  Trajectory optimization is an efficient approach for solving optimal control
problems for complex robotic systems. It relies on two key components: first
the transcription into a sparse nonlinear program, and second the corresponding
solver to iteratively compute its solution. On one hand, differential dynamic
programming (DDP) provides an efficient approach to transcribe the optimal
control problem into a finite-dimensional problem while optimally exploiting
the sparsity induced by time. On the other hand, augmented Lagrangian methods
make it possible to formulate efficient algorithms with advanced
constraint-satisfaction strategies. In this paper, we propose to combine these
two approaches into an efficient optimal control algorithm accepting both
equality and inequality constraints. Based on the augmented Lagrangian
literature, we first derive a generic primal-dual augmented Lagrangian strategy
for nonlinear problems with equality and inequality constraints. We then apply
it to the dynamic programming principle to solve the value-greedy optimization
problems inherent to the backward pass of DDP, which we combine with a
dedicated globalization strategy, resulting in a Newton-like algorithm for
solving constrained trajectory optimization problems. Contrary to previous
attempts of formulating an augmented Lagrangian version of DDP, our approach
exhibits adequate convergence properties without any switch in strategies. We
empirically demonstrate its interest with several case-studies from the
robotics literature.
