Observer Design for Needle Steering Using Task-Induced Symmetry and Reduction

Reduction induced either by the inherent dynamics of a system or control task at hand is often advantageous for analysis and control of the system. In the current work we consider state estimation for flexible, tip-steerable needle insertions. Specifically, we design observers that exploit the symmetry induced by the task of controlling the needle to a desired plane. These needles curve inside the tissue as they are inserted into it and have been modeled as a six degree of freedom nonholonomic system. Several planning and control algorithms have been proposed to automate needle insertions so as to improve their accuracy. Central to these algorithms is the estimation of the needle states (position and orientation of the needle tip) from only the needle-tip position measurements; needle-tip orientation cannot be directly measured since the needle shaft is very thin. We first show that the needle steering system is an observable system and then illustrate how the planar task decomposes the configuration space into a reduced space and a fiber space. Finally, we design an observer on the reduced space and an embedded-space observer on the fiber space.