mean}-P_{i}) \tag{8}\]  
\(\alpha\) is for bald eagle's own experience, which is itself is a parameter controlling position variation ranging from (1.5, 2), \(r\) is a random number between (0, 1), \(P_{best}\) represents the most adapted position among individual bald eagles, \(P_{mean}\) represents the average position of the bald eagle population, and \(P_{i}\) represents the position of the bald eagle \(i\).  
Step 2: Search space prey: The bald eagle selects a search space to search for prey and will search in and around the center point of the search space, moving in different directions within the spiral space to speed up the search, which can be expressed through the equation as:  
\[P_{i,new}=P_{i}+y(i)\times(P_{i}-P_{i+1})+x(i)\times(P_{i}-P_{mean}) \tag{9}\]  
\[x(i)=\frac{xr(i)}{\max(|xr|)},y(i)=\frac{yr(i)}{\max(|yr|)} \tag{10}\]  
\[xr(i)=r(i)\times\sin(\theta(i)),yr(i)=r(i)\times\cos(\theta(i)) \tag{11}\]  
\[\theta(i)=a\times\pi\times rand,r(i)=\theta(i)+R\times rand \tag{12}\]  
where \(\theta(i)\) represents the polar angle in polar coordinates, \(r(i)\) is the polar radius in polar coordinates; \(a\) and \(R\) are parameters to govern the helix search pattern of the bald eagle, varying over the ranges (0, 5) and (0.5, 2). rand represents a random number of (0, 1), and \(x(i)\) and \(y(i)\) represent the two coordinates of the bald eagle when converted to a plane rectangular coordinate system. \(P_{i+1}\) represents the position of bald eagle \(i+1\) before the population was updated.  
Step 3: Swooping for prey: The bald eagle finds prey and launches a swoop toward it. The bald eagle attacks the target prey from the optimum location in the search space, and all points go to the best point. The equation for the change in position of a bald eagle is:  
\[P_{i,new}=rand\times P_{best}+x_{1}(i)\times(P_{i}-c1\times P_{mean})+y_{1} \times(P_{i}-c2\times P_{best}) \tag{13}\]  
\[x_{1}(i)=\frac{xr(i)}{\max(|xr|)},y_{1}(i)=\frac{yr(i)}{\max(|yr|)} \tag{14}\]  
\[xr(i)=r(i)\times\sinh[\theta(i)],yr(i)=r(i)\times\cosh[\theta(i)] \tag{15}\]  
\[\theta(i)=\alpha\times\pi\times rand,r(i)=\theta(i) \tag{16}\]  
where \(c1\) and \(c2\) represent the motion intensity of the bald eagle to the optimal point and center point during the dive, and the value range is (1, 2).