On the uncertainty propagation: Why uncertainty on lie groups preserves monotonicity?

Researchers in the simultaneous localization and mapping (SLAM) community have taken for granted that uncertainty associated with the robot pose increases until the loop is closed. However, recently identified by [1], the monotonicity of uncertainty during exploration breaks when the robot returns to the initial position. In this paper, we propose a hypothesis that the monotonicity of pose uncertainty is preserved when the uncertainty is propagated on Lie groups rather than on Euclidean vector space. After deriving covariance propagated over Lie groups and Euclidean vector space, respectively, the monotonicity of uncertainty in each case is thoroughly investigated. Experiments with simulated and real-world scenarios on dead-reckoning validate our hypothesis on the monotonicity of uncertainty.