Stochastic Adaptive Estimation in Polynomial Curvature Shape State Space for Continuum Robots

In continuum robotics, real-time robust shape estimation is crucial for planning and control tasks that involve physical manipulation in complex environments. In this paper, we present a novel stochastic observer-based shape estimation framework designed specifically for continuum robots. The shape state space is uniquely represented by the modal coefficients of a polynomial, enabled by leveraging polynomial curvature kinematics (PCK) to describe the curvature distribution along the arclength. Our framework processes noisy measurements from limited discrete position, orientation, or pose sensors to estimate the shape state robustly. We derive a novel noise-weighted observability matrix, providing a detailed assessment of observability variations under diverse sensor configurations. To overcome the limitations of a single model, our observer employs the Interacting Multiple Model (IMM) method, coupled with Extended Kalman Filters (EKFs), to mix polynomial curvature models of different orders. The IMM approach, rooted in Markov processes, effectively manages multiple model scenarios by dynamically adapting to different polynomial orders based on real-time model probabilities. This adaptability is key to ensuring robust shape estimation of the robot's behaviors under various conditions. Our comprehensive analysis, supported by both simulation studies and experimental validations, confirms the robustness and accuracy of our methods.