# Three-Dimensional Dead-Reckoning Based on Lie Theory for Overcoming Approximation Errors
## 2 Limitation of Previously Used Dead-Reckoning Approach
Additionally, Lie theory-based DR performs calculations for rotation and translation of vehicles directly within the Lie group, dividing each rotation into small steps and minimizing approximation errors. In other words, because global optimization is performed through calculations within the Lie group, cumulative errors and approximation errors can be minimized, and accuracy can be maintained. Figure 1 illustrates the correspondence between Lie groups and Lie algebras and demonstrates the role of the \(\exp(\cdot)\) function, a key concept within Lie theory. The \(\exp(\cdot)\) function enables the mapping of elements from the Lie algebra to elements of the Lie group. If \(\mathbf{x}(t_{k})\) represents a point on the Lie group (\(\mathcal{M}\)), the velocity at this point (\(\frac{\mathbf{x}(t_{k})}{\mathbf{\sigma}t}\)) belongs to the tangent space of the Lie group at that point. The structure of this tangent space is consistently maintained throughout the manifold due to its natural smoothness. Notably, the tangent space at the identity element of the group corresponds to the Lie algebra (\(\mathfrak{m}\)). Elements within the Lie algebra are represented as skew-symmetric matrices. These matrices can be expressed using the operator \(\hat{\mathbf{\cdot}}\), which converts column vectors into matrices. The exponential map function \(\exp(\cdot)\) is employed to ensure a unique transformation from the Lie algebra to the Lie group matrix. This transformation allows for the mapping of elements from the Lie algebra to their corresponding elements in the Lie group. Mathematically expressing the relationship between the Lie group and Lie algebra, let \(\mathcal{M}\) be a Lie group and \(\mathfrak{m}\) be the Lie algebra corresponding to the Lie group \(\mathcal{M}\). The function \(\exp(\mathbf{\tau}^{\ast})\) maps the Lie algebra element \(\mathbf{\tau}^{\ast}\in\mathfrak{m}\) to the Lie group element \(\mathbf{\chi}\in\mathcal{M}\). The expression \(\exp(\mathbf{\tau})\) is defined differently depending on the group [33]. If \(\mathbf{\tau}^{\ast}\in\mathfrak{se}_{2}(3)\), then \(\exp(\mathbf{\tau}^{\ast})\) is as follows:\[\exp(\mathbf{\tau})= \begin{bmatrix}\exp(\mathbf{\theta}_{\times})&\mathbf{V}(\mathbf{\theta}) \mathbf{\xi}&\mathbf{V}(\mathbf{\theta})\mathbf{\rho}\\ \mathbf{0}_{1\times 3}&1&0\\ \mathbf{0}_{1\times 3}&0&1\end{bmatrix} \in SE_{2}(3),\ \mathbf{\tau}= \begin{bmatrix}\mathbf{\rho}\\ \mathbf{\xi}\\ \mathbf{\theta}\end{bmatrix} \in\mathbb{R}^{9}. \tag{14}\]  
\(\mathbf{V}(\mathbf{\theta})\) and \(\exp(\mathbf{\theta}_{\times})\) are calculated as  
\[ \begin{split}\mathbf{V}(\mathbf{\theta})=\mathbf{I}+\frac{1-\cos( \theta)}{\theta}[\mathbf{u}]_{\times}+\frac{\theta-\sin(\theta)}{\theta}[ \mathbf{u}]_{\times}^{2},\\ \exp(\mathbf{\theta}_{\times})=\mathbf{I}+\sin(\theta)[\mathbf{u}]_{ \times}+(1-\cos(\theta))[\mathbf{u}]_{\times}^{2},\end{split}  \tag{15}\]  
where \(\mathbf{\theta}=\theta\mathbf{u}\), \(\|\mathbf{u}\|=1\), and \(\left[\cdot\right]_{\times}\) is an operator that transforms a vector into a skew-symmetric matrix.