Pose tracking is fundamental to achieve preciseand safe insertion of a surgical tool for minimally invasiveinterventions. In this work, a method for the estimation of thefull pose of steerable needles is presented. Our approach uses aProgrammable Bevel Tip (PBN) needle with four-segment designas a case study. A novel 3D kinematic model of the PBN isdeveloped and used to predict the full needle pose during theinsertion. The pose prediction is estimated through an ExtendedKalman Filter using the position measurements provided byan electromagnetic sensor located at each tip of the needlesegments. The method estimates also the torsion of the needleshaft that can arise over the insertion of the needle becauseof the shear forces exerted between the needle and the insertionmedium. The feasibility of the proposed solution was validated ina number of experiments in gelatin demonstrating a small errorin position reconstruction (RMSE<0.6mm) and good accuracy incomparison to a bespoke geometric pose reconstruction method.