The problem of finding a fixed-endpoint motion of a rigid body in three space with angular velocity close to a given, arbitrary vector function omega is considered. In particular, if u is the angular velocity of the body in space coordinates, minimizers of parallel-tou-omegaparallel-to(p) on an admissible class consisting of smooth rigid body motions on [0, 1] with prescribed endpoints are sought for 1 less-than-or-equal-to p less-than-or-equal-to infinity. It is shown that, when working in an appropriate moving frame, each of these problems can be formulated as an autonomous problem that can be solved completely in closed form. While this moving frame must in general be obtained numerically, it can be obtained in advance, independently of the solutions; hence all of the extremals for this problem are identified and existence and uniqueness results are obtained.