  We study the problem of simultaneously reconstructing a polygonal room and a
trajectory of a device equipped with a (nearly) collocated omnidirectional
source and receiver. The device measures arrival times of echoes of pulses
emitted by the source and picked up by the receiver. No prior knowledge about
the device's trajectory is required. Most existing approaches addressing this
problem assume multiple sources or receivers, or they assume that some of these
are static, serving as beacons. Unlike earlier approaches, we take into account
the measurement noise and various constraints on the geometry by formulating
the solution as a minimizer of a cost function similar to \emph{stress} in
multidimensional scaling. We study uniqueness of the reconstruction from
first-order echoes, and we show that in addition to the usual invariance to
rigid motions, new ambiguities arise for important classes of rooms and
trajectories. We support our theoretical developments with a number of
numerical experiments.
