Appropriate spherical coordinate model for trocar port constraint in robotic surgery

This article deals with the representation of the pivot point created when inserting an endoscope in the human body during a surgery application. Several robotic-based surgical solutions use the well-known spherical coordinates to define the pose of the robotic endoscope tip, since it can only represents motions that satisfy this single insertion point constraint. We demonstrate in this article that the traditional spherical model may not be totally suitable for surgical applications. First of all, such model presents a representation singularity within the work space. Furthermore, control strategies based on that model may not produce optimal motions of the robotic arm holding the endoscope. We propose an adjustment of the traditional spherical model to alleviate these two issues. We also propose a very simple but efficient control task for servoing a robotic arm holding an endoscopic camera to center an instrument in the camera field of view. Simulations are provided for demonstrating the correctness and efficiency of the proposed model and control approach.