  This paper considers a combination of actuation tendons and measurement
strings to achieve accurate shape sensing and direct kinematics of continuum
robots. Assuming general string routing, a methodical Lie group formulation for
the shape sensing of these robots is presented. The shape kinematics is
expressed using arc-length-dependent curvature distributions parameterized by
modal functions, and the Magnus expansion for Lie group integration is used to
express the shape as a product of exponentials. The tendon and string length
kinematic constraints are solved for the modal coefficients and the
configuration space and body Jacobian are derived. The noise amplification
index for the shape reconstruction problem is defined and used for optimizing
the string/tendon routing paths, and a planar simulation study shows the
minimal number of strings/tendons needed for accurate shape reconstruction. A
torsionally stiff continuum segment is used for experimental evaluation,
demonstrating mean (maximal) end-effector absolute position error of less than
2% (5%) of total length. Finally, a simulation study of a torsionally compliant
segment 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.
