General probabilistic bounds for trajectories using only mean and variance

Two ideas have gained traction in research in the robotics planning community. Activity planning has become popular where a library of predefined manipulation of the vehicle state is accessible, and is commonly used for missions with complex goal specifications. Another focus has been chance-constrained programming as a method of providing robust motion planning, in which the probability of failure is bounded. A combination of the two would allow for robust satisfaction of complex directives. However, to perform chance-constrained activity planning, we must be able to provide probabilistic bounds on the trajectory of the vehicle. While this may be done through propagation of statistics, we would require information about the actuation noise for the vehicle dynamics. In addition to such parameters as mean and variance, we also need to know the appropriate function for the noise. In many cases, the exact distribution of the actuation noise may not be known, although researchers can easily approximate the first two moments through calibrations. In this work we look at statistics propagation when only the first two moments of the actuation uncertainty is known, assuming white noise. We show that for linear systems, propagation is exact. Further, by looking at the expected error squared as a stochastic process, we can show that it is a submartingale under certain assumptions, and thus derive error bounds for deviation from mean over the duration of the entire path. We empirically show that, for nonlinear dynamics, we may approximate the propagation with the unscented transform, and obtain the corresponding bounds.