The paper treats the model-based control of a nonlinear system composed of a continuum mechanism currently used as a neck of a humanoid robot. The structural flexibility of the continuum mechanism allows for compliant Cartesian motion of the link side, i.e. the head of the humanoid. The position control approach proposed in this work uses a reduced nonlinear model of the system. This reduced model consists of the rigid body dynamics and an experimentally obtained approximation of the nonlinear Cartesian spring characteristics of the continuum with multivariate polynomials. Two model-based controllers are set up, namely a partial feedback linearization and a passivity-based approach. As the system is underactuated, the six generalized coordinates are split up into four task and two remaining coordinates. The two proposed controllers are designed to regulate the task coordinates to desired positions. The proposed control approaches are tested and compared in simulation and experiments on the real system.