| 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 tip-steerable 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 of control to a plane. We conclude by giving the results of physical experiments for one of the simplified models.;The latter portion of the dissertation develops new methods of observing the full attitude of the tip of the needle given the limited measurement of only the bearing of the needle tip. The contribution herein consists of an observer described as a matrix differential equation on the manifold SO(3) directly, rather than an observer for a coordinatization or embedding. Coordinatizations and embeddings necessarily introduce geometric and kinematic singularities, which this observer avoids. This new class of space-preserving observers are called invariant observers. We give an exact proof of almost global convergence of the estimator---the estimator will converge to the true orientation for all but a set of initial conditions of measure zero given a mild constraint of persistent excitation. We conclude by discussing the possibilities of extending the invariant observer to an optimal estimator, in the sense of Kalman filtering minimization of error covariance. |
