## 3 Allan's Method of Variance  
Allan variance (AV) is a time-domain signal analysis method originally developed for oscillator stability studies [46; 47; 48]. Later, it was also adapted to analyze the stochastic drift properties of inertial sensors [23; 49; 50; 51]. In this method, the average relative deviation of the instantaneous value \(\overline{y}\) in the time interval \(\tau\) is determined (10), and then the standard deviation \(\sigma(\tau)\) is calculated in adjacent time intervals (10-13):  
\[\overline{y}_{i}=\frac{1}{\tau}\int_{t_{i}}^{t_{i+1}}x(t)dt\approx\frac{x_{i+ 1}-x_{i}}{\tau} \tag{10}\]  
where \(t_{i}=t_{0}+i\tau\), \(x_{i}=x(t_{i})\)'measurement data, \(t_{0}\)'start time. \[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{y}_{i+1}-\overline{y}_{i})}^{2}\rangle \tag{11}\]  
where \(\langle\ldots\rangle\)--mean value of the measurement set \(i=1\), \(2\), \(3\), \(...\), \(\infty\). The theory assumes a constant sampling time and an infinitely long measurement data vector. This condition is unachievable in practice and has to be approximated (12). However, taking into account the formula (11), the relation (13) is obtained, where the value \(N\) represents the length of the analyzed data sequence:  
\[\sigma(\tau)^{2}\cong\frac{1}{2(N-1)}\sum_{i=1}^{N-1}{(\overline{y}_{i+1}- \overline{y}_{i})}^{2} \tag{12}\]  
\[\sigma(\tau)^{2}\cong\frac{1}{2(N-1)\tau^{2}}\sum_{i=1}^{N-2}(x_{i+2}-2x_{i+1 }-x_{i})^{2} \tag{13}\]  
In the case of gyroscope studies, the Allan variance can be defined both with respect to the angular velocity \(\omega(t)\) and the angle \(\theta(t)\):  
\[\theta(t)=\int_{0}^{t}\omega(t)dt \tag{14}\]  
The average angular velocity for the period between \(t_{i}\) and \(t_{i+1}\) can be expressed by:  
\[\overline{\omega}_{i}(\tau)=\frac{\theta_{i+m}-\theta_{k}}{\tau} \tag{15}\]  
where \(\tau=m\tau_{0}\), \(\tau_{0}\)'sampling time. The Allan variance can be defined for the gyroscope data in the form of Relationship (16) or (17):  
\[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{\omega}_{i+m}-\overline{\omega }_{i})}^{2}\rangle \tag{16}\]  
\[\sigma(\tau)^{2}=\frac{1}{2\tau^{2}}\langle{(\theta_{i+2m}-2\theta_{i+m}+ \theta_{i})}^{2}\rangle \tag{17}\]  
\[\sigma(\tau)^{2}=\frac{1}{2(N-2m)\tau^{2}}\sum_{i=1}^{N-2m}(\theta_{i+2m}-2 \theta_{i+m}+\theta_{i})^{2} \tag{18}\]  
where \(N\)--length of the measurement set, \(m\)'number of samples analyzed in the range \(\tau\). When analyzing accelerometer data, the quantities \(\omega\) and \(\theta\) are replaced by \(a\) (acceleration) and \(v\) (linear velocity), respectively. In practice, the AV estimation is performed by recording a set of \(N\) samples. The set values are then corrected by removing the constant that occurs when the IMU device is turned on. After integrating the angular velocity \(\omega\) (or the acceleration a in the case of an accelerometer), AV is determined by (18) [51]. The variance AV of the quantization noise is described by equation (19). It was determined by substituting Function (4) with Formula (20), describing the relation between AV and power spectral density. Equation (20) allows the AV estimate to be derived from the power spectral density. The interpretation of this equation shows that the Allanvariance is proportional to the total noise power (in the case of gyroscope angular velocity measurements) filtered by the \(\sin 4(\pi f\tau)/(\pi f\tau)^{2}\) transition function. Note that the filter pass-band depends on the average period \(\tau\). This property makes it possible to separate disturbances of various natures by the appropriate selection of \(\tau\). Function (20) allows the determination of the parameters of stochastic disturbances, using the Allan variance plot directly.