Implicit Differential Dynamic Programming

Over the past decade, the Differential Dynamic Programming (DDP) method has gained in maturity and popularity within the robotics community. Several recent contributions have led to the integration of constraints within the original DDP formulation, hence enlarging its domain of application while making it a strong and easy-to-implement competitor against alternative methods of the state of the art such as collocation or multiple-shooting approaches. Yet, and similarly to its competitors, DDP remains unable to cope with high-dimensional dynamics within a receding horizon fashion, such as in the case of online generation of athletic motion on humanoid robots. In this paper, we propose to make a step towards this objective by reformulating classical DDP as an implicit optimal control problem, allowing the use of more advanced integration schemes such as implicit or variational integrators. To that end, we introduce a primal-dual proximal Lagrangian approach capable of handling dynamical and path constraints in a unified manner, while taking advantage of the time sparsity inherent to optimal control problems. We show that this reformulation enables us to relax the dynamics along the optimization process by solving it inexactly: far from the optimality conditions, the dynamics are only partially fulfilled, but continuously enforced as the solver gets closer to the local optimal solution. This inexactness enables our approach to robustly handle large time steps (100 ms or more), unlike other DDP solvers of the state of the art, as experimentally validated through different robotic scenarii.