# Total Least-Squares Iterative Closest Point Algorithm Based on Lie Algebra
## 4 Conclusions  
In this paper, we present two total least-squares ICP algorithms based on Lie algebra using a GHM framework. These methods can estimate a rigid transformation matrix from adjusted source coordinates and adjusted target coordinates. Our methods perform better than the common least-squares ICP algorithm because they consider random noise from the source coordinate, which results in a smaller sum of squared errors. The most important contribution of this paper is that our methods are more robust than [12; 19], which utilize the Euler angle to describe the rotation matrix. The nonlinearity in the cost function due to the rotation matrix results in there being no analytical solution to this problem. Thus, an iterative optimization algorithm is employed to solve the GHM model. The Euler angle is unsuitable for interpolation because of the gimbal lock problem. We replace the Euler angle with Lie algebra to execute the TLS optimization algorithm and give deduction of our method in detail. Conceptualization: [PERSON]; data curation: [PERSON] and H.Z. ; formal analysis: Q.W. ; methodology: [PERSON]; project administration: [PERSON]; resources: [PERSON]; software: [PERSON]; supervision: [PERSON]; validation: [PERSON]; visualization: H.Z. ; writing--original draft: [PERSON]; writing--review and editing: [PERSON] and [PERSON]. This research was funded by the projects of the National Key Research and Development Plan of China, grant number 2016 YFB0502103, the National Natural Science Foundation of China, grant number 61601123, and the Natural Science Foundation of Jiangsu Province of China, grant number BK20160696. This work was partially supported by State Key Laboratory of Smart Grid Protection and Control of China. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Estimated** & **Euler** & **Estimated** & **Method 1** & **LS** & **Estimated** & **Method 2** & **LS** \\
**Parameter** & **Method** & **Parameter** & & & & & \\ \hline \(Z\)-axis & & & & & & & \\ angle (rad) & 2.516 & \(\phi_{1}\) & 0.0207793 & 0.02066 & \(\xi_{1}\) & 151.836 & 151.834 \\ \(Y\)-axis & & & & & & \\ angle (rad) & \(-\)3.137 & \(\phi_{2}\) & \(-\)0.011339 & \(-\)0.0112 & \(\xi_{2}\) & 175.062 & 175.066 \\ \(X\)-axis & & & & & & \\ angle & \(-\)3.119 & \(\phi_{3}\) & \(-\)0.625356 & \(-\)0.6254 & \(\xi_{3}\) & \(-\)17.5989 & \(-\)17.6049 \\ \(t_{1}\) & 195.23 & \(t_{1}\) & 195.231 & 195.23 & \(\xi_{4}\) & 0.02065 & 0.02066 \\ \(t_{2}\) & 118.064 & \(t_{2}\) & 118.064 & 118.067 & \(\xi_{5}\) & 0.0112623 & 0.01128 \\ \(t_{3}\) & \(-\)15.138 & \(t_{3}\) & \(-\)15.1442 & \(-\)15.143 & \(\xi_{6}\) & \(-\)0.62536 & \(-\)0.6253 \\
**SSE** & **1.14** & **SSE** & 0.0564 & 68.577 & **SSE** & 0.005056 & 68.577 \\ \hline \end{tabular}
\end{table} 
Table 4: The comparision between Lie algebra TLS, Euler Angle TLS and common LS algorithm. **Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results. **Appendix A**  
**Target:** We want to calculate Jacobian matrix of \(Rp\) with respect to Lie algebra of \(SO\) (3).