# Self-Organizing Control of Mega Constellations for Continuous Earth Observation
## 2 Materials and Methods
### Stability Proof of Satellite Control
#### 2.3.2 Stability Proof of Relative Motion-Bound Control  
For relative motion-bound control, \(\mathbf{X}=\mathbf{X}_{i}-\mathbf{X}_{t}=\left[\widehat{d}_{ij}-d_{\mathrm{ maxt}},\widetilde{d}_{ij}-d_{\mathrm{mint}}\right]^{\mathrm{T}}\). Different from RAAN and semi-major axis control, the relative motion-bound control relates to the controlled satellite \(i\) and its adjacent coplanar satellite \(j\). Therefore, the time derivative of \(\mathbf{X}_{i}\) will be different. Here we give the stability proof of relative motion-bound control under self-organizing control rules. Assuming that \(\phi_{i}=k_{\mathrm{bound}}\Big{(}\widehat{d}_{i,i+1}-d_{\mathrm{maxt}}\Big{)}^{2}\). It can be inferred from Equation (43) that \(\widehat{d}_{i,i-1}\leq d_{\mathrm{maxt}}\), and \(\widehat{d}_{i,i+1}>d_{\mathrm{maxt}}\). In this case, \(\mathbf{X}_{i}=\widehat{d}_{i,i+1}\). \(\phi_{i}\) is positive definite. The time derivative of \(\mathbf{X}_{i}\) can be written as \[ \begin{split}&\frac{\mathrm{d}\mathbf{X}_{i}}{\mathrm{d}t}=\frac{ \partial\mathbf{X}_{i}}{\partial\mathbf{R}_{i}^{\mathrm{T}}}\frac{\mathrm{d} \mathbf{R}_{i}}{\mathrm{d}t}+\frac{\partial\mathbf{X}_{i}}{\partial\mathbf{R}_ {i+1}^{\mathrm{T}}}\frac{\mathrm{d}\mathbf{R}_{i+1}}{\mathrm{d}t}+\frac{ \partial\mathbf{X}_{i}}{\partial\mathbf{V}_{i}^{\mathrm{T}}}\frac{\mathrm{d} \mathbf{V}_{i}}{\mathrm{d}t}+\frac{\partial\mathbf{X}_{i}}{\partial\mathbf{V}_ {i+1}^{\mathrm{T}}}\frac{\mathrm{d}\mathbf{V}_{i+1}}{\mathrm{d}t}\\ &=\left(\frac{\mathrm{d}\mathbf{X}_{i}}{\mathrm{d}t}\right)_{two -body}+\frac{\partial\mathbf{X}_{i}}{\partial\mathbf{V}_{i}^{\mathrm{T}}} \mathbf{u}_{i}+\frac{\partial\mathbf{X}_{i}}{\partial\mathbf{V}_{i+1}^{\mathrm{ T}}}\mathbf{u}_{i+1}\\ &=\frac{\partial\mathbf{X}_{i}}{\partial\mathbf{V}_{i}^{\mathrm{ T}}}\mathbf{u}_{i}+\frac{\partial\mathbf{X}_{i}}{\partial\mathbf{V}_{i+1}^{ \mathrm{T}}}\mathbf{u}_{i+1}\end{split}  \tag{51}\]  
The time derivative of \(\mathbf{X}_{\mathrm{t}}\) is  
\[\frac{\mathrm{d}\mathbf{X}_{\mathrm{t}}}{\mathrm{d}t}=\left(\frac{\mathrm{d} \mathbf{X}_{\mathrm{t}}}{\mathrm{d}t}\right)_{two-body}=\mathbf{0} \tag{52}\]  
The control effect working on satellite \(i\) is  
\[ \begin{split}\mathbf{u}_{i}&=-\frac{\partial\mathbf{ X}_{i}^{\mathrm{T}}}{\partial\mathbf{V}_{i}}\frac{\partial\phi_{i}}{\partial \mathbf{X}_{i}}\\ \end{split}  \tag{53}\]  
Since \(\widehat{d}_{i,i-1}\leq d_{\mathrm{max}}\) and \(\widehat{d}_{i,i+1}>d_{\mathrm{max}}\), the control APF working on satellite \(i+1\) is  
\[ \begin{split}\phi_{i+1}&=\left\{\begin{array}{l l l l l}k_{\mathrm{bound}}\left(\widehat{d}_{i,i+1}-d_{\mathrm{max}} \right)^{2}=\phi_{i},&if&\widehat{d}_{i,i+1}>d_{\mathrm{max}}\quad and& \widehat{d}_{i+1,i+2}\leq d_{\mathrm{max}}\\ 0,&if&\widehat{d}_{i,i+1}>d_{\mathrm{max}}\quad and&\widehat{d}_{i+1,i+2} >d_{\mathrm{max}}\end{array}\right.\end{split}  \tag{54}\]  
The control effect working on satellite \(i+1\) is  
\[ \begin{split}\mathbf{u}_{i+1}&=\left\{\begin{array}{l l l l l l}-\frac{\partial\mathbf{X}_{i}^{\mathrm{T}}}{\partial \mathbf{V}_{i+1}}\frac{\partial\phi_{i}}{\partial\mathbf{X}_{i}^{\mathrm{T}}} \quad&if&\widehat{d}_{i,i+1}>d_{\mathrm{max}}\quad and&\widehat{d}_{i+1,i+2} \leq d_{\mathrm{max}}\\ 0,&if&\widehat{d}_{i,i+1}>d_{\mathrm{max}}\quad and&\widehat{d}_{i+1,i+2} >d_{\mathrm{max}}\end{array}\right.\end{split}  \tag{55}\]  
Hence, the time derivative of \(\phi_{i}\) is  
\[