3D balance in legged locomotion

Abstract

This paper explores the notion that the motion of dynamically stable 3D legged systems can be decomposed into a planar part that accounts for large leg and body motions that provide locomotion, and an extra-planar part that accounts for subtle corrective motions that maintain planarity. The large planar motions raise and lower the legs to achieve stepping, and they propel the system forward. The extra-planar motions ensure that the legged system remains in the plane. A solution of this form is simple because 3D dynamics do not play an important role.We develop a model of a 3D one legged hopping machine that incorporates a springy leg of non-zero mass and a two axis hip. The hopping machine is modeled as an open loop linkage that has different configurations in flight and in stance. Behavior at transitions between phases is calculated by invoking conservation of momentum. We have decomposed control of the model into four parts that control hopping height, forward velocity, body attitude, and spin. Hopping height is controlled by regulating vertical energy. Velocity is controlled by placing the foot fore and aft during flight. Body attitude is controlled by torquing the hip during stance. Spin is controlled by placing of the foot outside the plane of motion. Simulation data are presented which show that these control algorithms result in good control of velocity, body attitude and spin, while traveling along a straight desired path.