# Self-Organizing Control of Mega Constellations for Continuous Earth Observation
## 2 Materials and Methods
### Self-Organizing Control Using Artificial Potential Functions
#### 2.2.1 Scale-Independent Relative Orbital Elements and Bound Formulations  
Scale-independent relative orbital elements are high-precision configuration parameters for relative motion with near-circular reference orbit and arbitrary intersatellite separation. SIROEs have been proposed in our preliminary work [23]. Six independent SIROEs are \(b_{1},b_{2},b_{3},c,\theta_{1},\theta_{2}\), shown in Figures 5 and 6. The relationship between SIROEs and relative position is given in Equation (21). We denote \(\hat{x},\hat{y},\hat{z}\) as the basic relative motion coordinates. The relative trajectory can be obtained by rotating the basic trajectory by \(\theta_{1}\), counterclockwise. The basic trajectory projected on the \(x-y\) plane is the combination of an ellipse (with a semi-major axis of \(2b_{1}\) and a semi-minor axis of \(b_{1}\)), a circle (with a radius of \(b_{2}\)), and an additional offset (with \(x\)-offset of \(b_{3}-a\cos\theta_{1}\) and \(y\)-offset of \(a\sin\theta_{1}\)). The relative motion on the \(z\)-axis is harmonic. \[ \begin{split}\left[\begin{array}{c}x\\ y\\ z\end{array}\right]&=\left[\begin{array}{ccc}\cos\theta_{1}&-\sin\theta_{1}&0 \\ \sin\theta_{1}&\cos\theta_{1}&0\\ 0&0&1\end{array}\right]\left[\begin{array}{ccc}-b_{1}\cos(\theta_{2}+\psi) +b_{2}\cos 2\psi+b_{3}-a\cos\theta_{1}\\ 2b_{1}\sin(\theta_{2}+\psi)-b_{2}\sin 2\psi+a\sin\theta_{1}\\ c\sin\psi\end{array}\right]\\ &=\left[\begin{array}{ccc}\cos\theta_{1}&-\sin\theta_{1}&0\\ \sin\theta_{1}&\cos\theta_{1}&0\\ 0&0&1\end{array}\right]\left[\begin{array}{c}\hat{x}\\ \hat{y}\\ \hat{z}\end{array}\right]\end{split}  \tag{21}\]  
Figure 5: Illustration of the scale-independent relative orbital elements (\(x-y\) plane). Figure 6: Illustration of the scale-independent relative orbital elements (\(x-z\) plane). Without regard to the minor differences of semi-major axes between constellation satellites, the relative velocities represented by SIROEs are written as  
\[\left[ \begin{array}{c}\dot{x}/n\\ \dot{y}/n\\ \dot{z}/n\end{array} \right]=\left[ \begin{array}{ccc}\cos\theta_{1}&-\sin \theta_{1}&0\\ \sin\theta_{1}&\cos\theta_{1}&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}b_{1}\sin(\theta_{2}+\psi)-2b_ {2}\sin 2\psi\\ 2b_{1}\cos(\theta_{2}+\psi)-2b_{2}\cos 2\psi\\ c\cos\psi\end{array} \right] \tag{22}\]  
where \(n\) is the mean angular velocity of the reference orbit. According to Equations (21) and (22), SIROEs can be represented by translational states, given in Equation (23). \[ \begin{array}{l}c=\sqrt{z^{2}+(z/n)^{2}}\\ \sin\psi=z/c,\cos\psi=\dot{z}/nc\\ b_{2}=\frac{1}{2}\Big{(}a^{\prime}-\sqrt{a^{\prime 2}-c^{2}}\Big{)}\\ b_{3}=\frac{1}{2}\Big{(}a^{\prime}+\sqrt{a^{\prime 2}-c^{2}}\Big{)}\\ \cos\theta_{1}=-\frac{2b_{3}(2\dot{y}+nx+na)+3b_{2}\sin 2\psi(2 ny-\dot{x})}{ (2\dot{x}-ny)(2 ny-\dot{x})-(2 nx+2 na+y)(2\dot{y}+nx+na)}\\ \sin\theta_{1}=\frac{2 nb_{3}-(2 nx+2 na+\dot{y})\cos\theta_{1}}{2 ny-\dot{x}}\\ b_{1}=\sqrt{[-(x+a)\cos\theta_{1}-y\sin\theta_{1}+b_{2}\cos 2\psi+b_{3}]^{2}+ \left(\frac{\dot{x}}{n}\cos\theta_{1}+\frac{\dot{y}}{n}\sin\theta_{1}+2b_{2} \sin 2\psi\right)^{2}}\\ \sin\beta_{1}=\frac{\dot{x}\cos\theta_{1}+\dot{y}\sin\theta_{1}+2 nb_{2}\sin 2 \psi}{nb_{1}}\\ \cos\beta_{1}=\frac{-(x+a)\cos\theta_{1}-y\sin\theta_{1}+b_{2}\cos 2\psi+b_{3}}{ b_{1}}\\ \sin\theta_{2}=\sin\beta_{1}\cos\psi-\cos\beta_{1}\sin\psi\\ \cos\theta_{2}=\cos\beta_{1}\cos\psi+\sin\beta_{1}\sin\psi\end{array}  \tag{23}\]  
According to Equation (21), the squared relative distance between satellites can be written as  
\[w=x^{2}+y^{2}+z^{2}=\frac{A_{3}s^{6}+B_{3}s^{5}+C_{3}s^{4}+D_{3}s^{3}+E_{3}s ^{2}+F_{3}s+G_{3}}{\left(1+s^{2}\right)^{3}} \tag{24}\]  
where  
\[s=\tan\frac{\psi}{2},\ \ \ \ \ \psi\in[0,2\pi) \tag{25}\]  
Parameters \(A_{3},B_{3},C_{3},D_{3},F_{3},G_{3}\) are polynomials about SIROEs \(b_{1},b_{2},b_{3},c,\theta_{1},\theta_{2}\). Therefore, \(A_{3},B_{3},C_{3},D_{3},F_{3},G_{3}\) are time-invariant, while \(s\) is the only time variable. Differentiating \(w\) with respect to \(s\) results in  
\[