  The probabilistic velocity obstacle (PVO) extends the concept of velocity
obstacle (VO) to work in uncertain dynamic environments. In this paper, we show
how a robust model predictive control (MPC) with PVO constraints under
non-parametric uncertainty can be made computationally tractable. At the core
of our formulation is a novel yet simple interpretation of our robust MPC as a
problem of matching the distribution of PVO with a certain desired
distribution. To this end, we propose two methods. Our first baseline method is
based on approximating the distribution of PVO with a Gaussian Mixture Model
(GMM) and subsequently performing distribution matching using Kullback Leibler
(KL) divergence metric. Our second formulation is based on the possibility of
representing arbitrary distributions as functions in Reproducing Kernel Hilbert
Space (RKHS). We use this foundation to interpret our robust MPC as a problem
of minimizing the distance between the desired distribution and the
distribution of the PVO in the RKHS. Both the RKHS and GMM based formulation
can work with any uncertainty distribution and thus allowing us to relax the
prevalent Gaussian assumption in the existing works. We validate our
formulation by taking an example of 2D navigation of quadrotors with a
realistic noise model for perception and ego-motion uncertainty. In particular,
we present a systematic comparison between the GMM and the RKHS approach and
show that while both approaches can produce safe trajectories, the former is
highly conservative and leads to poor tracking and control costs. Furthermore,
RKHS based approach gives better computational times that are up to one order
of magnitude lesser than the computation time of the GMM based approach.
