# Solving the Time-Varying Inverse Kinematics Problem for the Da Vinci Surgical Robot
## 5 Conclusions  
(1) In this paper, an inverse kinematics mathematical model of the 6-DOF active arm of the Da Vinci surgical robot established by using a coordinate transformation matrix method according to its mechanical motion diagram and working characteristics. By introducing a unitary matrix, the mathematical model on the real plane is transformed to a complex plane, consequently avoiding the relatively complex trigonometric calculations associated with the conventional solution procedure. Following the steps proposed, this approach can apply to other similar robot kinematic problems.  
(2) The dialytic elimination method serves to individually eliminate the joint variables and ultimately obtains a high-order nonlinear equation with only one unknown variable of the first joint. Subsequently, the Newton iteration method is introduced to solve the nonlinear equation on the basis of selecting multiple initial values reasonably in the feasible region. The test shows that carefully changing the step size and selecting initial values in the feasible region for the iteration solution of the nonlinear equation will significantly reduce the possibility of solution failure.  
Figure 6: The endpoint comparison between results of our proposed method and the separate P&R method. (**a**) The position result comparison; (**b**) the orientation result comparison.  
(3) In theory, the accuracy of the approach can reach an error of \(<\) 0.00004 degree (or mm) and achieve a solution time of 0.5 s (Intel (R) Core i5-4460 CPU, clocked at 3.2 GHz (memory: 4 GB)). Running the procedure on better hardware and software (based on the \"R12 CCPU-V\" MELSEC iQ-R series C language controller from Mitsubishi Electric and VxWorks 6.9 operating system) decreases the solution time to 20 ms, which fully satisfies the real-time performance requirement of surgical robots.  
(4) A method that combines dialytic elimination and the Newton iteration method is applied to solve the inverse kinematics problem of the slave manipulators of the master-slave surgical robot. The proposed approach achieves higher accuracy than other solutions based on position and orientation separation while the real-time performance is satisfied. Moreover, in contrast to other numerical solutions, the solution method is based on the position of the joint and is therefore barely not affected by any singularity. Further, the application of our approach removes the restriction of the design of the master manipulator configuration.  
Conceptualization, L.B. and J.Y.; methodology, X.C.; software, F.Z.; validation, J.Y., L.B., Y.S. and X.C.; formal analysis, J.Y.; investigation, P.J.; resources, F.L.; data curation, L.B.; writing--original draft preparation, J.Y.; writing--review and editing, L.B., X.C. and Y.S.; visualization, X.C.; supervision, L.B.; project administration, L.B.; funding acquisition, X.C.  
This research was funded by the Special Cooperation Program for Higher Education Institutions Collaborative Innovation (grant no. KH2016006), the Foundation for Sci & Tech Research Project of Chongqing Science & Technology Commission (grant nos. cstc2016 jcyjA0472, and cstc2017 zdcy-zdzxX0007), the National Natural Science Foundation of China (grant nos. 51705050, and 51709023), and Fundamental Research Funds for the Central Universities (grant no. 2018 CDGFJX0022, 2018 CDQYHK0029).  
The authors gratefully acknowledge [PERSON] for his valuable support with language editing.  
The authors declare no conflict of interest.