Uneven terrain necessarily transforms periodic walking into a non-periodic motion. As such, traditional stability analysis tools no longer adequately capture the ability of a bipedal robot to locomote in the presence of such disturbances. This motivates the need for analytical tools aimed at generalized notions of stability -- robustness. Towards this, we propose a novel definition of robustness, termed \emph{$\delta$-robustness}, to characterize the domain on which a nominal periodic orbit remains stable despite uncertain terrain. This definition is derived by treating perturbations in ground height as disturbances in the context of the input-to-state-stability (ISS) of the extended Poincar\'{e} map associated with a periodic orbit. The main theoretic result is the formulation of robust Lyapunov functions that certify $\delta$-robustness of periodic orbits. This yields an optimization framework for verifying $\delta$-robustness, which is demonstrated in simulation with a bipedal robot walking on uneven terrain.

翻译：不平整地形必然将周期步行运动转化为非周期运动。因此，传统稳定性分析工具不再能充分描述双足机器人在此类扰动下行走的能力。这促使我们亟需面向广义稳定性概念（即鲁棒性）的分析工具。为此，我们提出一种新颖的鲁棒性定义——称为"$\delta$-鲁棒性"——用于刻画标称周期轨道在地形不确定情况下仍能保持稳定的定义域。该定义将地面高度扰动视为扩展庞加莱映射输入状态稳定性中的扰动，而该映射与周期轨道相关联。主要理论成果是建立了可验证周期轨道$\delta$-鲁棒性的鲁棒李雅普诺夫函数，进而构建出验证$\delta$-鲁棒性的优化框架，并通过双足机器人在不平整地形行走的仿真实验进行验证。