0$, we obtain the feedback making $\Gamma$ invariant $$\label{eqn_inv_feedback}
\begin{aligned}
& u = \frac{h^{-1}g \sin \Phi}{\delta} - \frac{ \dot s^2
}{\delta} \Big[ \Phi'' + \frac 1 h ( (1-h \kappa \sin \Phi) \kappa
  \|\sigma'\| \\
& + b \kappa' +b\kappa {\sigma'}\trans\sigma'' / \|\sigma'\|^2)
  \|\sigma'\| \cos \Phi \Big].
\end{aligned}$$ Substituting feedback  in the $s$ dynamics, we get the reduced dynamics on $\Gamma$ $$\label{eq:motion_on_Gamma}
\ddot s = \Psi_1(s) + \Psi_2(s) \dot s^2,$$ with $$\label{eq:Psi}
\begin{aligned}
&\Psi_1 = \frac{h^{-1}g s_{\Phi}}{\delta} \\
& \Psi_2= - \frac{1}{\delta} \Big[ \Phi'' + \frac 1 h ( (1-h
    \kappa s_{\Phi}) \kappa\|\sigma'\| \\ & + b \kappa'
    + b \kappa \sigma'{}\trans \sigma'' / \|\sigma'\|^2 )
    c_{\Phi} \|\sigma'\| \Big].
\end{aligned}$$ System  describes the motion of system  on the constraint manifold $\Gamma$ in the following sense. For a given initial condition $(s(0),\dot s(0)) =
(s_0,\dot s_0) \in [\Re]_T \times \Re$, let $(s(t),\dot s(t))$ be the corresponding solution of , and let $(\varphi(t),\dot \varphi(t), s_1(t),\dot s_1(t))$ be the solution of  with initial condition $(\varphi(0),\dot
\varphi(0),s(0),\dot s(0)) = ( \Phi(s_0), \Phi'(s_0) \dot s_0,s_0,\dot
s_0) \in \Gamma$. Then, for all $t \geq 0$ $(s_1(t),\dot s_1 (t)) =
(s(t),\dot s(t))$, and $(\varphi(t),\dot \varphi(t)) =$ $(\Phi(s(t)),$ $\Phi'(s(t)) \dot s(t))$. In order for the VHC $\varphi=\Phi(s)$ to 