# Three-Dimensional Dead-Reckoning Based on Lie Theory for Overcoming Approximation Errors
## 4 Experiments and Results
### Attitude Results  
Figures 24-31 graphically compare the errors in attitude based on Table 4. It can be confirmed that the proposed method shows mostly smaller values in mean error, standard deviation, and RMS. Attitude and rotation cannot be perfectly represented and computed in linear space. The previously widely used DR inevitably includes errors that occur in the linearization process. However, the proposed method accurately represents attitude and handles operations in a nonlinear space called the Lie group. Therefore, it does not cause errors that force approximation into a linear space, and it was able to reduce the overall DR error. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Stat**} & \multirow{2}{*}{**Ti. **} & \multirow{2}{*}{**Md. **} & \multirow{2}{*}{**Di. **} & \multicolumn{6}{c}{**Attitude Error (deg)**} \\ \cline{5-8}  & & & & **(1)** & **(2)** & **(3)** & **(4)** \\ \hline \multirow{8}{*}{Mn} & \multirow{4}{*}{0.1 (s)} & \multirow{4}{*}{Conv. DR} & \( \begin{array}{c}\phi\\ \theta\end{array} \) & 0.0763 & 0.3077 & 0.2194 & 0.3015 \\  & & \( \begin{array}{c}\psi\\ \theta\end{array} \) & \(-\)0.0602 & 0.0373 (**B**) & 0.1218 & \(-\)0.3586 \\  & & \( \begin{array}{c}\psi\\ \theta\end{array} \) & 1.2329 & 2.3185 (**W**) & \(-\)0.6320 \\  & & \( \begin{array}{c}\phi\\ \theta\end{array} \) & 0.0481 & 0.3408 & 0.1801 & 0.2553 \\  & & \( \begin{array}{c}\phi\\ \theta\end{array} \) & \(-\)0.0574 & 0.0405 & 0.1014 & \(-\)0.0608 \\  & & \( \begin{array}{c}\psi\\ \theta\end{array} \) & 1.0800 & 1.2151 & 2.3165 & \(-\)0.3437 \\ \cline{2-8}  & & \( \begin{array}{c}\phi\\ \theta\end{array} \) & \(-\)0.8458 & 1.0514 & 2.3939 & \(-\)3.9552 \\  & & \( \begin{array}{c}\psi\\ \theta\end{array} \) & \(-\)2.8671 & 2.4405 & 1.4226 & 3.3844 \\  & & \( \begin{array}{c}\psi\\ \theta\end{array} \) & \(-\)1.2849 & 4.4015 & 2.4816 & 16.2129 (**W**) \\  & & \( \begin{array}{c}\phi\\ \theta\end{array} \) & \(0.0034\) (**B**) & \(-\)0.4830 & 0.5417 & 0.5554 \\  & & \( \begin{array}{c}\psi\\ \theta\end{array} \) & \(-\)0.9671 & \(-\)0.0283 & \(-\)0.3320 & \(-\)0.0539 \\  & & \( \begin{array}{c}\psi\\ \theta\end{array} \) & \(-\)1.5038 & 3.6899 & 2.7614 & 2.4823 \\ \cline{2-8}  & & \( \begin{array}{c}\phi\\ \theta\end{array} \) & 15.9446 & 12.3953 & 8.0376 & 2.7170 \\  & & \( \begin{array}{c}\phi\\ \theta\end{array} \) & 4.6852 & \(-\)1.0113 & 10.9729 & 5.5419 \\  & & \( \begin{array}{c}\psi\\ \theta\end{array} \) & \(-\)49.2481 (**W**) & 21.3984 & 0.8525 & \(-\)1.1596 \\  & & \( \begin{array}{c}\phi\\ \theta\end{array} \) & 9.8586 & 3.6791 & 0.9710 & 1.4649 \\  & & \( \begin{array}{c}\phi\\ \theta\end{array} \) & 2.8610 & 1.2904 & 0.3972 (**B**) & 1.7685 \\  & & \( \begin{array}{c}\psi\\ \theta\end{array} \) & 4.9715 & 5.9009 & 3.8896 & 3.4412 \\ \hline \multirow{8}{*}{0.1 (s)} & \multirow{4}{*}{0.1 (s)} & \multirow{4}{*}{Conv. DR} & \( \begin{array}{c}\phi\\ \theta\end{array} \) & 0.6347 & 1.8853 & 1.2082 & 13.7041 (**W**) \\  & & \( \begin{array}{c}\phi\\ \theta\end{array} \) & 0.6135 & 2.7149 & 1.2460 & 12.0352 \\  & & \( \begin{array}{c}\psi\\ \theta\end{array} \) & 1.1821 & 1.2830 & 2.2789 & 3.3131 \\  & & \( \begin{array}{c}\phi\\ \theta\end{array}