Limit Cycle Gaits
Limit cycle walking is an approach to control of legged locomotion robots not as a robot arm fixed on the floor but as a limit cycle with impulse effects and is known as the most useful way to achieve energy-efficient robotic dynamic walking. This chapter mainly describes the mathematical basis for modeling of planar walkers, and introduces several major approaches to generation of limit cycle gaits on level ground. First, a model of a planar fully actuated compass-like biped robot with flat feet is introduced. The 2-DOF equation of motion for the single-support phase is developed according to the Lagrange’s method. The 6-DOF collision equation corresponding to the extended generalized coordinates is also developed on the assumption that the rear leg leaves the ground immediately after landing of the fore leg. Second, several methods for generating energy-efficient limit cycle gaits are introduced. Different biped models are considered depending on the methods, but the idea of restoring the lost kinetic energy at impact efficiently is common to all. With the introduction of these methods, energy efficiency calculation method and output-following control law are also explained. Third, the principles and analysis methods of limit cycle stability recently developed are described using passive and underactuated rimless wheel models. The limit cycle gait is treated as a linear time-invariant system with instantaneous state jumps, and the gait stability can be analytically determined without conducting numerical simulations.
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