generator*]{}. We show that if the path is sufficiently long compared to the height of the bicycle’s centre of mass and the wheel base, then the VHC can be chosen so that on the constraint manifold the bicycle traverses the entire curve with bounded speed, and its speed profile is periodic in steady-state. Finally, we design a controller that enforces the VHC, and recovers the asymptotic properties of the bicycle on the constraint manifold.

![Getz’s bicycle model.[]{data-label="fig:bicycle"}](FIGURES/bicycle){width=".5\textwidth"}

The concept of VHC is a promising paradigm for motion control. It is one of the central ideas in the work of Grizzle and collaborators on biped locomotion (e.g., [@PleGriWesAbb03] and [@WesGriKod03]), where VHCs are used to encode different walking gaits. The work of Shiriaev and collaborators in [@Can04; @ShiPerWit05; @FreRobShiJoh08] investigates VHCs for Lagrangian control systems, i.e., systems of the form [@Blo03] $$\label{eq:lagrangian}
\frac{d}{dt} \frac{\partial L}{\partial \dot q} - \frac{\partial
  L}{\partial q} = G u,$$ with control input $u$ and smooth Lagrangian $L(q,\dot q) = (1/2) \dot
q\trans D(q) \dot q - V(q)$, with $D = D\trans$ positive definite. In [@ShiPerWit05], the authors consider systems of the form  with degree of underactuation one. They find an integral of motion for the constrained 