# Three-Dimensional Dead-Reckoning Based on Lie Theory for Overcoming Approximation Errors
## 4 Experiments and Results
### Position Results  
The error of the calculated position in the ENU coordinate frame was compared with the error of the previously used DR. The performance of the method is compared based on the distance error. The distance error indicates the distance between the calculated position \(\hat{\textbf{p}}_{t}= \begin{bmatrix}\hat{x}_{t}&\hat{y}_{t}&\hat{z}_{t}\end{bmatrix} ^ {T}\) and the reference position \(\textbf{p}_{t}= \begin{bmatrix}x_{t}&y_{t}&z_{t}\end{bmatrix} ^{T}\) as shown in Equation (18). \[d(\hat{\textbf{p}}_{t},\textbf{p}_{t})=\sqrt{(\hat{x}_{t}-x_{t})^{2}+(\hat{y}_ {t}-y_{t})^{2}+(\hat{z}_{t}-z_{t})^{2}} \tag{18}\]  
Here, the reference position was obtained using RTK-GPS, which shows the highest accuracy. Figures 4-7 show the predicted position trajectories on the \(xy\) plane for the experiments conducted in the four regions indicated in Figure 3. The pink graph represents the trajectory measured by RTK-GPS, which is used as the reference in this figure. The red graph indicates the results calculated using the previous method, and the green graph shows the results  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Stat. **} & \multicolumn{4}{c}{**Data**} \\ \cline{2-5}  & **(1)** & **(2)** & **(3)** & **(4)** \\ \hline Driving time (s) & 537.1492 & 483.2752 & 287.5341 & 470.5266 \\ \hline Total distance (m)  
from the proposed method. In each figure, (a) represents the results when the data reception period is 0.1 s, (b) represents the results for a data reception period of 0.5 s, and (c) represents the results for a data reception period of 1 s.  
Both methods perform only prediction without correction through external measurements, leading to inevitable shifts and rotations in the calculated trajectories, with calculation errors accumulating over time and travel distance. However, the proposed method consistently exhibits a smaller calculation error than the previously widely used DR method, even as the data reception period increases, ensuring that the trajectory does not significantly deviate from the reference trajectory. Comparing the figures, it can be observed that the proposed method (green graph) consistently shows closer alignment to the reference trajectory (pink graph) compared to the previous method (red graph). This difference becomes more pronounced as the data reception period increases. As the data reception period increases from 0.1 s (a) to 1 s (c), both methods show increasing deviations from the reference trajectory, but the deviation in the proposed method is significantly smaller. Additionally, while the degree of deviation varies slightly across the four regions, the overall trend remains consistent, indicating the robustness of the proposed method across different environments. These observations highlight the advantages of the proposed method in reducing cumulative errors and maintaining trajectory accuracy over extended periods and varying data reception intervals. In Table 2, we compare the mean error (Mn), standard deviation of the error (Std), and root mean square (RMS) of the distance errors in the ENU coordinate frame. The proposed method consistently shows smaller errors than the previously widely used DR in most experimental results. This is because the proposed method operates directly on the Lie group, exactly representing the pose and velocity of the vehicle without approximation.