# Self-Organizing Control of Mega Constellations for Continuous Earth Observation
## 2 Materials and Methods
### Self-Organizing Control Using Artificial Potential Functions
#### 2.2.2 Design of Artificial Potential Functions for Single Satellite Control  
Referring to the concept of relative motion bound, Equation (20) can be rewritten as \(\widetilde{d}_{ij}\leq d_{\max}\), \(\widetilde{d}_{ij}\geq d_{\min}\). Together with Equation (18), the proposed continuous coverage constraint relates to RAAN differences and relative motion bounds. Therefore, the control effects will be performed in cases if \(\delta\Omega_{i}>\delta\Omega_{\max}\) or \(\delta\Omega_{i}<\delta\Omega_{\min}\) or \(\widetilde{d}_{ij}>d_{\max}\) or \(\widetilde{d}_{ij}<d_{\min}\). The following artificial potential functions are all designed in quadratic form, given in Equation (31). \[\phi_{i}=\left(\mathbf{X}_{i}-\mathbf{X}_{t}\right)^{\text{T}}\mathbf{K}( \mathbf{X}_{i}-\mathbf{X}_{t}) \tag{31}\]  
where \(\mathbf{X}_{i}\) is the control variable. \(\mathbf{X}_{t}\) is the expected value of \(\mathbf{X}_{i}\). \(\mathbf{K}\) is the control coefficient matrix, which is a positive definite symmetric matrix. In the geocentric inertial frame, the control effect generated from \(\phi_{i}\) is given as  
\[\Delta\mathbf{V}_{i}=-\frac{\partial\mathbf{X}_{i}^{T}}{\partial\mathbf{V}_{i }}\frac{\partial\phi_{i}}{\partial\mathbf{X}_{i}}=-\left[ \begin{array}{c} \frac{\partial\mathbf{X}_{i}^{T}}{\partial\mathbf{X}}\frac{\partial\phi_{i}} {\partial\mathbf{X}_{i}}\\ \frac{\partial\mathbf{X}_{i}^{T}}{\partial\mathbf{Y}}\frac{\partial\phi_{i}} {\partial\mathbf{X}_{i}}\\ \frac{\partial\mathbf{X}_{i}^{T}}{\partial Z}\frac{\partial\phi_{i}}{\partial \mathbf{X}_{i}}\end{array} \right] \tag{32}\]  
where \(\mathbf{V}_{i}=\left[\dot{X},\dot{Y},Z\right]^{\text{T}}\) represents the geocentric velocity vector of satellite \(i\). Hence, the velocity increments of \(\phi_{i}\) are the negative gradient of \(\phi_{i}\) with respect to satellite \(i\)'s geocentric velocity vector \(\mathbf{V}_{i}\). The stability proof will be given in Section 4. (1) RAAN Control  
To limit the RAAN difference, the artificial potential function is designed in Equation (33). The control variable takes \(\mathbf{X}_{i}=\left[\sin i_{i}\sin\Omega_{i},\sin i_{i}\cos\Omega_{i}\right]^ {\text{T}}\). The control coefficient matrix takes \(\mathbf{K}=\text{diag}(k_{i\Omega},k_{i\Omega})\). \(\delta\Omega_{i}=\Omega_{i}-\Omega_{\text{f}}\). \(i_{\text{f}}\) and \(\Omega_{\text{f}}\) represent the target inclination and RAAN values under \(l_{2}\) perturbation of the designed orbital plane, respectively. Therefore, \(\phi_{i\Omega}\) limits the RAAN difference within the expected range and corrects the inclination meanwhile. \[\partial\left(\widetilde{d}_{ij},\widetilde{d}_{ij}\right)\Big{/}\partial\dot{x}, \ \partial\left(\widetilde{d}_{ij},\widetilde{d}_{ij}\right)\Big{/}\partial\dot{y}, \ \partial\left(\widetilde{d}_{ij},\widetilde{d}_{ij}\right)\Big{/}\partial\dot{z}\] can be formulated based on Equations (23), (24), (29) and (30). \[\mathbf{Q}_{j}\] represents the transformation matrix from the geocentric inertial frame to satellite \[j\]'s orbital frame. \[\mathbf{Q}_{j}=\left[ \begin{array}{cc}-\sin\Omega_{j}\cos i \mathrm{sinu}_{j}+\cos\Omega_{j}\mathrm{cosu}_{j}&\cos\Omega_{j}\cos i\mathrm{ sinu}_{j}+\sin\Omega_{j}\mathrm{cosu}_{j}&\sin i_{j}\mathrm{sinu}_{j}\\ -\sin\Omega_{j}\cos i_{j}\mathrm{cosu}_{j}-\cos\Omega_{j}\mathrm{sinu}_{j}& \cos\Omega_{j}\cos i_{j}\mathrm{cosu}_{j}-\sin\Omega_{j}\mathrm{sinu}_{j}&\sin i _{j}\mathrm{cosu}_{j}\\ \sin\Omega_{j}\sin i_{j}&-\cos\Omega_{j}\mathrm{sin}i_{j}&\cos i_{j}\end{array} \right] \tag{38}\]  
* Semi-Major Axis Control The differences in semi-major axes between constellation satellites will lead to secular drift, and hence the relative motion-bound constraint will never be satisfied. Moreover, the semi-major axis will keep declining because of the influence of atmospheric drag. From this perspective, semi-major axis control is essential to mega constellation maintenance. The artificial potential function is designed to be a quadratic function about \(a_{i}\), given in Equation (39). \(\mathbf{X}_{i}=a_{i}\).