Vertically transverse functions as an extension of the transverse function control approach for second-order systems

In recent joint work, P. Morin and C. Samson have advocated the use of transverse functions on tori to design practical stabilizers (even asymptotic stabilizers whenever possible) and tracking controllers for driftless, control-affine systems. By design, those stabilizers and controllers retain their performance even in the presence of known disturbance drift terms and, moreover, they exhibit additional robustness features that derive from their smoothness in terms of the state. As yet, however, no systematic procedure has been devised to address the same problem in the case of systems whose drift vector fields are required to ensure local accessibility, including, for instance, second-order and simple mechanical systems. In this paper we show that one way to extend the usefulness of transverse functions to the realm of second-order systems is by taking their associated tangent mappings, a process that results in "vertically transverse functions." We believe that, although work remains to be done in this direction, this extension provides an initial step towards a more thorough theory that addresses both practical stabilization and tracking for such systems. Included is an example that illustrates how these ideas may be put to use for control purposes.