is the new control input. With this transformation, the motion of the bicycle when its rear wheel is made by feedback control to follow $\C$ reads as $$\label{eq:sys}
\begin{aligned}
& \ddot \varphi = h^{-1}g {s_\varphi}- h^{-1}\Big[ (1 - h \kappa(s)
    {s_\varphi})\kappa(s) \|\sigma'(s)\| + b \kappa'(s) \\
&  +\frac{b \kappa(s)}{\|\sigma'(s)\|^2} {\sigma'(s)}\trans
   \sigma''(s)\Big]{c_\varphi}\|\sigma'(s)\| \dot s^2 - a(s) {c_\varphi}u \\
& \ddot s = u,
\end{aligned}$$ where $a(s) = b h^{-1} \kappa(s) \|\sigma'(s)\|$. In the above equation, $u$ is the new control input, and it represents the acceleration of the curve parameter $s$. We will denote $\cX := \{
(q,\dot q) \in S^1 \times [\Re]_T \times \Re^2\}$ the state space of .

[If $\sigma(s)$ is a unit speed parametrization of $\C$ (i.e. it satisfies $\|\sigma'(s)\|=1$), then the first differential equation in  reduces to $h \ddot \varphi = g {s_\varphi}- \Big[ (1 - h \kappa(s) {s_\varphi})\kappa(s) + b
 \kappa'(s)\Big]{c_\varphi}\dot s^2 - b \kappa(s) {c_\varphi}u$. ]{}

[Maneuvering Problem]{} Find a feedback $u(q,\dot q)$ for system  such that there exists a set of initial conditions $\Omega$ with the property that, for all $( q(0),\dot q(0) ) \in \Omega$, the bicycle does not overturn, i.e., $|\varphi(t)| < \pi/2$ for all $t \geq 0$, and traverses the entire 