## 5 Conclusions and Future Work  
This article presented an efficient approach for inverse kinematics that is suitable for any configurations of robots with either revolute or prismatic joints with any arbitrary number of degrees of freedom. Compared with the traditional method, this paper transformed the inverse kinematics problem into the objective function optimization problem, which is based on the least-squares form of the angle error and the position error expressed by the PoE formula. Moreover, the necessary gradient  
Figure 8: The position and angle errors of the initial selected joint angles of the UR5 versus the number of iterations of the CCD, ICCD, and NICCD methods.  
Figure 7: The position and angle errors of the initial selected joint angles of the UR5 versus the convergence times of the CCD, ICCD, and NICCD methods.  
Figure 9: The evolution process of NICCD method.  
and Hessian matrix required to solve the formula are given. Finally, the NICCD method was proposed to solve the proposed objective function optimization problem.  
Five simulations were given in order to illustrate the effectiveness, accuracy, robustness, and generality of the proposed method. Simulation I verified the effectiveness of the proposed method, the results of which suggest that it has less fluctuation and is more effective than the NR method. Simulation II tests the accuracy of the proposed method. The results of which suggest that the approach proposed in this paper can accurately calculate the error of position and the angle to reach \(10^{-13}\) and \(10^{-15}\) respectively, but the NR method provides the error of the angle within \(10^{-11}\). This also proves the validity of the objective function constructed in Section 3.1. Simulation III demonstrated the fact that the proposed approach is not sensitive to the initial joint variables. Indeed, the correct rate for inverse kinematics calculations with respect to 3P, 3R, SCARA, UR5, and WAM7R are approximately 100%, 100%, 100%, 94.5%, and 93.1%. Simulation IV illustrated the fact that the proposed approach has offline programming abilities and that it is advantageous with respect to the inverse kinematics of continuous trajectories since each iteration only takes 0.0057 s to calculate. Simulation V verified that the ICCD method drops much more rapidly than the CCD method in positional and angle errors in the initial iteration, and the NICCD method combines the respective advantages of the ICCD and Newton's methods while trying to avoid their shortcomings.  
The future work will mainly focus on the following aspects. First of all, we will introduce the heuristic adaptive truncation criterion [27] into the NICCD method to improve the convergence rate. Next, we will improve the NICCD method to become a more effective method with respect to avoiding a singularity location and local minimum, which, in turn, will enhance the accuracy and robustness of the method. Last but not least, the NICCD method will be applied to practical applications such as trajectory planning and motion control of the robot.  
Y.C. and X.L.: necessary formula and algorithm design; B.H. and Y.J.: programming; G.L. and X.W.: simulations and data analysis.  
This research was funded by National Key R & D Program of China (2016 YFC0803000, 2016 YFC0803005)  
**Acknowledgments:** Thanks to [PERSON] and [PERSON] for providing language help and all our colleagues for providing all types of help during the preparation of this manuscript.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} NICCD & Newton-improved cyclic coordinate descent \\ ICCD & improved cyclic coordinate descent \\ CCD & cyclic coordinate descent \\ NR & Newton-Raphson \\
6R & six revolute \\ PoE & product-of-exponentials \\
3R & three-link planar arm \\ SCARA & selective compliance assembly robot arm \\
3P & Cartesian manipulator \\ \end{tabular}