Abstract
This research introduces a novel approach to the analysis of relative motion modelling of free and tethered satellites. For relative motion of free satellites the variational Kepler problem is considered and a geometrical method is developed in which the general solution of the variational Kepler problem is solved in the physically relevant relative position coordinates and expressed only in terms of the constants of motions. The method also allows for variations of parameters which is relevant in dust comet tail modelling where the different sizes of the dust particles relative to the nucleus in turn gives rise to a difference in mass parameter. In the part of the research devoted to tethered satellites, different conservative models of tethered satellites are related mathematically to provide a unified framework and it is established in what limit they may provide useful insight into the underlying dynamics. First, the infinite dimensional model is regularised through the resistance against bending and then linked to a finite dimensional model, the slack-spring model, through a conjecture on the singular perturbation of tether thickness. Using a developed variational, symplectic integrator of the regularised system, numerical evidence is provided for the validity of the conjecture. Moreover, numerical computations of an orbiting tether system document that bending may be significant in regions of phase space. The slack-spring model is then naturally related to a billiard model in the limit of an inextensible spring. Next, the motion of a dumbbell model, which is lowest in the hierarchy of models, is identified within the motion of the billiard model through a theorem on the existence of invariant curves by exploiting Moser’s twist map theorem. Numerical computations provide insight into the dynamics of the billiard model. To investigate the slack-spring limit further, a Galerkin approximation of the full massive tether model is considered. Here it is shown that the the slack-spring dynamics can be identified with the slow dynamics on a normally elliptic slow manifold with bifurcations. Using averaging and a blow-up near the bifurcation it is in this thesis proven that the slow manifold persists adiabatically. It is believed that extending and generalising this result to more degrees of freedoms would attract considerable interest within both academia and industry. The research also focuses attention on optimal attitude control. The research in this direction develops a novel geometrical and coordinate-independent approach to variational attitude dynamics and obtains within the linear approximation explicit, analytic expressions for the constrained L2-optimal torque. The optimal torque is applied to two different formation flying missions scenarios where the range of validity of the linear approximation was also quantified. The results demonstrate an error of ~1 for a net rotation of 25. A feedback law is also suggested in which the optimal control is updated via measurements of the instantaneous attitude and angular velocities. Finally, this research considers a gravitational two-body problem where one of the bodies is modelled as a pseudo-rigid body. The other body is assumed to be a rigid sphere. Due to the rotational and “re-labelling” symmetries, the system is shown to possess conservation of angular momentum and “circulation”. By following the classical reduction procedure undertaken in the study of the two-body problem of a general rigid body and a rigid sphere, a similar reduced non-canonical Hamiltonian system is computed. The classical two-body problem then becomes a natural subsystem. Then relative equilibria of the system are considered and it is shown that the notions of locally central and planar equilibria coincide. Finally, it is shown that Riemann’s theorem on pseudo-rigid bodies has an extension to planar relative equilibrium of this system.