Stabilization of rigid body dynamics by internal and external torques (Q1194942)

The stabilization of a rigid body dynamics by external torques (gas jets) and internal torques (momentum wheels) is discussed. The starting point is a generalization of the stabilizing quadratic feedback law for a single external torque recently analyzed in a previous paper by two of the authors [\textit{A. M. Bloch} and \textit{J. E. Marsden}, Syst. Control Lett. 14, No. 4, 341-346 (1990; Zbl 0701.93083)]. It is shown 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 the other two authors [\textit{P. S. Krishnaprasad}, Nonlinear Anal., Theory Methods Appl. 9, 1011-1035 (1985; Zbl 0626.70028); and \textit{G. Sánchez de Alvarez}, Ph. D. Diss. (1986)]. Stabilization with a single rotor by using the energy-Casimir method is established. It is shown also how to realize the external torque feedback equations using internal torques. Finally, a formula for the amplitude drift for the rigid body- rotor system, when it is perturbed away from a stable equilibrium, is given and it is shown how to compensate for this.