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.
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