Rendiconti di Matematica e delle Sue Applicazioni (Jan 2014)
Some models for the description of the attitude dynamics
Abstract
The study of rotational dynamics is of seminal importance for the description of the motion of natural and artificial bodies. In this work with are mainly interested to the attitude of spacecraft, possibly including dissipative e↵ects, which must be accurately accounted during mission design as they might drastically change the attitude, and lead to instability and mission failure. In this work we review some mechanical models of rotational dynamics and provide interesting applications of some mathematical theories. The first one is the spin-orbit problem describing the motion of a satellite rotating around an internal spin-axis and moving on a Keplerian orbit around a planet. A noticeable application of KAM theory to this model will be shortly reviewed ([6]). We will also discuss dissipative tidal e↵ects, which might act on the system. The second model is the so-called pitch-yaw-roll problem. In particular, we consider the pitch model, in which the yaw and roll angles are constantly zero; we shall also assume that one of the moments of inertia depends on time and that the atmospheric drag acts on the system. Following [21], we provide an instructive application of Melnikov’s method to establish the onset of chaos by evaluating the existence of heteroclinic intersections. The third model concerns the sloshing e↵ect acting within a spacecraft; assuming a linear motion of the fluid in the spacecraft, this problem will be mathematically described using an equivalent mechanical model by suitably combining springs, pendulums and dampers ([33]). The last model describes the effect of a variable mass (e.g., due to fuel consumption) on the attitude of the spacecraft. Following [14], this model admits an explicit solution in the case in which the container has a cylindrical shape.
