Stabilization of Rigid Body Dynamics by Internal and External Torques

In this paper we discuss the stabilization of the rigid body dynamics by external torques (gas jets) and internal torques (momentum wheels). Our starting point is a generalization of the stabilizing quadratic feedback law for a single external torque recently analyzed in Bloch and Marsden [Proc. 27th IEEE Conf. Dec. and Can., pp. 2238-2242 (1989b); Sys. Can. Letts., 14,341-346 (1990)] with quadratic feedback torques for internal rotors. We show that with such torques, the equations for the rigid body with momentum wheels are Hamiltonian with respect to a Lie-Poisson bracket structure. Further, these equations are shown to generalize the dual-spin equations analyzed by Krishnaprasad [Nonlin. Ana. Theory Methods and App., 9, 1011-1035 (1985)] and Sanchez de Alvarez [Ph.D. Diss. (1986)]. We establish stabilization with a single rotor by using the energy-Casimir method. We also show how to realize the external torque feedback equations using internal torques. Finally, extending some work of Montgomery [Am. J. Phys., 59, 394-398 (1990)J, we derive a formula for the attitude drift· for the rigid body-rotor system when it is perturbed away from a stable equilibrium and we indicate how to compensate for this