On the Kinematics of Robotic-assisted Minimally Invasive Surgery

Minimally invasive surgery is characterized by the insertion of the surgical instruments into the human body through small insertion points called trocars, as opposed to open surgery which requires substantial cutting of skin and tissue to give the surgeon direct access to the operating area. To avoid damage to the skin and tissue, zero lateral velocity at the insertion point is crucial. Entering the human body through trocars in this way thus adds constraints to the robot kinematics and the end-effector velocities cannot be found from the joint velocities using the simple relation given by the standard Jacobian matrix. We therefore derive a new Jacobian matrix which gives the relation between the joint variables and the endeffector velocities and at the same time guarantees that the velocity constraints at the insertion point are always satisfied. We denote this new Jacobian the Remote Center of Motion Jacobian Matrix (RCM Jacobian). The main contribution of this paper is that we address the problem at a kinematic level and that we through the RCM Jacobian can guarantee that the insertion point constraints are satisfied which again allows for the controller to be implemented in the end-effector workspace. By eliminating the kinematic constraints from the control loop we can derive the control law in the end-effector space and we are therefore able to apply Cartesian control schemes such as compliant or hybrid control.