the fact that VHCs are enforced through forces produced by feedback control. For underactuated systems such as the constrained Getz’s bicycle model in , the control forces produce work. ]{}

[In the context of walking robots, [@WesGriKod03] proved that under certain conditions, the hybrid zero dynamics subject to a VHC exhibit an exponentially stable hybrid limit cycle. The mechanism enabling this exponential stability property is the jump map representing the impact of the robot’s foot with the ground. Proposition \[prop:internal\_dynamics\_general\_properties\] shows that VHCs may induce stable limit cycles even when the control system has no jumps.]{}

Consider the differential equation $$\label{eq:constrained_system_lifted}
\ddot x = \tilde \Psi_1 (x) + \tilde \Psi_2(x) \dot x^2.$$ Let $\bar \pi: \Re \times \Re \to [\Re]_T \times \Re$ be defined as $\bar \pi(x,\dot x) = (\pi(x),\dot x)$. Then, it is readily seen that the vector fields in  and  are $\bar
\pi$-related [@Lee13]. It follows that if $(x(t),\dot x(t))$ is a solution of , then $(s(t),\dot
s(t)) = (\pi(x(t)),\dot x(t))$ is a solution of . We will show that there exists a smooth $T$-periodic function $\tilde \nu : \Re \to (0,+\infty)$ such that the set $\tilde \cR = \{(x,\dot x) \in \Re^2: \dot x = \tilde
\nu(x)\}$ 