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In this paper, the problem of tip position tracking control of a rigid-flexible manipulator is investigated. The backstepping design scheme is developed for this tracking control problem. The proposed backstepping controller stabilizes the flexible robotic manipulator while driving the trajectory tracking error and the tip-deflection asymptotically to zero. As a case study, the needle insertion problem is considered where the manipulator-needle system is modeled as a rigid-flexible link system. Needle deflection and tissue deformation are major problems for accurate needle insertion. In this paper, flexible modes of the needle are estimated by means of a recurrent neural network. Simulation results are given to illustrate the efficient performance of the proposed control scheme.