This paper is motivated by the problem of asymptotically stabilizing invariant sets in the state space of control systems by means of output feedback. The sets considered are smooth embedded in submanifolds and the class of system is nonlinear, finite-dimensional, autonomous, deterministic, single-input, and control-affine. Given an invariant set and a control system with fixed output, necessary and sufficient conditions are presented for feedback equivalence to a normal form that facilities the design of output feedback controllers that stabilize the set using existing design techniques.