0-Step Capturability, Motion Decomposition and Global Feedback Control of the 3D Variable Height-Inverted Pendulum

One common method for stabilizing robots after a push is the Instantaneous Capture Point, however, this has the fundamental limitation of assuming constant height. Although there are several works for balancing bipedal robots including height variations in 2D, the amount of literature on 3D models is limited. There are optimization methods using variable Center of Pressure (CoP) and reaction force to the ground, although they do not provide the physical region where a robot can step and require a precomputation for the analysis. This work provides the necessary and sufficient conditions to maintain balance of the 3D Variable Height Inverted Pendulum (VHIP) with both, fixed and variable CoP. We also prove that the 3D VHIP with Fixed CoP is the same as its 2D version, and we generalize controllers working on the 2D VHIP to the 3D VHIP. We also show the generalization of the Divergent Component of Motion to the 3D VHIP and we provide an alternative motion decomposition for the analysis of height and CoP strategies independently. This allow us to generalize previous global feedback controllers done in the 2D VHIP to the 3D VHIP with a Variable CoP.