  This paper presents a novel approach using sensitivity analysis for
generalizing Differential Dynamic Programming (DDP) to systems characterized by
implicit dynamics, such as those modelled via inverse dynamics and variational
or implicit integrators. It leads to a more general formulation of DDP,
enabling for example the use of the faster recursive Newton-Euler inverse
dynamics. We leverage the implicit formulation for precise and exact contact
modelling in DDP, where we focus on two contributions: (1) Contact dynamics in
acceleration level that enables high-order integration schemes; (2) Formulation
using an invertible contact model in the forward pass and a closed form
solution in the backward pass to improve the numerical resolution of contacts.
The performance of the proposed framework is validated (1) by comparing
implicit versus explicit DDP for the swing-up of a double pendulum, and (2) by
planning motions for two tasks using a single leg model making multi-body
contacts with the environment: standing up from ground, where a priori contact
enumeration is challenging, and maintaining balance under an external
perturbation.
