Simultaneous Estimation of Shape and Body Contact Force for Continuum Robots Under Mixed External Geometric Constraints

It is challenging to simultaneously estimate the shape and contact force of miniature continuum surgical robots due to the limited sensing capabilities, deformable structure, and complexity of the external environment they operate in. In this work, the integration of a quaternion-based multicontact Cosserat model with contact force regularization enables a dual-pronged approach for simultaneous shape and body contact force estimation using minimum shape measurement input (i.e., tip position). In addition to internal actuation, cable friction, and material nonlinearity, the proposed state estimation approach also accounts for multiple environmental contacts with clinically relevant mixed geometric constraints (point, plane, and curved surface), as these complex and variable anatomical interactions critically influence the robot's behavior. The solving procedure is then reformulated as a nonlinear large-scale optimization problem to improve the state estimation robustness. Finally, both simulations and experiments on a cable-driven variable-stiffness notched-tube continuum robot were conducted to validate the proposed state estimation approach, demonstrating a comparable level of accuracy to state-of-the-art methods.