  External contact force is one of the most significant information for the
robots to model, control, and safely interact with external objects. For
continuum robots, it is possible to estimate the contact force based on the
measurements of robot configurations, which addresses the difficulty of
implementing the force sensor feedback on the robot body with strict dimension
constraints. In this paper, we use local curvatures measured from fiber Bragg
grating sensors (FBGS) to estimate the magnitude and location of single or
multiple external contact forces. A simplified mechanics model is derived from
Cosserat rod theory to compute continuum robot curvatures. Least-square
optimization is utilized to estimate the forces by minimizing errors between
computed curvatures and measured curvatures. The results show that the proposed
method is able to accurately estimate the contact force magnitude (error:
5.25\% -- 12.87\%) and locations (error: 1.02\% -- 2.19\%). The calculation
speed of the proposed method is validated in MATLAB. The results indicate that
our approach is 29.0 -- 101.6 times faster than the conventional methods. These
results indicate that the proposed method is accurate and efficient for contact
force estimations.
