# An Improved Pied Kingfisher Optimizer for Maritime UAV Path Planning
## 2 Principle of the Improved PKO Algorithm
### Improved PKO Diving Phase Update Formula  
The proposed algorithm introduces an enhanced variable helix sine-cosine search approach, seamlessly integrating the Pied Kingfisher's diving phase update formula. This augmentation is designed to bolster the exploitation capability during the diving phase by harnessing the sinuous prowess of the sine-cosine algorithm and employing a variable helix strategy for meticulous local exploration. However, it is crucial to address that during this phase, there might be a tendency for the algorithm to venture into unexplored areas while inadvertently neglecting the effective exploitation of familiar areas, thereby leading to insufficient exploitation capability. The Sine-Cosine Algorithm (SCA) is a swarm intelligence-based optimization algorithm proposed by [PERSON] [25] in 2016, designed to tackle complex global optimization problems. Drawing inspiration from the cyclical and undulating patterns of sine and cosine waves, the SCA effectively combines global and local search by simulating the dynamic changes of these functions within the solution space. Primarily utilizing the oscillatory characteristics of sine and cosine functions guides the search process. These functions exhibit periodicity, symmetry, and oscillation traits that enable comprehensive exploration and fine-tuned exploitation in the search space. It is precisely this robust capability for exploration and exploitation that allows for the seamless integration of strengths when combined with the Pied Kingfisher's development phase. The mathematical model underlying the SCA is delineated as follows:  
\[X_{i}^{t+1}=\left\{ \begin{array}{ll}X_{i}^{t}+r_{1}\cdot\sin(r_{2})\cdot \left|r_{3}\cdot P_{i}^{t}-X_{i}^{t}\right|&\text{if $r_{4}<0.5$}\\ X_{i}^{t}+r_{1}\cdot\cos(r_{2})\cdot\left|r_{3}\cdot P_{i}^{t}-X_{i}^{t}\right|& \text{if $r_{4}\geq 0.5$}\end{array} \right. \tag{14}\]  
where \(X_{i}^{t+1}\) represents the position of individual \(i\) at generation \(t+1\), \(X_{i}^{t}\) denotes the position of individual \(i\) at generation \(t\), and \(P_{i}^{t}\) is the reference point or target position of individual \(i\) at generation \(t\). \(r_{1}\) serves as a variable scaling factor that regulates the magnitude of steps taken during the search process, typically decreasing as the number of iterations increases. \(r_{2}\) is a random angle within the interval \([0,2\pi]\), determining the oscillatory behavior of the position update. \(r_{3}\) and \(r_{4}\) are random numbers, both uniformly distributed in \([0,1]\), introduced to add randomness and diversity into the update formula. The Sine-Cosine Search Algorithm initially enhances its global search capability by conducting an extensive exploration. However, this approach may encounter the issue of having an excessively large search range, which can lead to boundary violations and an increased number of ineffective searches. To address this concern, a variable helix factor is introduced to dynamically adjust the search range at different stages in order to improve search effectiveness. As illustrated in Figure 3, during the initial phase of the algorithm, an intentionally configured broad search range takes advantage of the oscillatory characteristics inherent in the sine and cosine functions, resulting in a spiral-like path that facilitates exploration across a vast solution space. At this stage, the algorithm focuses on searching for potential optimal solutions throughout the entire solution space by utilizing larger step sizes that cover more areas and increase the likelihood of discovering the global optima. As the search progresses, the helix factor gradually decreases to improve search precision and local optimization capability. At this stage, the search path of the algorithm begins to contract, resulting in a more focused search range. By progressively narrowing down the search range, the algorithm can delve deeper into previously identified promising regions, thereby augmenting its effectiveness in locating local optima.