## 2 Materials and Methods
### Adaptive DBO for DPE
#### 2.3.2 Adaptive DBO Algorithm  
In order to search for navigational solutions for DPEs more efficiently, high-performance optimization algorithms are essential. Therefore, we improve the algorithm for DBO to further enhance its search performance. In this section, the basic principle of DBO is first briefly introduced. Subsequently, an improved ADBO algorithm is proposed to enhance its adaptability and search capability in complex optimization problems. The Standard DBO Algorithm  
The DBO algorithm updates the population's positions by simulating four natural behaviors of dung beetles: ball-rolling, spawning, foraging, and stealing. These behaviors correspond to four types of dung beetles: rolling, spawning, foraging, and stealing dung beetles, with the four dung beetles constituting a fixed proportion of the population of 20%, 20%, 25%, and 35%. Each type of dung beetle uses a distinct positional update strategy, working together to collaboratively search for the optimal solution. Figure 1: Optimization problem in (12) as a function of the unknown two-dimensions receiver position \(\varepsilon_{x}\) and \(\varepsilon_{y}\). Figure 2: Flowchart of the proposed DPE-SI framework. Ball-rolling behavior: In nature, the dung beetles use the sun for navigation to maintain a straight path while rolling their dung balls. The position of the ball-rolling dung beetle is updated according to (13) when it encounters no obstacles  
\[x_{l}(t+1)=x_{l}(t)+a\times k\times x_{l}(t-1)+b\times\Delta x, \tag{13}\] \[\Delta x=|x_{l}(t)-X^{w}|,\]  
where \(t\) denotes the iteration number, \(x_{l}(t)\) represents the position of the dung beetle at iteration \(t\). The coefficient \(a\) determines whether the dung beetle deviates from its original direction (1 indicates no deviation, \(-1\) indicates deviation). \(k\in(0,0.2]\) is the defect factor, originally set to 0.1, and \(b\in[0,1]\) is a constant, set to 0.3 in the original code. \(X^{w}\) denotes the global worst position. \(\Delta x\) simulates solar illumination, with a larger \(\Delta x\) indicating the dung beetle is further from the light source. If an obstacle is encountered, the dung beetle must reorient itself by dancing on the dung ball to find a new rolling route, updating its position as shown in (14):  
\[x_{l}(t+1)=x_{l}(t)+\tan(\theta)\times|x_{l}(t)-x_{l}(t-1)|, \tag{14}\]  
where \(\theta\in[0,\pi]\) is the deflection angle, and when \(\theta\in\{0,\pi/2,\pi\}\), it means that the position is not updated. Reproductive behavior: Dung beetles roll their dung balls to a safe area for females to lay eggs and protect their offspring. The safe area is defined by  
\[Lb^{*}=max(X^{*}\times(1-R),Lb),\] \[Ub^{*}=min(X^{*}\times(1+R),Ub), \tag{15}\]  
where \(X^{*}\) denotes the current local optimal position, and \(Ub\) and \(Ub\) denote the lower and upper bounds of the optimization problem, respectively. \(Lb^{*}\) and \(Ub^{*}\) denote the lower and upper bounds of the spawning region, \(R=1-t/T_{\max}\) is the convergence factor, and \(T_{\max}\) is the maximum number of iterations. Afterward, female dung beetles will each lay one egg per iteration within the brood balls in this safety area, updating the position as  
\[B_{l}(t+1)=X^{*}+b_{1}(B_{l}(t)-Lb^{*})+b_{2}(B_{l}(t)-Ub^{*}). \tag{16}\]  
If the position exceeds the safety area, it is adjusted as  
\[B_{l}=\left\{ \begin{array}{ll}Lb^{*},&B_{l}<Lb^{*}\\ Ub^{*},&B_{l}>Ub^{*},\end{array} \right. \tag{17}\]  
where \(B_{l}(t)\) is the position of the \(l\)-th brood ball at the \(t\)-th iteration, and \(b_{1}\) and \(b_{2}\) are independent random vectors with the same dimension as the optimization problem. Foraging behavior: The eggs will grow into dung beetles and come out from the ground to forage. Each dung beetle will choose an optimal foraging area, defined as follows:  
\[Lb^{b}=\max(X^{b}\times(1-R),Lb),\] \[Ub^{b}=\min(X^{b}\times(1+R),Ub), \tag{18}\]where \(X^{b}\) denotes the global optimal position, and \(Lb^{b}\) and \(Ub^{b}\) denote the lower and upper bounds of the optimal foraging area, respectively.