View
218
Download
1
Category
Preview:
Citation preview
0278-0046 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2701783, IEEETransactions on Industrial Electronics
Signal Processing in Cyber-Physical MEMS Sensors:
Inertial Measurement and Navigation Systems Sergey Edward Lyshevski, IEEE Member
Department of Electrical and Microelectronic Engineering,
Rochester Institute of Technology, Rochester, NY 14623, USA
E-mail: Sergey.Lyshevski@mail.rit.edu URLs: http://people.rit.edu/seleee www.rit.edu/kgcoe/staff/sergey-lyshevski
Abstract – Motivated by industry needs, this paper focuses
on statistical models, descriptive probabilistic data analysis and
data-prescriptive signal processing in smart inertial sensors.
These multimode sensors combine physical and cyber
components such as solid-state and micromachined motion
sensing elements, processing and interfacing integrated circuits,
middleware, software, etc. We develop consistent algorithms and
tools based upon cross-cutting engineering science along with
substantiation and validation. Fundamental, applied and
experimental results are reported. Our multidisciplinary findings
advance fundamental knowledge and enhance transformative
technologies. We empower synergetic system-level integration of
diverse device physics with descriptive, predictive and
prescriptive analyses. This paper contributes to design and
deployment of next generation of smart sensors which utilize
front-end microelectronic, microelectromechanical system and
processing technologies.
Index Terms – inertial sensors; MEMS; navigation systems;
signal processing; smart sensors
I. INTRODUCTION
Low-power solid-state and micromachined smart sensors
with integrated circuits (ICs) are widely used in aerospace,
automotive, communication, energy, healthcare,
manufacturing, medical, naval, navigation, robotic, security,
virtual reality and other systems [1-4]. These sensors should
meet the IEEE-1451, MIL-STD-1553, MIL-STD-1760 or
other standards. Affordability, enhanced functionality,
compliance, enabled integration and multiple sensing
modalities are empowered by microelectromechanical systems
(MEMS) [3, 5, 6]. The hierarchical spatiotemporal distributed
smart sensor arrays are the key components of cyber-physical
systems (CPS) which enable adaptability, interoperability,
modularity, reliability, resiliency, scalability and usability [7].
The integrated MEMS-technology sensors ensure compliance
and consistency in CPS, internet of things (IoT) systems, and
supervisory control and data acquisition (SCADA) systems.
Data analytics is of importance to quantitatively evaluate
data quality, enable data fusion, support information
management, ensure situation awareness, support cognition,
etc. Data validity, data conformity and data completeness are
enabled by smart sensors which ensure the overall system-
level reasoning, decision and control in high-assurance CPS.
Multiple sensing modalities are assured by multi-degree-of-
freedom MEMS-technology inertial measurement units
(IMUs). These IMUs with system-on-chip high-performance
microcontrollers constitute inertial navigation systems (INSs).
The IMUs and INSs are the most advanced and complex smart
cyber-physical sensors. Functionality, performance and
capabilities improvements by IMUs and INSs are of a
particular importance for control, communications,
intelligence, surveillance, target acquisition and
reconnaissance (C2ISTAR) platforms. These applications
imply a broad range of strengthen specifications and
requirements. The cyber-physical sensors perform sensing,
data fusion and data processing by MEMS, solid-state sensors,
application-specific ICs (ASICs) and software solutions.
The spanned multimode smart sensors are facing
formidable integration, data fusion and other challenges. We
consider IMUs which comprise of a triaxial accelerometer,
gyroscope and magnetometer, as well as pressure sensor
(altimeter) [6, 8-15]. These IMUs are used in various
applications, including when global positioning system (GPS)
signals are unavailable, jammed or disturbed by interferences.
For C2ISTAR platforms, IMUs may enable uncompromised
navigation, guidance, control and other tasks reducing
dependence on GPS [16-20]. Redundant hybrid navigation
systems use IMUs. Adequate performance and capabilities
must be guaranteed. Advanced sensing and microfabrication
technologies, processing schemes and software-hardware
design lead to high-confidence systems advancing Internet of
Things (IoT) and CPS [21]. High-assurance CPS and IoT
imply adequate functionally, security and safety. Semi-
automated checkable hardware-specific and physical-layer-
consistent software are under developments.
To advance the medium- and long-term microelectronics-
MEMS technology for smart sensors and distributed
networked sensor arrays, there is a need to extend knowledge
and develop signal processing paradigms which are consistent
with device physics and data domains. The commercialized
proprietary signal processing schemes are device-specific.
While the Kalman, Wiener and other filters are used in some
sensors, many concepts cannot be adequately applied to IMUs
and INSs. The statistical signal processing schemes may be
enabled by the proposed data-descriptive, predictive and
prescriptive adaptive signal processing inroad. Practical
solutions of the addressed signal processing problems enable
MEMS-technology smart sensor, thereby contributing to high-
confidence information management, decision making,
integrity monitoring, reliability, authentication, data
aggregation, estimation, calibration and characterization.
Formative statistical models are developed. By fostering
probabilistic concepts and tools, we enable data-analytic
signal processing applied to filtering, data fusion and data
acquisition. For the proposed Inertial Measurement Platform
with IMU–Microcontroller (IMU–C), the experimentally-
substantiated findings enable spatial and temporal resolutions,
as well as ensure accuracy and precision.
0278-0046 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2701783, IEEETransactions on Industrial Electronics
II. INERTIAL MEASUREMENT SENSORS
The physical quantities y are measured by sensing
elements yielding ȳ. In cyber-physical sensors, ȳ is processed
by ASICs which realize analog-to-digital and digital-to-analog
conversions, filtering, compensation, estimation and other
tasks, yielding the output vector ŷ. The inertial sensors exhibit
multispectral noise, cross-axis coupling, nonuniformity,
temperature sensitivity, misalignment and other impediments
[3-6, 21]. The sensors outputs ȳ and IMU outputs ŷ are
superimposed with noise n, errors and distortions of
different origin. The measurement and processing schemeOutputs IMUFusion Data
Processing ASICs
OutputsSensor
Quantities MeasuredQuantities Physical Acting
yyy yields the
measurements ȳ=y+nȳ+ȳ+ȳ and outputs ŷ=y+nŷ+ŷ+ŷ.
Noise n, errors and distortions are due to
transductions, quantization, sampling, nonlinearities, bias,
drift, interference and other phenomena in sensing elements,
microelectronic devices, ASICs, interconnect, etc. The IMU
outputs ŷ can be further processed yielding ỹ. This implies the
design of IMUs and INSs with additional microcontroller (C)
implementing the following sensing and processing scheme Outputs μC-IMU Analytics Data
Processing μC
Outputs IMUFusion Data
Processing ASICs
OutputsSensor
Quantities MeasuredQuantities Physical Acting
~ˆ yyyy (1)
with ȳ=y+nȳ+ȳ+ȳ, ŷ=y+nŷ+ŷ+ŷ and ỹ=y+nỹ+ỹ+ỹ.
Data-centric processing calculus and practical algorithms
are needed to attenuate noise n, minimize error and
minimize distortions . The information quality is affected by
sensing paradigms, device physics, microelectronics,
fabrication technologies, processing calculus, etc. Low-power
consumer, industrial and military-grade IMUs and INSs are
designed and fabricated by Analog Devices, Bosch Sensortec,
Epson Electronics, Fairchild Semiconductor, Honeywell,
InvenSense, Northrop Grumman, Panasonic, Silicon Sensing,
STMicroelectronics, Systron Donner and others. These IMUs
and INSs are the core components of aerospace, automotive,
marine, robotic, surgical and others inertial guidance,
navigation and position systems. Accuracy, precision,
resolution, bandwidth and information management are of
importance in hybrid navigation systems of aircraft, missiles,
satellites, submarines, unmanned aerial vehicles, autonomous
ground vehicles, manipulators, etc. The navigation-grade
IMUs imply the arcminute/s and g accuracy, stability, high
bandwidth and linearity.
Noise attenuation, error minimization and information
losses reduction enable intelligence, mission effectiveness,
functional verification, reasoning, situation awareness,
cognition, etc. Consider multi-degree-of-freedom inertial
sensors illustrated in Figures 1. Due to Newton, Coriolis,
centrifugal, Euler and other forces which act on proof-masses,
the resulting linear accelerations and angular velocities
y=[(ax,ay,az),(,,)] are measured by sensing elements
which yield )],,(),,,[( zyx aaay . Using the signal
processing algorithms, the vector ȳ is processed by IMU’s
ASICs. The IMU outputs are )]ˆ,ˆ,ˆ(),ˆ,ˆ,ˆ[(ˆ zyx aaay . In
MEMS-technology accelerometers and gyroscopes, reported
in Figures 1, analog, digital and hybrid ASICs implement
different proprietary processing calculi and signal processing
algorithms. The used linear time-invariant filters, estimators
and observers may not ensure optimality due to nonlinearities,
heteroscedasticity, nonstationarity, etc. [12,17]. Our goal is to
enable overall capabilities of the strategic-, navigation-,
tactical- and consumer-grade IMUs by data-analytic post-
processing of ŷ yielding )]~,~,~(),~,~,~[(~ zyx aaay .
(a) (b) Figure 1. (a) Two-axis Analog Devices ADXL210EB and EVAL-ADXRS450
iMEMS-technology accelerometer and gyroscope which exhibit nonlinearities
and multimodal distributions; (b) InvenSense MEMS-technology IMU MPU-9250 to measure
y=[(ax,ay,az),(,,),(Bx,By,Bz)], and, process measurements on-chip yielding
the IMU output )]ˆ,ˆ,ˆ(),ˆ,ˆ,ˆ(),ˆ,ˆ,ˆ[(ˆzyxzyx BBBaaa y .
III. STATISTICAL MODELS AND QUANTITATIVE ANALYSIS
The formative analysis is applied to examine physical and
processed data. This will enable dynamic range, data accuracy,
data conformity, data completeness and data validity. For the
IMU output ŷ, the sample space is finite. A statistical model is
a set of probability models defined by a pair (S,P), where S is
the set of observations, and, P is a set of probability
distributions over the sample space S. An identifiable
parametric statistical model P={P:},2121 PP
should be parametrized by finding the unknown , where
d is the parameter space (set of model parameters), and,
d is an integer of the model dimension.
A parametric statistical model P is characterized by a
family F={fX(x;):} of probability density functions
(pdf) fX(x;) defined on S. Letting F be the family of normal
N(,2) distributions, one has X~N(,2
). The compact
parametric model is F={N(,2):,2
>0}.
The notation X~DP() means that X is a DP-distributed
random variable, and, X is fully characterized by P or F. A
compact, parametric and identifiable statistical model
F={fX(x;):} (2)
must be consistent with the physical stochastic process X with
a corresponding physical PX. Using the measured {Xt:tT},
one must find F ensuring FPX.
Example 3. 1. To find a statistical model (2), the
cumulative distribution function (cdf) FX(x): or
probability density function (pdf) ƒX(x): are
parametrized. For a random variate X with the real-valued
continuous cdf FX():, one has
)()( xFdx
dxf XX , 1)(
dxxfX
, fX(x)≥0, x.
A variate X means a random variable which satisfies a
well-defined compact probabilistic distribution. Let the data
evolves within a single-variable Gaussian distribution
X~N(,2), F={ 2
2
2
)(
2
2
1),;(
x
X exf : ,2>0},
0278-0046 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2701783, IEEETransactions on Industrial Electronics
where is the mean; and 2 are the standard deviation and
variance.
The model dimension d is 2. For X~N(,2), FX(x) and
fX(x) are parametrized by finding and 2. The probability
that X lies in the range a≤b is b
aX dxxfbXa )(Pr . For
normal distribution,
2erf
2
1
2
1)(
xxFX
, x[–∞ ∞].
For =0 and 2=1, the distribution function gives the
probability that a standard normal variate X assumes a value in
[0 z], and
2erf
2
1
2
1)(
0
2
2
zdxezF
z x
X
. ■
The normal and extreme value distributions X~N(,2)
and X~EV(,) are frequently assumed. The corresponding
single random variable pdfs are
2
2
2
)(
2
2
1),;(
x
X exf , x(–∞,∞), , 2>0, (3)
x
e
x
X eexf1
),;( , x(–∞,∞), , >0. (4)
For X~DP(), conventional bimodal exponential, skew-
symmetric and other distributions may result in inadequacy,
and, PPX. To find F, nonmonotonic cdfs FX(x) and pdfs
fX(x) must be found. For the exhibited multimodal
distributions, we introduce the generalized multimodal normal
distribution MN in variates X~MN(,2,an), as well as the
polynomial extreme value distribution PEV for
X~PEV(,,an,bn) which meet the Kolmogorov axioms. The
pertained pdfs which map multimodal asymmetric probability
distributions are [4]
02
)(2
1
2
2
1),,;( n
nn xa
nX eaxf
, 1)(
dxxfX
, (5)
x(–∞,∞), , 2>0, an,
1
1
1
)(1
1),,,;(
n
nxnb
n
nn
exa
nnX eebaxf
, 1)(
dxxfX
,(6)
x(–∞,∞), , >0, an, bn,
where an and bn are the parameters.
IV. EXPERIMENTAL AND ANALYTIC PROBABILISTIC ANALYSES
The sensing paradigms, ASICs, fabrication technologies,
performance and capabilities of inertial sensors are different.
Technology-pertinent, device physics relevant, and design-
specific statistical models of multispectral noise and errors are of
a great importance. There are challenges in probabilistic analysis
[4, 6]. Nonlinear additive and multiplicative errors, error-noise
correlation, heterogeneous multisource noise sources
(interferences, heat, temperature variations, fluctuations,
nonuniformity, quantization, etc.) and other factors are difficult
to identify, quantify and analyze. In inertial sensors, inertia
reduction, axes decoupling and closed-loop compensation
schemes are implemented on-chip by ASICs. Various quantities
are characterized by histograms, probability distributions,
spectral densities, nonlinear regression statistics, etc. The
random-walk error with drift is
ntnt–1+t–1+wt–1, wt2=E(nt)=t
2+E(wt–1
2).
The experiments are performed for the InvenSense MPU-
6050 and MPU-9250. The GY-521 board is interfaced with an
Arduino Mega sampling the IMU outputs using the inter-
integrated circuit (I2C) bus at 400 kHz. Our goal is to find
compact, parametric and statistically significant models for the
measured physical variables. For a motion with constant
accelerations, the measured noise tuples
)],,(),,,[( ˆˆˆˆˆˆˆ nnnnnnzyx aaayn are depicted in Figures 2. The
histograms for ],,[ ˆˆˆˆ zyx aaa nnnyn are reported in Figures 3.
The statistical models (2) are found and parametrized using
the measured observables nŷ. For Xj~N(·) and Xj~EV(·)j=x,y,z,
the unimodal pdfs (3) and (4) are parameterized as reported in
Figures 3 and Table 1.
][m/s , and , Measured 2
ˆˆˆ zyx aaa nnn [rad/s] , and , Measured ˆˆˆ nnn
xan ˆ
yan ˆ
zan ˆ
n
n
n
Figure 2. Measured noise )],,(),,,[( ˆˆˆˆˆˆˆ nnnnnnzyx aaayn .
Histograms and Unimodal PDFs for ],,[ ˆˆˆˆ zyx aaa nnnyn
][m/s , Noise 2
ˆxan
][m/s , Noise 2
ˆxan
][m/s , Noise 2
ˆ yan
][m/s , Noise 2
ˆ yan
][m/s , Noise 2
ˆzan
][m/s , Noise 2
ˆzan
Figure 3. Histograms and pdfs to characterize noise in the IMU output
accelerometer channels. For Xj~N(·) and Xj~EV(·)j=x,y,z, the corresponding
pdfs plotted by solid and dashed lines.
0278-0046 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2701783, IEEETransactions on Industrial Electronics
TABLE 1. Parameterized Normal and Extreme Value Distributions
Accelerometer Axes x, y and z
Normal Distribution,
2
2
2
)(
2
1
x
X ef
Extreme Value Distribution,
x
e
x
X eef1
Noise in âx =0.00054, =0.0413 = –0.0088, =0.0389
Noise in ây =0.00012, =0.0433 = –0.0108, =0.0404 Noise in âz = –0.00075, =0.0529 = –0.0134, =0.0496
The normal and extreme value distributions with
monotonic cdfs FX(x) consistently describe the probabilistic
characteristics for ],,[ ˆˆˆˆ zyx aaa nnnyn with FPX. However,
the unimodal normal, extreme value, exponential, lognormal
and other conventional distributions [22-25] cannot be applied
in probabilistic analysis of many stochastic processes on
physical phenomena and quantities.
Consider the cross-coupled gyroscope channels. The
histograms for ],,[ ˆˆˆˆ nnnyn are depicted in Figures 4. For
Xj~N(·)j=,,, the corresponding pdfs fX(x) are reported in
Figures 4. It is evident that PN(·)PX. For the exhibited
bimodal distributions, we find consistent F using the
generalized multimodal normal and polynomial extreme value
distributions X~MN(,2,an) and X~PEV(,,an,bn). The pdfs
(5) and (6) consistently map the multimodal asymmetric
distributions. The compact parametrized models are found, and,
FMN(·)PX and FPEV(·)PX. Table 2 reports the resulting pdfs.
The plots for fX(x) are reported in Figures 4 by solid and dashed
lines. Our statistical models are validated by using ten data sets
for which the variations of , , ani and bni do not exceed 5%. Histograms and PDFs for ],,[ ˆˆˆˆ nnnyn
[rad/s] , Noise ˆ
n
[rad/s] , Noise ˆn
[rad/s] , Noise ˆ
n
[rad/s] , Noise ˆn
[rad/s] , Noise ˆ
n
[rad/s] , Noise ˆn
Figure 4. Histograms and pdfs to characterize noise in the gyroscope channels.
For distributions Xj~MN(·) and Xj~PEV(·)j=,,, FMN(·)PX and FPEV(·)PX.
The multimodal normal and polynomial extreme value pdfs are reported by the
solid and dashed curves.
TABLE 2. Parameterized Normal, Generalized Multimodal Normal and
Polynomial Extreme Value Distributions
Gyroscope
Axes
, and
Normal
Distribution,
Xj~N(,2),
2
2
2
)(
2
1
x
X ef
Generalized Multimodal
Normal Distribution,
Xj~MN(,2,an),
022
1
2
1n
nn xa
X ef
Multimodal Polynomial
Extreme Value
Distribution,
Xj~PEV(,,an,bn),
1
1
1
1
1 n
nxnb
n
nn
exa
X eef
Noise in
=0.003,
=0.0805
=0.0376, =0.0845,
a1= –0.0904, a2=0.25,
a3=31.3, a4=193.6
=0.0375, =0.079,
a1= –1.14, a2= –13.5,
b1= –0.764, b2= –9.43
Noise in
=0.0037,
=0.092
=0.0397, =0.099, a1= –0.168, a2= –0.37,
a3=35.6, a4=219.4
=0.0604, =0.0812, a1= –1.63, a2= –13.04,
b1= –1.16, b2= –9.41
Noise in
=0.00193,
=0.0893
=0.0116, =0.115, a1= –0.11, a2= –3.48,
a3=18.4, a4=354.2
= –0.0018, =0.105, a1= –0.06, a2= –21.8,
b1= –0.00034, b2= –11.4
V.ADAPTIVE FILTER DESIGN: DATA-PRESCRIPTIVE PROCESSING
Filtering, data fusion and information management,
accomplished using adequate processing calculus by robust
algorithms, are aimed to ensure noise n attenuation, error
reduction, as well as distortions cancelation or rejection. The
distortions (inconsistencies, disruptions, erroneous
measurements, confounding and extraneous factors, etc.) arise
due to interference, transverse sensitivity, vibrations, acoustic
perturbations, etc. Low signal-to-noise ratio and multispectral
noise with high noise power are exhibited by IMUs. Multisource
noise nŷ results in low accuracy, inconsistencies, uncertainties
and information losses.
The extended Kalman filter may ensure optimal
estimation assuming known nonlinear system models with
additive independent white noise. The governing model and
measured output equations are [27, 28]
x[n]=f(x[n–1])+w[n–1], y[n]=h(x[n])+v[n], (7)
where f and h are the nonlinear maps; w[n] and v[n] are the
multivariate Gaussian noises with the known covariance Qn
and Rn. Finding the Kalman gain, one may estimate the
residual, covariance and predicted state estimates [27, 28].
However, in IMUs, f and h are unknown, parameters are
varying, and, noise is not Gaussian. In multi-degree-of-
freedom MEMS-technology IMUs, filtering and denoising
(attenuation of a specific noise frequency) may not be
accomplished using Kalman, Kalman-Bucy, vector, wavelet,
graph, singular-value decomposition and other filters [12, 26]
due to: (1) Nonlinear device physics; (2) Nonlinear models
with time-varying parameters, uncertainties and unmodelled
dynamics; (3) Non-Gaussian noise and errors with varying
covariance; (4) Nonlinear axes cross-coupling.
Filtering of multisource noise nŷ must be accomplished by
using advanced concepts and practical algorithms. An IMU–C
platform is designed and evaluated. A microcontroller
implements linear infinite impulse response digital filters
,)(
)()(
0
0
M
l
l
l
N
k
k
k
za
zb
zX
zYzH ,][][][
00
N
k
k
M
l
l knxblnyany (8)
where x[n] and y[n] are the filter input and output; M and N are
the feedback and feedforward filter orders; al and bk are the
feedback and feedforward coefficients which can be adjusted in
real-time by identifying the dynamic modes [4].
0278-0046 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2701783, IEEETransactions on Industrial Electronics
The Bessel, Cauer, Chebyshev, elliptical, state variable
and other filters are examined. The Butterworth and notch
filters have advantages because the magnitude |H|dB is constant
or monotonically decreasing function of frequency at all
frequencies. The adaptive low-pass Butterworth filters are
designed and implemented. The cut-off frequencies are found
in the operating frequency envelope fE[fmin fmax] with fmax>10
Hz. We examine: (i) System and IMU dynamic ranges and
bandwidths; (ii) Multispectral noise power and frequency; (iii)
Microcontroller capabilities to design and reconfigure
adaptive filters in near-real-time using the dynamic modes
within fE, stop and pass band frequencies s and p, etc. The
passband envelope Ep[pmin pmax] depends on fE. In fE, the
acceleration magnitudes should not be attenuated, and,
minimal phase delay must be ensured.
The second- and high-order low-pass adaptive
Butterworth filters are designed with the pass band gain
|H|dBmax=0 dB in the passband frequency envelope Ep with
pmin=100 rad/s. The stop band gain |H|dB min varies in the stop
band frequency envelope Es[smin smax]. The adaptive
filters (8) are implemented, and, the feedback and feedforward
coefficients al and bk are adjusted in real-time,
The IMU is positioned on a free end of an elastic constant
circular cross-section 1-meter-long beam. The fixed end of a
steel beam is secured on a six-degree-of-freedom platform.
The linear elastic, isotropic, homogeneous beam exhibits very
complex lateral, vertical and torsional bending depending on
platform motion and initial conditions. The IMU outputs
)]ˆ,ˆ,ˆ(),ˆ,ˆ,ˆ[(ˆ zyx aaay are reported in Figures 5. The
multispectral noise is attenuated, distortions are minimized
and errors are reduced by post-processing. The IMU–C
yields )]~,~,~(),~,~,~[(~ zyx aaay which ensures accuracy
and precision in estimation of velocities, positions, orientation,
trajectory, etc. Filters redesign, interpolation and
reconfiguration take ~0.003 sec guarantying high bandwidth. âx and ãx [m/s2]
âx
ãx
Time [sec]
4
2
0
–2
–4
–6
0 2 4 6 8 10
ãy
ây
Time [sec]
0 2 4 6 8 10
0.5
–0.5
0
ây and ãy [m/s2]
âz and ãz [m/s2]
âz
ãz
6
4
2
0
–2 Time [sec]
0 2 4 6 8 10 (a)
and [rad/s]
5
0
–5
Time [sec]0 2 4 6 8 10
~
~ and [rad/s]
0.5
0
–0.5
–1
1
Time [sec]0 2 4 6 8 10
~
~
and [rad/s]
Time [sec]
2
–2
0
0 2 4 6 8 10
–4
~
~
(b)
Figure 5. IMU outputs )]ˆ,ˆ,ˆ(),ˆ,ˆ,ˆ[(ˆ zyx aaay (blue lines), and, post-
processed accelerations )]~,~,~(),~,~,~[(~ zyx aaay (red lines):
(a) Linear accelerations ]ˆ,ˆ,ˆ[ˆzyx aaaay and ]~,~,~[~
zyx aaaay [m/s2];
(b) Angular velocities ]ˆ,ˆ,ˆ[ˆ ωy and ]~,~,~[~
ωy [rad/s].
VI. POSITION AND TRAJECTORY ESTIMATIONS
In the proposed Inertial Measurement Platform with
IMU–C, the processing scheme is
yOutputsollerMicrocontrIMUProcessingFilteringAdaptive
yOutputsIMU ~
Velocities and onsAccelerati Angular andLinear Processed-Post
ˆ
ASICs sIMU'by Processed:Outputs IMU
Elements Sensing sIMU'by Velocities and onsAccelerati MeasuredonsAcceleratiAngular andLinear Acting
).~,~,~(),~,~,~()ˆ,ˆ,ˆ(),ˆ,ˆ,ˆ(
),,(),,,(),,(),,,(
zyxzyx
zyxzyx
aaaaaa
aaaaaa
(9)
The IMU–C outputs )]~,~,~(),~,~,~[(~ zyx aaay are used
to estimate velocities, position, orientation and trajectory as
PositionsAngular andLinear Estimated
n Integratio Quadrature Adaptive
andion Extrapolat ion,Interpolat :Calculus Estimation
VelocitiesAngular andLinear Estimated
n Integratio Quadrature Adaptive
andion Extrapolat ion,Interpolat :Calculus Estimation
Outputsoller Microcontr-IMU Processed-Post
).~,~
,~
(),~,~,~()~,~,~(),~,~,~(
)~,~,~(),~,~,~(
zyxvvv
aaa
zyx
zyx
(10)
The lateral and vertical displacement estimates ]~,~[ zx are
found by performing interpolation and adaptive quadrature integration of the post-processed ỹ. A free end of a circular beam exhibits a coupled longitudinal and lateral damped asymmetric motions depending on platform motions and initial displacement [x0,y0,z0] with equilibrium x
0=y
0=z
0=0. The
experimental results are illustrated in Figures 6.
0278-0046 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2701783, IEEETransactions on Industrial Electronics
Figure 6. Estimated lateral x~ and vertical z~ displacements of a beam tip
with IMU for asymmetric dynamic bending motions: Tip displacement
estimates ]~,~[ zx during free vibrations with the initial deflections
[x0,z0]=[0.05 0.09] and [x0,z0]=[0.04 0.095] m.
From (10), one finds the estimates
)~,~
,~
(),~,~,~(
)~,~,~(),~,~,~(
)~,~,~(),~,~,~(~
zyx
vvv
aaa
zyx
zyx
y which can be compared with the
physical quantities )],(),,(),,[( rxωvαay . For physical
accelerations ],[ αay , using the IMU output vector ŷ, the
post-processed estimates )]~,~(),~,~(),~,~[(~ rxωvαay are found.
VII. PERFORMANCE MEASURES AND METRICS:
FINDINGS AND EXPERIMENTAL SUBSTANTIATION Nonlinear physical transductions by sensors and ICs Tỹ,
distortions, uncertainties, noise and other factors ỹ affect
accuracy Aỹ, precision Pỹ, error ỹ and information losses. One
has Aỹ=(Tỹ,nỹ,ỹ,ỹ), Pỹ=(Tỹ,nỹ,ỹ,ỹ) and
ỹ=f(Tỹ,nỹ,ỹ,ỹ). Using the industrial performance metrics, the measured noise spectral density and root mean square (RMS) are reported in Table 3. The experimental results for
IMU and Inertial Measurement Platform IMU–C are conducted. The bandwidth of the data-perspective signal processing is 325 Hz. TABLE 3. Noise Attenuation in an IMU–C Versus of-the-Shelf IMU
Commercial off-the-Self IMU Outputs )]ˆ,ˆ,ˆ(),ˆ,ˆ,ˆ[(ˆ zyx aaay and
Post-Processed )]~,~,~(),~,~,~[(~ zyx aaay in the IMU–C
Noise nỹ Spectral Density Noise nỹ RMS Linear Accelerations
[g/√Hz] Angular Velocities
[º/s/√Hz] Linear Accelerations
[mg] Angular Velocities
[º/s]
IMU âx,ây,âz
IMU–C ãx,ãy,ãz
IMU
ˆ,ˆ,ˆ IMU–C
~,~,~ IMU
âx,ây,âz IMU–C ãx,ãy,ãz
IMU
ˆ,ˆ,ˆ IMU–C
~,~,~
416 152 0.0095 0.0029 9.4 2.8 0.0031 0.0012
Accuracy, precision, resolution and other key
performance quantities are examined. Using the Lp-metrics
and p-norm, the accuracy and dynamic precision measures are
pp
yyy
~1D , T p
dtp
yyy
~1tD , p≥1. (11)
For p=1,
yyy
~1D and T
dtyyy
~1tD .
These D and Dt provide quantitative performance and
capabilities estimates. The accelerations (D[a,],Dt,[a,]),
velocities (D[v,],Dt,[v,]) and positions (D[x,r],Dt,[x,r]) estimates
are found. For example
pp
]~,~[],[],[
1],[ rxrx
rxrx D , T pt dt
p
]~,~[],[],[
1],[, rxrx
rxrxD .(12)
If, hypothetically, there are no static and dynamic errors,
D=0 and Dt=0. To minimize errors and information losses, one
minimizes n, and other affecting factors. To guarantee
accuracy and precision, we minimize
),(),,(minmin),(),(
εnεnεnεn
tt
DDDD
. The goal is to ensure D≤ and
Dt≤t with the specified 0 and t0. A principal component
analysis is applied. Using an orthogonal transformation, a set
of statistically-dependent and correlated (n,) is mapped to
adaptive filter structure, order and coefficients to evaluate and
minimize information losses. Due to complexity, the
minimization problem may not be solved in near-real-time for
rapidly-changing heterogeneous a(t) and (t). If real-time
adaptation cannot be ensured, the processing algorithms are
configured to the worst-case scenario implementing robust
schemes using fE and Ep as reported in Section 6.
Noise, distortions and errors affect the systematic error
and information losses in ỹ=y+nỹ+ỹ+ỹ. One has f(nỹ+ỹ).
Let X and Y be two independent random variables with pdfs
fX(x) and fY(y). For the random variable Z=X+Y, the pdf fZ(z) is
the convolution of fX(x) and fY(y),
dxxfxzfdyyfyzfzff XYYXYX )()()()()(* . For
2
2
1
2
1)(
x
X exf
and 2
21
2
1)(
y
Y eyf
, we have
2
412
212
21
2
1)(
2
1)(
zyyz
Z edyeezf
. The probability of
errors, propagating error, and other IMU statistical
characteristics are studied to assess performance metrics. In
commercial-grade IMUs and INSs, there are significant errors
in angular velocities and displacements. The experiments are
conducted using ten medium-g data sets with aE[amin amax],
amax=40 m/s2, E[min max], max=10 rad/s
2, fE[fmin fmax],
fmax=10 Hz for t=[0 100] sec. For rapidly-changing a(t) and
(t), the post-processing ensures the angular velocity and
position errors 2.3 º/s and r9.4%, while in IMU, min is
~4 º/s and rmin>25% of the traveled distance. The noise and
distortions attenuation resulted in the error ỹ reduction. The
errors in estimation of the linear and angular velocities
)]~,~,~(),~,~,~[( zyx vvv and positions )]~,~
,~
(),~,~,~[( zyx are
reduced by 3.3 and 2.1 times, respectively.
The accuracy and precision measures D and Dt are found
using the estimates (11) and (12) for p=1 and p=2. For the
data, reported in Figures 5 and 6, the adaptive data-
prescriptive signal processing ensures Da0.018, Dv0.057
and Dr0.15.
The designed and implemented processing schemes, data-
prescriptive adaptive filter design, robust algorithms and
solutions guarantee: (1) Accuracy and precision; (2)
Minimization of information losses; (3) Noise attenuation and
impaired signal-to-noise ratio; (4) Errors and distortions
reduction; (5) Adequate bandwidth and dynamic range; (6)
Robustness and stability; (7) Compliance, applicability and
usability to other classes of smart sensors.
0278-0046 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2701783, IEEETransactions on Industrial Electronics
VIII. CONCLUSIONS
With recent fundamental and technology progress in
microfabrication and microelectronics, smart sensors may
ensure beyond-state-of-the-art capabilities. We performed
descriptive, predictive and prescriptive data analyses by
deriving statistically significant statistical models, performing
quantitative probabilistic analysis and developing robust
algorithms. New consistent families of distributions and
formative probability densities are found. Compact statistical
models are found, justified and substantiated using
experimental data. Data-prescriptive adaptive signal
processing and algorithmic solutions were developed. The
reconfigurable filters and processing algorithms were
designed, implemented and tested to attenuate noise, reduce
errors and minimize information losses. The proposed solution
guarantees attenuation, cancelation and rejection of data
inconsistencies, disruption, gaps, erroneous measurements,
confounding and extraneous factors, etc. Our results enabled
the overall data consistency, data completeness and data validity.
The hardware-adequate and software-consistent data-centric
calculus and algorithms were developed and applied. We
emphasized innovative inroads to further enable performance
and capabilities of cyber-physical sensors which are the key
components of the C2ISTAR, CPS, IoT and SCADA
platforms. We focused on synergetic discoveries of theoretical
engineering science and applied engineering design towards
high-impact transformative technologies.
Acknowledgements – The author sincerely appreciates
anonymous reviewers’ comments, suggestions and feedback
which were very helpful to revise the manuscript.
REFERENCES 1. B. Betts, “Smart sensors,” Spectrum, vol. 43, issue 4, pp. 50-53, 2006.
2. V. C. Gungor, B. Lu and P. Hancke, “Opportunities and challenges of
wireless sensor networks in smart grid,” IEEE Trans. Industrial Electronics, vol. 57, issue 10, pp. 3557-3564, 2010.
3. S. E. Lyshevski, MEMS and NEMS: Systems, Devices and Structures,
CRC Press, Boca Raton, FL, 2008. 4. S. E Lyshevski, Mechatronics and Control of Electro-mechanical
Systems, CRC Press, Boca Raton, FL, 2017.
5. International Technology Roadmap for Semiconductors, 2005, 2007, 2009, 2011 and 2013 Editions, Semiconductor Industry Association,
Austin, Texas, USA, 2017. 6. International Technology Roadmap for Semiconductors, 2011 and 2013
Edition, Micro-Electromechanical Systems (MEMS), Semiconductor
Industry Association, Austin, TX, USA, 2017. 7. R. Alur, Principles of Cyber-Physical Systems, The MIT Press, MA,
2015.
8. B. Alandry, L. Latorre, F. Mailly and P. Nouet, “A CMOS-MEMS
inertial measurement unit,” Proc. IEEE Conf. Sensors, pp. 1033-1036, 2010.
9. G. Chatterjee, L. Latorre, F. Mailly, et. all., “MEMS based inertial
measurement units for strategic applications”, Proc. Symp. Design Test, Integration and Packaging of MEMS and MOEMS, pp. 1-5, 2015.
10. K. Froyum, S. Goepfert, J. Henrickson and J. Thorland, “Honeywell
micro electro mechanical systems (MEMS) inertial measurement unit (IMU),” Proc. Symp. Position Location and Navigation, pp. 831-836,
2012.
11. E. Kanso, A. J. Szeri and A. P. Pisano, “Cross-coupling errors of micromachinned gyroscopes,” J. Microelectromechanical Systems, vol.
13, no. 2, pp. 323-331, 2004.
12. D. K. Shaeffer, "MEMS inertial sensors: A tutorial overview," IEEE Communication Magazine, vol. 51, issue 4, pp. 100-109, 2013.
13. S. Zotov, M. Rivers, C. Montgomery, A. Trusov and A. Shkel,
"Chipscale IMU using folded-MEMS approach," Proc. IEEE Sensors Conf., pp. 1043-1046, 2010.
14. S. Zimmermann, J. Bartholomeyczik, U. Breng, et. all., “Single axis
gyroscope prototype based on a micromechanical Coriolis rate sensor,” Proc. Symp. Gyro Technology, Stuttgart, Germany, pp. 9.0-9.9, 2006.
15. N. Yazdi, F. Ayazi and K. Najafi, “Micromachined inertial sensors,”
Proc. IEEE, vol. 86, no. 8, pp. 1640-1659, 1998. 16. J. Baziw and C.T. Leondes, "In-flight alignment and calibration of
inertial measurement units: Part I and Part II," IEEE Trans. Aerospace
and Electronic Systems, vol. 8, issue 4, pp. 439-449, pp. 450-465, 1972. 17. F. Caron, E. Duflos, D. Pomorski and P. Vanheeghe, "GPS/IMU data
fusion using multisensor Kalman filtering: Introduction of contextual aspects," Information Fusion, vol. 7, pp. 221-230, 2006.
18. S. Guerrier, "Improving accuracy with multiple sensors: Study of
redundant MEMS-IMU/GPS configurations," Proc. Int. Tech. Meeting of The Satellite Division of the Institute of Navigation, pp. 3114-3121,
2009.
19. A. Lawrence, Modern Inertial Technology: Navigation, Guidance, and Control, Springer, NY, 1998.
20. A. G. Quinchia, C. Ferrer, G. Falco, E, Falletti and F. Dovis, “Analysis
and modelling of MEMS inertial measurement unit,” Proc. Conf. Localization and GNSS, pp. 1-7, 2012.
21. B. Abbott and S. E. Lyshevski, “Signal processing in MEMS inertial
measurement units for dynamic motional control”, Proc. IEEE Conf. Electronics and Nanotechnology, pp. 309-314, 2016.
22. J. L. Devore, Probability and Statistics, Cengage Learning, NY, 2011.
23. M. H. DeGroot and M. J. Schervish, Probability and Statistics, Person, NY, 2011.
24. W. Feller, An Introduction to Probability Theory and Its
Applications, vol. 1 and 2, Wiley, NY, 1968 and 1971. 25. D. Freedman, R. Pisani and R. Purves, Statistics, W. W. Norton and Co.,
NY, 2007.
26. S. Guerrier, "Improving accuracy with multiple sensors: Study of redundant MEMS-IMU/GPS configurations," Proc. Int. Technical
Meeting of The Satellite Division of the Institute of Navigation, pp.
3114-3121, 2009. 27. G. A. Einicke and L. B. White, "Robust extended Kalman filter," IEEE
Trans. Signal Processing, vol. 47, issue 9, pp. 2596-2599, 1999.
28. K. Reif and R. Unbehauen, "The extended Kalman filter as an exponential observer for nonlinear systems," IEEE Trans. Signal
Processing, vol. 47, issue 8, pp. 2324-2328, 1999.
Recommended