# An Improved Cuckoo Search Algorithm and Its Application in Robot Path Planning
## 5 The Application in Robot Path Planning Problems  
To substantiate the algorithm's applicability, BASCS alongside CS and its four variants were deployed to address the robot path planning challenge. Essential in robotics, effective path planning for mobile robots is imperative [38]. The objective is to identify the most viable optimal path for a robot, navigating from its origin to the target within intricate environments. Robot path planning encompasses two key elements: environmental modeling and the evaluation function. Crucial for efficient navigation and path planning, environmental modeling involves constructing a computer-interpretable model of the robot's surroundings. The evaluation function, in turn, gauges path length and smoothness, providing the fitness metric for the algorithm's optimization process. The environment map was constructed using a grid-based method to delineate the robot's search space, as shown in Figure 22. The red origin in Figure 22 is at (0, 0), and the green destination is at (9, 9). White regions signify traversable areas, while black regions indicate obstacles. The blue curve represents the optimal robot path planning. The evaluation function assesses the feasibility of paths by scrutinizing all potential routes to identify the optimal one. It evaluates paths based on two criteria. First, the path length is paramount; a shorter operational route for the robot is preferable. Second, path smoothness is essential; a smoother trajectory enhances robotic movement efficiency. The path's smoothness is determined by the angles between adjacent path segments. Larger angles at turns indicate a smoother path with greater distances between the points. Conversely, smaller angles result in a more convoluted path with shorter distances between points. Taking both length and smoothness into account, the fitness \(F\) is calculated as Equation (19):  
\[F=W_{1}\times F_{1}+W_{2}\times F_{2}, \tag{19}\]The fitness function integrates two components: \(F_{1}\) for path length and \(F_{2}\) for path smoothness, each weighted by \(W_{1}\) and \(W_{2}\), with \(\frac{1}{W_{1}}+W_{2}=1\). This weighting principle ensures normalization and balance between the two components, keeping the contributions of the weighted terms \(\frac{1}{F_{1}}\) and \(F_{2}\) proportional. The normalization prevents one component from overpowering the other due to imbalanced weights, ensuring both make meaningful contributions to the final result. By maintaining the sum of the weights as 1, the authors achieve a harmonious influence of \(\frac{1}{F_{1}}\) and \(F_{2}\) on the overall outcome. \(F_{1}\) and \(F_{2}\) are calculated using Equations (20) and (21), where n denotes the total number of points along the path, and \((x_{i},y_{i})\) signifies the coordinates of each path point:  
\[F_{1}=\frac{1}{\sum_{i=1}^{n-1}\sqrt{\left(x_{i+1}-x_{i}\right)^{2}+\left(y_{i +1}-y_{i}\right)^{2}}}, \tag{20}\]  
\[F_{2}=\sum_{i=1}^{n-2}\sqrt{\left(x_{i+2}-x_{i}\right)^{2}+\left(y_{i+2}-y_{i} \right)^{2}}, \tag{21}\]  
Based on Equations (19)-(21), it is evident that when the fitness \(F\) is relatively large, the path is shorter and smoother. On the contrary, a smaller fitness F is associated with a longer and more convoluted path. BASCS is compared with CS and its four variants, all employing parameters listed in Table 3. To mitigate algorithmic randomness, each was executed 30 times, with the minimum, maximum, and average results documented in Table 8. It reveals that BASCS consistently attained the best results across minimum, maximum, and mean performances among the six, thereby reinforcing the algorithm's practicality and superiority.