The hybrid Cram\xe9r-Rao lower bound for simultaneous self-localization and room geometry estimation

Abstract

This paper addresses the problem of tracking a moving source, e.g., a robot, equipped with both receivers and a source, that is tracking its own location and simultaneously estimating the locations of multiple plane reflectors. We assume a noisy knowledge of the robot’s movement. We formulate this problem, which is also known as simultaneous localization and mapping (SLAM), as a hybrid estimation problem. We derive the extended Kalman filter (EKF) for both tracking the robot’s own location and estimating the room geometry. Since the EKF employs linearization at every step, we incorporate a regulated kinematic model, which facilitates a successful tracking. In addition, we consider the echo-labeling problem as solved and beyond the scope of this paper. We then develop the hybrid Cramér-Rao lower bound on the estimation accuracy of both the localization and mapping parameters. The algorithm is evaluated with respect to the bound via simulations, which shows that the EKF approaches the hybrid Cramér-Rao bound (CRB) (HCRB) as the number of observation increases. This result implies that for the examples tested in simulation, the HCRB is an asymptotically tight bound and that the EKF is an optimal estimator. Whether this property is true in general remains an open question.