On stochastically observable directions of the estimation theoretic SLAM state space

The theory of stochastic observability is vital in describing the performance of Simultaneous Localization and Mapping (SLAM) as a nonlinear stochastic state estimation problem quantifying effects of random noise on its observability. We show that the eigen space corresponding to the stochastically unobservable states of the state error covariance matrix of the SLAM problem initialized with unknown initial conditions are in the null space of the information matrix associated with observations of the SLAM problem. We establish by using theoretical analysis and examples that the stochastically unobservable directions of the SLAM state space can be changed by modifying the observation model of the SLAM problem. We then use simulations and experiments to show that stochastically observable directions of state space and their degree of stochastic observability can be modified as required in a particular application (such as surveying, mapping and surveillance) by changing the vehicle path with respect to the landmarks in the environment by selecting which landmarks to observe and by modifying the observation model.