# Modeling and Control of Satellite Formations: A Survey
## 4 Leader-Follower Formation Architecture
Paper [10] is apparently the first attempt to use hybrid optimization with a genetic algorithm to find reconfiguration trajectories, although the genetic algorithm has often been used for other trajectory optimization problems. In [10],the relative motion in the LVLH reference frame is described using the HCW equations of motion. The satellite constellation consists of a head satellite and surrounding slave satellites. The orbit of the lead satellite is considered a reference orbit, and the relative orbits of the follower are considered to be PCO. PCO predicted distances between the master and slave satellites [129] are constant. The HCW equation describes the relative motion between the master and slave satellites in the local reference frame LVLH with the origin on the master satellite. In this system, the \(OX\) axis is defined in the radial direction of the main orbit, and the \(OY\) axis is defined as the longitudinal direction of the main orbit. The \(OZ\) axis is defined to complete the right coordinate system. The relative position vector \(\mathbf{r}\) and the relative velocity vector \(\mathbf{v}\) of the slave satellite are defined in the LVLH reference system. State vector \(\mathbf{x}\) of relative dynamics is defined as \(\mathbf{x}= \begin{bmatrix}\mathbf{r}^{\mathrm{T}}&\mathbf{v}^{\mathrm{T}} \end{bmatrix} ^{\mathrm{T}}\), where \(\mathbf{r}= \begin{bmatrix}x&y&z\end{bmatrix} ^{\mathrm{T}}\), \(\mathbf{v}= \begin{bmatrix}\dot{x}&\dot{y}&\dot{z}\end{bmatrix} ^{\mathrm{T}}\). Then, [10] displays the linearized equation of relative motion for two bodies. The distances between the master and slave satellites are ignored during linearization since they are significantly less than the distance between the main satellite and the center of gravity (Earth, when moving in the LEO). Thus, the linear equations of state are obtained  
\[\dot{\mathbf{x}}(t)= \begin{bmatrix}\mathbf{O}_{3}&\mathbf{I}_{3}\\ \mathbf{A}&\mathbf{B}\end{bmatrix} \mathbf{x}(t),\quad\mathbf{A}=n^{2} \begin{bmatrix} 3&0&0\\ 0&0&0\\ 0&0&-1\end{bmatrix} ,\quad\mathbf{B}=n \begin{bmatrix}0&2&0\\ -2&0&0\\ 0&0&0\end{bmatrix} , \tag{57}\]  
where \(\mathbf{O}_{3},\mathbf{I}_{3}\) are the zero and unit matrices of order 3, \(n=\sqrt{GM/a^{3}}\) is the average speed of the lead spacecraft, \(G\) is the gravitational constant, \(M\) is the mass of the central body (for the Earth, \(GM=398,603\cdot 10^{9}\mathrm{m}^{3}\cdot\mathrm{c}^{-2}\)), \(a\) is the semi-major axis of the orbit of the lead satellite. The homogeneous Equations (57) are then analytically integrated using the Cauchy formula \(\mathbf{x}(t)=\Phi\mathbf{x}(0)\), where \(\Phi=\exp( \begin{bmatrix}\mathbf{O}_{3}&\mathbf{I}_{3}\\ \mathbf{A}&\mathbf{B}\end{bmatrix} t)\) is the fundamental matrix of the system (57), \(t\geqslant 0\) stands for the current time, \(\mathbf{x}(0)\) denotes the initial state vector. (There are some inaccuracies in this expression for matrix \(\Phi\) in [10].) [PERSON] et al. [10] explored the three types of reconfiguration maneuvers described in [73] namely resizing, reassigning, and reorienting. Resizing is the change in distance between spacecraft in the formation. In [10], a hybrid optimization strategy with a genetic algorithm and basis vector theory is used for fuel-optimal reconfiguration of a satellite group. In hybrid optimization, the genetic algorithm performs a global search to create two-impulse trajectories at each transition time, and vector analysis of basis vectors serves for local optimization with two-impulse trajectories as initial approximations. The final optimal trajectory has multiple pulses and also lower fuel consumption than two-pulse trajectories. The mission planner must investigate the trade-off between fuel saved by multiple impulses and the increased mission complexity due to additional impulses. In many missions, the trajectory obtained by the genetic algorithm will have an advantage for the transition due to its simplicity and almost optimal fuel consumption.