# Self-Organizing Control of Mega Constellations for Continuous Earth Observation
## 2 Materials and Methods
### Continuous Coverage Constraint of Mega Constellation
#### 2.1.1 Definition of Continuous Coverage  
Walker constellation has been applied in many mega constellation missions because of its simple design and regular configuration. The spatial configuration of the Walker constellation can be represented by the total number of satellites \(N\), the number of orbital planes \(P\), and the phase parameters \(F\), where \(2\pi F/N\) represents the phase difference between two satellites in adjacent orbits. All satellites are placed in circular orbits at the same altitude and at the same inclination.  
The latitude range that can be covered by the constellation is determined by the designed inclination and orbital altitude, and the distribution density of constellation satellites at higher latitudes is much larger than that at lower latitudes. Hence, it is convincible that continuous coverage for the designed latitude range can be achieved as long as low latitude regions can be covered continuously.  
The coverage geometry of a single satellite is shown in Figure 1, where the angular radius of the coverage range is \(\alpha_{\max}\), the orbital altitude is \(H\), the Earth's radius is \(R_{\rm e}\), and the minimum elevation angle is \(E_{\rm min}\). The relationship between \(\alpha_{\max}\), \(H\), and \(E_{\rm min}\) is given by Equation (1).  
\[\alpha_{\max}=\arccos\biggl{(}\frac{R_{\rm e}}{R_{\rm e}+H}\cos E_{\rm min} \biggr{)}-E_{\rm min} \tag{1}\]  
The coverage range of adjacent coplanar satellites overlaps with each other, forming a street-of-coverage (SOC), shown in Figure 2. The width of the SOC \(\sigma\) depends on the coverage range \(\alpha_{\max}\) and the geocentric angle between adjacent coplanar satellites \(\theta\), written in Equation (2).  
\[\sigma=\arccos\biggl{(}\frac{\cos 2\alpha_{\max}}{\cos\theta}\biggr{)} \tag{2}\]  
Figure 1: Satellite coverage geometry.  
Since mega Walker constellation satellites are located at the same inclination, the SOC of adjacent orbital planes are approximately parallel to each other at low latitudes, shown in Figure 3. If the SOC of adjacent orbital planes could overlap with each other, continuous coverage can be achieved at low latitudes. It is noted that adjacent orbital planes could be co-rotating planes or counter-rotating planes, where satellites in the co-rotating planes move in the same direction (\(\Delta\Omega\) is close to zero), while satellites in the counter-rotating planes move in the opposite direction (\(\Delta\Omega\) is close to \(\pi\)). Therefore, the relative motion relationships between co-rotating satellites are relatively fixed compared to counter-rotating satellites. From the perspective of self-organization, the adjacent orbital planes in the coverage constraint are specified as co-rotating planes.  
To sum up, the continuous coverage constraint of the mega constellation is defined as follows: The continuous coverage can be guaranteed for a designed latitude range if the  
Figure 3: Streets-of-coverage geometries of adjacent orbital planes.  
Figure 2: Street-of-coverage geometry of one orbital plane.  
SOC of every two adjacent co-rotating orbital planes overlaps with each other. Theoretically, the existence of adjacent counter-rotating orbital planes can provide more coverage multiplicity.