  Differential Dynamic Programming (DDP) is a popular technique used to
generate motion for dynamic-legged robots in the recent past. However, in most
cases, only the first-order partial derivatives of the underlying dynamics are
used, resulting in the iLQR approach. Neglecting the second-order terms often
slows down the convergence rate compared to full DDP. Multi-Shooting is another
popular technique to improve robustness, especially if the dynamics are highly
non-linear. In this work, we consider Multi-Shooting DDP for trajectory
optimization of a bounding gait for a simplified quadruped model. As the main
contribution, we develop Second-Order analytical partial derivatives of the
rigid-body contact dynamics, extending our previous results for fixed/floating
base models with multi-DoF joints. Finally, we show the benefits of a novel
Quasi-Newton method for approximating second-order derivatives of the dynamics,
leading to order-of-magnitude speedups in the convergence compared to the full
DDP method.
