  We address the issue of safe optimal path planning under parametric
uncertainties using a novel regularizer that allows trading off optimality with
safety. The proposed regularizer leverages the notion that collisions may be
modeled as constraint violations in an optimal control setting in order to
produce open-loop trajectories with reduced risk of collisions. The risk of
constraint violation is evaluated using a state-dependent relevance function
and first-order variations in the constraint function with respect to
parametric variations. The approach is generic and can be adapted to any
optimal control formulation that deals with constraints under parametric
uncertainty. Simulations using a holonomic robot avoiding multiple dynamic
obstacles with uncertain velocities are used to demonstrate the effectiveness
of the proposed approach. Finally, we introduce the car vs. train problem to
emphasize the dependence of the resultant risk aversion behavior on the form of
the constraint function used to derive the regularizer.
