Accurate closed-loop control of continuum manipulators requires integration of both models that describe their motion and methods to evaluate manipulator shape. This work presents a model that approximates the continuous shape of a continuum manipulator by a serial chain of rigid links, connected by flexible rotational joints. This rigid-link model permits a description of manipulator shape under different loading conditions. A kinematic controller, based on the manipulator Jacobian of the proposed rigid-link model, is implemented and realizes trajectory tracking, while using the kinematic redundancy of the manipulator to perform a secondary task of avoiding obstacles. The controller is evaluated on an experimental testbed, consisting of a planar tendon-driven continuum manipulator with two bending segments. Fiber Bragg grating (FBG) sensors are used to reconstruct 3-D manipulator shape, and is used as feedback for closed-loop control of the manipulator. Manipulator steering is evaluated for two cases: the first case involving steering around a static obstacle and the second case involving steering along a straight path while avoiding a moving obstacle. Mean trajectory tracking errors are 0.24 and 0.09 mm with maximum errors of 1.37 and 0.52 mm for the first and second cases, respectively. Finally, we demonstrate the possibility of FBG sensors to measure interaction forces, while simultaneously using them for shape sensing.