Achieving elastic stability of concentric tube robots through optimization of tube precurvature

Minimally invasive surgery can involve navigating inside small cavities or reaching around sensitive tissues. Robotic instruments based on concentric tube technology are well suited to these tasks since they are slender and can be designed to take on shapes of high and varying curvature along their length. One limitation of these robots, however, is that elastic instabilities can arise when rotating one pre-curved tube inside another. While prior work has considered tubes of piecewise-constant pre-curvature, this paper proposes varying tube pre-curvature as a function of arc length as a means to enhance stability. Stability conditions for a planar tube pair are derived and used to define an optimal design problem. This framework enables solving for pre-curvature functions that achieve a desired tip orientation range while maximizing stability and respecting bending strain limits. Analytical and numerical examples of the approach are provided.