allanvar (version 1.1)

avari: Allan Variance (from integrated values).

Description

The function avari computes the Allan Variance of a set of values with a given constant sampling frequency. The diferent with avar function is that the input values are the integral value of sensor output (i.e: rate/acceleration). That means angle from gyros and velocity from accelerometers. In this version of the Allan variance computation the number and size of cluster n has been computed as in avar function $n=2^l, l=1,2,\ldots$(Allan 1987).

Usage

avari(values, freq)

Arguments

values

Integration of the calibrated sensor output (i.e: angel or velocity)

freq

Sampling frequency rate in Hertz

Value

• Return an object of class data.frame containing the Allan Variance computation with the following fields:
• time
  Value of the cluster time.
• av
  The Allan variance value: variance among clusters of same size.
• error
  Error on the estimation: Uncertainty of the value.

Details

The Allan variance can also be defined either in terms of the output rate as defined in avar by $\Omega(t)$ or using the output angle/velocity as this function does. Defining:

$$\theta(t) = \int^t \Omega(t^{'})dt^{'}$$

The lower integration limit is not specified as only angle differences are employed in the definition. Angle measurement are made at discrete times given by $t = nt_0, n = 1,2,\ldots, N$. The notation is simplify by writing $\theta_k = \theta (k t_0)$. The cluster average is now defined as:

$\bar{\Omega}_k(\tau)= (\theta_{k+n} - \theta_k)/\tau$ and $\bar{\Omega}_{k+1}(\tau)= (\theta_{k+2n} - \theta_{k+n})/(\tau)$

The equivalent Allan Variance estimation is defined as:

$$\theta^2(\tau) = 1/(2\tau^2 (N - 2n)) \sum_{k=1}^{N-2n} [\theta_{k+2n} - 2\theta_{k+n} + \theta_k ]^2$$

References

Allan, D. W. (1966) Statistics of Atomic Frequency Standards Proceedings of IEEE, vol. 54, no. 2, pp. 221-230, Feb, 1966.

IEEE Std 952-1997 IEEE Standard Specification Format Guide and Test Procedure for Single Axis Interferometric Fiber Optic Gyros.

El-Sheimy, N.; Haiying Hou.; Xiaoji, Niu (2008) Analysis and Modeling of Inertial Sensors Using Allan Variance IEEE Transaction on Instrumentation and Measurement.

Examples

#Load data 
data(gyroz)

#Allan variance computation using avari
#Simple integration of the angular velocity
igyroz <- cumsum(gyroz@.Data * (1/frequency(gyroz)))
igyroz <- ts (igyroz, start=c(igyroz[1]), delta=(1/frequency(gyroz)))
avigyroz <- avari(igyroz@.Data, frequency(igyroz))
plotav(avigyroz)
abline(1.0+log(avigyroz$time[1]), -1/2, col="green", lwd=4, lty=10)
abline(1.0+log(avigyroz$time[1]), 1/2, col="green", lwd=4, lty=10)
legend(0.11, 1e-03, c("Random Walk"))
legend(25, 1e-03, c("Rate Random Walk"))