  Trajectory optimization is a fundamental problem in robotics. While
optimization of continuous control trajectories is well developed, many
applications require both discrete and continuous, i.e., hybrid, controls.
Finding an optimal sequence of hybrid controls is challenging due to the
exponential explosion of discrete control combinations. Our method, based on
Differential Dynamic Programming (DDP), circumvents this problem by
incorporating discrete actions inside DDP: we first optimize continuous
mixtures of discrete actions, and, subsequently force the mixtures into fully
discrete actions. Moreover, we show how our approach can be extended to
partially observable Markov decision processes (POMDPs) for trajectory planning
under uncertainty. We validate the approach in a car driving problem where the
robot has to switch discrete gears and in a box pushing application where the
robot can switch the side of the box to push. The pose and the friction
parameters of the pushed box are initially unknown and only indirectly
observable.
