Line-Search Filter Differential Dynamic Programming for Optimal Control with Nonlinear Equality Constraints

We present FilterDDP, a differential dynamic programming algorithm for solving discrete-time, optimal control problems (OCPs) with nonlinear equality constraints. Unlike prior methods based on merit functions or the augmented Lagrangian class of algorithms, FilterDDP uses a step filter in conjunction with a line search to handle equality constraints. We identify two important design choices for the step filter criteria which lead to robust numerical performance: 1) we use the Lagrangian instead of the cost as one of the filter criterion and, 2) for the stopping criteria and backward pass Hessians, we replace the value function gradient with an estimated dual variable of the dynamics constraints. Both choices are rigorously justified, for 2) in particular by a formal proof of local quadratic convergence. We validate FilterDDP on three contact implicit trajectory optimisation problems which arise in robotics.