We study control problems for a specific mechanical system consisting of a rigid base body with an unactuated internal degree of freedom. The key assumptions are that the translational and rotational motion of the base body can be completely controlled by external forces and moments, while the internal degree of freedom is unactuated. This specific example is representative of a class of underactuated mechanical control systems that are not linearly controllable. This class of control problems presents major theoretical and practical difficulties, and such models do arise in important terrestial and spacecraft applications. Our focus in this paper is on the simplest mechanical example in this class; it illustrates important features and difficulties associated with the class of such problems.
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