  The equilibrium shape of a continuum robot is resulted from both its internal
actuation and the external physical interaction with a surrounding environment.
A fast and accurate shape estimation method (i) can be used as a feedback to
compensate for more accurate motion; and (ii) can reveal rich information about
physical interactions (e.g. instrument-anatomy contacts / forces during a
surgery). From a prior work that demonstrated an offline calibration of
continuum robots, we adopt its shape modal representation and error propagation
models that include identification Jacobians. In this work, we present an
iterative observer approach to enable online shape estimation. We develop a
dual Extended Kalman Filter (EKF) to estimate both the robot state and the
shape modal parameters. The dual EKF provides robust estimation on (i) the
configuration space variables that are controllable and driven by internal
actuation; and (ii) the modal coefficients representing homotopies of shape
families that are governed by the physical interactions with the environment.
We report results from simulation studies in this work, and plan to investigate
methods in the future to use the proposed approach for predicting physical
interactions.
