# Three-Dimensional Dead-Reckoning Based on Lie Theory for Overcoming Approximation Errors
## 1 Introduction  
In the realm of pose (position and attitude) estimation, dead-reckoning (DR) plays a pivotal role, particularly in Kalman filter (KF)-based methods where it is utilized during the prediction stage. The accuracy of DR is paramount in improving the performance of pose estimation algorithms. As DR is employed to forecast the future state of a vehicle based on its current state and motion dynamics, any inaccuracies in DR computations can significantly impact the accuracy of the estimation process. Therefore, ensuring precise DR calculations is essential to enhance the overall estimation performance [1; 2; 3; 4]. However, most previously used DR approaches are prone to errors due to inevitable approximations in numerical integration and approximations in attitude-related operations. In other words, they face issues such as decreasing accuracy of position with increasing time of numerical integration and problems in handling attitude and rotation operations in Cartesian space. DR involves predicting the position and attitude of a vehicle through integration, which introduces approximation errors due to sensor-measured data integration over the motion prediction time [5]. These errors accumulate over time, leading to significant discrepancies between the predicted and actual poses [6; 7]. Although shorter prediction intervals may reduce errors, precision is not guaranteed. Consequently, implementing DR with approximate integration decreases prediction accuracy and increases uncertainty, especially in recursive filters like the KF. Additionally, as the prediction cycle of the vehicle increases, the method's accuracy diminishes, making it unsuitable for position prediction with long velocity measurement cycles. Therefore, ensuring precise DR calculations is essential to enhance the overall prediction performance. The rotation and attitude of a vehicle cannot be perfectly represented and computed within the linear space of Cartesian coordinates [8; 9]. When the attitude is approximated and calculated within this linear space, errors due to approximation arise, leading to a decrease in position accuracy. Numerous researchers have assumed that attitudes, represented by Euler angles, form a linear space and have performed calculations based on this assumption. However, Euler angles can lead to gimbal lock, where a particular rotation axis becomes aligned with another axis [10; 11]. Euler angles indeed encounter difficulties in accurately representing all attitudes, particularly when the pitch axis nears \(\pm\)90 degrees, thereby complicating processing and representation tasks [12; 13]. However, when it comes to DR, the challenges are more nuanced and stem from the limitations of Euler angles in expressing and operating rotations. Specifically, the use of Euler angles introduces singularities, such as gimbal lock, which can disrupt the continuity of orientation measurements and lead to inaccuracies in pose estimation during DR calculations. Furthermore, the nonlinear nature of Euler angle transformations complicates the integration process, making it susceptible to cumulative errors over time. These issues highlight the inadequacy of Euler angles in robustly representing motion dynamics, particularly in dynamic environments where precise pose estimation is crucial for navigation and localization tasks. The primary limitation of previously used DR methods stems from approximating the motion of the vehicle during a single sampling period as linear, leading to errors in position calculation. Another significant issue arises from the linear piecewise approximation of the continuous and coupled changes in pose, which introduces uncertainties. These uncertainties include errors resulting from simply integrating angular rates to estimate changes in attitude, the assumption of attitudes forming a linear space, and the occurrence of gimbal lock. Research aimed at minimizing cumulative errors to enhance the accuracy of DR is actively being conducted across various fields [14; 15; 16; 17; 18]. [PERSON] et al.