A stability region criterion for flat-footed bipedal walking on deformable granular terrain

Achieving stable bipedal robotic walking on deformable terrain is an open and challenging problem at the intersection of robotics and physics. Ground deformation introduces underactuation; uncertainty in terrain dynamics further complicates dynamical modeling and control methods. This work provides a stability criterion for flat-footed bipedal locomotion and allows model-based control methods to function on homogeneous deformable granular terrain (e.g. sand and dirt). By characterizing static reaction forces from granular materials, in conjunction with granular resistive force theory (RFT), we model and validate a static stability region for the center of mass (CoM) projection of a biped on a granular surface. We show that this stability region approximates the admissible Zero Moment Point (ZMP) region for walking, rendering common Linear Inverted Pendulum Model (LIPM) methods valid with our foot placement strategy. By interpreting the stability region as the maximum reaction moment of the terrain, we formulate walking as a hybrid dynamical system and utilize the partial hybrid zero dynamics (PHZD) based methodology to generate walking gaits. Finally, we experimentally validate both the ZMP and PHZD walking gaits on a planar bipedal robot, showing that the stability region criterion permits stable dynamic walking on homogeneous granular terrain.