Closed-Form Differential Kinematics for Concentric-Tube Continuum Robots with Application to Visual Servoing

Active cannulas, so named because of their potential medical applications, are a new class of continuum robots consisting of precurved, telescoping, elastic tubes. As individual component tubes are actuated at the base relative to one another, an active cannula changes shape to minimize stored elastic energy. For the first time, we derive the differential kinematics of a general n tube active cannula while accounting for torsional compliance, which can strongly affect the accuracy of robot tip pose prediction. Our derivation makes several explicit assumptions that have never been vetted for a robotic task, so we experimentally validate the Jacobian using a three-link prototype in a simple stereo visual servoing scheme. Our visual servoing experiments validate the Jacobian and also demonstrate the feasibility of using active cannulas under image guidance—a key step toward realizing their potential to reach dexterously through small, winding environments, which is of particular importance in medical applications.