There exists a class of nonlinear systems which fail to have a well-defined relative degree but have a robust relative degree. We have removed the full relative degree assumption which the previous results required, and have provided a local state observer for nonlinear systems that have robust relative degree γ n and have detectability property in some sense. The proposed observer utilizes the coordinate change which transforms the system into an approximate normal form. Using the proposed method, we constructed an observer for the ball and beam system on a vibrating frame. Simulation results reveal that substantial improvement in the performance is achieved compared with other local observers.