# Roto-Translational Control of Spacecraft in Low Earth Orbit Using Environmental Forces and Torques
## 4 Simulation Results and Discussion
The observed increase was expected due to the additional requirement of achieving a specific cross-sectional areas for the relative orbit maneuver. The required control inputs are presented in Figure 9 for the target and chaser. Saturation due to the physical limitations of the DMD lengths was applied in the simulation. The DMD lengths reach saturation levels at the beginning of the maneuver where compensation for the initially large errors was required. Significant differences in the DMD  
Figure 4: Relative states for the roto-translational maneuver. Figure 5: Parameter estimates from the relative maneuvering controller. length profiles could be observed between target and chaser. The chaser spacecraft required changing the total cross-sectional and simultaneously generate the required torques using only the DMD surfaces, which can be observed in Figure 9. The DMD surfaces, in the case of the chaser spacecraft, were significantly more active than those of the target. This behavior was especially noticeable when the requirements of the relative orbit controller were demanding (i.e., required cross-sectional area at or near saturation levels), the changes between maximum and minimum area could be observed during the first 40 h. To avoid instantaneous and fast changes of the DMD lengths due to the 30 s interval between computations, the points were joined with splines and passed through a low-pass filter with cut-off frequency \(\omega_{c}\in\mathbb{R}_{>0}\) of 0.017 Hz. The maximum resulting deployment rate required from an actuator during the peak of control demand was 2.5 m per minute, as illustrated in Figure 10.  
Figure 6: Quaternions \(\boldsymbol{q}\) and \(\boldsymbol{q_{d}}\) for the target (**left**) and chaser (**right**), first 20 h of simulation. Figure 7: Angular velocities \(\boldsymbol{\omega}\) and \(\boldsymbol{\omega_{d}}\) for the target (**left**) and chaser (**right**), last 20 h of simulation. The parameter estimates for the attitude controller of the target are presented in Figure 11 the parameters associated with the aerodynamic drag, and Figure 12 those associated with the aerodynamic drag. Figure 8: Attitude of the target (**left**) and chaser (**right**) spacecraft expressed in Euler angle representation. Figure 10: Zoomed-in view of the DMD lengths applied to the chaser spacecraft. Interval between 16 and 17 h.  
Figure 9: Required DMD surfaces lengths of the target (**left**) and chaser (**right**) for the roto-translational maneuver. the aerodynamic lift, respectively. Similarly, in the case of the chaser, Figure 13 shows the estimated parameters associated with aerodynamic drag and Figure 14 presents those associated with aerodynamic lift, respectively. The estimates were adjusted by the adaptive update law to compensate for the uncertainties. Some estimates exhibited a diverging trend as compared to others, this behavior can be explained by the fact that the ultimately bounded results obtained from both controllers ensured convergence of the states but not necessarily of the estimates. Inside the ultimate bound, an increase of the Lyapunov function could indicate that the norm of the estimates was increasing, generating the behavior observed in some estimates. Although this behavior did not affect the stability result, in case the estimates kept growing, the continuous projection algorithm used in the adaptive update laws would keep them bounded. However, during the 70 h of simulation, these bounds were never reached. Figure 4:0 Parameter estimates from the attitude controller associated with the aerodynamic drag of the target. Components \(\bar{B}_{1}\widehat{C}_{Dj}\) of \(\widehat{\mathbf{\Theta}}_{1,t}\) with \(j=1,2,3,4\) (**left**), remaining components of \(\widehat{\mathbf{\Theta}}_{1,t}\) (**right**).