# Roto-Translational Control of Spacecraft in Low Earth Orbit Using Environmental Forces and Torques
## 4 Simulation Results and Discussion

Table 1: Spacecraft physical parameters, subscripts 1 and \(t\) denote chaser 1 and target, respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(\phi_{0,k}\) [deg] & 13.5 \\ \(\theta_{0,k}\) [deg] & \(-12\) \\ \(\psi_{0,k}\) [deg] & 10 \\ \(\phi_{0,k}\) [deg/s] & \(5\times 10^{-3}\) \\ \(\theta_{0,k}\) [deg/s] & \(-1.5\times 10^{-2}\) \\ \(\psi_{0,k}\) [deg/s] & \(1.8\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial Euler angles and angle rates for the roto-translational maneuver, subscript \(k=1\), \(t\) denotes chaser 1 and target, respectively. Figure 4 presents the relative states of the target expressed in the LVLH frame with origin at the CoM of the chaser spacecraft. In this simulation, the chaser required approximately 55 h to enter the ultimate bounds of \(\pm 10\) and \(\pm 60\) meters along the directions \(\Delta\mathbf{\hat{x}}\) and \(\Delta\mathbf{\hat{y}}\), respectively, as compared to the convergence to a circle of 10 m radius around the desired position obtained with the individual, decoupled counterpart in [29]. The resulting residual error was expected due to the ultimately bounded stability result obtained in the coupled attitude-orbit case. The uncertain parameters for the relative maneuvering controller are presented in Figure 5, where dynamic variation of the estimates to compensate for the uncertainties can be observed. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Semi-Major Axis [km] & 6778 \\ Eccentricity & 0 \\ Inclination [deg] & 51.94 \\ RAAN [deg] & 206.26 \\ Arg. of perigee [deg] & 201.07 \\ True Anomaly [deg] & 108.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Initial orbital parameters for the roto-translational maneuver. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(\phi_{0,t}\) [deg] & \(-25\) \\ \(\theta_{0,t}\) [deg] & 0 \\ \(\psi_{0,t}\) [deg] & 15 \\ \(\phi_{0,1}\) [deg] & 30 \\ \(\theta_{0,1}\) [deg] & 0 \\ \(\psi_{0,1}\) [deg] & \(-10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Desired Euler angles and angle rates for the roto-translational maneuver, subscripts \(1\) and \(t\) denote chaser 1 and target, respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(W_{att}\) & \(1\) \\ \(W_{orb}\) & \(5\times 10^{-5}(1-\overline{\alpha})\) \\ \(K_{1}\) (\(\times 10^{-3}\)) & \(\text{diag}(3,3,3)\) \\ \(\beta\) (\(\times 10^{-3}\)) & \(\text{diag}(1.5,5,5)\) \\ \(\beta_{1}\) (\(\times 10^{-6}\Figure 6 shows the actual and desired quaternion and Figure 7 presents the actual and desired angular velocities. Note that although the required orientation was fixed with respect to LVLH, the desired \(\mathbf{q_{d}}\) was time-varying and \(\mathbf{\omega_{d}}\) was different from zero; therefore, the controller was really performing a tracking maneuver. Figure 8 shows the resulting orientations in Euler angle representation for the target and chaser. Although the orientation of both spacecraft had the same initial conditions, and the control parameters were also the same, the resulting orientation had different transients and ultimate bounds. This behavior can be explained by the fact that the chaser spacecraft was the only one in charge of satisfying the required cross-sectional area for the relative orbit maneuver. The only goal of the target was to regulate its attitude and to broadcast its cross-sectional area and ECI states to the chaser so that it could compute (33). While the target required 4.8 h to enter the ultimate bounds of \(\pm 2,\ \pm 3\) and \(\pm 2\) degrees of error in roll, pitch and yaw, respectively, the chaser required 43 h to enter the ultimate bounds of \(\pm 2,\ \pm 6.5\) and \(\pm 5.5\) degrees of error in roll, pitch and yaw, respectively. The resulting ultimate bounds showed an increase with respect to the individual, decoupled counterpart in [30] of \(\pm 3,\ \pm 1.5\) and \(\pm 3\) in roll, pitch and yaw, respectively.