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:不平坦的地形必然将周期性步行转化为非周期性运动。因此,传统的稳定性分析工具不能充分捕捉双足机器人在此类扰动存在下的运动能力。这促使我们需要面向稳健性的分析工具。为此,我们提出了一个新的稳健性定义,称之为$\delta$-稳健性,用于描述标称周期轨道在存在不确定地形时保持稳定的定义域。这个定义是通过将地面高度扰动作为输入到状态稳定性(ISS)的表示周期轨道的扩展Poincaré映射的扰动来推导出来的。主要的理论结果是得出了能够证明周期轨道具有$\delta$-稳健性的稳健李雅普诺夫函数的公式。这就形成了验证$\delta$-稳健性的优化框架,其中我们用一个双足机器人在不平坦的地形上行走的模拟来展示。