Aerospace and Electronic Systems - August 2018 - 32

Online Calibration Platform with Applications to Inertial Sensors
Given that these quantities should be in some sense close to
each other, the GMWM attempts to inverse the mapping between
the WV and the model parameters θ by finding the solution to the
following minimization problem:

θˆ = arg min (νˆ −ν (θ ) ) Ω (νˆ −ν (θ ) ) ,
T

θ ∈Θ

Figure 3.

Example log-log-plot showing the influences of the different models
with their shape. Red dotted: sum of theoretical noise models ν(θ).

it is relatively simple to detect and associate one type of model,
but, when several models jointly explain the errors, their identification becomes more problematic. Going back to Table 1, this
collects all the models that the aforementioned software can deal
with; in particular, the last column shows the parameters that we
are interested in estimating (i.e., the values that in some way explain the behavior of the measurements) that we have denoted as θ.
Once a model is identified by the user, it is possible to obtain
a known form for the WV (called theoretical WV), which depends
on the parameters θ and which will be denoted as:

ν 2j (θ ) = Var W j,t (θ )  ,

(2)

where Wj,t(θ) represents the wavelet coefficients issued from the
jth scale of the wavelet decomposition, which are a function of the
parameter vector θ, whereas Var[·] represents the variance operator (for more details see [10] where a similar notion is discussed
for the AV). With this in mind, given a certain model (defined by
the user) and its parameter values, it is therefore possible to obtain
the theoretical WV ν 2j (θ ) . However, in reality the parameter vector θ is obviously unknown, whereas the only quantity that can be
observed is the estimated (or empirical) WV vˆ which should be
"close" to the vector of theoretical model-based WV denoted as
ν (θ ) = ν 2j (θ ) 
whose parameters are unknown.
j =1,..., J

Table 1.

List of Models (Acronym and Process Name) that
Can be Included in a Complex Composite Model
Describing the Stochastic Error Signal
Acronym

Process Name

Slope

Parameter

Quantization
Noise

-1

White Noise

-0.5

σ2

Random Walk

+0.5

γ2

Drift

Gauss-Markov

[-0.5,
0.5]

β, ς2

where Ω is a positive-definite weighting matrix chosen in an appropriate manner [8]. Hence, the GMWM tries to find the values of
θ that allow the theoretical WV (defined by the user model) to be
close to the empirical (observed) WV. The statistical properties of
the GMWM estimator θˆ have been proven to show that the estimator is consistent and asymptotically normally distributed.
Using the properties of the estimator, the GMWM allows to
deliver a series of very useful tools for the sensor calibration procedure [11], [12]:
1. a large variety of noise-models mentioned in Table 1 can be
combined to deliver the needed complex models;
2. a so-called robust version of the GMWM allows to perform the
same procedure also when the observed data is affected by disturbances that could otherwise negatively impact the analysis
and estimation;
3. confidence intervals for the estimated parameters which provide a range of values within which there is a high probability
of finding the true parameter θ of interest;
4. a goodness-of-fit test that allows to determine if the estimated
model well describes the observed errors;
5. the Wavelet Variance Information Criterion (WVIC) which allows to classify different models and determine which model is
the best in terms of prediction;
6. an automatic model selection procedure based on the WVIC.

EXAMPLE: IMU SENSOR DATA
An applied example will now guide the reader through several features of
the platform and their usage in the GUI (Figure 4). The implementation
of the methods and features presented in the previous section are shown
in more detail from a practical point of view. In a first step, the data is selected, its empirical WV is thus calculated and plotted. In the latter plot it
is also possible to overlay the noise-model defined in the datasheet, which
is the (simple) model that is identified and estimated by the supplier of
the IMU. In a second step, the underlying model is manually identified
through a visual comparison generally represented by Figure 3. Once the
model is identified, the parameters of the latter are estimated and the costvalue verified. These steps are iterative as the user keeps a certain flexibility in the choice of the models. Nonetheless, the final model selection will
be made at the end based on the aforementioned criteria.

GUI OVERVIEW

NOTE: The slope is referring to the log-log plot of Figure 3.

(3)

When starting the program, the user will have a general view on the
GUI as shown in Figure 4. The window is divided into four parts.
The first part, located in the upper half, displays the graphical and

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Aerospace and Electronic Systems - August 2018

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