Learning the Forward and Inverse Kinematics of a 6-DOF Concentric Tube Continuum Robot in SE(3)

Recent physics-based models of concentric tube continuum robots are able to describe pose of the tip, given the preformed translation and rotation in joint space of the robot. However, such model-based approaches are associated with high computational load and highly non-linear modeling effort. A data-driven approach for computationally fast estimation of the kinematics without requiring the knowledge and the uncertainties in the physics-based model would be an asset. This paper introduces an approach to solve the forward kinematics as well as the inverse kinematics of concentric tube continuum robots with 6-DOF in three dimensional space SE(3). Two artificial neural networks with ReLU (rectified linear unit) activation functions are designed in order to approximate the respective kinematics. Measured data from a robot prototype are used in order to train, validate, and test the proposed approach. We introduce a representation of the rotatory joints by trigonometric functions that improves the accuracy of the approximation. The results with experimental measurements show higher accuracy for the forward kinematics compared to the state of the art mechanics modeling. The tip error is less then 2.3 mm w.r.t. position (1% of total robot length) and 1.1 degrees w.r.t. orientation. The single artificial neural network for the inverse kinematics approximation achieves a translation and rotation actuator error of 4.0 mm and 8.3 degrees, respectively.