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This article investigates the problem of enforcing a virtual holonomicconstraint (VHC) on a mechanical system with degree of underactuation one whilesimultaneously stabilizing a closed orbit on the constraint manifold. Thisproblem, which to date is open, arises when designing controllers to inducecomplex repetitive motions in robots. In this paper, we propose a solutionwhich relies on the parameterization of the VHC by the output of adouble-integrator. While the original control inputs are used to enforce theVHC, the control input of the double-integrator is designed to asymptoticallystabilize the closed orbit and make the state of the double-integrator convergeto zero. The proposed design is applied to the problem of making a PVTOLaircraft follow a circle on the vertical plane with a desired speed profile,while guaranteeing that the aircraft does not roll over for suitable initialconditions.

Dynamic Virtual Holonomic Constraints for Stabilization of Closed Orbits in Underactuated Mechanical Systems