## 2 Power Spectral Density of Selected Disturbances
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Noise** \\ **Coefficients of Gyroscopes** \\ \end{tabular} } & \multicolumn{2}{c}{**Angular Velocity \(P\)**} & \multicolumn{2}{c}{**Angular Velocity \(Q\)**} & \multicolumn{2}{c}{**Angular Velocity \(R\)**} \\ \cline{2-9}  & **PSD Method** & **AV** & **GMWM** & **PSD** & **AV** & **GMWM** & **PSD** & **AV** & **GMWM** \\  & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** \\ \hline \hline \end{tabular}  \ \begin{tabular}{c} Measurements for gyroscopes during IMU self-heating \\ \hline N (ARW) & 0.00350 & 0.00305 & 0.00308 & 0.00400 & 0.00324 & 0.00328 & 0.00400 & 0.00337 & 0.00340 \\ \(\mathrm{[deg/s]}\) & 0.00200 & 0.00180 & 0.00185 & 0.00300 & 0.00212 & 0.00216 & 0.00250 & 0.00182 & 0.00182 \\ K (KRW) & 0.00085 & - & 0.0006 * & 0.00005 & 0.00140 & 0.00140 & 0.00040 & 0.00107 * & 0.0008 * \\ \hline \hline \end{tabular} 
\begin{tabular}{c} Measurements for gyroscopes at a fixed temperature of the IMU \\ \hline N (ARW) & 0.00400 & 0.00311 & 0.00316 & 0.00400 & 0.00330 & 0.00332 & 0.00400 & 0.00344 & 0.00346 \\ \(\mathrm{[deg/s]}\) & 0.00100 & 0.00047 & 0.00046 & 0.00050 & 0.00091 & 0.00091 & 0.00250 & 0.00112 & 0.00112 \\ K (RRW) & 0.00015 * & 0.00010 & 0.00008 & 0.0 Figure 11: One-sided PSD for \(10^{6}\) lateral and vertical acceleration samples (PCIM-01 IMU) at a stabilized sensor temperature value. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Accelerometer Noise Coefficients**} & \multicolumn{3}{c}{\(a_{s}\)} & \multicolumn{3}{c}{\(a_{s}\)} & \multicolumn{3}{c}{\(a_{s}\)} \\ \cline{2-10}  & **PSD Method** & **AV** & **GMWM** & **PSD** & **AV** & **GMWM** & **PSD** & **AV** & **GMWM** \\  & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** \\ \hline \hline \multicolumn{10}{c}{Measurements for accelerometers during IMU self-heating} \\ \hline N (VRW) & 0.00180 & 0.00128 & 0.00140 & 0.00160 & 0.00125 & 0.00140 & 0.00250 & 0.00169 & 0.00187 \\ \(\left[(m/s^{2})/\sqrt{s}\right]\) & 0.0021 & 0.00074 & 0.00029 & 0.00150 & 0.00057 & 0.00034 & 0.00500 & 0.00140 & 0.00031 \\ \(\left[(m/s^{2})/(s/\sqrt{s})\right]\) & 0.00080 & 0.00025 & 0.00025 & 0.00050 & 0.00018 & 0.00019 & 0.00490 & 0.00122 & 0.00125 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Accelerometer noise coefficients determined by the following methods: PSD, AV, and GMWM. Figure 10: One-sided PSD for \(10^{6}\) samples of longitudinal acceleration (PCIM-01 IMU) after device start (**top**) and with stabilized sensor temperature (**bottom**). Noise coefficients (sequence 1/2): K = 0.001, B = 0.0025, N = 0.002/K = -, B = 0.001, N = 0.0015. The PSD chart is useful for finding the time constant \(T\) of the complementary filter. Figures 12 and 13 present the one-sided PSD of an angle calculated with the use of data from a gyroscope and an ideal measurement of the correction angle. Ideal measurement of correction angle means that only errors of rate gyros are taken into account--properties of gyroscopes are crucial for AHRS behavior. The time constants \(T\) of the complementary filter were 1, 10, 100. Temperature was not stabilized, and bias was significant. As shown in Figures 12 and 13, PSD analysis is useful for the design of an algorithm for the desired system. If the time constant is too high, significant low-frequency (semi-constant) errors of estimated values occur. With a very small time constant, the whole system exhibits high levels of noise. The properties of a whole control system do not depend exclusively on the quality of sensors but are also significantly influenced by the applied algorithm. Figure 14 presents the PSD analysis for the calculated roll angle, with different time constants of the complementary filter. In contrast to Figures 12 and 13, the influence of all sensors was taken into account.