Robust image-based computation of the 3D position of RCM instruments and its application to image-guided manipulation

In this paper, we address the 3D position control of RCM-constrained instruments with monocular cameras. To compute the instrument's position from a single 2D image, we develop an innovative gradient descent algorithm which rotates and translates a line segment (over the plane spanned by the imaged instrument and the optical centre) until it best aligns with the manipulated tool. In contrast with other approaches in the literature, our algorithm only requires to simultaneously observe two feature points; the proposed iterative algorithm is not based on the exact solution, therefore it can still work with noisy image measurements. We derive a kinematic controller that uses the proposed position estimator to guide the 3D motion of a robotic instrument with a monocular camera. We evaluate the performance of our approach with numerical simulations and experiments.