Incorporating tube-to-tube clearances in the kinematics of concentric tube robots

Mechanics-based formulations of concentric tube robots incorporate tube bending and twisting, but do not include other phenomena that could model observed hysteretic behavior in which tube configurations reached by rotating tubes in different directions achieve different tip positions. As a step toward incorporating hysteretic tube-on-tube friction, this paper derives a model that enables computation of the contact forces applied by the tubes on each other along their lengths. To do so, it is necessary to include the small, but finite clearances between the tubes. Recasting the constrained energy minimization problem as its dual problem enables numerically efficient solution for the clearance-constrained centerlines of each tube as well as their contact forces. These variables are investigated through numerical examples and it is shown that, even without considering friction, the assumption of zero clearance can introduce tip position errors of several millimeters for clinically relevant robot lengths.