a two-step computationally eﬃcient procedure for imu ... · procedure for imu classiﬁcation and...

A Two-Step Computationally EfficientProcedure for IMU Classification and Calibration

Gaetan Bakalli1

joint work withAhmed Radi2, Prof. Stephane Guerrier3,Yuming Zhang3, Dr. Roberto Molinari3

& Prof. Sameh Nassar2

1 University of Geneva2 University of Calgary

3 Pennsylvania State University

IEEE/ION PLANS , Monterey 2018

April 25, 2018G.Bakalli MGMWM April 25, 2018 1 / 29

Introduction Outline

Outline

1 Introduction1 Current Framework2 The GMWM

3 Near-Stationarity1 Motivation2 Definition3 Test and calibration4 Model Selection

4 Implementation1 mgmwm package and shiny

web-app2 Case Study

5 Conclusion

G.Bakalli MGMWM April 25, 2018 2 / 29

Introduction Current Framework

Wavelet Variance (Allan Variance) log-log plot

Scale τ

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Var

ianc

e ν2

Haar Wavelet Variance Representation

21 23 25 27 29 211

10-4

10-3

10-2

10-1

100

Empirical WV ν2

CI(ν2, 0.95)


Introduction GMWM

The GMWM estimator and its Model Selection Criterion

Definition: GMWM EstimatorThe GMWM estimator is the solution of the following optimizationproblem

θ = argminθ∈Θ

‖ν − ν(θ)‖2Ω,

in which Ω, a positive definite weighting matrix, is chosen in a suitablemanner such that the above quadratic form is convex.

The Wavelet Variance Information Criterion

WVIC = E[E0[‖ν0 − ν(θ)‖2

Ω

]],

where E[·] and E0[·] represent specific probabilistic expectations whichmeasure how well the WV implied by the estimated model fits the WVobserved on a future replicate.


Near-Stationarity Motivation

What we observe in reality

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ADIS 16405 IMU 100 Hz Gyroscope Y-Axis

10-1 100 101 102 103

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Run 1Run 2Run 3Run 4Run 5Run 6

Figure: Data coming from an ADIS16405 IMU.

Remarks:Several sequences ofthe error signalsissued from an IMU.



Current Assumption on IMU’s Error Signal

2



What we observe in reality

2


Near-Stationarity Definition

Near stationarity

ContextLower grade IMUs.Several recording of IMU error signal.Static conditions.

Near stationarityWe define a nearly stationary time series, as one which exhibit thefollowing properties:

1 Same model, but with different parameter values for eachsequences.

2 The vector of parameter θ has a probability distribution G(θ0),where E[θ] = θ0.

3 The distribution G(θ0) can be interpreted as the internal sensormodel, which may account for unobserved factors (e.g. temperature).


Near-Stationarity Test and calibration

Near-Stationarity test

2 2

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Near-Stationarity Test and calibration

Multisignal GMWM (MGMWM)

MGMWMConsidering k = 1, . . . ,K , the number replicates recorded from the same IMU in staticcondition, we define the Multisignal GMWM estimator as the solution of the followingminimization problem:

θ = argminθ∈Θ

‖ν − ν(θ)‖2Ω,

where ν represent the stack of WV computed and each sequences, and Ω a blockdiagonal weighting matrix.

MGMWM considering independent signals

θ = argminθ∈Θ

1K

K∑k=1

‖νk − ν(θ)‖2Ωk .

Let νjk , respectively νj (θ) be the j th elements of the vectors νk , the empirical WVcomputed on the k th replicates of size j = 1, . . . , Jk .


Near-Stationarity Model Selection

Back to the WVIC

The Wavelet Variance Information Criterion

WVIC = E[E0[‖ν0 − ν(θ)‖2

Ω

]],

where E[·] and E0[·] represent specific probabilistic expectations whichmeasure how well the WV implied by the estimated model fits the WVobserved on a future replicate.


Near-Stationarity Model Selection

WVIC with Multiple Signal Replicates


Implementation Package and Shiny web-app

mgmwm.smac-group.com


http://mgmwm.smac-group.com

Implementation Case Study

Empirical WV

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2


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Figure: Empirical WV of 6 replicates coming from an ADIS 16405 IMU gyroscopes Y-axisrecorded a 100hrz



First Model estimated

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Implied WVAR1AR1AR1RWWN

Figure: Empirical and Implied WV (in orange) of 6 replicates coming from an ADIS 16405 IMUgyroscopes Y-axis recorded a 100hrz. Model fitted is composed of 3 AR1 + RW + WN. Plainlines represent the contribution of each individual process.



Selected model trough WVIC

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Implied WVAR1AR1AR1

Model selected

Figure: Empirical and Implied WV (in orange) of 6 replicates coming from an ADIS 16405 IMUgyroscopes Y-axis recorded a 100hrz. Model selected is composed of 3 AR1. Plain linesrepresent the contribution of each individual process.



WVIC comparison for all nested models.

CV-WVIC (Model) - CV-WVIC (3*AR1)

CI for CV-WVIC

1 100 10000

2*AR1+RW

WN+3*AR1

WN+2*AR1+RW

2*AR1

3*AR1+RW

WN+3*AR1+RW

WN+2*AR1

WN+AR1+RW

AR1+RW

WN+AR1

AR1

WN+RW

WN

RW

|

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||

Figure: Comparison of the WVIC criteria for every model nested in 3 AR1 + RW + WN withthe selected model (log scale). The dots represent the value of the WVIC, the lines alongside itsrespective confidence interval. Model in green represent ”equivalent” models with less orequivalent model complexity.



Implied WV for equivalent models

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2*AR12*AR13*AR12*AR13*AR12*AR1+RW

2*AR13*AR12*AR1+RWWN+2*AR1+RW

2*AR13*AR12*AR1+RWWN+2*AR1+RWWN+2*AR1

Figure: Fit comparison for ”equivalent” models.



WVIC comparison for equivalent nested models.

CV-WVIC (Model) - CV-WVIC (3*AR1)

CI for CV-WVIC

-0.2 0.0 0.2 0.4 0.6

2*AR1+RW

WN+2*AR1+RW

2*AR1

WN+2*AR1

| |

| |

| |

| |

Figure: Comparison of the WVIC criteria for ”equivalent” model nested in 3 AR1 + RW + WNwith the selected model.


Implementation Uncertainty Evaluation

Uncertainty Evaluation: Near-Stationary Case

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Uncertainty Evaluation: Near-Stationary Case

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Confidence Intervals

0.9990 0.9992 0.9994 0.9996 0.9998

Sm

oot

hed

Den

sity

1



Confidence Intervals

0.17 0.18 0.19 0.20 0.21 0.22

Sm

oot

hed

Den

sity

3


Conclusions

Conclusion and Further Development

ConclusionsConcept of near stationarity.Framework for Multisignal calibration and model selection.Implementation in mgmwm package and web-app.

DevelopmentsSpeed-up the computation in the R package.Create a data-loader for various imu’s.Extend this set-up for GNSS data.


Conclusions

Related Presentations

Session A6An Optimal Virtual Inertial Sensor Framework using WaveletCross Covariance: Yuming Zhang, Pennsylvania State University,USA; Haotian Xu, University of Geneva, Switzerland; Ahmed Radi,University of Calgary, Canada; Roberto Molinari, Stephane Guerrier,Pennsylvania State University, USA; Naser El-Sheimy, University ofCalgary, Canada.Construction of Dynamically-Dependent Stochastic ErrorModels: Philipp Clausen, Swiss Federal Institute of Technology inLausanne, Switzerland; Samuel Orso, University of Geneva,Switzerland; Jan Skaloud, Swiss Federal Institute of Technology inLausanne, Switzerland; StÃľphane Guerrier, Pennsylvania StateUniversity, USA.


Conclusions Thanks!

Thank you very much for your attention!

A special thanks toProf. Naser El-Sheimy (U. of Calgary)Justin Lee (Penn State University)

Any questions?

More info...

SMAC-group.comgithub.com/[email protected]@SMAC_Group


SMAC-group.com

github.com/SMAC-Group

mailto:[email protected]

@SMAC_Group

Conclusions Thanks!

Two Estimators: Average GMWM vs MGMWM

Average GMWM vs MGMWM

We define θ as the Average GMWM (AGMWM) defined as:

θ = 1K

K∑k=1

θk ,

with θk = argminθk∈Θ ‖νk − ν(θk)‖2Ωk

. Remeber that we define theMultisignal GMWM (MGMWM) as

θ? = argminθ∈Θ

1K

K∑k=1‖νk − ν(θ)‖2

Ωk .


Conclusions Thanks!

Two Estimators: Average GMWM vs MGMWM

PropertiesIt turns out that the MGMWM appears far more appropriate than the AGMWM for twomain reasons:

The MGMWM is more efficient than the AGMWM, i.e.

tr(

minΩi var[θ])

tr(

minΩi var[θ?]) P7−−→ c > 1.

From Jensen inequality implies that

If ν(θ) is linear, i.e for stochastic processes WN, DR and QN, or ifG(θ0) is a Dirac function,

θ? − θP7−−→ 0.

If ν(θ) is not linear i.e for stochastic processes RW and AR1, than

θ? − θP7−−→ δ 6= 0.


a two-step computationally eﬃcient procedure for imu ... · procedure for imu classiﬁcation and...

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