Differential Dynamic Programming (DDP) is a popular technique used to generate motion for dynamic-legged robots in the recent past. However, in most cases, only the first-order partial derivatives of the underlying dynamics are used, resulting in the iLQR approach. Neglecting the second-order terms often slows down the convergence rate compared to full DDP. Multi-Shooting is another popular technique to improve robustness, especially if the dynamics are highly non-linear. In this work, we consider Multi-Shooting DDP for trajectory optimization of a bounding gait for a simplified quadruped model. As the main contribution, we develop Second-Order analytical partial derivatives of the rigid-body contact dynamics, extending our previous results for fixed/floating base models with multi-DoF joints. Finally, we show the benefits of a novel Quasi-Newton method for approximating second-order derivatives of the dynamics, leading to order-of-magnitude speedups in the convergence compared to the full DDP method.

翻译：微分动态规划（DDP）是近年来用于动态腿式机器人运动生成的流行技术。然而，在大多数情况下，仅使用底层动力学的第一阶偏导数，形成了iLQR方法。与完整的DDP相比，忽略二阶项往往会降低收敛速度。多段射击是另一种提高鲁棒性的常用技术，尤其在动力学高度非线性时。本研究考虑将多段射击DDP应用于简化四足模型边界步态的轨迹优化。作为主要贡献，我们开发了刚体接触动力学的二阶解析偏导数，将我们先前关于多自由度关节的固定/浮动基模型结果进行了扩展。最后，我们展示了一种新颖的拟牛顿方法在近似动力学二阶导数时的优势，与完整DDP方法相比，该方法在收敛速度上实现了数量级的提升。