# Optimal Progressive Pitch for OneWeb Constellation with Seamless Coverage
## 4 Numerical Simulation and Results
### The Convergence of the Genetic Algorithm  
Figure 4 shows the relationship between the number of iterations and the fitness. It can be seen that the fitness of the best individual reaches convergence after 50 iterations. According to the chromosome of the optimal individual of the population after reaching convergence, we can obtain the interference avoidance strategy of the optimal individual by decoding this chromosome, and the curve of the pitch angle and the number of closed beams of this interference avoidance strategy is shown in Figure 5a,b.  
In order to demonstrate the effectiveness, several progressive pitch schemes have been compared. A brief introduction of the proposed schemes is as the follows:  
* Interference-only: An interference avoidance strategy that considers only interference constraints as given in the literature [30] ignores the seamless coverage.
* Brutal force: This method could achieve the seamless coverage with the interference constraints. Two parameters of latitude need to be determined: one parameter identi  
Figure 4: Convergence of the genetic algorithm.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Item** & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** \\ \hline \(N\) & 40 & 42 & 44 & 46 & 48 \\ overlap (deg) & 0.72 & 1.15 & 1.54 & 1.89 & 2.22 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Coverage overap between adjacent satellites in the same orbit.  
fies the latitude where the LEO satellites start to tilt their attitude, the another is the latitude that the pitch angle starts to be same with the interference-only scheme.
* GA: The genetic algorithm scheme with random initialisation could ensure the seamless coverage without generating harmful interference to the GSO satellite system, but it has no advantages compared to the brutal force scheme. The action space for \(A\) is \(\{0,+0.5,+1.5,+1\}\), of which the previous three elements represent the changes of pitch angle, and the last element represents the changes of the number of off beams.
* GA + brutal force: This method fine tunes the brutal force scheme with a genetic algorithm to further extend the capacity of the LEO satellite. The initial scheme of GA + brutal force is the same with the brutal force scheme. The action space for \(A\) is \(\{0,+0.2,-0.2,+0.6,-0.6,+1,-1\}\), of which the previous five elements represent the changes of pitch angle and the last two elements represent the changes of the number of off beams.  
Figure 5a,b show the variations of the satellite pitch angle and the number of off beams along the latitude in the northern hemisphere, and the interference avoidance strategy in the southern hemisphere is symmetric with the northern hemispere. The maximum values of the number of off beams are the same for the above methods, while the maximum value of the pitch angle in the approaches based on a genetic algorithm is smaller than the approaches of brutal force and interference-only. As LEO satellites orbit to lower latitudes, both pitch angle and the number of off beams increase monotonically, except for the GA + brutal force approach. Combining with Figure 5a,b, the pitch angle of the LEO satellite in the interference-only progressive pitch approach is smaller than other approaches before the satellites turn off the beams. In the interference-only approach, the seamless coverage constraint could not be ensured, while in other methods the seamless coverage is satisfied. In order to enable the seamless coverage, the coverage of LEO satellite needs to be shifted in advance toward the equator to provide more overlap between the adjacent satellite in a lower latitude.