Lyapunov-Based Adaptive Feedback for Spacecraft Planar Relative Maneuvering via Differential Drag

I N 1989, Leonard et al. [1] introduced the concept of using differential drag at low Earth orbits (LEOs) for propellantless inplane spacecraft relative motion control. This method consists of varying the aerodynamic drag experienced by different spacecraft, by opening or closing a set of drag surfaces, or varying the attitude of asymmetrical spacecraft, thus generating differential accelerations between them. Since there is no propellant exhaust and no plume impingement, highly sensitive onboard sensors may operate in a cleaner environment. Moreover, since the relative accelerations generated by the drag forces are small, equipment sensitive to shocks or vibrations may benefit from the use of differential drag, assuming that drag control devices operate without exciting vibration modes of the spacecraft. The main limitation of using differential drag for relative motion control is that one must operate at a relatively low LEO. In this regime, differential drag forces can be made large enough to achieve effective control. However, this increased drag force results in faster orbit decay, and thus a more limited mission life. However, these formation-flying orbits are of interest since they can be used for communications, astronomical, atmospheric, and Earth observation applications [2,3]. In this work, a chaser and a target spacecraft are considered. The reference frame commonly used for spacecraft relative motion is the local-vertical/local-horizontal (LVLH) reference frame, centered at the target spacecraft, where x points from Earth to the target spacecraft, y points along the track of the target spacecraft, and z completes the right-handed frame. The state of the system consists of the position and velocity, in the LVLH frame, of the chaser spacecraft relative to the target spacecraft. Atmospheric differential drag is projected on the alongtrack direction and can provide effective control only in the orbital plane (x and y). The control law is based on the assumption that the control is either positive maximum ( 1), which implies chaser maximizing (opening) its drag surface and target minimizing (closing) it; negative maximum (−1), which implies chaser maximizing (opening) its drag surface and target minimizing (closing) it; or zero (0), which implies same surface on chaser and target: that is, no differential acceleration, as previously done in [4–6]. In previous work [7], a Lyapunov controller was developed for maneuvering using differential drag. An analytical expression for the magnitude of the differential drag acceleration that ensures stability was also found. Partial derivatives of this critical value in terms ofQ (Lyapunov equation matrix) and Ad (reference linear dynamics matrix) were presented in [8,9] for the case in which the controller acts as a regulator. Furthermore, an adaptation that chooses an appropriate positive definite matrix P in a quadratic Lyapunov function, by modifying the Q and Ad matrices based on the partial derivatives, was developed. Nonetheless, the adaptation was limited to regulation maneuvers, since the partial derivatives were developed for that case only. The foremost contribution of this work consists of the complete analytical expressions for the mentioned partial derivatives for the general case in which the spacecraft are tracking a linear reference model, which can also be used for tracking a guidance trajectory or a desired final state (regulation). Simulations validate the adaptive Lyapunov controller for a fly-around maneuver followed by a longterm formation-keeping period and a rendezvous maneuver via the Systems Tool Kit (STK®). An assessment of the performances of the designed adaptive Lyapunov controller and a comparison versus a nonadaptive Lyapunov controller [7] are shown.