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：不平坦地形必然将周期性行走转化为非周期性运动。因此，传统稳定性分析工具难以充分描述双足机器人在此类干扰下运动的能力。这推动了针对广义稳定性概念——鲁棒性——的分析工具需求。为此，我们提出了一种新颖的鲁棒性定义，称为$\delta$-鲁棒性，用以刻画在不确定地形条件下名义周期轨道保持稳定的域。该定义通过将地面高度扰动视为扩展庞加莱映射（与周期轨道相关）在输入-状态稳定性（ISS）框架下的干扰而导出。主要理论结果是构建了验证周期轨道$\delta$-鲁棒性的鲁棒李雅普诺夫函数。这形成了一个用于验证$\delta$-鲁棒性的优化框架，并通过双足机器人在不平坦地形上行走的仿真进行了演示。