  The attitude of a rigid body evolves on the three-dimensional special
orthogonal group, and it is often estimated by measuring reference directions,
such as gravity or magnetic field, using an onboard sensor. As a single
direction measurement provides a two-dimensional constraint, it has been widely
accepted that at least two non-parallel reference directions should be
measured, or the reference direction should change over time, to determine the
attitude completely. This paper uncovers an intriguing fact that the attitude
can actually be estimated by using multiple measurements of a single, fixed
reference direction, provided that the angular velocity and the direction
measurements are resolved in appropriate frames, respectively. More
specifically, after recognizing that the attitude uncertainties propagated over
the left-trivialized stochastic kinematics are distinct from those over the
right-trivialized one, stochastic attitude observability with single direction
measurements is formulated by an information theoretic analysis. These are
further illustrated by numerical simulations and experiments.
