A Passive Hopper With Lossless Collisions

A passive vertical hopping robot is here highly idealised as two vertically arranged masses acted on by gravity and coupled by a linear spring. The lower mass makes dead (e = 0) collisions with the rigid ground. The equations of motion can be reduced to a one dimensional map. Fixed points of the map are found in which case the robot hops incessantly. For these conservative solutions the lower mass collides with the ground with zero impact velocity. The interval of attraction for these conservative fixed points depends on system parameters.