Torsional Kinematic Model for Concentric Tube Robots

A recent approach to steerable needle design is based on combining pre-curved tubes concentrically. By rotating and extending the tubes with respect to each other, the position and orientation of the needle tip, as well as the shape of the inserted length, can be controlled. Prior models neglected torsional twisting in the curved portions of the tubes. This paper presents a mechanics model that includes torsion, applies to any number of tubes and allows curvature and stiffness to vary with arc length. While the general model is comprised of differential equations, an analytic solution is given for two tubes of constant curvature. This solution enables analytic prediction of "snap through" instability based on a single dimensionless parameter. Simulation and experiments are used to illustrate the results.