# Three-Dimensional Dead-Reckoning Based on Lie Theory for Overcoming Approximation Errors
## 1 Introduction
(2022) proposed pedestrian DR to address the drift issue over time caused by errors and noise in inertial sensor measurements in the navigation of quadcopters [14]. [PERSON] et al. (2024) developed a position and direction estimation algorithm for active driving of wheelchairs using ultra-wideband (UWB) sensors [15]. [PERSON] et al. (2023) proposed an accurate underwater DR mathematical model based on the motion model of autonomous underwater vehicles (AUVs) to recursively compute ground truth and associated errors [16]. [PERSON] et al. (2023) proposed a method for tracking upper limb segments using a single inertial sensor to reduce drift in DR-based position tracking [17]. [PERSON] et al. (2023) proposed a DR method using only wheel speeds to achieve accurate path tracking in extreme situations where location sensors are absent [18]. To further enhance the robustness and accuracy of DR systems, recent studies have explored the integration of factor graphs [19; 20]. [PERSON] et al. (2024) optimized AUV navigation by integrating side-scan sonar (SSS) data and inertial measurement units (IMU) using factor graphs, demonstrating significant improvements in positioning accuracy under challenging conditions [19]. Similarly, [PERSON] et al. (2021) developed a multi-source information fusion algorithm based on factor graphs for AUVs, achieving enhanced computational efficiency and navigation performance by processing asynchronous heterogeneous measurements [20]. Additionally, information geometry has been employed to optimize navigation algorithms by enhancing their robustness and accuracy [21; 22]. [PERSON] et al. (2022) utilized street geometry information to improve visual navigation and path tracking for drones. This approach compensated for visual odometry drift through scene matching and servo control, leveraging known street geometries to enhance the precision of image alignment [21]. Moreover, information geometry principles have been applied to improve unmanned aerial vehicle (UAV) circular control by optimizing Fisher information, thus enhancing the stability and accuracy of the navigation system [22]. These advanced approaches address the limitations of traditional methods by effectively handling attitude and rotation in nonlinear spaces and reducing numerical approximation errors. To overcome these limitations, this paper proposes a DR approach by representing the pose and velocity of the vehicle in the three-dimensional special Euclidean group, an extended Lie group (\(SE_{2}(3)\)), based on Lie theory [23; 24; 25]. The proposed method resolves the approximation error issue of previously used DR by computing the attitude of objects moving in nonlinear spaces. Additionally, as the attitude of the vehicle is also represented by the Lie group, the 3D special orthogonal group (\(SO(3)\)), there is no need to approximate the attitude in a linear space, thus addressing the issue of model uncertainty concerning attitude. Lie theory-based DR handles the integration of the attitude of vehicles accurately and without approximation, significantly reducing the influence of time intervals compared to previously used DR methods [26; 27]. Constructing the state as a Lie group in Lie theory-based DR provides highly accurate spatial representations, overcoming the limitations of previously used DR methods and achieving improved performance in terms of accuracy, consistency, and stability. The main contributions and novel aspects of this study can be summarized as follows:  
1. Enhanced Accuracy in Pose Prediction: The proposed method significantly improves the accuracy of pose prediction for vehicle motion. By employing Lie theory-based methods, the approach enables rigorous DR calculations without approximations, precisely modeling translations and rotations, especially in three-dimensional space. This results in more accurate vehicle pose determination. 2. Elimination of Linear Space Assumptions: Utilizing Lie groups, the method eliminates the necessity to assume that vehicle rotation or attitude forms a linear space.