Helices, relative equilibria, and optimality on the special Euclidean group

We consider a left-invariant optimal control problem on the six-dimensional special Euclidean group. Helical trajectories are extremals of the optimal control problem, and we derive an explicit parameterization of these helices. Using this parameterization of the helical extremals, we compute the relative equilibria of the Hamiltonian system associated with the optimal control problem. For a particular choice of system parameters, we use Jacobi's sufficient condition to determine which of the relative equilibria correspond to local optima of the optimal control problem. We show that the optimality of a relative equilibrium is completely determined by the curvature and torsion of the helix that the trajectory traces.