Research in minimally invasive surgery has placed great focus on percutaneous intervention, leading to a booming research interest in steerable needles. Such systems, compared with rigid needles, enjoy the advantage of following a curvilinear trajectory towards a target that lies under anatomical tissue surrounded by vessels while avoiding obstacles along the path. Amongst these, a bio-inspired programmable bevel-tip steerable needle (PBN) has been developed, which can steer in any direction in three-dimensional space without causing tissue damage due to torsion; its steering behaviour has been described by a mechanics-based model. To further investigate needle tracking performance under the newly-developed needle model, this paper presents a tailored closed-loop controller which characterizes needle movement by applying an underactuated Autonomous Underwater Vehicle (AUV) model.