Subriemannian metric induced by single-input locally accessible control-affine systems

A subriemannian metric, defined by a distribution and an inner product on the distribution, on the state space of single-input locally accessible control systems is constructed based on the vector fields of the control system and their higher order derivatives. Conditions for the existence of such a metric are derived based on a feedback invariant quantity and a normal form for this class of control systems is provided. The construction is illustrated in a simple car control problem.