Transverse Anisotropy Stabilizes Concentric Tube Robots

Abstract

We propose a new method for improving the stability of concentric tube robots. Prior work sought to improve stability through laser-cut patterns that reduced <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$EI/GJ$</tex-math></inline-formula> , the ratio of flexural rigidity to torsional rigidity, but this strategy has always entailed a steep trade-off in overall robot stiffness. Instead, we show that stability can be improved much more efficiently by allowing transverse anisotropy (direction-dependent flexural rigidity, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$EI_x\ne EI_y$</tex-math></inline-formula> ). We provide a generalized robot model with this property and give its associated stability criterion. In the two-tube case, this provides a new version of the well-known <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$EI/GJ$</tex-math></inline-formula> criterion, now generalized to reveal the independent effects of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$EI_x$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$EI_y$</tex-math></inline-formula> . It shows that the most efficient strategy to improve stability while preserving stiffness is to reduce the flexural rigidity about the axis of precurvature while maintaining high off-axis flexural rigidity and torsional rigidity. We validate the approach with a balanced-stiffness pair of laser-machined Nitinol tubes, demonstrating model accuracy and significant stability improvement while requiring less stiffness reduction than prior methods.