Dynamics and control of multibody systems in central gravity

This dissertation studies the dynamics and control of multibody systems, and their numerical simulation, in a central gravitational field. Initially, the dynamics of multibody systems moving in a plane in a central gravity is studied. There is a cyclic orbital coordinate, and the corresponding conjugate momentum is conserved. The dynamics is reduced to eliminate this cyclic variable. A general development for analyzing stability of relative equilibria of the full dynamics, corresponding to equilibria of the reduced dynamics, is obtained. This is applied to some examples of multibody spacecraft in planar motion. A control scheme, based on averaging theory, is developed for orbit transfer from one relative equilibrium to another. This scheme is applied to a planar dumbbell-shaped rigid body in central gravity. The dynamics of such systems in three-dimensional motion also has a cyclic coordinate, and the associated conjugate momentum is conserved. The dynamics is reduced to eliminate this degree of freedom. Stability analysis of the relative equilibria of a rigid dumbbell-shaped body is carried out. Potential shaping with attitude feedback is used to stabilize the unstable relative equilibria of the dumbbell body.; For numerical simulations of the dynamics of free and controlled multibody systems in central gravity, numerical integration algorithms obtained from discrete variational mechanics are used. These algorithms exactly preserve the symplectic form and conserved momenta of such systems. They also nearly preserve the total energy of conservative systems over long simulation times. These properties are usually not present in other numerical integration algorithms. Variational integration algorithms for the full and reduced dynamics of multibody systems in a potential field are obtained, and applied to the specific examples treated in this dissertation. Comparison with a standard Runge-Kutta fourth order integrator, for an example problem, shows the better performance of the variational integrators in long time simulations. These numerical simulations also confirm stability results obtained analytically. The controlled motion of the planar dumbbell model with averaging feedback control is simulated with a variational integrator to confirm analytically obtained results for this control scheme.