Attitude maneuvers which avoid a forbidden direction

In this paper we consider the problem of slewing a rigid spacecraft from rest to rest between prescribed attitudes while avoiding a forbidden direction during the slew. Using methods of differential geometric control theory, we derive a control law which minimises a cost functional penalizing both high angular velocities and proximity to the forbidden direction. The system of differential equations for the optimal angular velocities, arrived at by an application of Pontryagin's principle, turns out to be completely integrable. The constants of integration (which depend on the target attitude) can be determined analytically for a suitable choice of the cost functional; in the general case, a numerical scheme based on the integration of the variational equations is proposed.