# Optimization of Multi-Mission CubeSat Constellations with a Multi-Objective Genetic Algorithm
## 3 Problem and Optimization Formulation
### Objective Functions Formulation  
The main goal of this optimization design is to minimize ART for both missions, minimize the number of satellites, and maximize the total global coverage. The optimization variables are inclination \((i)\), the number of orbital planes \((P)\), and the number of satellites in each orbital plane \((S)\). In this study, a total of four objective functions are formulated. The first objective is to minimize the total number of satellites \(T\) in the constellation given as:  
\[Min\ f_{1}=P\times S \tag{1}\]  
The second and third objectives are to minimize the ART for global and regional coverage, respectively. These two objective functions are formulated using target areas defined in Section 2.2. CubeSat sensors FOV geometry is used to calculate the coverage of a given target point. The payload in the CubeSat design has a nadir FOV that points directly beneath the CubeSat. Figure 3 shows the general FOV geometry of the sensors where \(R_{E}\) represents the Earth radius, while \(h\) represents the altitude of the CubeSat. The CubeSat elevation angle \(\varepsilon\), measured between the local horizontal and coverage end zone, can be calculated from the payload half-angle \(\eta\) as:  
\[\frac{R_{E}}{\sin\eta}=\frac{R_{E}+h}{\sin(\varepsilon+\frac{\pi}{2})}\quad \implies\quad\sin(\varepsilon+\frac{\pi}{2})=\cos\varepsilon=\sin\eta\cdot \frac{R_{E}+h}{R_{E}} \tag{2}\]  
The angle \(\beta\), which indicates the effective coverage angle of the CubeSat, measured at the center of the Earth from the nadir point to the end of coverage:  
\[\beta=\frac{\pi}{2}-\eta-\varepsilon \tag{3}\]  
Let \(r_{Cj}^{ECEF}\) and \(r_{Ni}^{ECEF}\) represent the position vectors in the Earth-centered, Earth-fixed (ECEF) coordinate system of the \(j^{th}\) CubeSat and \(i^{th}\) target, respectively. Where, \(j\) represents a CubeSat from the total of \(T\) CubeSats in the constellation. Furthermore, \(i\) represents a target point form given \(N\) number of target area points. Then, the angle \(\alpha\) between the two vectors is:  
\[\alpha=\cos^{-1}\frac{r_{Cj}^{ECEF}\cdot r_{Ni}^{ECEF}}{\left|r_{Cj}^{ECEF} \right|\cdot\left|r_{Ni}^{ECEF}\right|} \tag{4}\]  
From the geometry of the FOV, the criteria for judging if the \(i^{th}\) target is covered by the \(j^{th}\) CubeSat will be \(\alpha\leq\beta\). When we have coverage for a given target, we will save the corresponding coverage start time as \(t_{start}\). The coverage criteria will be continuously checked for every time interval of the given calculation period, saving the start time for each access to the target. ART is calculated for a given target per CubeSat that has two or  
Figure 3: Geometry representation of the sensor FOV assuming spherical Earth with constant radius \(R_{E}\). more accesses. Therefore, the ART of \(i^{th}\) target for a total \(k_{j}\) number of accesses by the \(j^{th}\) CubeSat is given as:  
\[ART_{i,j}=\frac{\sum_{l=2}^{k_{j}}t_{l+1,\ start}-t_{l,\ start}}{k_{j}-1} \tag{5}\]  
When calculating the ART for the \(i^{th}\) target by the constellation, the access start time of all CubeSats in the constellation will be considered. Therefore, the ART for the \(i^{th}\) target by the constellation with a total \(k\) number of accesses will be given as:  
\[ART_{i}=\frac{\sum_{l=2}^{k}t_{l+1,\ start}-t_{l,\ start}}{k-1} \tag{6}\]  
For global coverage, ART is calculated for total \(N_{G}\) number of grid points accessed by the constellation from global target points. The second objective function is given as:  
\[Min\ f_{2}=\frac{\sum_{i=1}^{N_{G}}ART_{i}}{N_{G}} \tag{7}\]  
For regional coverage, ART is calculated for \(N_{L}\) number of grid points accessed by the constellation from regional data centers. The third objective function is given as:  
\[Min\ f_{3}=\frac{\sum_{i=1}^{N_{L}}ART_{i}}{N_{L}} \tag{8}\]  
The fourth objective is to maximize the total percentage coverage of the global grid points.