A Stochastic Framework for Continuous-Time State Estimation of Continuum Robots

State estimation techniques for continuum robots (CRs) typically involve using computationally complex dynamic models, simplistic shape approximations, or are limited to quasi-static methods. These limitations can be sensitive to unmodelled disturbances acting on the robot. Inspired by a factor-graph optimization paradigm, this work introduces a continuous-time stochastic state estimation framework for continuum robots. We introduce factors based on continuous-time kinematics that are corrupted by a white noise Gaussian process (GP). By using a simple robot model paired with high-rate sensing, we show adaptability to unmodelled external forces and data dropout. The result contains an estimate of the mean and covariance for the robot's pose, velocity, and strain, each of which can be interpolated continuously in time or space. This same interpolation scheme can be used during estimation, allowing for inclusion of measurements on states that are not explicitly estimated. Our method's inherent sparsity leads to a linear solve complexity with respect to time and interpolation queries in constant time. We demonstrate our method on a CR with gyroscope and pose sensors, highlighting its versatility in real-world systems.