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The probabilistic velocity obstacle (PVO) extends the concept of velocityobstacle (VO) to work in uncertain dynamic environments. In this paper, we showhow a robust model predictive control (MPC) with PVO constraints undernon-parametric uncertainty can be made computationally tractable. At the coreof our formulation is a novel yet simple interpretation of our robust MPC as aproblem of matching the distribution of PVO with a certain desireddistribution. To this end, we propose two methods. Our first baseline method isbased 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 ofrepresenting arbitrary distributions as functions in Reproducing Kernel HilbertSpace (RKHS). We use this foundation to interpret our robust MPC as a problemof minimizing the distance between the desired distribution and thedistribution of the PVO in the RKHS. Both the RKHS and GMM based formulationcan work with any uncertainty distribution and thus allowing us to relax theprevalent Gaussian assumption in the existing works. We validate ourformulation by taking an example of 2D navigation of quadrotors with arealistic noise model for perception and ego-motion uncertainty. In particular,we present a systematic comparison between the GMM and the RKHS approach andshow that while both approaches can produce safe trajectories, the former ishighly conservative and leads to poor tracking and control costs. Furthermore,RKHS based approach gives better computational times that are up to one orderof magnitude lesser than the computation time of the GMM based approach.

Reactive Navigation under Non-Parametric Uncertainty through Hilbert Space Embedding of Probabilistic Velocity Obstacles