Accurate shape estimation of concentric tube robots (CTRs) using mathematical models remains a challenge, reinforcing the need to develop techniques for accurate and real-time shape sensing of CTRs. In this paper, we develop a fusion algorithm that predicts the robot's shape by combining a mathematical model of the CTR with a measurement of the Cartesian coordinates of the robot's tip using an electro-magnetic sensor. We experimentally validated our method in static and dynamic scenarios with and without external loading. Results demonstrated that the fusion algorithm improves the error of model-based shape prediction by an average of 44.3%, corresponding to 2.43% of the robot's arc length. Furthermore, we demonstrate that our method can be used in real-time to simultaneously track the robot's tip position and predict its shape.

The Cosserat rod equations are used to model continuum and soft robots. Solving these equations are computationally expensive, particularly due to mixed boundary values and kinematic constraints. In this paper, we present a novel nonlinear observer that can rapidly estimate the solution of the Cosserat rod equations. We present details of the observer design and analyse its convergence and stability. Furthermore, we compare the accuracy and performance of the observer with common solvers used in the literature. Our results show that the proposed observer can significantly improve the computational efficiency of continuum robots’ models and estimates the solution of the Cosserat rod equations 7 times faster than common solvers.

Concentric Tube Robots (CTR) have been gaining ground in minimally-invasive robotic surgeries due to their small footprint, compliance, and high dexterity. CTRs can assure safe interaction with soft tissue, provided that precise and effective motion control is achieved. Controlling the motion of CTRs is still challenging. Commonly used model-based control approaches often employ simplified geometric/dynamic assumptions, which could be very inaccurate in the presence of unmodelled disturbances and external interaction forces. Additionally, application of emerging data-driven algorithms in real-time control of CTRs is limited due to the fact that these controllers require considerable amount of time to let the algorithm develop enough to reach a desired accuracy and relevancy. In this paper, we present a hybrid approach to overcome the aforementioned difficulties. This hybrid solution uses the solution of a kinematic model of the robot to estimate initial values for a model-free data-driven method. The proposed algorithm combines both model-based and data-driven algorithms to provide real-time motion control of CTRs interacting with an unknown external environment. Three different simulations studies were performed to thoroughly evaluate the efficacy of the proposed hybrid control approach as compared to two common model-based and data-driven control techniques. The results demonstrate superior performance of the proposed method. The root-mean-square error of the proposed hybrid approach is less than 1.1 mm, which is 9 times less than a common model-based controller.