Existence and stability of a limit cycle in the model of a nonuniform rimless wheel rolling down a slope

Abstract

The rimless wheel model appears in engineering literature as a zero dynamics reduction of the model of a biped walking down a slope. Nonuniformity in the rimless wheel (i.e. unequal lengths of spokes and unequal angles between successive spokes) can be viewed as nonuniformity of the walking terrain. Existence of a limit cycle for nonuniform rimless wheel model was established in the earlier literature but stability was addressed just briefly. The present paper formulates conditions for asymptotic stability of the limit cycle and verifies the findings with numeric simulations (Wolfram Mathematica notebook is uploaded with this paper as supplementary material).