Stochastic State Estimation for Simultaneous Localization and Map Building in Mobile Robotics

1The study of stochastic models for Simultaneous Localization and Map Building (SLAM) in mobile robotics has been an active research topic for over fifteen years. Within the Kalman filter (KF) approach to SLAM, seminal work (Smith and Cheeseman, 1986) suggested that as successive landmark observations take place, the correlation between the estimates of the location of such landmarks in a map grows continuously. This observation was later ratified (Dissanayake et al., 2001) with a proof showing that the estimated map converges monotonically to a relative map with zero uncertainty. They also showed how the absolute accuracy of the map reaches a lower bound defined only by the initial vehicle uncertainty, and proved it for a one-landmark vehicle with no process noise. From an estimation theoretic point of view, we address these results as a consequence of partial observability. We show that error free reconstruction of the map state vector is not possible with typical measurement models, regardless of the vehicle model chosen, and show experimentally that the expected error in state estimation is proportional to the number of landmarks used. Error free reconstruction is only possible once full observability is guaranteed. Explicit solutions to the continuous time SLAM problem for a one-dimensional vehicle called the monobot appear in (Gibbens et al., 2000, Kim, 2004). Both give closed form asymptotic values of the state error covariance P. Kim observed that, for the case when not all landmarks are observed at all times, the asymptotic value on the determinant of P reaches a constant value greater than zero. Gibbens et al. on the other hand, observed that the rate of convergence of P is proportional to the number of landmarks used, and that its asymptotic value is independent of the plant variance. In their solution to the 1-d Brownian motion case, the state error covariance is linked to the total number of landmarks in the form of the total Fisher information 2