is smooth if and only if $h$ is smooth. Finally, $\image(h)$ denotes the image of a function $h$.

Problem formulation {#sec:problem}
===================

Consider the bicycle model depicted in Figure \[fig:bicycle\], with the following variable conventions (taken from [@HauSacFre04]):

-   $(x,y)$ - coordinates of the point of contact of the rear wheel

-   $\varphi$ - roll angle (a positive $\dot \varphi$ implies that the bicycle leans to the right)

-   $\psi$ - yaw angle (a positive $\dot \psi$ means that the bicycle turns right)

-   $\alpha$ - projected steering angle on the $(x,y)$ plane

-   $b$ - distance between the projection of the centre of mass and the point of contact of the rear wheel

-   $p$ - wheel base

-   $h$ - pendulum length

-   $v$ - forward linear velocity of the bicycle

-   $f$ - thrust force.

We denote $\bar \kappa = (\tan \alpha) / p = \dot \psi / v$. For a given velocity signal $v(t)$ and steering angle signal $\alpha(t)$, $\bar \kappa(t)$ represents the curvature of the path $(x(t),y(t))$ traced by the point of contact of the rear wheel. In [@GetMar95], the bicycle of Figure \[fig:bicycle\] was modelled by writing the Lagrangian of the unconstrained bicycle, incorporating 