Torsional Dynamics and Rotational Estimation of Tip-steerable Needles

In this dissertation, we present improvements to modeling of tip-steerable needles and new estimation techniques with applications to tip-steerable needles. Previous models of tip-steerable needles assumed the angles and velocities at the tip and base of the needle were equivalent. Such assumptions were simplifying, but not accurate. Additionally, to accurately control the trajectories of tip-steerable needles, the control algorithms must have accurate estimates of the position and orientation of the needle tip. In the beginning of this dissertation, we derive the torsional dynamics of tipsteerable needles during continuous insertion. This is coupled with an existing kinematic model of tip motion to achieve a more faithful model. We describe the needle under torsion with a partial differential equation derived via the Newton force formulation. We use proper orthogonal decomposition to change coordinates to a different basis and Galerkin projection reduces this infinite dimensional system to a discrete number of ordinary differential equations in time only. We present several different practical modeling assumptions with associated controllers for attaining the objective ii