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 in the step acceptance criterion and, 2) in the backward pass, we perturb the value function Hessian. Both choices are rigorously justified, for 2) in particular by a formal proof of local quadratic convergence. In addition to providing a primal-dual interior point extension for handling OCPs with both equality and inequality constraints, we validate FilterDDP on three contact implicit trajectory optimisation problems which arise in robotics.

翻译：本文提出FilterDDP，一种用于求解带非线性等式约束的离散时间最优控制问题的微分动态规划算法。与以往基于价值函数或增广拉格朗日类算法的方法不同，FilterDDP采用步长滤波器结合线搜索来处理等式约束。我们确定了步长滤波器准则中两个关键的设计选择，以实现鲁棒的数值性能：1）在步长接受准则中使用拉格朗日函数而非成本函数；2）在后向传播中对价值函数的Hessian矩阵进行扰动。这两种选择均得到严格论证，特别是针对第2）点，我们提供了局部二次收敛性的形式化证明。除了给出处理同时包含等式与不等式约束最优控制问题的原对偶内点法扩展外，我们在机器人学中出现的三个接触隐式轨迹优化问题上验证了FilterDDP的有效性。