Post on 31-Jul-2020
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
Wav
elet
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)
G.Bakalli MGMWM April 25, 2018 3 / 29
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.
G.Bakalli MGMWM April 25, 2018 4 / 29
Near-Stationarity Motivation
What we observe in reality
Scale(s)
Wav
elet
Var
iance
2
ADIS 16405 IMU 100 Hz Gyroscope Y-Axis
10-1 100 101 102 103
10-5
10-4
10-3
10-2
10-1
Run 1Run 2Run 3Run 4Run 5Run 6
Figure: Data coming from an ADIS16405 IMU.
Remarks:Several sequences ofthe error signalsissued from an IMU.
G.Bakalli MGMWM April 25, 2018 5 / 29
Near-Stationarity Motivation
Current Assumption on IMU’s Error Signal
2
G.Bakalli MGMWM April 25, 2018 6 / 29
Near-Stationarity Motivation
What we observe in reality
2
G.Bakalli MGMWM April 25, 2018 7 / 29
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).
G.Bakalli MGMWM April 25, 2018 8 / 29
Near-Stationarity Test and calibration
Near-Stationarity test
2 2
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Ha : H0 is false<latexit sha1_base64="VuD0dw1/P9GQrwGWEAe7WeBF82k=">AAACFnicbVC7SkNBEJ3r2/iKWmqxGASrcK+NjypoY6lgVEhCmLuZq4t7H+zOFcMlf2Hjn4iNhYqt2Nn5KW4SC18HFs6eM8PMnDDTyrLvv3sjo2PjE5NT06WZ2bn5hfLi0olNcyOpLlOdmrMQLWmVUJ0VazrLDGEcajoNL/f7/ukVGavS5Ji7GbViPE9UpCSyk9rlajNGvpCoi4NeG3fF968vmkzXXAhlRYTakui1yxW/6g8g/pLgi1Rqq3dHHwBw2C6/NTupzGNKWGq0thH4GbcKNKykpl6pmVvKUF7iOTUcTTAm2yoGd/XEulM6IkqNewmLgfq9o8DY2m4cusr+2va31xf/8xo5R9utQiVZzpTI4aAo14JT0Q9JdJQhybrrCEqj3K5CXqBByS7Kkgsh+H3yX1LfrO5Ug6OgUtuDIaZgBdZgAwLYghocwCHUQcIN3MMjPHm33oP37L0MS0e8r55l+AHv9RMGbaGi</latexit><latexit sha1_base64="S7E+Rs8RU9w0hfGeFOVGpYg72JQ=">AAACFnicbVDLSgNBEJz1bXxFPSoyGARPYdeLj1PQS44RjApJCL2T3mTI7IOZXjEsOfoHXvwT8eJBxat48xv8CSeJB40WDNRUddPd5SdKGnLdD2dicmp6ZnZuPrewuLS8kl9dOzdxqgVWRaxifemDQSUjrJIkhZeJRgh9hRd+92TgX1yhNjKOzqiXYCOEdiQDKYCs1MwX6yFQR4DKyv0mHPGfX5fXCa8p49LwAJRB3m/mC27RHYL/Jd43KZQ2708/b7buK838e70VizTEiIQCY2qem1AjA01SKOzn6qnBBEQX2lizNIIQTSMb3tXnO1Zp8SDW9kXEh+rPjgxCY3qhbysHa5txbyD+59VSCg4amYySlDASo0FBqjjFfBASb0mNglTPEhBa2l256IAGQTbKnA3BGz/5L6nuFQ+L3qlXKB2zEebYBttmu8xj+6zEyqzCqkywW/bAntizc+c8Oi/O66h0wvnuWWe/4Lx9AeM9owg=</latexit><latexit sha1_base64="S7E+Rs8RU9w0hfGeFOVGpYg72JQ=">AAACFnicbVDLSgNBEJz1bXxFPSoyGARPYdeLj1PQS44RjApJCL2T3mTI7IOZXjEsOfoHXvwT8eJBxat48xv8CSeJB40WDNRUddPd5SdKGnLdD2dicmp6ZnZuPrewuLS8kl9dOzdxqgVWRaxifemDQSUjrJIkhZeJRgh9hRd+92TgX1yhNjKOzqiXYCOEdiQDKYCs1MwX6yFQR4DKyv0mHPGfX5fXCa8p49LwAJRB3m/mC27RHYL/Jd43KZQ2708/b7buK838e70VizTEiIQCY2qem1AjA01SKOzn6qnBBEQX2lizNIIQTSMb3tXnO1Zp8SDW9kXEh+rPjgxCY3qhbysHa5txbyD+59VSCg4amYySlDASo0FBqjjFfBASb0mNglTPEhBa2l256IAGQTbKnA3BGz/5L6nuFQ+L3qlXKB2zEebYBttmu8xj+6zEyqzCqkywW/bAntizc+c8Oi/O66h0wvnuWWe/4Lx9AeM9owg=</latexit><latexit sha1_base64="S08iFJm3LbAsifM4mu3OaVySKdc=">AAACFnicbVC5TsNAEF1zEy4DJc2KCIkqsmk4qgialCARiJRY0XgzTlasD+2OEZGVv6DhV2goANEiOv6GTXARjiet9Pa9Gc3MCzMlDXnepzMzOze/sLi0XFlZXVvfcDe3rkyaa4FNkapUt0IwqGSCTZKksJVphDhUeB3enI3961vURqbJJQ0zDGLoJzKSAshKXbfWiYEGAlTRGHXhhE9/Pd4hvKOCS8MjUAb5qOtWvZo3Af9L/JJUWYnzrvvR6aUijzEhocCYtu9lFBSgSQqFo0onN5iBuIE+ti1NIEYTFJO7RnzPKj0epdq+hPhEne4oIDZmGIe2cry2+e2Nxf+8dk7RUVDIJMsJE/E9KMoVp5SPQ+I9qVGQGloCQku7KxcD0CDIRlmxIfi/T/5Lmge145p/4Vfrp2UaS2yH7bJ95rNDVmcNds6aTLB79sie2Yvz4Dw5r87bd+mMU/Zssx9w3r8A+KmfXQ==</latexit>
G.Bakalli MGMWM April 25, 2018 9 / 29
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 .
G.Bakalli MGMWM April 25, 2018 10 / 29
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.
G.Bakalli MGMWM April 25, 2018 11 / 29
Near-Stationarity Model Selection
WVIC with Multiple Signal Replicates
G.Bakalli MGMWM April 25, 2018 12 / 29
Implementation Package and Shiny web-app
mgmwm.smac-group.com
G.Bakalli MGMWM April 25, 2018 13 / 29
Implementation Case Study
Empirical WV
Scale(s)
Wav
elet
Var
iance
2
ADIS 16405 IMU 100 Hz Gyroscope Y-Axis
10-1 100 101 102 103
10-5
10-4
10-3
10-2
10-1
Run 1Run 2Run 3Run 4Run 5Run 6
Figure: Empirical WV of 6 replicates coming from an ADIS 16405 IMU gyroscopes Y-axisrecorded a 100hrz
G.Bakalli MGMWM April 25, 2018 14 / 29
Implementation Case Study
First Model estimated
Scale(s)
Wav
elet
Var
iance
2
ADIS 16405 IMU 100 Hz Gyroscope Y-Axis
10-1 100 101 102 103
10-5
10-4
10-3
10-2
10-1
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.
G.Bakalli MGMWM April 25, 2018 15 / 29
Implementation Case Study
Selected model trough WVIC
Scale(s)
Wav
elet
Var
iance
2
ADIS 16405 IMU 100 Hz Gyroscope Y-Axis
10-1 100 101 102 103
10-5
10-4
10-3
10-2
10-1
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.
G.Bakalli MGMWM April 25, 2018 16 / 29
Implementation Case Study
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.
G.Bakalli MGMWM April 25, 2018 17 / 29
Implementation Case Study
Implied WV for equivalent models
Scale(s)
Wav
elet
Var
iance
2
ADIS 16405 IMU 100 Hz Gyroscope Y-Axis
10-1 100 101 102 103
10-5
10-4
10-3
10-2
10-1
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.
G.Bakalli MGMWM April 25, 2018 18 / 29
Implementation Case Study
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
| |
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Figure: Comparison of the WVIC criteria for ”equivalent” model nested in 3 AR1 + RW + WNwith the selected model.
G.Bakalli MGMWM April 25, 2018 19 / 29
Implementation Uncertainty Evaluation
Uncertainty Evaluation: Near-Stationary Case
Scale(s)
Wav
elet
Var
iance
2
ADIS 16405 IMU 100 Hz Gyroscope Y-Axis
10-1 100 101 102 103
10-5
10-4
10-3
10-2
10-1
Run 1Run 2Run 3Run 4Run 5Run 6
G.Bakalli MGMWM April 25, 2018 20 / 29
Implementation Uncertainty Evaluation
Uncertainty Evaluation: Near-Stationary Case
Scale(s)
Wav
elet
Var
iance
2
ADIS 16405 IMU 100 Hz Gyroscope Y-Axis
10-1 100 101 102 103
10-5
10-4
10-3
10-2
10-1
Run 1Run 2Run 3Run 4Run 5Run 6
G.Bakalli MGMWM April 25, 2018 21 / 29
Implementation Uncertainty Evaluation
Confidence Intervals
0.9990 0.9992 0.9994 0.9996 0.9998
Sm
oot
hed
Den
sity
1
G.Bakalli MGMWM April 25, 2018 22 / 29
Implementation Uncertainty Evaluation
Confidence Intervals
0.17 0.18 0.19 0.20 0.21 0.22
Sm
oot
hed
Den
sity
3
G.Bakalli MGMWM April 25, 2018 23 / 29
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.
G.Bakalli MGMWM April 25, 2018 24 / 29
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.
G.Bakalli MGMWM April 25, 2018 25 / 29
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.
G.Bakalli MGMWM April 25, 2018 26 / 29
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/SMAC-Groupgaetan.bakalli@unige.ch@SMAC_Group
G.Bakalli MGMWM April 25, 2018 27 / 29
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 .
G.Bakalli MGMWM April 25, 2018 28 / 29
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.
G.Bakalli MGMWM April 25, 2018 29 / 29