This paper presents a novel approach using sensitivity analysis for generalizing Differential Dynamic Programming (DDP) to systems characterized by implicit dynamics, such as those modelled via inverse dynamics and variational or implicit integrators. It leads to a more general formulation of DDP, enabling for example the use of the faster recursive Newton-Euler inverse dynamics. We leverage the implicit formulation for precise and exact contact modelling in DDP, where we focus on two contributions: (1) Contact dynamics in acceleration level that enables high-order integration schemes; (2) Formulation using an invertible contact model in the forward pass and a closed form solution in the backward pass to improve the numerical resolution of contacts. The performance of the proposed framework is validated (1) by comparing implicit versus explicit DDP for the swing-up of a double pendulum, and (2) by planning motions for two tasks using a single leg model making multi-body contacts with the environment: standing up from ground, where a priori contact enumeration is challenging, and maintaining balance under an external perturbation.

中文翻译：

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使用隐式微分动态规划的富接触运动轨迹优化

本文提出了一种使用灵敏度分析的新方法，用于将差分动态编程（DDP）推广到以隐式动力学为特征的系统，例如通过逆动力学和变分或隐式积分器建模的系统。它导致了DDP的更一般化的表述，例如可以使用更快的递归牛顿-欧拉逆动力学。我们利用隐式公式在DDP中进行精确和精确的接触建模，其中我们集中在两个方面：（1）加速级的接触动力学实现了高阶积分方案；（2）在前向传递中使用可逆接触模型并在后向传递中使用闭式解的公式化，以提高接触的数值分辨率。