Stiff yet Bendy: Tubular Transmissions for Driving Surgical Robots through Flexible Endoscopes

Concentric push-pull robots delivered through flexible endoscopes work best if their laser-cut transmission tubes have high axial stiffness, high torsional stiffness, and low bending stiffness. This paper simultaneously addresses all three output stiffness values in the transmission design problem, explicitly considering axial stiffness, whereas prior work on laser-cut tube design has focused on the bending/torsional stiffness ratio. We demonstrate an inherent trade-off present in existing laser-cut patterns: it is difficult to simultaneously achieve high axial stiffness and low bending stiffness because these properties are very tightly correlated. To break this correlation and design all three stiffness independently, we propose a new type of laser material removal pattern that leverages local stiffness asymmetry ( $E{I}_x\ne E{I}_y$ ) in discrete bending segments separated by segments of solid tube. These discrete asymmetric segments are then rifled down the tube to achieve global stiffness symmetry. We parameterize the design and provide a study of the properties through finite-element analysis. We also consider the effect of interference between the tubes when the discrete segments are not aligned. Results show that our discrete asymmetric segment concept can achieve high axial stiffness and torsional stiffness better than previously suggested laser patterns while maintaining equally low bending stiffness. We also experimentally validated the proposed design's properties and actuation performance with professionally manufactured prototype Nitinol tubes for use in an endoscopic robot system.