Persistent passive hopping and juggling is possible even with plastic collisions

We describe simple one-dimensional models of passive (no energy input, no control), generally dissipative, vertical hopping and one-ball juggling. The central observation is that internal passive system motions can conspire to eliminate collisions in these systems. For hopping, two point masses are connected by a spring and the lower mass has inelastic collisions with the ground. For juggling, a lower point-mass hand is connected by a spring to the ground and an up-per point-mass ball is caught with an inelastic collision and then re-thrown into gravitational free flight. The two systems have identi-cal dynamics. Despite inelastic collisions between non-zero masses, these systems have special symmetric energy-conserving periodic motions where the collision is at zero relative velocity. Additionally, these special periodic motions have a non-zero sized, one-sided re-gion of attraction on the higher-energy side. For either very large or very small mass ratios, the one-sided region of attraction is large. These results persist for mildly non-linear springs and non-constant gravity. Although non-collisional damping destroys the periodic mo-tions, small energy injection makes the periodic motions stable, with a two-sided region of attraction. The existence of such special energy conserving solutions for hopping and juggling points to possibly use-ful strategies for both animals and robots. The lossless motions are demonstrated with a table-top experiment. KEY WORDS—hopping robot, juggling robot, passive-dynamic, collisions, efficiency, periodic, ma