a constructive methodology to find VHCs for Getz’s bicycle that meet the requirements of the maneuvering problem. In Section \[sec:constrained\_motion\] we analyze the motion of Getz’s bicycle on the virtual constraint manifold. In Proposition \[prop:internal\_dynamics\_general\_properties\] we provide a general result with sufficient conditions under which an unforced second-order system of a certain form possesses an asymptotically stable closed orbit. In Proposition \[prop:internal\_dynamics\_bicycle\] we apply this result to Getz’s bicycle model. In Section \[sec:solution\] we bring these results together and solve the maneuvering problem. Finally, in Section \[sec:numerics\] we make remarks on numerical implementation of the proposed controller.

[**Notation.**]{} Throughout this paper we use the following notation. If $x$ is a real number and $T>0$, the number $x$ modulo $T$ is denoted by $[x]_T$. We let $[\Re]_T : = \{[x]_T: x \in \Re\}$. The set $[\Re]_T$ is diffeomorphic to the unit circle. We let $\pi: \Re
\to [\Re]_T$ be defined as $\pi(t) = [t]_T$. Then, $\pi$ is a smooth covering map (see [@Lee13 p.91]). If $M$ is a smooth manifold, and $h: [\Re]_T \to M$ is a smooth function, we define $\tilde h:=h
\circ \pi : \Re \to M$. This is a $T$-periodic function. Moreover, by [@Lee13 Theorem 4.29], $\tilde h$ 