# Roto-Translational Control of Spacecraft in Low Earth Orbit Using Environmental Forces and Torques
## 3 Controller Design
### Stability Analysis  
Stability of the relative orbit and attitude subsystems when applying the adaptive control and update laws in (25), (28), (33) and (34) has been verified by Lyapunov-based stability analysis in [29; 30], respectively. In this section, the requirement of performing simultaneous attitude and relative orbit maneuver is included, and the modified stability result is presented. The attitude control law computes the required torques for achieving the attitude tracking objective, and an intermediate step where the a suitable set of DMD surfaces lengths are computed to generate such torques was proposed using a numerical algorithm to solve (27). Considering that the proposed strategy to integrate the attitude and orbit controllers is to modify the minimization problem as in (35), and that the influence of the residual mismatch \(\chi\) between the desired and obtained torques was explicitly included, the stability proof of the attitude controller does not change. The ultimately bounded attitude tracking result is presented for completeness in Theorem 1. **Theorem 1**. : _Consider the spacecraft attitude dynamics governed by the non-linear system in (11). The auxiliary controller in (25) and the adaptive update law in (28) ensure uniformly ultimately bounded attitude tracking in the sense that_  
\[\|\boldsymbol{e}_{\boldsymbol{v}}\|\leq\epsilon_{1}\text{exp}\{-\epsilon_{2} \}+\epsilon_{3}, \tag{36}\]  
_where \(\epsilon_{1},\,\epsilon_{2},\,\epsilon_{3}\in\mathbb{R}_{>0}\) are known bounding constants and provided that \(\boldsymbol{\eta}(0)\in\mathcal{S}\), with \(\boldsymbol{\eta}\triangleq[\boldsymbol{e}_{\boldsymbol{v}}^{T}\boldsymbol{ \mu}^{T}]^{T}\in\mathbb{R}^{6}\). The set \(\mathcal{S}\), and the constants \(\epsilon_{1},\,\epsilon_{2},\,\text{and}\,\,\,\epsilon_{3}\) are explicitly defined in [30]._  
The relative maneuvering control law computes the required cross-sectional area required by the chaser spacecraft to achieve regulation of the in-plane relative states in \(\boldsymbol{X}\). In [29], the cross-sectional area could be computed analytically under the assumption of attitude stabilized spacecraft. For the roto-translational maneuver, variation of the drag coefficient due to time-varying attitude, and the use of (35) as an intermediate step for obtaining the total cross-sectional area must be considered. To facilitate the stability analysis of the relative maneuvering controller accounting for the influence of a simultaneous attitude maneuver, some definitions are introduced. Let the constants \(\epsilon_{4},\,\epsilon_{5},\,\epsilon_{6}\in\mathbb{R}_{>0}\) be defined as \(\epsilon_{4}\triangleq\sqrt{\frac{\lambda\|\boldsymbol{X}(0)\|^{2}+\overline{ \gamma}}{\lambda}}\), \(\epsilon_{5}\triangleq\frac{\lambda_{1}}{2\lambda}\), \(\epsilon_{6}\triangleq\sqrt{\frac{\overline{\gamma}-\overline{\gamma}}{ \lambda}+\frac{\overline{\lambda}\xi_{4}}{\lambda_{1}}}\), where \(\zeta_{4},\,\lambda,\,\,\overline{\lambda},\,\,\underline{\gamma},\,\, \overline{\gamma}\in\mathbb{R}_{>0}\) are known bounding constants. **Theorem 2**. : _Given the spacecraft in-plane relative dynamics in (8) and (9), the controller in (33) and the adaptive update law in (34) ensure uniformly ultimately bounded regulation of the relative state \(\boldsymbol{X}\) in the sense that_  
\[\|\boldsymbol{X}\|\leq\epsilon_{4}\text{exp}\{-\epsilon_{5}t\}+\epsilon_{6}. \tag{37}\]  
Proof.