  This work considers the problem of estimating the unscaled relative positions
of a multi-robot team in a common reference frame from bearing-only
measurements. Each robot has access to a relative bearing measurement taken
from the local body frame of the robot, and the robots have no knowledge of a
common or inertial reference frame. A corresponding extension of rigidity
theory is made for frameworks embedded in the \emph{special Euclidean group}
$SE(2) = \mathbb{R}^2 \times \mathcal{S}^1$. We introduce definitions
describing rigidity for $SE(2)$ frameworks and provide necessary and sufficient
conditions for when such a framework is \emph{infinitesimally rigid} in
$SE(2)$. Analogous to the rigidity matrix for point formations, we introduce
the \emph{directed bearing rigidity matrix} and show that an $SE(2)$ framework
is infinitesimally rigid if and only if the rank of this matrix is equal to
$2|\mathcal{V}|-4$, where $|\mathcal{V}|$ is the number of agents in the
ensemble. The directed bearing rigidity matrix and its properties are then used
in the implementation and convergence proof of a distributed estimator to
determine the {unscaled}{} relative positions in a common frame. Some
simulation results are also given to support the analysis.
