## 3 Improved Rotational Modulation Scheme Based on Tri-Axis RINS
### Error Characteristics of Different Rotation Schemes
Although the equivalent drift \(\Delta\overset{\rightarrow}{\Delta\omega}_{is}^{n}=\Delta\overset{ \rightarrow}{\theta}_{is}^{n}/t\) will gradually decrease with time, it is still harmful to navigation accuracy, especially in the early stages of navigation. To be more visual, a simple numerical calculation is made below. We assume that the gyro scale factor error is 20 ppm and the installation error is \(10^{\prime\prime}\). Then, when \(T_{s}=1/2T\), the error angles caused by the scale factor errors and the installation errors of FOG in the geographic system are plotted in Figure 9, and the average values of the three rotation schemes are \([0,0,12.96^{\prime\prime}]^{T}\), \([5^{\prime\prime},-25^{\prime\prime},25.92^{\prime\prime}]^{T}\) and \([0,0,0]^{T}\). For example, for a 12-h navigation test, \(25.92^{\prime\prime}\) corresponds to a drift of about \(6.0\times 10^{-4\circ}/\text{h}\) and such an error would cover the advantage of Scheme B, especially for RINS with FOG. The smallest average value of the error angle of Scheme C caused by the gyroscope scale factor errors and the installation errors also means that a high demand for calibration accuracy is not required, which is one of the advantages of the improved rotational modulation scheme presented in this manuscript. Although Scheme C does not completely suppress gyro drifts like Scheme B, it is better at suppressing oscillatory errors caused by gyroscope scale factor errors and installation errors, which is very essential for FOG. Scheme C, proposed in this manuscript, is considered one of the best rotational modulation schemes for improving navigation performance. Such an advantage will be more evident for shorter navigation times, and simulations and experiments based on the actual specifications of the inertial devices can be designed to further illustrate the advantages of the new rotation scheme. It is important to note that the quantitative effects of each error cannot be accurately known, so the above analysis of the rotational modulation scheme proposed in this manuscript actually serves as a sensitivity analysis. According to Table 5, for Scheme A, the most important error source is the zero-bias error. For Scheme B, the gyro scale factor error and installation error have a significant influence. However, Scheme C is able to comprehensively suppress the influence of these three kinds of errors, making it the best choice for overall navigation performance among the three schemes. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Position** & **Oscillation Error Angles** \\ \hline Position 1–3 & (1): \([2\pi\Lambda K_{gx},0,0]^{T},[-2\pi\Lambda K_{gx},0,0]^{T}\) & (2): \([0,2\pi\Lambda K_{gy},0]^{T},[0,-2\pi\Lambda K_{gy},0]^{T}\) \\  & (3): \([0,0,\Delta K_{gx}\pi]^{T}\) \\ \hline Position 4–6 & (4): \([-2\pi\Lambda K_{gx},0,0]^{T},[2\pi\Lambda K_{gx},0,0]^{T}\) & (5): \([0,-2\pi\Lambda K_{gy},0]^{T},[0,2\pi\Lambda K_{gy},0]^{T}\) \\  & (2): \([2\pi\Lambda K_{gx},0]^{T},[-2\pi\Lambda K_{gy},0,0]^{T}\) & (6): \([0,0,-\Delta K_{gy}\pi]^{T}\) \\ \hline Position 7–9 & (7): \([2\pi\Lambda K_{gx},0,0]^{T},[-2\pi\Lambda K_{gx},0,0]^{T}\) & (8): \([0,2\pi\Lambda K_{gy},0]^{T},[0,-2\pi\Lambda K_{gy},0]^{T}\) \\  & (9): \([0,0,-\Delta K_{gx}\pi]^{T}\) \\ \hline Position 10–12 & (10): \([-2\pi\Lambda K_{gx},0,0]^{T},[2\pi\Lambda K_{gx},0,0]^{T}\) & (11): \([0,-2\pi\Lambda K_{gy},0]^{T},[0,2\pi\Lambda K_{gy},0]^{T}\) \\  & (12): \([0,0,\Delta K_{gx}\pi]^{T}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Error Angles in the n-frame of Scheme C.