In order to perform highly dynamic and agile maneuvers, legged robots typically spend time in underactuated domains (e.g. with feet off the ground) where the system has limited command of its acceleration and a constrained amount of time before transitioning to a new domain (e.g. foot touchdown). Meanwhile, these transitions can have instantaneous, unbounded effects on perturbations. These properties make it difficult for local feedback controllers to effectively recover from disturbances as the system evolves through underactuated domains and hybrid impact events. To address this, we utilize the fundamental solution matrix that characterizes the evolution of perturbations through a hybrid trajectory and its 2-norm, which represents the worst-case growth of perturbations. In this paper, the worst-case perturbation analysis is used to explicitly reason about the tracking performance of a hybrid trajectory and is incorporated in an iLQR framework to optimize a trajectory while taking into account the closed-loop convergence of the trajectory under an LQR tracking controller. The generated convergent trajectories are able to recover more effectively from perturbations, are more robust to large disturbances, and use less feedback control effort than trajectories generated with traditional optimization methods.

翻译：为了实现高度动态和敏捷的动作，四足机器人通常需要在欠驱动阶段（例如，脚离开地面）停留一段时间，此时系统对加速度的控制能力有限，并且在新阶段（例如，脚触地）过渡前受限于一定的时间。同时，这些过渡可能对扰动产生瞬时、无界的影响。这些特性使得局部反馈控制器难以在系统经历欠驱动阶段和混合冲击事件时有效恢复扰动。为此，我们利用基本解矩阵来描述混合轨迹中扰动的演化过程及其2-范数，该2-范数表征扰动的最坏情况增长。本文采用最坏情况扰动分析来显式推理混合轨迹的跟踪性能，并将其融入iLQR框架中，以在考虑LQR跟踪控制器下轨迹的闭环收敛性的同时优化轨迹。与采用传统优化方法生成的轨迹相比，生成的收敛性轨迹能够更有效地恢复扰动，对较大扰动具有更强的鲁棒性，并且所需反馈控制努力更少。