Optimizing Tube Precurvature to Enhance Elastic Stability of Concentric Tube Robots

Robotic instruments based on concentric tube technology are well suited to minimally invasive surgery since they are slender, can navigate inside small cavities and can reach around sensitive tissues by taking on shapes of varying curvature. Elastic instabilities can arise, however, when rotating one precurved tube inside another. In contrast to prior work that considered only tubes of piecewise constant precurvature, we allow precurvature to vary along the tube's arc length. Stability conditions for a planar tube pair are derived and used to formulate an optimal design problem. An analytic formulation of the optimal precurvature function is derived that achieves a desired tip orientation range while maximizing stability and respecting bending strain limits. This formulation also includes straight transmission segments at the proximal ends of the tubes. The result, confirmed by both numerical and physical experiment, enables designs with enhanced stability in comparison to designs of constant precurvature.