## 2 Problem Formulation and Modeling
### Time-Invariance of Motion Parameters  
At the beginning, an aircraft is employed as a representative case to investigate which parameters are treated as invariant during motion state extrapolation. Furthermore, a systematic evaluation of the benefits and limitations associated with these invariant parameters in dead reckoning applications is presented. Let \(P_{0}\) be the position vector of the extrapolation reference point in the world coordinate system, let \(V_{b}\) be the velocity vector in the body coordinate system, and let \(A_{b}\) represent the tangential acceleration. \(\Omega\) is the skew-symmetric matrix form (Lie algebra) of the angular velocity vector. The corresponding DR algorithm is shown in Equation (2). Obviously, \(V_{b}\), \(A_{b}\), and \(\Omega\) are the default invariants in dead reckoning algorithms. \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(\mathbf{P}(t)\) & Position vector in global coordinates \\ \([R]_{w\to b}\) & Rotation matrix from world frame to body frame \\ \(\Phi\) & Rotation vector (axis-angle representation) \\ \(\Omega\) & Skew-symmetric matrix of angular velocity \\ \(V_{b}\), \(A_{b}\) & Linear velocity/acceleration in body frame \\ \(\Delta t\) & Discrete time step increment \\ \((\dot{\cdot})\) & Time derivative operator \\ \(J\) & Total cost function \\ \(\Theta(\cdot)\) & Terminal state error penalty \\ \(F(\cdot)\) & Energy consumption function \\ \(X\) & State vector \\ \(A\), \(B\) & Coefficients for calculating intersection coordinates of two parametric circles. \\ \(U\) & Control input vector \\ \(x_{f}\), \(y_{f}\) & Expected endpoint coordinate position on a 2-D coordinate plane \\ \(q(t)\) & Bearing angle: \(\tan^{-1}\frac{y_{f}-y(t)}{x_{f}-x(t)}\) \\ \(r\) & Radial distance: \(\sqrt{(x_{f}-x(t))^{2}+(y_{f}-y(t))^{2}}\) \\ \(\theta_{f}\) & Final desired heading angle \\ \(t_{0}\), \(t_{f}\) & Initial and final time instants \\ \(\tau\) & Integration time variable \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols and notations used in this paper. \[[R]_{w\to b} =e^{[\Phi(t_{0}+\Delta t)]}[R_{b}]_{w\to b} \tag{2a}\] \[P(t_{0}+\Delta t) =P_{\mathrm{e}}+[R]_{w\to b}^{-1}\underbrace{\left(\int_{0}^{\Delta t} \varepsilon^{\mathrm{t}\Omega}V_{b}\mathrm{d}t+\int_{0}^{\Delta t}\varepsilon^{ \mathrm{t}\Omega}A_{b}\mathrm{d}t\right)}_{\text{displacement relative to }P_{\mathrm{e}}^{\mathrm{e}} \mathrm{d}t_{0}+\Delta t} \tag{2b}\]  
Among them, the exponential mapping in Equation (2a) is expanded as follows:  
\[e^{[\Phi(t_{0}+\Delta t)]}=\cos(|\omega|\Delta t)I+(1-\cos(|\omega|\Delta t)) \frac{\omega\omega^{\mathrm{T}}}{|\omega|^{2}}-\frac{\sin|\omega|\Delta t}{| \omega|}\Omega\]  
By setting a fixed state synchronization period to correct the dead reckoning error, a three-dimensional space track and dead reckoning trajectory can be obtained, as shown in Figure 1. The red trajectory curve is the actual motion path of the aircraft. The continuous yellow line segments are the predicted (extrapolated) trajectories at equal time intervals. The segments where the transition from the yellow to the red trajectories occurs are the hard synchronizations when the error threshold is surpassed. By projecting the trajectory onto a plane consisting of the vertical and x-axis, Figure 2 can be obtained, and the relationship between the line segments (the blue dashed lines) and the smooth trajectory (the black curves) calculated according to the second-order dead position can be seen. It can be seen that after the imminent large overload maneuver at \(x=18,\!000\) m, the maximum deviation in the vertical direction of dead reckoning reaches \(75\) m because the acceleration in the vertical direction is changing rapidly, even in reverse. Figure 1: DR algorithm applied to the aircraft. The red trajectory curve is the actual trajectory of the aircraft. The continuous yellow lines are the extrapolated segments at equal time intervals.