  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, it is challenging to ensure that the
constraints are not violated. In this paper, we propose Safe-CDDP, a safe
trajectory optimization and control approach 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 evaluated on three different robot dynamics with up to 12 degrees
of freedom in simulation. The computational feasibility and applicability of
the approach are demonstrated with a physical hardware implementation.
