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This paper considers a combination of actuation tendons and measurementstrings to achieve accurate shape sensing and direct kinematics of continuumrobots. Assuming general string routing, a methodical Lie group formulation forthe shape sensing of these robots is presented. The shape kinematics isexpressed using arc-length-dependent curvature distributions parameterized bymodal functions, and the Magnus expansion for Lie group integration is used toexpress the shape as a product of exponentials. The tendon and string lengthkinematic constraints are solved for the modal coefficients and theconfiguration space and body Jacobian are derived. The noise amplificationindex for the shape reconstruction problem is defined and used for optimizingthe string/tendon routing paths, and a planar simulation study shows theminimal number of strings/tendons needed for accurate shape reconstruction. Atorsionally stiff continuum segment is used for experimental evaluation,demonstrating mean (maximal) end-effector absolute position error of less than2% (5%) of total length. Finally, a simulation study of a torsionally compliantsegment demonstrates the approach for general deflections and string routings.We believe that the methods of this paper can benefit the design process,sensing and control of continuum and soft robots.

Lie Group Formulation and Sensitivity Analysis for Shape Sensing of Variable Curvature Continuum Robots with General String Encoder Routing