curve $\C$ in one direction, i.e., there exists $\bar t>0$ such that $| \dot s(t) | >0$ for all $ t \geq \bar t$. Moreover, the speed $\dot s(t)$ of the bicycle on $\C$ should remain bounded.

Our solution of this problem relies on the notion of VHC.

A [**virtual holonomic constraint (VHC)**]{} for system  is a relation $\varphi = \Phi(s)$, where $\Phi: [\Re]_T \to S^1$ is smooth and the set $\Gamma = \{(q,\dot q)\in \cX: \varphi = \Phi(s), \ \dot \varphi = \Phi'(s)
\dot s\}$ is controlled invariant. That is, there exist a smooth feedback $u(q,\dot q)$ such that $\Gamma$ is invariant for the closed-loop system. The set $\Gamma$ is called the [**constraint manifold**]{} associated with the VHC $\varphi = \Phi(s)$.

The definition above formalizes the notion of VHC used in [@WesGriKod03] in the context of biped locomotion. The constraint manifold $\Gamma$ is a two-dimensional submanifold of $\cX$ parametrized by $(s,\dot s)$, and therefore diffeomorphic to the cylinder $[\Re]_T \times \Re$. It is the collection of all those phase curves of  that can be made to satisfy the constraint $\varphi = \Phi(s)$ via feedback control. In order to solve the maneuvering problem, we look for VHCs $\varphi = \Phi(s)$ 