such that $| \Phi (s) | < \pi/2$ for all $s \in [\Re]_T$, and then stabilize the associated virtual constraint manifold. In [@HauSacFre04], Hauser-Saccon-Frezza find “bounded roll trajectories,” i.e., controlled trajectories of  along which the roll angle $\varphi$ is bounded in the interval $(-\pi/2,\pi/2)$. In our context, each VHC $\varphi = \Phi(s)$ provides a [*family*]{} of bounded roll trajectories. Once $\Gamma$ has been made invariant via feedback control, bounded roll trajectories can be obtained by picking arbitrary $(s_0,\dot s_0) \in
[\Re]_T \times \Re$, and picking as initial condition in , $(\varphi(0),s(0)) = (\Phi(s_0),s_0)$, $(\dot
\varphi(0),\dot s(0)) = (\Phi'(s_0)\dot s_0,\dot s_0)$. The resulting solution $(q(t),\dot q(t))$ will satisfy $\varphi(t) = \Phi(s(t))$, implying that the roll angle trajectory $\varphi(t)$ is bounded in the interval $(-\pi/2,\pi/2)$.

The virtual constraint generator {#sec:vcg}
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In this section we show that VHCs for  can be generated as solutions of a first-order differential equation, which we call the [*VHC generator.*]{} This idea was first presented in our previous work [@ConMag10_1]. We begin with a sufficient condition for a relation $\Phi$ to be a VHC for .

\[lem:feasibility\] A relation $\varphi = \Phi(s)$, where $\Phi:[\Re]_T \to S^1$ is smooth, is a VHC for system  if $$\label{eq:feasibility}
(\forall s 