Adaptive image-based positioning of RCM mechanisms using angle and distance features

In this paper, we address the positioning problem of remote centre of motion (RCM) mechanisms with uncalibrated image feedback from a monocular camera. Nowadays, RCM mechanisms are widely used in minimally invasive robotic surgery due to their ability to distally rotate a tool around a fixed entry port; note that in most surgical applications, the tools are typically controlled by manual/teleoperated motion commands given by a human user. In this paper, we depart from the traditional manual control scheme and derive sensor-based methods to automatically position the manipulated tool using real-time image feedback. To this end, we first characterise the mechanism´s 3-DOF configuration with the angle of the image projected tool and scalar distances between feature points. To cope with uncertainty in the camera´s calibration parameters, we propose two gradient descent estimators that adaptively compute the unknown Jacobian matrix; the stability of these algorithms is proved with Lyapunov theory. Finally, we derive a kinematic image-based controller and evaluate its performance with several positioning experiments.