# Three-Dimensional Dead-Reckoning Based on Lie Theory for Overcoming Approximation Errors
## 4 Experiments and Results
### Position Results
Figure 4: Dead-reckoned trajectory calculated by the local data at region (1): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
Figure 5: Dead-reckoned trajectory calculated by the local data at region (2): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
Figure 6: Dead-reckoned trajectory calculated by the local data at region (3): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
To highlight the performance differences, we have marked the best (B) and worst (W) cases within each group defined by the data reception period and methods. Specifically, for each combination of data reception period (0.1 s, 0.5 s, 1 s) and methods, we identified the minimum and maximum values across the columns (1), (2), (3), and (4). The minimum values were marked as the best cases (B), and the maximum values were marked as the worst cases (W). This approach ensures a clear comparison of performance metrics across different configurations. Whereas the previous method performs well only when the data reception period is small, the results indicate that as the data reception period increases, the proposed method consistently outperforms the previous method. Figures 8-15 graphically compare the distance errors of position prediction based on Table 2. The path predicted by the proposed method is closer to the true path than that predicted by the previous method. Any trajectory error in the proposed method arises from errors in measured velocity and inherent computation. Unlike Equations (9), (11) and (12), the motion Equation (16) does not involve any approximation. Additionally, the proposed method can reduce cumulative errors, thereby improving accuracy even when the data reception period is long. These results demonstrate the overall effectiveness of the proposed method under various conditions. Figure 7: Dead-reckoned trajectory calculated by the local data at region (4): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
Figure 11: Comparison of distance error, mean, and standard deviation in region (4): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
Figure 8: Comparison of distance error, mean, and standard deviation in region (1): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
Figure 10: Comparison of distance error, mean, and standard deviation in region (3): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
Figure 3.0 Comparison of distance error, mean, and standard deviation in region (2): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
Figure 3.1 Comparison of distance error, including RMS, in region (3): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
Figure 3.2 Comparison of distance error, including RMS, in region (1): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
Figure 3.3 Comparison of distance error, including RMS, in region (2): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.  
Figure 3.4 Comparison of distance error, including RMS, in region (4): (**a**) has a data reception period of 0.1 s; (**b**) has a data reception period of 0.5 s; and (**c**) has a data reception period of 1 s.