Hopping in legged systems — Modeling and simulation for the two-dimensional one-legged case

A two-dimensional one-legged hopping machine is modeled and simulated in order to better understand legged systems that hop and run. The analysis is focused on balance, dynamic stability, and resonant oscillation for the planar case. A springy leg with nonzero mass, a simple body, and an actuated hinge-type hip are incorporated in the model. Control of the model is decomposed into a vertical hopping part, a horizontal velocity part, and a body attitude part. Estimates of total system energy are used in regulating hopping height in order to initiate hopping, to maintain level hopping, to change from one hopping height to another, and to terminate hopping. Balance and control of forward velocity are explored with three algorithms. The feasibility of decomposing the control of running into a height control part, a forward velocity control part, and an attitude control part is verified by simulations.