This article proposes a solution to the path following problem for the planar vertical take-off and landing aircraft (PVTOL) applicable to a class of smooth Jordan curves. Our solution relies on the stabilization of two nested embedded submanifolds of the state space that are defined based on the path one wishes to follow. The stabilization of these sets is performed using the ideas of transverse feedback linearization and finite-time stabilization. Our path following controller enjoys the two crucial properties of output invariance of the path (i.e., if the PVTOL's centre of mass is initialized on the path and its initial velocity is tangent to the path, then the PVTOL remains on the path at all future time) and boundedness of the roll dynamics. Further, our controller guarantees that, after a finite time, the time average of the roll angle is zero, and the PVTOL does not perform multiple revolutions about its longitudinal axis.
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