Aerospace and Electronic Systems - August 2018 - 31

As highlighted above, these measurement errors can in part be
explained by known (external) factors such as biases in the sensor
or their alignment with respect to a reference. However, an important part of these errors is often explained by the dependence of the
measurements with the previous measurements through time and,
hence, the stochastic properties of the different measurements play
an important role when integrating all the information within the
navigation filter. In the case of high-end sensors, these stochastic
noise properties are fairly easy to characterize while the low-cost
sensors have an extremely complex stochastic structure, which
requires more complicated models to characterize them. Figure 2
gives an example of the complexity of the stochastic noise. The
shape and the amount of the stochastic properties of the XSENS
MTi-G compared to the Navchip are clearly visible. Their individual characterization is important as they are not equal at all.
In order to understand and estimate the parameters of these
complex models, different (statistical) methods have been adapted
and/or developed. To cite a few, there is the Maximum Likelihood
approach estimated via the so-called Expectation-Maximization
algorithm [1]-[3], as well as a linear regression approach based
on the log-log-representation of a quantity called Allan Variance
(AV) [4]-[7]. However, these methods suffer from various limitations, including numerical instability, computational inefficiency,
and statistical inconsistency. For this reason, a recently proposed
approach has been used to build a new computational platform for
sensor calibration, which makes use of the quantity called Wavelet
Variance (WV) in order to deliver an estimation framework whose
name is Generalized Method of Wavelet Moments (GMWM) [8].
The latter method allows not only to estimate considerably complex stochastic models, but also to do so in a numerically stable,
computationally efficient, and statistically consistent manner for
any kind of signal or measurement.
In this article, we illustrate how the GMWM can be employed
through the newly developed computational platform provided by
the GMWM package (programmed in C++), which can be found in
the open-source statistical environment R. In order to facilitate the
use of this tool for practitioners interested in sensor calibration, a
Graphical-User-Interface (GUI) called gui4gmwm has been created.
To introduce these tools, we will highlight some basic statistical
theory behind the GMWM and its related features. These are then
put into practice to study and model the noise structure of the two
IMUs mentioned in Figure 1 and Figure 2 via the GUI available
online.
AUGUST 2018

THEORY AND IMPLEMENTATION OF THE GMWM
The WV, similarly to the AV, is commonly represented through a
log-log-plot. In general terms, to obtain this quantity, the observed
errors are subject to a kind of weighted-average over different
"scales" of observations (i.e., the averages are applied to a certain
number of observations at a time). If we refer to these scales with
the letter j, then the estimated (or empirical) WV, denoted as vˆ 2j , can
be directly calculated with the formula

νˆ 2j =

1
Mj

W
t =1

2
j,t

(1)

where Mj is the number of weighted-averages, formally called wavelet
coefficients and denoted as Wj,t, issued from the scale of decomposition
j. In general, the total number of scales is defined as J = log 2 (T )  − 1
with T indicating the total number of data points [9].
Supposing we have J scales (or levels) using a technique called
the maximum-overlap wavelet decomposition (which is a way of
applying the averages to the data) and using weights given by the
so-called Haar wavelet filter, then we can define the vector of em.
pirical WV as νˆ = νˆ 2j 
j =1,, J

The first step to explain the stochastic behavior of the errors is
to identify the kind of stochastic model that can best describe them.
This can be identified using the log-log plot of the WV; Figure 3
visualizes some of these models (their sum is represented with a
red-dotted line) that are also listed in Table 1. This plot shows that

Figure 2.

Comparison of the gyroscope noise-level between the MEMS IMUs
Navchip (blue) and the MTi-G (orange).

IEEE A&E SYSTEMS MAGAZINE

Aerospace and Electronic Systems - August 2018

Table of Contents for the Digital Edition of Aerospace and Electronic Systems - August 2018