Contact force is one of the most significant feedback for robots to achieve accurate control and safe interaction with environment. For continuum robots, it is possible to estimate the contact force based on the feedback of robot shapes, which can address the difficulty of mounting dedicated force sensors on the continuum robot body with strict dimension constraints. In this paper, we use local curvatures to estimate the magnitude and location of single or multiple contact forces based on Cosserat rod theory. We validate the proposed method in a thin elastic tube and calculate the curvatures via Fiber Bragg Grating (FBG) sensors or image feedback. For the curvature feedback obtained from multicore FBG sensors, the overall force magnitude estimation error is $0.062 \pm 0.068$ N and the overall location estimation error is $3.51 \pm 2.60$ mm. For the curvature feedback obtained from image, the overall force magnitude estimation error is $0.049 \pm 0.048$ N and the overall location estimation error is $2.75 \pm 1.71$ mm. The results demonstrate that the curvature-based force estimation method is able to accurately estimate the contact force.

A novel nonlinear teleoperation algorithm for simultaneous inertial parameters and force estimation at the master and slave sides of the teleoperation system is proposed. The scheme, called Extended Active Observer (EAOB), is an extension of the existing active observer. It provides effective force tracking at the master side with accurate position tracking at the slave side in the presence of inertial parameter variation and measurement noise. The proposed method only requires the measurement of robot position, and as a result significantly reduces the difficulty and cost of implementing bilateral teleoperation systems. The approach is described and its stability is analytically verified. The performance of the method is validated through computer simulation and compared with the Nicosia observer-based controller. According to the results, EAOB outperforms the Nicosia observer method and effectively rejects noise.