  Differential Dynamic Programming (DDP) is an efficient trajectory
optimization algorithm relying on second-order approximations of a system's
dynamics and cost function, and has recently been applied to optimize systems
with time-invariant parameters. Prior works include system parameter estimation
and identifying the optimal switching time between modes of hybrid dynamical
systems. This paper generalizes previous work by proposing a general
parameterized optimal control objective and deriving a parametric version of
DDP, titled Parameterized Differential Dynamic Programming (PDDP). A rigorous
convergence analysis of the algorithm is provided, and PDDP is shown to
converge to a minimum of the cost regardless of initialization. The effects of
varying the optimization to more effectively escape local minima are analyzed.
Experiments are presented applying PDDP on multiple robotics systems to solve
model predictive control (MPC) and moving horizon estimation (MHE) tasks
simultaneously. Finally, PDDP is used to determine the optimal transition point
between flight regimes of a complex urban air mobility (UAM) class vehicle
exhibiting multiple phases of flight.
