# Self-Organizing Control of Mega Constellations for Continuous Earth Observation
## 2 Materials and Methods
### Stability Proof of Satellite Control
#### 2.3.3. Selection Criteria of Quadratic Potential Function Variables  
In past studies, the control variables of quadratic APFs can be position states \(X,Y,Z\)[16; 18; 25], velocity states \(\dot{X},\dot{Y},\dot{Z}\)[19], relative orbital elements [20; 21; 22], classical orbital elements [26], etc. The stabilities of quadratic APFs in terms of different control variables are discussed as follows. 1. Using \(\mathbf{R}\) and \(\mathbf{V}\) as Control Variables  
In this case, \(\mathbf{X}_{i}=[\mathbf{R}_{i};\mathbf{V}_{i}]\), \(\mathbf{X}_{\mathbf{t}}=[\mathbf{R}_{i};\mathbf{V}_{\mathbf{t}}]\). The control effects aim to make the geocentric position and velocity of the controlled satellite converge to that of the target one. The difference between the time derivative of \(\mathbf{X}_{\mathbf{i}}\) and that of \(\mathbf{X}_{\mathbf{t}}\) under the two-body condition is  
\[\left(\frac{\mathrm{d}\mathbf{X}_{i}}{\mathrm{d}t}-\frac{\mathrm{d}\mathbf{X}_ {\mathbf{t}}}{\mathrm{d}t}\right)_{two-body}=\left[ \begin{array}{c}\frac{ \mathrm{d}\mathbf{R}_{i}}{\mathrm{d}t}-\frac{\mathrm{d}\mathbf{R}_{\mathbf{t} }}{\mathrm{d}t}\\ \frac{\mathrm{d}\mathbf{V}_{i}}{\mathrm{d}t}-\frac{\mathrm{d}\mathbf{V}_{ \mathbf{t}}}{\mathrm{d}t}\end{array} \right]=\left[ \begin{array}{c}\mathbf{ V}_{i}-\mathbf{V}_{\mathbf{t}}\\ -\frac{\mu\mathbf{R}_{i}}{R_{i}^{3}}+\frac{\mu\mathbf{R}_{\mathbf{t}}}{R_{i}^{ 3}}\end{array} \right] \tag{57}\]  
When the relative distance between the controlled satellite and the target one is much closer than the orbital altitude, the difference in velocity and two-body gravity in Equation (57) is minor and is ignored. Referring to Equation (49), \(\left(\mathrm{d}\mathbf{X}_{i}/\mathrm{d}t-\mathrm{d}\mathbf{X}_{\mathbf{t}}/ \mathrm{d}t\right)_{two-body}=0\), and the time derivative of \(\phi_{i}\) is negative definite. The control effect is stable. However, the orbital dynamics differ a lot with the increase in the intersatellite distance. \(\left(\mathrm{d}\mathbf{X}_{i}/\mathrm{d}t-\mathrm{d}\mathbf{X}_{\mathbf{t}}/ \mathrm{d}t\right)_{two-body}\) can be positive or negative, and the stability cannot be guaranteed anymore. Therefore, quadratic APF control in terms of \(\mathbf{R}\) and \(\mathbf{V}\) is stable only if the relative distance is small, or if the gravity is weak, for example, on the ground or in deep space. 1. Using Orbital Elements as Control Variables  
In this case, \(\mathbf{X}_{i}=[a_{i},e_{i},i_{i},\Omega_{i},\omega_{i},M_{i}]^{\mathrm{T}}\), \(\mathbf{X}_{\mathbf{t}}=[a_{\mathbf{t}},e_{\mathbf{t}},i_{\mathbf{t}},\Omega _{\mathbf{t}},\omega_{\mathbf{t}},M_{\mathbf{t}}]^{\mathrm{T}}\). Under two-body Keplerian motion condition, it can be inferred that  
\[\left(\frac{\mathrm{d}\mathbf{X}_{i}}{\mathrm{d}t}-\frac{\mathrm{d}\mathbf{X}_ {\mathbf{t}}}{\mathrm{d}t}\right)_{two-body}=\left[ \begin{array}{c}\mathrm{ d}a_{i}/\mathrm{d}t-\mathrm{d}a_{\mathbf{t}}/\mathrm{d}t\\ \mathrm{d}e_{i}/\mathrm{d}t-\mathrm{d}e_{\mathbf{t}}/\mathrm{d}t\\ \mathrm{d}i_{i}/\mathrm{d}t-\mathrm{d}i_{\mathbf{t}}/\mathrm{d}t\\ \mathrm{d}\Omega_{i}/\mathrm{d}t-\mathrm{d}\Omega_{\mathbf{t}}/\mathrm{d}t\\ \mathrm{d}\omega_{i}/\mathrm{d}t-\mathrm{d}\omega_{\mathbf{t}}/\mathrm{d}t\\ \mathrm{d}M_{i}/\mathrm{d}t-\mathrm{d}M_{\mathbf{t}}/\mathrm{d}t\end{array}  \right]=\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ n_{i}-n_{\mathbf{t}}\end{array} \right] \tag{58}\]  
According to Equation (49), if the slow variables of orbital elements, i.e., \(a,e,i,\Omega,\omega\), are chosen as APF control variables, the control effect is stable. 1. Using Relative Orbital Elements as Control Variables  
In this case, \(\mathbf{X}_{i}=[b_{1},b_{2},b_{3},c,\theta_{1},\beta_{1},\psi]^{\mathrm{T}}\), \(\mathbf{X}_{\mathbf{t}}=[b_{1\mathbf{t}},b_{2\mathbf{t}},b_{3\mathbf{t}},c_{ \mathbf{t}},\theta_{1\mathbf{t}},\beta_{1\mathbf{t}},\psi_{\mathbf{t}}]^{ \mathrm{T}}\). There exists  
\[\left(\frac{\mathrm{d}\mathbf{X}_{i}}{\mathrm{d}t}-\frac{\mathrm{d}\mathbf{X}_ {\mathbf{t}}}{\mathrm{d}t}\right)_{two-body}=\left[