  We propose a Stochastic MPC (SMPC) formulation for path planning with
autonomous vehicles in scenarios involving multiple agents with multi-modal
predictions. The multi-modal predictions capture the uncertainty of urban
driving in distinct modes/maneuvers (e.g., yield, keep speed) and driving
trajectories (e.g., speed, turning radius), which are incorporated for
multi-modal collision avoidance chance constraints for path planning. In the
presence of multi-modal uncertainties, it is challenging to reliably compute
feasible path planning solutions at real-time frequencies ($\geq$ 10 Hz). Our
main technological contribution is a convex SMPC formulation that
simultaneously (1) optimizes over parameterized feedback policies and (2)
allocates risk levels for each mode of the prediction. The use of feedback
policies and risk allocation enhances the feasibility and performance of the
SMPC formulation against multi-modal predictions with large uncertainty. We
evaluate our approach via simulations and road experiments with a full-scale
vehicle interacting in closed-loop with virtual vehicles. We consider distinct,
multi-modal driving scenarios: 1) Negotiating a traffic light and a fast,
tailgating agent, 2) Executing an unprotected left turn at a traffic
intersection, and 3) Changing lanes in the presence of multiple agents. For all
of these scenarios, our approach reliably computes multi-modal solutions to the
path-planning problem at real-time frequencies.
