  The task of inducing, via continuous static state-feedback, an asymptotically
stable heteroclinic orbit in a nonlinear control system is considered in this
paper. The main motivation comes from the problem of ensuring convergence to a
so-called point-to-point maneuver in an underactuated mechanical system, that
is, to a smooth curve in its state--control space that is consistent with the
system dynamics and which connects two stabilizable equilibrium points. The
proposed method uses a particular parameterization, together with a state
projection onto the maneuver's orbit as to combine two linearization techniques
for this purpose: the Jacobian linearization at the equilibria on the
boundaries and a transverse linearization along the orbit. This allows for the
computation of stabilizing control gains offline by solving a semidefinite
programming problem. The resulting nonlinear controller, which simultaneously
asymptotically stabilizes both the orbit and the final equilibrium, is
time-invariant, locally Lipschitz continuous, requires no switching and has a
familiar feedforward plus feedback--like structure. The method is also
complemented by synchronization function--based arguments for planning such
maneuvers for mechanical systems with one degree of underactuation. Numeric
simulations of the non-prehensile manipulation task of a ball rolling between
two points upon the "butterfly" robot demonstrates the efficacy of the full
synthesis.
