A Jacobian-based algorithm for motion planning for an underactuated system that is a rigid-body operated by two input-rotations is discussed in this paper. The rigid body undergoes a four-rotation fully-reversed (FR) sequence of rotations which consists of a series of initial two rotations about the axes of a coordinate frame attached to the rigid body and subsequent two rotations that undo the proceeding rotations. Due to the insufficient degrees of freedom of four-rotation FR sequences required to achieve all possible orientations, the rigid body cannot achieve some orientations. In order to best approximate these infeasible orientations, the Jacobian-based algorithm is implemented in the sense of least squares. As some orientations can never be attained by a single four-rotation FR sequence, two different four-rotation FR sequences are exploited alternately to ensure the convergence of the proposed algorithm. Assuming the orientation is supposed to be manipulated using three input-rotations, the switching-Jacobian algorithm proposed in this paper has significant practical importance for motion planning for aerospace and underwater vehicles maneuvered using only two input-rotations due to the failure of one of torque-generation mechanisms.

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