  This paper is concerned with the problem of estimating (interpolating and
smoothing) the shape (pose and the six modes of deformation) of a slender
flexible body from multiple camera measurements. This problem is important in
both biology, where slender, soft, and elastic structures are ubiquitously
encountered across species, and in engineering, particularly in the area of
soft robotics. The proposed mathematical formulation for shape estimation is
physics-informed, based on the use of the special Cosserat rod theory whose
equations encode slender body mechanics in the presence of bending, shearing,
twisting and stretching. The approach is used to derive numerical algorithms
which are experimentally demonstrated for fiber reinforced and cable-driven
soft robot arms. These experimental demonstrations show that the methodology is
accurate (<5 mm error, three times less than the arm diameter) and robust to
noise and uncertainties.
