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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: [email protected] 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.

mailto:[email protected]

http://people.rit.edu/seleee/

http://www.rit.edu/kgcoe/staff/sergey-lyshevski



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},



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.



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].



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.



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.



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.

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