  This article investigates the problem of enforcing a virtual holonomic
constraint (VHC) on a mechanical system with degree of underactuation one while
simultaneously stabilizing a closed orbit on the constraint manifold. This
problem, which to date is open, arises when designing controllers to induce
complex repetitive motions in robots. In this paper, we propose a solution
which relies on the parameterization of the VHC by the output of a
double-integrator. While the original control inputs are used to enforce the
VHC, the control input of the double-integrator is designed to asymptotically
stabilize the closed orbit and make the state of the double-integrator converge
to zero. The proposed design is applied to the problem of making a PVTOL
aircraft follow a circle on the vertical plane with a desired speed profile,
while guaranteeing that the aircraft does not roll over for suitable initial
conditions.
