# Roto-Translational Control of Spacecraft in Low Earth Orbit Using Environmental Forces and Torques
## 1 Introduction  
CubeSats have seen an increase of popularity in recent years. Born as an educational platform inside university courses [1], in the last decade they became a low-cost high reward platform to perform in-orbit technology development [2], scientific missions [3], and interplanetary exploration [4]. The increasing popularity of this kind of platform has generated interest on developing miniaturized technologies to extend their capabilities. One of the key areas of interest is providing them with maneuverability, momentum control, and orbit adjustment. Traditionally, propulsion systems have been utilized for attitude control and for delta-V maneuvers; however, most CubeSats do not include on-board propulsion systems due to volume, mass and safety limitations. For this reason, alternative solutions are under investigation for orbital and re-entry applications, among others. One of the possible solutions is based on the concept of exploiting forces and torques that are usually considered disturbances. Similar to the idea of using the magnetic field of the Earth for attitude control; atmospheric drag can be exploited in Low Earth Orbit (LEO) by using variable shape devices to modify the experienced aerodynamic acceleration and torque. Changes in orbital velocity enable the capability of performing maneuvers in the same orbital plane, like formation flying or rendezvous and docking, without using on-board propulsion systems. The introduction of differential drag for formation flying control dates back to 1989, when the Clobessy-Wilshire (CW) linear equations for relative motion between two spacecraft were used to design an algorithm to control the relative in-plane motion [5]. to regulate the states with a discrete input. An improved solution based on the Schweighart-Sedwick (SS) linear equations for relative motion that include the \(J_{2}\) perturbation were introduced in [6]. In [7], the generation and use of lift to enable out-of-plane maneuvering capability was implemented. A Lyapunov-based control strategy was used in [8] to achieve spacecraft rendezvous using differential drag. In [9], the attitude of a spacecraft was used to change the experienced drag instead of dedicated actuators for drag surfaces. An adaptive sliding mode strategy was used in [10] to control therelative dynamics using a continuous differential drag input. In [11], a constrained least squares problem is formulated to find the best achievable set of individual inputs to control a set of spacecraft consisting of multiple chasers and a single target under mutual constraints and actuator saturations. Drag-based algorithms and techniques have been successfully demonstrated in orbit by the ORBCOMM constellation of satellites [12] to save propellant in thruster-based formation keeping maneuvers, and the Planet Labs satellites constellation [13] for propellant-less phasing maneuvers along the same orbit. Variable shape devices consisting of several independent surfaces can be installed on the spacecraft, locating the center of pressure at distances with respect to the Center of Mass (CoM) such that significant torques can be applied. This feature allows exploiting aerodynamic forces for attitude control maneuvers. In [14], a variable shape device has been used to control the attitude for an earth observation satellite mission; while in [15], an on-line parameter estimation procedure has been used to improve the performance in the presence of environmental and spacecraft uncertainties. In [16], several independent surfaces have been used to control the attitude of a spacecraft also during the re-entry phase, using a direct force control method in substitution of the classic bank angle modulation control. Variable shape devices can also be used to de-orbit satellites from Low Earth Orbit [17]. They can also be used to perform a controlled re-entry maneuver to target a precise location on the ground [18], and be able to sustain the aero-thermodynamic loads [19] typical of a deployable capsule re-entry profile [20].