A model for concentric tube continuum robots under applied wrenches

Continuum robots made from telescoping precurved elastic tubes enable base-mounted actuators to specify the curved shapes of robots as thin as standard surgical needles. While free space beam mechanics-based models of the shape of these `active cannulas´ exist, current models cannot account for external forces and torques applied to the cannula by the environment. In this paper we apply geometrically exact beam theory to solve the statics problem for concentric-tube continuum robots. This yields the equivalent of forward kinematics for an active cannula with general tube precurvature functions and arbitrarily many tubes, under loading from a general wrench distribution. The model achieves average experimental tip errors of less than 3 mm over the workspace of a prototype active cannula subject to various tip forces.