# A Neural Network Based Approach to Inverse Kinematics Problem for General Six-Axis Robots
## 6 Conclusions  
Inverse kinematics is a fundamental problem in robotics. Since the degree of non-linearity of the kinematic equations increases sharply with the number of DoFs, seeking inverse kinematics solutions for general robots with high orders is extremely challenging. In this paper, a novel NN-based approach composed of three key parts--the prediction system, the classification system, and the optimization system--is presented to solve the IKP for the general six-axis robot. Within our framework, for the prediction system, the joint space is first divided into some smaller subspaces and a sets of MLP are applied to fit the inverse kinematics function in pieces. For each pair of neural networks, a five-output MLP is used to predict values of the first five joints, while another single output MLP is trained to approximate the last joint separately. Subsequently, to reduce the computational cost of making predictions, a classification system is trained to determine which foregoing MLP, representing difference joint spaces, should be selected given a desired pose. Considering some tasks requiring significantly high precision, such as micrometer level, the initial solution from the prediction system is further refined by a numerical iterative process originating from Newton-Raphson method. The feasibility of our algorithm is first tested by 4800 random desired poses with a following task of drawing a curve in the workspace to simulate the practical applications. Comparison are made with other approaches in literature to reveal the advantages of our algorithm.  
The results from the random pose and trajectory tracking test show that with our workflow, the system can make reliable and consecutive predictions with the preset goal--a position and orientation error less than 0.001 mm and 0.001 rad respectively. Meanwhile, thanks to the classification components, the processing speed of our approach, 0.0218 s per target on average, is satisfying. Furthermore, the proposed NR-based error minimization  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(F_{singular}\) & \(F_{limit}\) & **Total Steps** & **Run Time (s)** & **Time/Solution (s)** \\ \hline Test 01 & 53 & 18927 & 23780 & 481.52 & 0.1003 \\ Test 02 & 74 & 18293 & 23167 & 475.345 & 0.099 \\ Test 03 & 79 & 18423 & 23302 & 476.243 & 0.0992 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Results of NR with random goal poses tests.  
method is verified to be extremely robust towards the singularity problem with an efficiency being dozens of times that of the heuristic methods reported in literature.  
There is some room for the methodology improvement and simplification of the training process is the preoccupation. At this moment, the training process of our methodology is a little tedious since the step size for the workspace division is selected by experiments. We will focus on a systematic approach to select this hyper-parameter efficiently in the future. The future work also includes experimental test for the methodology validation.  
[PERSON] performed the algorithm, implemented all simulation work and drafted the manuscript. [PERSON] and [PERSON] are responsible for supervising the research work, organizing the manuscript sequence alignment, proofreading and revising the manuscript. All authors read and approved the final form of the manuscript.  
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) through an Alliance COVID-19 special program (No. 214159) and in part by VP Startup Fund from Memorial University under Grant 212948.  
Not applicable.  
The support from the Natural Sciences and Engineering Research Council of Canada (NSERC) through an Alliance Covid-19 special program (No. 214159) and the VP Startup Fund from Memorial University (No. 212948) to the second author is dutifully acknowledged. The fund from China Scholarship Council to the first author is also acknowledged.  
The authors declare no conflict of interest.