## 2 Hybrid Path Planning Based on Improved ABC and BAS Algorithms
### ABC Algorithm for Global Path Exploration Optimization
#### 2.1.2 Improvement Strategies  
The Tent chaotic function [26] is adopted to enrich the population diversity, prevent the ABC algorithm from falling into a premature convergence situation during the optimization process, and improve the algorithm's global search capability. The chaotic mapping function has strong traversal and randomness and can be mathematically mapped to transform chaotic sequences into individual spatial search capabilities. Among them, Tent chaos has a faster iteration speed and better uniformity, but there may be errors on immovable points. Therefore, random variables are introduced into the Tent chaos function to control the random values within a certain range and keep the algorithm's regularity. Firstly, during the initialization of the bee population, a random initial individual is generated as a b-dimensional vector, where each dimension is within the interval [0,1]. Subsequently, the next individual is generated by substituting the value of each dimension of the current individual into the Tent chaotic formula sequentially to update the corresponding dimension of the new individual. This process of generating new individuals is repeated until (\(n-1\)) new individuals are produced. All generated individuals are then mapped to the variable range, resulting in the Tent chaotic initial bee population. The initial population is generated as  
\[z_{d+1}=\left\{\begin{array}{l}\frac{2z_{d}+\text{rand}(0,1)}{N},0\leq z_{d} \leq\frac{1}{2}\\ \frac{2(1-z_{d})+\text{rand}(0,1)}{N},\frac{1}{2}<z_{d}\leq 1\end{array} \right., \tag{5}\]  
where \(N\) is the number of particles within the sequence, and rand (0,1) takes a random number in that range.  
A non-uniform mutation operator is introduced to enhance solution accuracy. The non-uniform mutation operator [27] can introduce large perturbations in the early iterations of the algorithm, thereby enhancing the exploration of the search space. In the later stages, it reduces the perturbation magnitude, promoting the algorithm's convergence to the vicinity of the global optimal solution and improving the quality of the global solution. For each bee generation after position update, the \(d\)-dimensional vector is randomly perturbed using a variance probability \(l_{m}\), which decreases dynamically with the number of iterations. The mutation formulas for the \(o\)-th dimensional component are shown in  
\[c_{d}^{o+1}=\left\{\begin{array}{l}c_{d}^{o}+\Delta\big{(}e,p_{o}-c_{d}^{o} \big{)},\alpha=0\\ c_{d}^{o}-\Delta\big{(}e,c_{d}^{o}-q_{o}\big{)},\alpha=1\end{array}\right., \tag{6}\]  
\[\left\{\begin{array}{l}\Delta(o,r)=r\times\Big{(}1-s^{l_{m}^{-1}}\Big{)}\\ l_{m}=1-\frac{e}{E_{\max}}\end{array}\right., \tag{7}\]  
where \(p_{0}\) and \(q_{0}\) are the maximum and minimum values of the vector range, respectively; \(\alpha\) randomly takes the value 0 or 1; \(e\) is the current number of iterations; \(E_{max}\) is the maximum number of iterations; \(s\) is the number of random numbers in the interval [0,1]; and \(t\) determines the dependence degree of random number perturbation on the iteration count \(e\). In the early stage of the algorithm, \(e\) is small, and the operator performs uniform search in the space. In the later stage, as \(e\) increases, the new solution generated by mutation infinitely approximates the optimal real solution, yielding the best nectar source.