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：我们提出FilterDDP，一种求解离散时间、含非线性等式约束的最优控制问题（OCP）的微分动态规划算法。与基于罚函数或增广拉格朗日类算法的传统方法不同，FilterDDP采用步长滤波器结合线搜索来处理等式约束。我们识别出步长滤波器准则的两项关键设计选择，这些选择能带来稳健的数值性能：1）在步长接受准则中使用拉格朗日函数而非代价函数；2）在反向递推中对价值函数Hessian矩阵添加扰动。这两项选择均具有严格的理论依据，特别对于选择2），我们给出了局部二次收敛的正式证明。除了提供处理同时含等式与不等式约束OCP的原-对偶内点扩展外，我们在机器人学中的三类接触隐式轨迹优化问题中验证了FilterDDP的有效性。