Before revision:  However, these methodsare not fast enough for many robotics applications such as real-time humanoid motor control . Exploiting the sparse structure of optimal control problem , such as in Differential DynamicProgramming (DDP), has proven to significantly boost the computational efficiency, and recent works have been focused on handling arbitrary constraints. Despite that , DDP has been associated with poor numerical convergence , particularly when considering long time horizons. One of the main reasons is due to system instabilities and poor warm-starting (only controls). This paper presents control-limited Feasibility-driven DDP (Box-FDDP), a solver that incorporates a direct-indirect hybridization of the control-limited DDP algorithm. Concretely, the forward and backward passes handle feasibility and control limits. We showcase the impact and importance of our method on a set of challenging optimal control problems against the Box-DDP and squashing-function approach .
After revision:  However, these methodsare not fast enough for many robotics applications such as real-time humanoid motor control . Exploiting the sparse structure of optimal control problem . Our feasibility search emulates the numerical resolution of a direct transcription problem with only dynamics constraints, namely a multiple shooting formulation. We show that our approach has better numerical convergence than BOX-DDP (a shooting method), and that its convergence rate and runtime performance are competitive with state-of-the-art direct transcription formulations solved using the interior point and active set algorithms available in KNITRO. We further show that our approach decreases the dynamic feasibility error monotonically -- as in state-of-the-art nonlinear programming algorithms. We demonstrate the benefits of our hybrid approach by generating complex and athletic motions for quadruped and humanoid robots .