An efficient yet accurate model of the continuum robot is the main component for its real-time control, simulation as well as localization. Previous models of the continuum robot, based on rod theory, suffer from high computational burden. The models also require a priori knowledge of the robot environment. This paper presents an efficient static model for the planar continuum robot that experiences external forces at the tip as a result of contact with its surroundings (measured by the built-in force sensors), thus no a priori information about the environment is required. The typical example of such robots is steerable catheters used in medical operations. The proposed approach involves discretizing the robot backbone curve to elastic arc elements. After deriving the equilibrium equations for the infinitesimal elements, a recursive algorithm with the time complexity of O(n) is proposed for realizing the shape of the robot as a result of the external force. Accuracy of the proposed method is evaluated both theoretically and experimentally for a case study, i.e., an intracardiac ablation catheter. Results validate the accuracy and time-efficiency of the proposed approach for real-time applications.

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