# Fault Reconstruction for a Giant Satellite Swarm Based on Hybrid Multi-Objective Optimization
## 2 Construction of On-Orbit Reconstruction Indicators for a Giant Satellite Swarm
### Relative Motion Modeling  
To better describe the position and velocity of the satellite, a coordinate system \(\mathbf{R}\) is established with the virtual center \(\mathbf{O}_{t}\) of the satellite swarm as the origin. The \(\mathbf{X}_{t}\) axis is perpendicular to the orbit where \(\mathbf{O}_{t}\) is located; the \(\mathbf{Y}_{t}\) axis is perpendicular to the \(\mathbf{X}_{t}\) axis and tangent to the \(\mathbf{O}_{t}\) orbit, and the direction of movement is positive. The \(\mathbf{Z}_{t}\) axis is perpendicular to the orbital plane and forms a right-handed coordinate system with the \(\mathbf{X}_{t}\) and \(\mathbf{Y}_{t}\) axes. In this coordinate system, the reconstructed satellite CS moves toward the faulty satellite TS, as shown in Figure 3.  
Due to the small relative distance of the satellite swarms, the motion of the satellite swarms satisfies the C-W equation [31]. Then, the satellite reconstruction problem is:  
\[\left\{\begin{array}{l}\ddot{x}-2n\dot{y}-3n^{2}x=u_{x}\\ \ddot{y}+2n\dot{x}=u_{y}\\ \ddot{z}+n^{2}z=u_{z}\end{array}\right. \tag{1}\]  
where \(x\), \(y\) and \(z\) are the position components of the reconstructed satellite in the \(\mathbf{R}\) coordinate system; \(u_{x}\), \(u_{y}\), and \(u_{z}\) are the orbit control accelerations of the reconstructed satellite; and \(n\) is the orbital angular velocity of the faulty satellite.  
Let the state sequence be \(\mathbf{x}(t)=\begin{bmatrix}x&y&z&\dot{x}&\dot{y}&\dot{z}\end{bmatrix}^{ \mathrm{T}}\) and the orbital acceleration be \(\mathbf{u}(t)=\begin{bmatrix}u_{x}&u_{y}&u_{z}\end{bmatrix}^{\mathrm{T}}\). The relative motion Equation (1) is written in matrix form to obtain Equation (2):  
\[\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t) \tag{2}\]  
where  
\[\mathbf{A}=\left[\begin{array}{ccc}\mathbf{0}_{3\times 3}&\mathbf{I}_{3\times 3} \\ \mathbf{A}_{01}&\mathbf{A}_{02}\end{array}\right]\text{, }\mathbf{B}=\begin{bmatrix} \mathbf{0}_{3\times 3}&\mathbf{I}_{3\times 3}\end{bmatrix}^{T}\]  
\[\mathbf{A}_{01}=\begin{bmatrix}3n^{2}&0&0\\ 0&0&0\\ 0&0&-n^{2}\end{bmatrix}\text{, }\mathbf{A}_{02}=\begin{bmatrix}0&2n&0\\ -2n&0&0\\ 0&0&0\end{bmatrix}\]  
Figure 3: Relative motion model.  
According to optimal control theory, considering the finite time, the optimal energy is:  
\[J=\frac{1}{2}\int_{t_{0}}^{t_{f}}u^{\mathrm{T}}u\mathrm{d}t \tag{3}\]  
In Equation (3), \(t_{0}\) and \(t_{f}\) are the initial and final times, respectively. The energy-optimal position-velocity equation is as follows:  
\[\left\{\begin{array}{l}x=\frac{-t}{2t}c_{1}-\frac{3t^{2}}{2 nt}c_{2}+\frac{5 tb }{2 nt}c_{4}+\frac{5 ta}{12n^{2}}c_{5}+bc_{10}+ac_{11}\\ y=\frac{3t^{2}}{8 nt}c_{1}-\left(\frac{2t}{n^{2}}-\frac{3t}{4}\right)c_{2}-\left( \frac{b}{2n^{3}}+\frac{5 ta}{6n^{2}}\right)c_{4}-\left(\frac{a}{2n^{3}}-\frac{5 tb}{6n^{2}}\right)c_{5}-\frac{3}{2}ntc_{7}+c_{8}-2 ac_{10}+2 bc_{11}\\ z=\frac{-nt}{4 nt}c_{3}+\frac{1b}{4 nt}c_{6}+bc_{9}+ac_{12}\\ \dot{x}=\frac{-1}{4n}c_{1}-\frac{3t}{4n}c_{2}+\left(\frac{5b}{2 nt}-\frac{5 tb}{2 nt}\right)c_{4}+\left(\frac{5t}{12n^{2}}+\frac{5 tb}{2 nt}\right)c_{5}-nac_{10} +nbc_{11}\\ \dot{y}=\frac{3t}{4n}c_{1}+\left(\frac{9t^{2}}{4}-\frac{2}{n^{2}}\right)c_{2} -\left(\frac{a}{3n^{2}}+\frac{5 tb}{6n}\right)c_{4}+\left(\frac{b}{3n^{2}}- \frac{5 ta}{6 nt}\right)c_{5}-\frac{3}{2}nc_{7}-2 nbc_{10}-2 nac_{11}\\ \dot{z}=-\left(\frac{a+ntb}{4 nt}\right)c_{3}+\left(\frac{b-nta}{4t}\right)c_{6} -nac_{9}+nbc_{12}\end{array}\right. \tag{4}\]  
In the formula, \(c_{1}\sim c_{12}\) are constants, which can be obtained by substituting the known position and speed states of CS and TS into Equation (4); in addition, \(a=\sin nt\) and \(b=\cos nt\).