The right way to ride the wrong bike: An exploration of Klein’s ‘unridable’ bicycle

Abstract

Professor Richard Klein and his students built a bicycle with a rather interesting feature: no one was able to ride it. A prize was offered. Hundreds of students and cycling enthusiasts attempted it. Years passed, and the prize money grew. Klein’s rear-steered bicycle became a canonical example of how non-minimum phase systems can be difficult and sometimes nearly impossible to control. It has been lauded as a particularly effective educational example in which students can experience the loss of controllability in a seemingly simple, albeit unorthodox bicycle. The primary result of the work reported here is a demonstration that it is possible for a human of modest athletic ability to ride Klein’s unridable bicycle, to keep it balanced, and to control its direction of travel. There is a secret to riding Klein’s rear-steer bicycle. The secret is revealed through an exploration of the dynamics and control of the bike that contains three elements: (1) modeling the physics of the actively steered bicycle as an inverted pendulum riding atop a carriage; (2) recognizing that the steer kinematics leads to competing physical mechanisms which an aspiring rider might exploit; and (3) examining limitations of controllability and stabilizability of the system from a state space perspective. From this vantage point, one can devise a novel strategy, based on a component of lateral acceleration that dominates at low speed, for riding the so-called “unridable” bike and solving Klein’s puzzle. The work adds a new chapter on the dynamics and control of the rear-steered bicycle, a problem of academic interest.