Controllability of a bicycle

The bicycle is an intriguing machine as it is laterally unstable at low speed and stable, or easy to stabilize, at high speed. During the last decade a revival in the research on dynamics and control of bicycles has taken place [1]. Most studies use the so-called Whipple [2] model of a bicycle. In this model a rigid rider is rigidly connected to the rear frame. However, from experience it is known that some form of control is required to stabilize the bicycle and/or carry out tracking operations. This control is either carried out by steering or by performing some sort of upper body motions. Note that in both cases the system is underactuated. The precise control used by the rider is still under study [3]. This paper addresses the question whether the underactuated bicycle is controllable by only steering or upper body motion in the forward speed range of 0 to 36 km/h. Whipple-like models are studied with either steering or upper body control. It is shown that at certain specific forward speeds some modes are uncontrollable. However, either the forward speed is extremely low or the uncontrollable modes are all stable modes and are therefore of no concern to the rider.