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New results on the self-stability of bicycles

Oleg Kirillov's picture

https://arxiv.org/abs/1806.03741v1

Bicycle is easy to ride but surprisingly difficult to model. Refinement of the mathematical model of a bicycle
continues over the last 150 years with contributions from Rankine, Boussinesq, Whipple, Klein, Sommerfeld, Appel, Synge and many others.
A canonical, commonly accepted nowadays model goes back to the 1899 work by Whipple. The Whipple bike is a system consisting of four rigid bodies with knife-edge wheels making it non-holonomic, i.e. requiring for its description more configuration coordinates than the number of its admissible
velocities. Due to the non-holonomic constraints even the bicycle tire tracks have a nontrivial and beautiful geometry that has deep and unexpected links to integrable systems, particle traps, and the Berry phase.

A fundamental empirical property of real bicycles is their self-stability without any control at a sufficiently high speed.
Understanding the passive stabilization is expected to play a crucial part in formulating principles of design of energy-efficient wheeled and bipedal robots.
However, the theoretical explanation of self-stability has been highly debated throughout the history of bicycle dynamics.
The reason to why ''simple questions about self-stabilization of bicycles do not have straightforward answers'' lies in the symbolical complexity of the Whipple model that contains 7 degrees of freedom and depends on 25 physical and design parameters.
In recent numerical simulations self-stabilization has been observed for some benchmark designs of the Whipple bike. These results suggested further simplification of the model yielding a reduced model of a bicycle with vanishing radii of the wheels (that are replaced by skates), known as the two-mass-skate (TMS) bicycle. Despite the self-stable TMS bike has been successfully realized in the recent laboratory experiments, the reasons for its self-stability still wait for a theoretical explanation.

In this paper, we will show how localization of complex and real exceptional points allows to find hidden symmetries in the model suggesting further reduction of the parameter space and, finally, providing explicit relations between the parameters of stability-optimized TMS bikes.

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