# Three-Dimensional Dead-Reckoning Based on Lie Theory for Overcoming Approximation Errors
## 2 Limitation of Previously Used Dead-Reckoning Approach  
In this section, we describe the previously widely used DR approaches and discuss their limitations. The state at time \(t\) consists of the position \(\mathbf{p}_{t}\), velocity \(\dot{\mathbf{p}}_{t}\), and attitude \(\boldsymbol{\theta}_{t}\) as follows:  
\[\mathbf{X}_{t}= \begin{bmatrix}\mathbf{p}_{t}&\dot{\mathbf{p}}_{t}&\boldsymbol{ \theta}_{t}\end{bmatrix} ^{\mathsf{T}} \tag{1}\]  
In Equation (1), the position \(\mathbf{p}_{t}\) and velocity \(\dot{\mathbf{p}}_{t}\) of the vehicle belong to a three-dimensional real vector space. Both position and velocity are described in the east-north-up (ENU) coordinate frame. \[\mathbf{p}_{t}= \begin{bmatrix}x_{t}&y_{t}&z_{t}\end{bmatrix} ^{\mathsf{T}}. \tag{2}\]  
\[\dot{\mathbf{p}}_{t}= \begin{bmatrix}\dot{x}_{t}&\dot{y}_{t}&z_{t}\end{bmatrix} ^ {\mathsf{T}}. \tag{3}\]  
The attitude \(\boldsymbol{\theta}_{t}\) is expressed as Euler angles as follows:  
\[\boldsymbol{\theta}_{t}= \begin{bmatrix}\boldsymbol{\phi}_{t}&\boldsymbol{ \theta}_{t}&\boldsymbol{\psi}_{t}\end{bmatrix} ^{\mathsf{T}}. \tag{4}\]  
Previously widely used DR predicts the pose and velocity of a vehicle through a motion model from the state \(\mathbf{X}_{t_{k}}\) and the measured input \(\boldsymbol{U}_{t_{k}}\) as follows:  
\[\mathbf{X}_{t_{k+1}}=f(\mathbf{X}_{t_{k}},\boldsymbol{U}_{t_{k}})+\mathbf{w} \tag{5}\]  
where \(f(\cdot)\) denotes a time-varying nonlinear function, and \(\mathbf{w}\) is the process noise. The expression \(\boldsymbol{U}_{t_{k}}\) is the measured input and is expressed as follows:  
\[\boldsymbol{U}_{t_{k}}= \begin{bmatrix}\mathbf{v}_{t_{k}}\\ \mathbf{a}_{t_{k}}\\ \boldsymbol{\omega}_{t_{k}}\end{bmatrix} . \tag{6}\]  
In Equation (6), \(\mathbf{v}_{t_{k}}= \begin{bmatrix}u_{t_{k}}&v_{t_{k}}&w_{t_{k}}\end{bmatrix} ^ {T}\) is the linear velocity of the vehicle, \(\mathbf{a}_{t_{k}}= \begin{bmatrix}a_{xt_{k}}&a_{y_{t_{k}}}&a_{zt_{k}}\end{bmatrix}  ^{T}\) is the acceleration of the vehicle, and \(\boldsymbol{\omega}_{t_{k}}= \begin{bmatrix}p_{t_{k}}&q_{t_{k}}&r_{t_{k}}\end{bmatrix}  ^{T}\) is the angular velocity of the vehicle. The position, velocity, and attitude of the vehicle at the \(k+1\)th time step can be calculated as follows:  
\[\mathbf{X}_{t_{k+1}}=\mathbf{X}_{t_{k}}+\delta\mathbf{X}_{t_{k}} \tag{7}\]  
Equation (7) is rephrased as  
\[ \begin{bmatrix}\mathbf{p}_{t_{k+1}}\\ \dot{\mathbf{p}}_{t_{k+1}}\end{bmatrix} = \begin{bmatrix}\mathbf{p}_{t_{k}}\\ \dot{\mathbf{p}}_{t_{k}}\end{bmatrix} + \begin{bmatrix}\delta\dot{\mathbf{p}}_{t _{k}}\\ \delta\dot{\mathbf{p}}_{t_{k}}\\ \delta\boldsymbol{\theta}_{t_{k}}\end{bmatrix}  \tag{8}\]  
The change in position, \(\delta\mathbf{p}_{t_{k}}\), is calculated  
where  
\[ \begin{split}\textbf{R}_{k_{k}}\text{(1,1)}&=\cos(\theta_{t _{k}})\cdot\cos(\psi_{t_{k}})\\ \textbf{R}_{t_{k}}\text{(1,2)}&=\sin(\phi_{t_{k}}) \cdot\sin(\theta_{t_{k}})\cdot\cos(\psi_{t_{k}})-\cos(\phi_{t_{k}})\cdot\sin( \psi_{t_{k}})\\ \textbf{R}_{t_{k}}\text{(1,3)}&=\cos(\phi_{t_{k}}) \cdot\sin(\theta_{t_{k}})\cdot\cos(\psi_{t_{k}})+\sin(\phi_{t_{k}})\cdot\sin( \psi_{t_{k}})\\ \textbf{R}_{t_{k}}\text{(2,1)}&=\cos(\theta_{t_{k}}) \cdot\sin(\psi_{t_{k}})\\ \textbf{R}_{t_{k}}\text{(2,2)}&=\sin(\phi_{t_{k}}) \cdot\sin(\theta_{t_{k}})\cdot\sin(\psi_{t_{k}})+\cos(\phi_{t_{k}})\cdot\cos( \psi_{t_{k}})\\ \textbf{R}_{t_{k}}\text{(2,3)}&=\cos(\phi_{t_{k}}) \cdot\sin(\theta_{t_{k}})\cdot\sin(\psi_{t_{k}})-\sin(\phi_{t_{k}})\cdot\cos( \psi_{t_{k}})\\ \textbf{R}_{t_{k}}\text{(3,1)}&=-\sin(\theta_{t_{k}}) \\ \textbf{R}_{t_{k}}\text{(3,2)}&=\sin(\phi_{t_{k}}) \cdot\cos(\theta_{t_{k}})\\ \textbf{R}_{t_{k}}\text{(3,3)}&=\cos(\phi_{t_{k}}) \cdot\cos(\theta_{t_{k}})\\ \end{split} \]  
The change in velocity, \(\delta\dot{\textbf{p}}_{t_{k}}\), is calculated from the acceleration \(\textbf{a}_{t_{k}}\) and \(\triangle t\).