Mechanics-based modeling of bending and torsion in active cannulas

Active cannulas are meso-scale continuum robots, enabling dexterity in diameters from hundreds of microns to tens of centimeters. Constructed from telescoping, concentric, precurved, superelastic tubes, they exhibit ldquosnake-likerdquo dexterity with a form factor similar to a needle, making them well-suited for applications in minimally invasive surgery. Such applications are facilitated by an accurate kinematic model. The accuracy of prior models has been limited by the assumption of infinite torsional rigidity beyond initial straight transmissions. In this paper, we consider both bending and torsion, describing the total elastic energy stored in two curved concentric tubes. We apply relevant constraints to reduce the energy integral to a classical variational form, in terms of a single unknown function relating tube torsion angles. We then determine the function that minimizes the stored energy analytically, which yields the shape of the active cannula. Experiments demonstrate that this framework can be used predict the shape of a 3-link active cannula more accurately than previous models, reducing tip error by 72% over a bending-only model, and 35% over a model that includes only transmissional torsional. An implication of our work is that the general shape of an active cannula not piecewise constant curvature (as basic assumptions of previous models have implied), but that under certain conditions it may approximate piecewise constant curvature.