## 2 Problem Description and Methodology
### Optimal Design of LEO Navigation Constellation
#### 2.1.3 Optimization Algorithms and Models  
Compared with the first-generation algorithm, the NSGA-II algorithm introduced a fast non-dominated sorting method to reduce computation time, adopted an elite strategy to preserve outstanding individuals, discarded the shared radius, and implemented a crowding method to enhance uniformity of the solution set in the target space. Crowding distance represents the density of solutions within a frontier and maintains population diversity by calculating the crowding of individuals in the target space. The routine of crowding distance sorting is as follows: Firstly, objective functions are sorted for individuals in every non-domination layer. The maximum and minimum for the m-th objective function are denoted as \(f_{m}^{\text{max}}\) and \(f_{m}^{\text{min}}\), respectively. Next, crowding distance \(i_{d}\) is calculated as \(\left(f_{m}^{i+1}-f_{m}^{i-1}\right)/\left(f_{m}^{\text{max}}-f_{m}^{\text{min}}\right)\)[13].  
To simplify the model, a satellite is placed into a circular orbit and is set as the seed satellite: \(e=\Omega=\omega=M=0\). The parameter for the satellite is denoted as \(\{a,i\}\) with a Walker constellation phase factor \(F=1\), and the constellation parameter is represented by \(\{P,N/P\}\). Consequently, the optimization parameter set for the constellation can be expressed as follows:  
\[X=\{a,i,P,N/P\} \tag{6}\]  
The objective function of this study incorporates the number of visible satellites and PDOP, applying the \(2\sigma\) principle to mitigate data outliers as much as possible [22]. The goal is to minimize the total number of satellites while ensuring a minimal PDOP in order to reduce costs. To avoid polarized optimization results, the PDOP performance of an LEO navigation constellation is constrained by the mid-latitude PDOP from four GNSS constellations, with minimized satellite orbital altitude and inclination.  
\[\begin{array}{c}\text{mean}(PDOP)\\ \text{min}(Total_{ constellation})\\ \text{$s.t.$}\quad\begin{array}{c}\text{$\underset{-1\text{\emph{\emph{ \emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph 1}}}}}}}}}}}}}}$}$\\ \text{$\underset{-1\text{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{ \emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{ }}}}}}}}}}}}}}}}}}$ $}$\\ \text{$\underset{-1\text{\emph{\emph{\emph{\where \(S\)= 2 represents the number of Walker configurations, \(N_{C}\) denotes the count of satellites in the \(C_{\text{th}}\) Walker constellation, and \(i_{C}\) signifies the inclination of satellites in the \(C_{\text{th}}\) Walker constellation.  
Equation (8) is an underdetermined equation, and the only geometrical condition to determine the minimum GDOP combined configuration needs to be given with certain conditions. Therefore, based on the optimized results of the LEO tilt constellation, we determine the optimized range of the corresponding orbital inclination of the LEO polar constellation through Equation (8). After the introduction of one Walker configuration, another Walker configuration should make appropriate adjustments to the satellite orbit parameters (orbit inclination). When optimizing the polar constellation, refer to Equation (8) and change the index in the formula from GDOP to PDOP.