Kinematics of wristed laparoscopic instruments

A method, comprising:

• (a) inserting a robotic arm into a trocar positioned at an incision site of a patient, wherein the robotic arm comprises a surgical instrument and a tool disposed thereon, wherein the tool comprises an elbow joint and a wrist joint;
• (b) determining (i) a three-dimensional (3D) position of the wrist joint based on a three-dimensional position and orientation of the tool and (ii) a three-dimensional position of the elbow joint based on the three-dimensional position of the wrist joint;
• (c) determining (iii) a first angle corresponding to a twist of the tool, (iv) a second angle corresponding to an elbow angle of the tool, (v) a third angle corresponding to a wrist angle of the tool, and (vi) a fourth angle corresponding to an open/close angle of the tool, based at least in part on the three-dimensional position of the wrist joint and the three dimensional position of the elbow joint;
• and
• (d) generating a virtual representation of a surgical scene that is accessible via the incision site, wherein the virtual representation comprises a virtual model of the surgical instrument and a virtual anatomy of one or more anatomical structures in the surgical scene, wherein the virtual representation provides a real-time visualization of a procedure involving the surgical instrument and the one or more anatomical structures.