## 2 Hybrid Path Planning Based on Improved ABC and BAS Algorithms  
### ABC Algorithm for Global Path Exploration Optimization  
#### 2.1.1 The Optimized ABC Algorithm  
The ABC algorithm model is optimized as an \"employed-onlooker-scout\" model, which makes full use of the contribution of each type of bee in the global nectar search process and determines the location of the next path point based on the fitness value. Given that the ABC algorithm excessively relies on the optimization capability of leading bees, it tends to bypass the exploration of fitness values in unknown regions by scout bees, which leads to stagnation in the later stages of the search. To address this, the Tent chaotic perturbation mechanism and non-uniform mutation operator are introduced, respectively, aiming to enrich population diversity while enhancing the exploration capability of the search space, thereby generating more global waypoints available for selection by the algorithm.  
Assuming there are \(A\) bees in a B-dimensional search space, the position of each bee is shown as  
\[\mathbf{C}=\begin{bmatrix}c_{1,1}&c_{1,2}&\dots&c_{1,b}&\dots&c_{1,B}\\ c_{2,1}&c_{2,2}&\dots&c_{2,b}&\dots&c_{2,B}\\ \vdots&\vdots&\ddots&\vdots&\iddots&\vdots\\ c_{d,1}&c_{d,2}&\dots&c_{d,b}&\dots&c_{d,B}\\ \vdots&\vdots&\iddots&\vdots&\ddots&\vdots\\ c_{A,1}&c_{A,2}&\dots&c_{A,b}&\dots&c_{A,B}\end{bmatrix}, \tag{1}\]  
where \(\mathrm{d}=1,2,\dots,\mathrm{A}\); \(\mathrm{b}=1,2,\dots,\mathrm{B}\); and \(c_{d,b}\) is the position of the \(d\)-th bee in dimension \(B\).  
The employed bees guide the movement of the entire honey colony and can search for nectar sources anywhere in the region. Their updated position is shown in  
\[c_{d,b}^{e+1}=\left\{\begin{array}{l}c_{d,b}^{e}\exp\left(\frac{-b}{a\cdot \mathrm{iter}_{max}}\right),0.5\leq A\leq 1\\ c_{d,b}^{e}+\mathrm{FG},0\leq A<0.5\end{array}\right., \tag{2}\]  
where \(e\) is the number of model iterations; \(iter_{max}\) is the maximum number of iterations; \(\alpha\) is a random number with \(0<\alpha\leq 1\); \(F\) is a random value obeying the normal distribution function; and \(\mathbf{G}\) is a \(1\times\mathrm{b}\) matrix where each element is \(1\). \(\mathrm{A}\in[0,1]\), which denotes the concentration of nectar in the region. When \(0<\mathrm{A}<0.5\), the current concentration of nectar in the region is low, and the search range needs to be expanded. When \(\mathrm{A}\geq 0.5\), the current concentration of nectar is high, and all the honeybees need to fly to the region quickly.  
The onlooker bees follow the employed bees in search of nectar and may harvest nectar together with the employed bees to enhance the nectar concentration of the entire population. Their updated position is shown in  
\[c_{d,b}^{e+1}=\left\{\begin{array}{l}\mathrm{F}\exp\left(\frac{c\mathrm{ worse}_{b}^{e}-c_{d,b}^{e}}{d^{2}}\right),d>\frac{\mathrm{A}}{2}\\ c\mathrm{best}_{b}^{e+1}+\frac{1}{\mathrm{B}}\sum\limits_{b=1}^{B}\left( \left|c_{d,b}^{e}-c\mathrm{best}_{b}^{e+1}\  
the current fitness value of the bees; \(k_{b}\) is the current optimal fitness value in the global search; \(k_{w}\) is the worst fitness value; and \(\varepsilon\) is a non-zero minimum constant. \(k_{d}\
eq k_{b}\) indicates that the scout bee is located at the edge of the population and \(k_{d}=k_{b}\) indicates that the scout bee has found a better source of nectar and needs to move to a new region.