Needle steering systems are a topic of increasing research interest due to the many potential advantages associated with the ability to reach deep-seated targets while avoiding obstacles. Existing embodiments, such as those designed around a fixed bevel tip, are necessarily disruptive to the substrate, with the potential to cause a target to move away from the insertion trajectory, as well as potentially increasing the extent of tissue trauma at the needle interface, when compared to straight needles. To alleviate these issues, we proposed a biologically inspired design, which can steer without the need for duty-cycle spinning along the insertion axis or any active mechanisms at the tip. In this work, we demonstrate for the first time that our needle is able to steer within a deformable substrate, along with a user-defined trajectory in three-dimensional space. A simplified kinematic model is reported, which is subsequently used to design an adaptive strategy enabling the tracking of arbitrary curvatures along any given reference plane. Experimental results in gelatin are used to validate our model, as well as the performance of the controller under laboratory conditions.