Stochastic model-based contact force estimation for concentric tube robots

This work utilizes a Wiener process to describe a fluctuating load on a Concentric tube robot (CTR) or its smooth motion through fixed obstacles leading to a kineto-static piecewise constant strain (KPCS) model. We then propose an observer based on the KPCS model, represented as a stochastic switched system, to estimate the externally applied forces by knowing the position of discrete points on the backbone. We have demonstrated that the observer describes the behavior of CTRs in unknown constraint environments and slow movement. The experiments carried out using electromagnetic sensors on the CTR prototype validate the feasibility of the proposed approach for shape and tip-force estimation. The average time required for force estimation is about 3.8, 6.2, and 20 ms per sample, respectively, when the estimated external forces are applied on a three-tube CTR's tip point and three and fifteen equally spaced points along its length. The implementation results reveal the estimated force reduces the error of the robot tip position by half (about 1.5% of the robot length on average) compared to the known external force.