Differential Dynamic Programming with Nonlinear Safety Constraints Under System Uncertainties

Safe operation of systems such as robots requires them to plan and execute trajectories subject to safety constraints. When those systems are subject to uncertainties in their dynamics, ensuring that the constraints are not violated is challenging. In this paper, we propose a safe trajectory optimization and control approach (Safe-CDDP) for systems under additive uncertainties and non-linear safety constraints based on constrained differential dynamic programming (DDP). The safety of the robot during its motion is formulated as chance-constraints with user-chosen probabilities of constraint satisfaction. The chance constraints are transformed into deterministic ones in DDP formulation by constraint tightening. To avoid over conservatism during constraint tightening, linear control gains of the feedback policy derived from the constrained DDP are used in the approximation of closed-loop uncertainty propagation in prediction. The proposed algorithm is empirically demonstrated on three different robot dynamics with up to 12 states and the results show the applicability of the approach for safety-aware applications.