## 2 Dynamic Model of Underactuated SFF  
Consider the chief spacecraft in the formation flying in elliptical orbits. As depicted in Figure 1, two reference frames are used to describe the relative motion between the chief and deputy, \(O_{E}X_{I}Y_{I}Z_{I}\) is the Earth-centered inertial (ECI) frame with \(O_{E}\) being the center of Earth, and \(O_{C}xyz\) is the local-vertical local-horizontal (LVLH) frame fixed at the center of the chief \(O_{C}\), where the x axis is pointing in the radial direction of the chief, the z axis is normal to the orbital plane, and the y axis completes the right-handed frame.  
Therefore, the orbital equations of underactuated SFF can be written in the state space form as [10].  
\[\dot{\mathbf{X}}=\mathbf{A}\mathbf{X}+m_{d}^{-1}(\mathbf{B}_{i}\mathbf{U}_{i}+\mathbf{DF}),\ i=1,2, \tag{1}\]  
with  
\[\mathbf{A}=\left[\begin{array}{cccccc}0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ \omega_{C}^{2}+2n_{C}^{2}&\alpha_{C}&0&0&2\omega_{C}&0\\ -\alpha_{C}&\omega_{C}^{2}-n_{C}^{2}&0&-2\omega_{C}&0&0\\ 0&0&-n_{C}^{2}&0&0&0\\ \end{array}\right],\mathbf{B}_{i}=\left[\begin{array}{c}\mathbf{0}_{3\times 2}\\ \mathbf{B}_{22}\end{array}\right], \tag{2}\]  
\[\mathbf{B}_{12}=\left[\begin{array}{cc}0&0\\ 1&0\\ 0&1\end{array}\right],\mathbf{B}_{22}=\left[\begin{array}{cc}1&0\\ 0&0\\ 0&1\end{array}\right],\mathbf{D}=\left[\begin{array}{cc}0&0&0\\ 0&0&0\\ 1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right].\]  
Denote the relative state vector of the deputy as \(\mathbf{X}=\left[\rho^{\mathrm{T}}\quad\dot{\rho}^{\mathrm{T}}\right]^{\mathrm{T}}\), where \(\mathbf{\rho}=\left[\begin{matrix}x&y&z\end{matrix}\right]^{\mathrm{T}}\) and \(\dot{\mathbf{\rho}}=\left[\begin{matrix}\dot{x}&\dot{y}&\dot{z}\end{matrix}\right] ^{\mathrm{T}}\) are the relative position vector and relative velocity vector, respectively, \(\omega_{C}=n_{C}\sqrt{(1+e\cos\theta)}\) is the angular velocity of the chief and \(\alpha_{C}=-2n_{C}^{2}\sin\theta\) is the angular acceleration of the chief, \(n_{C}=\sqrt{\mu/R_{C}^{3}}\), \(\mu\) is the Earth's gravitational constant, \(R_{C}\) is the radius of the deputy \(e\) and \(\theta\) are, respectively, the eccentricity and true anomaly, \(m_{D}\) is the mass of the deputy spacecraft, \(\mathbf{F}\in\mathbb{R}^{3}=\left[\begin{matrix}d_{x}&d_{y}&d_{z}\end{matrix} \right]^{\mathrm{T}}\) denotes the vector of total disturbances including the nonlinear terms and perturbations. For the case without radial control, the control input is denoted as \(\mathbf{U}_{1}=\left[\begin{matrix}U_{y}&U_{z}\end{matrix}\right]^{\mathrm{T}}\). For the case without in-track control, the control input is denoted as \(\mathbf{U}_{2}=\left[\begin{matrix}U_{x}&U_{z}\end{matrix}\right]^{\mathrm{T}}\).  
The controllability of the underactuated systems and the feasibility of reconfiguration have been well investigated in the literature [13] and lead to the following lemma:  
Figure 1: Definition of coordinate frames.  
**Lemma 1**.: _For the case without radial control, formation reconfiguration in elliptical orbit is still feasible without any supplementary conditions. For the case without in-track control, it is conditionally feasible provided that the condition \(\widetilde{x}_{1}(0)=\widetilde{x}_{\rm II}(0)\) holds, where \(\widetilde{x}_{1}(0)\) and \(\widetilde{x}_{\rm II}(0)\) are the coordinate transformations of the relative radial positions of the chief and deputy spacecraft in configuration I and configuration II respectively at the initial moment, and can be expressed as:_  
\[\left\{\begin{array}{l}\widetilde{x}_{\rm I}(0)=(1+e\cos\theta)x_{I}(0),\\ \widetilde{x}_{\rm II}(0)=(1+e\cos\theta)x_{II}(0).\end{array}\right. \tag{3}\]  
Since the uncontrollable state of the system \(\widetilde{x}_{\rm I}(0)=(1+e_{\rm C}\cos\theta_{\rm C})x_{I}(0)\) keeps its initial value constant during the reconfiguration process, the preconditions for achieving configuration II are naturally satisfied when the above conditions are met, so the reconfiguration is still feasible in the case of in-track underactuated.