3D bipedal walking with knees and feet: A hybrid geometric approach

Abstract

Motivated by the goal of obtaining moreanthropomorphic walking in bipedal robots, this paper considers a hybrid model of a 3D hipped biped with feet and locking knees. The main observation of this paper is that functional Routhian Reduction can be used to extend twodimensional walking to three dimensions-even in the presence of periods of underactuation-by decoupling the sagittal and coronal dynamics of the 3D biped. Specifically, we assume the existence of a control law that yields stable walking for the 2D sagittal component of the 3D biped. The main result of the paper is that utilizing this controller together with "reduction control laws" yields walking in three dimensions. This result is supported through simulation.

Extending two-dimensional human-inspired bipedal robotic walking to three dimensions through geometric reduction

“…The first controller shapes the Lagrangian [32, Thm. 1] to into an almost-cyclic Lagrangian [32,Eq. (10)] which is amenable to reduction.…”

“…FUNCTIONAL ROUTHIAN REDUCTION Classical Routhian reduction [30] is a type of geometric reduction which leverages conserved momentum in the form of a constant to eliminate cyclic variables in a dynamical system. Functional Routhian reduction (first introduced in [31]) is a specific type of geometric reduction used in the presence of external forces (see [32]) in which the momentum map is set to a function rather than a constant. This reduction will decouple the sagittal and coronal dynamics; then, stable walking is obtained in three dimensions by applying control laws which give stable walking in the sagittally-restricted counterpart whose Lagrangian takes the structure of a functional Routhian [32,Eq.…”

Speed regulation in 3D robotic walking through motion transitions between Human-Inspired partial hybrid zero dynamics

“…Restrictions are imposed via control which render both feet flat throughout the gait; for non flat-foot models, more complex hybrid systems must be considered [7], [8]. The configuration space, QR, of the system is given in coordi nates by:…”

“…These ideas have been used to design passivity-based control laws in 2D [5], [6]. These 2D control laws have been extended to 3D through geometric reduction [7], [8], yet these methods have only recently been realized experimentally [9]. Therefore, there exists a gap between formal methods and experimental realization for three-dimensional robotic walking.…”

Planar multi-contact bipedal walking using hybrid zero dynamics

“…Capture point has been successfully realized experimentally on the humanoid M2V2 [17] both walking and push recovery. Other more mathematically formal methods that have been developed more recently include geometric reduction [6], [20], control symmetries [21], and hybrid zero dynamics [2], [6]. Hybrid zero dynamics based control has seen great success on multiple platforms such as RABBIT [5], MABEL [22], ATRIAS [7], AMBER 1 [18], and AMBER 2 [15].…”