the input matrix. Now consider a $C^3$ closed Jordan curve $\C$ in the $(x,y)$ plane with regular parametrization $\sigma: [\Re]_T \to \Re^2$, not necessarily unit speed. Let $\kappa:[\Re]_T \to \Re$ denote the signed curvature of $\C$. Throughout this paper, we assume the following.

\[assump\_convex\] The curve $\C$ is strictly convex, i.e., $\kappa(s) > 0$ for all $s
\in [\Re]_T$.

In this paper we investigate the dynamics of the bicycle when the point $(x,y)$ is made to move along the curve $\C$ by an appropriate choice of the steering angle. In order to derive the constrained dynamics, suppose that $( x(0), y(0) ) \in \C$, i.e., $(x(0),y(0)) =
\sigma(s_0)$, for some $s_0 \in [\Re]_T$. A point $\sigma(s(t))$ moving on $\C$ has linear velocity $$\label{eqn_for_v}
v(t) = \|\sigma'(s(t))\| \dot s(t)$$ and acceleration $$\label{eqn_for_v_dot}
\dot v(t) = \|\sigma'(s(t))\| \ddot s(t) + \frac{\dot s^2(t)}{
 \|\sigma'(s(t))\|} \sigma'(s(t))\trans \sigma''(s(t)).$$

For an arbitrary input signal $f(t)$, $(x(t), y(t))$ traverses $\C$ with velocity $v(t)$ if and only if $(x(0),y(0)) \in \C$, $(\dot
x(0),\dot y(0))$ is tangent to $\C$, $\bar \kappa(0) = \kappa(s_0)$, and the input signal $\tau(t)$ is chosen to be $\tau(t) = \kappa'(s(t)) \dot s(t)$, where $\kappa'(s(t)) =
\frac{d\kappa} {ds} (s(t))$. With this choice, we obtain $$\label{eqn_for_bar_kappa}
