Positive definiteness analysis of the covariance matrix in EKF-SLAM

This paper presented an analysis of the positive definite character of the covariance matrix in the extended Kalman filter based simultaneous localization and mapping (EKFSLAM) algorithm. It was shown that during a finite-time two-dimensionalSLAM with point landmarks observed using a range-and-bearing sensor, the positive definiteness holds on condition that three certain Jacobian matrices keep full rank and the initial covariance matrix is positive definite. The popular motion and observation models were investigated and indicated that the conditions of Jacobians can be generally satisfied. And we argued that a positive definite initialization is more reasonable than a zero or completely correlative one. The positive definiteness was verified by Monte Carlo tests. Furthermore, we showed that ignoring correlations between landmarks, a popular way of reducing the computational complexity, will usually result in non-positive definite covariance matrix.