# Three-Dimensional Dead-Reckoning Based on Lie Theory for Overcoming Approximation Errors
## 2 Limitation of Previously Used Dead-Reckoning Approach
However, because the acceleration \(\textbf{a}_{t_{k}}\) is the acceleration in the body frame of the vehicle, the coordinate transformation to the ENU coordinate frame must be performed as follows:  
\[\delta\dot{\textbf{p}}_{t_{k}}=\textbf{R}_{k_{k}}\cdot\textbf{a}_{t_{k}}\cdot \triangle t \tag{11}\]  
The attitude change vector \(\delta\theta_{t_{k}}\) is calculated using the angular velocity vector \(\boldsymbol{\omega}_{t_{k}}\). This expresses the relationship between angular velocity and attitude, modeling the time variation of attitude. The connection between angular velocity and attitude change is represented by the derivative of the rotation matrix with respect to the Euler angles. \[\delta\theta_{t_{k}}= \begin{bmatrix}1&\sin(\phi_{t_{k}})\cdot\tan(\theta_{t _{k}})&\cos(\phi_{t_{k}})\cdot\tan(\theta_{t_{k}})\\ 0&\cos(\phi_{t_{k}})&-\sin(\phi_{t_{k}})\\ 0&\sin(\phi_{t_{k}})/\cos(\theta_{t_{k}})&\cos(\phi_{t_{k}})/\cos(\theta_{t_{k }})\end{bmatrix} \cdot\boldsymbol{\omega}_{t_{k}}\cdot\triangle t \tag{12}\]  
The previous method corresponds to a first-order approximation. \[ \begin{split} f(t+\triangle t)&=f(t)+\sum_{k=1}^{ \infty}\frac{f^{(k)}(t)}{k!}\cdot(\triangle t)^{(k)}\\ &=f(t)+f^{{}^{\prime}}(t)\cdot\triangle t+\frac{f^{{}^{\prime }}(t)}{2}\cdot(\triangle t)^{2}+\cdots\\ &\simeq f(t)+f^{{}^{\prime}}(t)\cdot\triangle t\end{split}  \tag{13}\]  
Here, Equations (9), (11) and (12) correspond to \(f^{{}^{\prime}}  
Lie groups are used to represent continuous transformations in DR systems. The position, velocity, and attitude of a vehicle are expressed as elements of a Lie group, and these elements are used to change over time and perform various transformations. Lie groups are also important tools for understanding the symmetry of DR systems, allowing for the study and analysis of transformations performed by the system. Lie groups are one of the key concepts in algebraic structure and differential geometry. Mathematically, it is a special set that is both a group and a smooth manifold. The concept of a smooth manifold is related to differential geometry, and the smoothness of a Lie group means that the operation between transformations is differentiable. Commonly used Lie groups in robotics include the n-dimensional special orthogonal group (\(SO(n)\)) [28], which represents rotation transformations, the n-dimensional special unitary group (\(SU(n)\)), which plays an important role in quantum mechanics, and the special Euclidean group (\(SE(3)\)), which represents rotations and translations in three-dimensional space [29; 30; 31]. In addition, there is \(SE_{n}(3)\) extended from \(SE(3)\) for rigid body transformations related to transformations or kinematic changes [26; 32; 33; 34]. Lie algebra is an important mathematical structure connected to Lie groups. Whereas Lie groups are special sets that are algebraically smooth and at the same time smooth manifolds, Lie algebras are algebraic objects that represent transformations around the identity element of the corresponding group. Lie algebras are used to provide information about small changes in Lie groups. Therefore, small dynamic changes or errors of a vehicle can be expressed as elements of a Lie algebra. Lie algebras are mainly defined as the tangent space of a Lie group, that is, the tangent space near the identity element of the group. In particular, Lie algebras representing small transformations in Lie groups are mainly expressed as a set of skew-symmetric matrices. Therefore, they are expressed as a set of matrices that satisfy \(A^{T}=-A\). The size of the matrix depends on the dimension of the Lie group. Therefore, in Lie theory-based DR, Lie groups and Lie algebras act as important tools for modeling the pose and velocity of vehicles and understanding transformations. These concepts are essential in the design and improvement of DR algorithms and play an important role in improving accuracy and stability.