  We present an optimization-based method to plan the motion of an autonomous
robot under the uncertainties associated with dynamic obstacles, such as
humans. Our method bounds the marginal risk of collisions at each point in time
by incorporating chance constraints into the planning problem. This problem is
not suitable for online optimization outright for arbitrary probability
distributions. Hence, we sample from these chance constraints using an
uncertainty model, to generate "scenarios", which translate the probabilistic
constraints into deterministic ones. In practice, each scenario represents the
collision constraint for a dynamic obstacle at the location of the sample. The
number of theoretically required scenarios can be very large. Nevertheless, by
exploiting the geometry of the workspace, we show how to prune most scenarios
before optimization and we demonstrate how the reduced scenarios can still
provide probabilistic guarantees on the safety of the motion plan. Since our
approach is scenario based, we are able to handle arbitrary uncertainty
distributions. We apply our method in a Model Predictive Contouring Control
framework and demonstrate its benefits in simulations and experiments with a
moving robot platform navigating among pedestrians, running in real-time.
