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.