# Automatic Estimation of Dynamic Lever Arms for a Position and Orientation System
## 4 Simulation Analysis
### Simulation Results of Lever Arm Estimation  
We use the reduced-order filter model with parameters set as in Appendix C to estimate the lever arms and position errors in two different trajectories. The simulation results of Trajectory 1 are illustrated in Figure 3, while the simulation results of Trajectory 2 are given in Figure 4.  
Figure 3a shows the estimation errors of lever arm \(L_{OI}^{b}\), Figure 3b,c illustrate the estimation errors of lever arm \(L_{OG}^{b_{c}}\) and dynamic lever arm \(L_{IG}^{b_{c}}\), while Figure 3d gives the position errors of SINS/GPS integrated navigation. As shown in Figure 3a, the heading gimbal of ISP rotates by \(90^{\circ}\) at 30 s, and the \(x\) and \(y\) components of lever arm \(L_{OI}^{b}\) are estimated correctly. When the roll gimbal of the ISP rotates by \(30^{\circ}\) at 60 s, the remaining component \(L_{OI}^{bz}\) is identified quickly. It can be seen that the rotations of ISP gimbals can distinguish the lever arm \(L_{OI}^{b}\), but have no effect on lever arm \(L_{OG}^{b_{c}}\). Therefore, there are still significant position errors in integrated navigation before 90 s as shown in Figure 3d. Once the vehicular attitude changes at 90 s and 120 s, the combined lever arm \(L_{IG}^{b_{c}}=L_{OG}^{b_{c}}-\mathbf{C}_{U}^{b_{c}}L_{OI}^{b}\) is estimated as in Figure 3c and the position errors converge immediately as shown in Figure 3d. Since the lever arm \(L_{OI}^{b}\) is identified, the constant lever arm \(L_{OG}^{b_{c}}\) is determined with the convergence of lever arm \(L_{IG}^{b_{c}}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Error Types** & **Error Sources** & **Value (Root Mean Square, RMS)** \\ \hline \multirow{8}{*}{Initial Errors} & Initial attitude errors & [0.01; 0.01; 0.1]\({}^{\circ}\) \\  & Initial velocity errors & [0.1; 0.1; 0.1] m/s \\  & Initial position errors & [5; 5; 5] m \\  & Gyro biases & 0.01\({}^{\circ}\)/h \\  & Accelerometer biases & 30 \(\mu\)g \\  & Scale factor errors of gyros & 3 ppm \\  & quadrature errors of gyros & 3 arc second \\  & Scale factor errors of accelerometers & 20 ppm \\  & quadrature errors of accelerometers & 3 arc second \\ \hline \multirow{2}{*}{Process noises} & Gyro random walk & 0.001\({}^{\circ}\)/\(\sqrt{\text{h}}\) \\  & Accelerometer random walk & 5 \(\mu\)g/\(\sqrt{\text{Hz}}\) \\ \hline \multirow{2}{*}{Measurement noises} & Positioning errors of CDGPS & 0.01 m (horizontal); 0.015 m (vertical) \\  & Encoder measurement noise & 0.05\({}^{\circ}\) \\ \cline{1-1}  & Time-asynchrony noises & 0.1 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specifications of the sensors. Figure 2: The simulated angular motion trajectories and (**a**) Trajectory 1; (**b**) Trajectory 2. Figure 4: The simulation results of Trajectory 2 and (**a**) estimation errors of the lever arm \(L_{OI}^{b}\); (**b**) estimation errors of the lever arm \(L_{OG}^{b}\); (**c**) estimation errors of the combined dynamic lever arm \(L_{IG}^{b}\); (**d**) position errors of SINS/GPS integrated navigation. Figure 3: The simulation results of Trajectory 1 and (**a**) estimation errors of the lever arm \(L_{OI}^{b}\); (**b**) estimation errors of the lever arm \(L_{OG}^{b}\); (**c**) estimation errors of the combined dynamic lever arm \(L_{IG}^{b}\); (**d**) position errors of strapdown inertial navigation system (SINS)/Global Positioning System (GPS) integrated navigation. As shown in Figure 4c, the combined lever arm \(L_{IG}^{b}\) is identified when the vehicular heading turns at 30 s and the roll rotates at 60 s. Clearly, the change of attitude matrix \(\mathbf{C}_{b}^{n}\) is helpful to estimate the equivalent lever arm \(L_{IG}^{b}\). Even though the constant lever arms \(\mathbf{L}_{OI}^{b}\) and \(\mathbf{L}_{OG}^{b_{c}}\) cannot be separated, it does not affect the positioning accuracy of SINS/GPS, as shown in Figure 4d. As the ISP gimbals begin to rotate at 90 s and 120 s, the lever arms \(\mathbf{L}_{OI}^{b}\) and \(\mathbf{L}_{OG}^{b_{c}}\) are further separated, as shown in Figure 4a,b.