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Constrained Angular MotionEstimation in a Gyro-Free IMU
EZZALDEEN EDWAN, Member, IEEESTEFAN KNEDLIK, Member, IEEEOTMAR
LOFFELD, Senior Member, IEEEZESSGermany
In this paper, we present an extended Kalman filter
(EKF)-based solution for the estimation of the angular
motion
using a gyro-free inertial measurement unit (GF-IMU) built
of
twelve separate mono-axial accelerometers. Using such a
GF-IMU
produces a vector, which we call the angular information
vector
(AIV) that consists of 3D angular acceleration terms and six
quadratic terms of angular velocities. We consider the
multiple
distributed orthogonal triads of accelerometers that consist
of three nonplanar distributed triads equally spaced from a
central triad as a specific case to solve. During research for
the
possible filter schemes, we derived equality constraints. Hence
we
incorporate the constraints in the filter to improve the
accuracy
of the angular motion estimation, which in turn improves the
attitude accuracy (direction cosine matrix (DCM) or
quaternion
vector).
Manuscript received July 8, 2008; revised June 5, 2009; released
forpublication October 10, 2009.
IEEE Log No. T-AES/47/1/940050.
Refereeing of this contribution was handled by D.
Gebre-Egziabher.
This work was partially supported by the German
ResearchFoundation (DFG) under Grant KN 876/1-2.
Authors’ address: Center for Sensorsystems (ZESS), Universityof
Siegen, Paul-Bonatz Str. 9-11, Siegen 57068, Germany,
E-mail:([email protected]).
0018-9251/11/$26.00 c° 2011 IEEE
I. INTRODUCTION
A conventional inertial measurement unit (IMU)consists of three
gyros for measuring rotationalmotion and three accelerometers for
measuring linearacceleration. In contrast to a conventional IMU,
agyro-free IMU (GF-IMU) uses a configuration ofaccelerometers only
to measure acceleration androtational motion of a rigid body in 3D
space. Inprinciple it benefits from an effect known as
lever-armeffect. To have this effect, the accelerometers mustbe
distributed in distance over the body. In orderto distinguish it
from a conventional gyro IMU,researchers use other names
interchangeably todescribe its functionality. The most commonly
usednames are GF-IMU [1—3], nongyroscopic inertialmeasurement unit
(NGIMU) [4, 5], accelerometerbased IMU, and all-accelerometer IMU
[6—8].There exists a variety of reasons for using
distributed accelerometers to estimate the angularvelocity
vector. Most of the GF-IMU publicationslist some of the
justifications [1, 3, 4, 6, 8—13]. Inshort, the gyroscope typically
has the disadvantageof complicated manufacturing techniques, high
cost,high power consumption, large weight, large volume,and limited
dynamic range [11]. In addition, low-costmicro-electro mechanical
system (MEMS) gyroscopeshave more inherent physical complexities
thanaccelerometers [1]. On the other hand, accelerometersare
smaller, less costly, and less power consumingthan comparable
gyros.Using certain configurations of twelve separate
mono-axial accelerometers produces an angularinformation vector
(AIV) that consists of a 3D angularacceleration vector and six
quadratic terms of angularvelocities, as is shown later.
Unfortunately the angularvelocity vector is not expressed
explicitly in the AIV.The motivation for this research is that
little attentionhas been given to the sensor fusion of the AIV.The
approach of using a double integration of theangular acceleration
as described in [14] to achievethe orientation has been proven to
be theoreticallyfeasible, but it has a faster error growth rate
comparedwith a conventional inertial navigation system (INS).The
advantage of only using the angular accelerationvector has a little
value compared with the need fora double integrator. Another
problem with such anapproach is the need to initialize the angular
velocityto be able to integrate the angular acceleration.Continuous
research efforts have been conductedto benefit from the other
components in the AIV.Recently Park [1] aided the angular
accelerationvector by three cross-quadratic terms of
angularvelocity using an extended Kalman filter (EKF),but his
scheme is not using all possible quadraticangular velocity terms as
it is possible to get threemore quadratic terms. Furthermore, such
a schemesuffers from an observability problem if one of the
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angular velocity terms is zero. Parsa [6] obtained
thenine-element AIV using twelve accelerometers. Inthat work, the
complete AIV is used, but the focusis on the optimal design of the
GF-IMU, avoidingthe use of tri-axial accelerometers, and not on
thefusion scheme. Zappa [15] presented the rules forusing twelve
separate mono-axial accelerometersto obtain the AIV without
inhibiting singularityof the coefficient matrix. Ying Kun [13] used
thesame configuration of twelve accelerometers thatwe use in this
paper to create a GF-INS fromaccelerometers that already exist in
an automobile’sonboard control and safety system. Again his
focuswas not on the sensor fusion scheme; rather he useda simple
numerical approach, which may result inan imaginary solution. In
our new EKF scheme, weuse all of the six quadratic terms of angular
velocityas suggested by Zappa [15] for the deterministiccase but
this time for the stochastic case, which isthe realistic one, and
we benefit from the angularacceleration vector at the same time. In
other words,we aid the integrated angular acceleration vector bysix
quadratic terms of angular velocities using anEKF.The choice of the
EKF for the fusion of the AIV is
a trivial choice because we get measurable terms thathave the
nonlinear function of the angular velocitycomponents. The recently
developed modificationsof the Kalman filter for nonlinear systems
like theunscented Kalman filter (UKF) are also applicable.During
the formulation of the EKF, we derivednonlinear equality
constraints that can improve theestimation performance of the EKF,
but the statemodel needs to be expanded as we explain
later.Constrained Kalman filtering has proven to be anefficient
technique for improving the estimated statevector if the
constraints are handled properly.This paper is organized as
follows. Section II
introduces the distributed orthogonal triad ofaccelerometers.
Section III describes the multipledistributed accelerometer triad
set configuration thatgives the AIV and presents a calibration
procedure.Section IV describes the underlying three-statemodel EKF
for estimating the angular change vector.Section V shows the
three-state model simulationresults. Section VI shows how to use
the angularchange vector to update attitude. Section VII derivesthe
constraints. Section VIII shows how to handle theconstraints.
Section IX shows simulation results forthe constrained model and
analysis. Section X givesconclusions and suggestions for future
work.
II. DISTRIBUTED ACCELEROMETER TRIADDEFINITION
Following [16] for the derivation of the lever-armeffect, we
assume that we have a rigid body with
Fig. 1. Distributed orthogonal triad of accelerometers at point
pb.
its body center at point ob = [0 0 0]T as shown inFig. 1. The
measurable acceleration of a distributedorthogonal triad of
accelerometers fixed at a rigidpoint pb = [px py pz]
T that is aligned with the bodyframe can be expressed as
abp = abo+ g
bo¡gb
p+ _!bib£pb+!bib £!bib £pb (1)
where
abo is the central acceleration vector,pb is the displacement
vector from ob to pb equalspb¡ ob,gbo¡gb
pis the local gravity difference vector,
!bibis the angular velocity vector,_!bibis the angular
acceleration vector,£ is the cross-product operation defined
asskew-symmetric form
!bib =
264!x!y!z
375 , !bib£=264 0 ¡!z !y!z 0 ¡!x¡!y !x 0
375 :(2)
In (1), (2), the superscript b denotes that the quantityis
expressed in the body frame. The gravitationaldifference vector can
be ignored for relatively smalldistribution distance, and hence (1)
simplifies to
abp ¼ abo + _!bib £pb +!bib£!bib£pb: (3)Since the angular
velocity vector is not expressedexplicitly in the above equation
and in reality theequation is stochastic, extra effort must be
takento extract the angular velocity. One approach is tointegrate
the angular acceleration to get the angularvelocity and to perform
another integration to getthe attitude. An example of such an
approach,where a Kalman filter is used with external
GPSobservations, is given in [9]. The drawback ofsuch an approach
is that attitude and positionerrors grow quickly because of error
growth withintegration. In the following equations, we omit thebody
superscript as all equations in this paper aregiven in the body
frame. We rewrite (3) in matrix
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Fig. 2. Configuration of multiple distributed
tri-axialaccelerometers.
form as264apxapyapz
375=264aoxaoyaoz
375
+
264¡!2y ¡!2z !x!y ¡ _!z !x!z + _!y
!x!y + _!z ¡!2z ¡!2x !y!z ¡ _!x!x!z ¡ _!y !y!z + _!x ¡!2x
¡!2y
375264pxpypz
375 :(4)
III. CONFIGURATION OF MULTIPLE DISTRIBUTEDTRI-AXIAL
ACCELEROMETERS
We focus on configurations consisting of twelvemono-axial
accelerometers that follow the rules listedby Zappa [15]. Without
loss of generality, we considerthe set of four tri-axial
accelerometers shown in Fig. 2as an example configuration that
follows Zappa’s rulesfor extracting the AIV without inhibiting
singularityof the coefficient matrix.Mainly we consider this
configuration because
a minimum of twelve accelerometers are needed todetermine the
magnitude of the angular velocity andits direction (algebraic sign
cannot be determineduniquely as explained in [15]). The greatest
amountof angular motion information, which is in thenine angular
terms that we show next, can beextracted from this configuration.
Moreover, thisconfiguration has, according to [3], a low
geometricdilution of precision (GDOP) factor for bothangular and
translational acceleration without thecentral accelerometer triad.
In [8] it is stated thata configuration of twelve accelerometers is
theonly known minimum number to both navigateand determine gravity
in a gravitational fieldwith a uniform gradient. Finally yet
importantly,it is the most practical configuration becauseIMUs
exist in triads of gyros and accelerometers.
This specific configuration was analyzed byAlgrain [17] where he
derived the followingrelations:
_!x = (aCz ¡ aAz )=2dC +(aAy ¡ aDy )=2dD
_!y = (aDx ¡ aAx )=2dD +(aAz ¡ aBz )=2dB
_!z = (aBy ¡ aAy )=2dB +(aAx ¡ aCx )=2dC
(5)
!x!y = (aBy ¡ aAy )=2dB +(aCx ¡ aAx )=2dC
!x!z = (aBz ¡ aAz )=2dB +(aDx ¡ aAx )=2dD
!y!z = (aCz ¡ aAz )=2dC +(aDy ¡ aAy )=2dD
(6)
!2x = (aBx ¡ aAx )=2dB +(aAy ¡ aCy )=2dC +(aAz ¡ aDz )=2dD
!2y = (aCy ¡ aAy )=2dC +(aAx ¡ aBx )=2dB +(aAz ¡ aDz )=2dD
!2z = (aDz ¡ aAz )=2dD +(aAx ¡ aBx )=2dB +(aAy ¡ aCy )=2dC:
(7)
The notation atriadaxis refers to the distributedaccelerometer
measurement with a superscriptreferring to the triad location and a
subscript referringto the axis index. For the case of a unique
distributiondistance d for every distributed triad, the
AIVbecomes:
_!x = (aCz ¡ aAz ¡ aDy + aAy )=2d
_!y = (aDx ¡ aAx ¡ aBz + aAz )=2d
_!z = (aBy ¡ aAy ¡ aCx + aAx )=2d
(8)
!x!y = (aBy ¡ aAy + aCx ¡ aAx )=2d
!x!z = (aBz ¡ aAz + aDx ¡ aAx )=2d
!y!z = (aCz ¡ aAz + aDy ¡ aAy )=2d
(9)
!2x = (aBx ¡ aAx ¡ aCy + aAy ¡ aDz + aAz )=2d
!2y = (aCy ¡ aAy ¡ aBx + aAx ¡ aDz + aAz )=2d
!2z = (aDz ¡ aAz ¡ aBx + aAx ¡ aCy + aAy )=2d:
(10)
All of the previous equations are linear combinationsof
accelerometers measurements.
A. GF-IMU Error Calibration
A model that describes the sensor’s outputrelated to the
reference value to be measured canapproximate the behavior of most
of the inertialsensors. A proper model can be established basedon
the data sheets provided with the sensor andusing the lab’s
calibration equipment. The calibrationissue of a GF-IMU has been
addressed in theliterature before. Park [1] gives a detailed
descriptionof calibrating uni-axial accelerometers in a
cubeconfiguration GF-IMU. However in our configurationwe recommend
calibrating each tri-axial independentlyfor the following
reasons.
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1) Uni-axial calibration does not take intoaccount the
cross-axis sensitivities, and hence anothercalibration for
misalignment must be performed.2) In our derivation, we assumed
that all
accelerometers are aligned with one frame called thebody frame.
Therefore, the most convenient way tocalibrate is to find the
misalignment of each of thefour triads from that common body
frame.
The calibration of every accelerometer triad will bedone faster
because we benefit from the symmetry inthe configuration by
collecting measurements for thefour-accelerometer triads in every
flipping position.Accordingly we suggest calibrating each
tri-axialof the accelerometers as a whole group rather
thancalibrating every single accelerometer independently.There are
two types of errors that result in the
measurable AIV in the GF-IMU. The first type oferror is due to
the tri-axial accelerometers’ errors, andthe second type of error
is due to errors in distributedtriad locations in the geometric
configuration. Theerror model of a tri-axial accelerometer is
composedof different sources of error that can be
generallyclassified into four parts: misalignment, scale
factor,bias, and noise. We can calibrate for three typesof errors:
scale factor parameters, misalignment ofthe accelerometers, and
three bias parameters in asingle calibration procedure of each
triad. We find the9-element sensitivity matrix whose diagonal
elementscontain the scale factors and whose off-diagonalelements
contain misalignment parameters. Likewisewe find the three bias
elements. In matrix form theaccelerometer triad output is related
to the referencespecific force as264zxzy
zz
375=264mxx mxy mxzmyx myy myzmzx mzy mzz
375264axayaz
375+264baxbaybaz
375+264vaxvayvaz
375 :(11)
There are many methods to solve for m and belements [18, 19].
Following a similar approach tothe one given in [19], we rearrange
(11) into thefollowing form:
z =Oμ+ va
μ =
26664m1
m2
m3
ba
37775 , m1 =264mxxmxymxz
375
m2 =
264myxmyymyz
375 , m3 =264mzxmzymzz
375va = [vax vay vaz]
T
ba = [bax bay baz]T:
(12)
The observation matrix for every measurement isgiven by
O =
264ax ay az 0 0 0 0 0 0 1 0 00 0 0 ax ay az 0 0 0 0 1 00 0 0 0 0
0 ax ay az 0 0 1
375 :(13)
We stack N observation matrices from N observationsto form the
total measurement vector and the totalobservation matrix shown
next:2664
z1
...
zN
3775=264O
1
...
ON
375μ+2664v1a...
vNa
3775 , zt =Otμ+ vta:(14)
To have an observable system, we need a minimum offour
measurements with proper values of the specificforce values. To
extract the m and b elements, we useleast square estimation
μ̂ = (OtTOt)¡1OtTzt: (15)
For separation distances calibration, a turntableis used to
measure the separation distance fromevery distributed accelerometer
triad to the centralaccelerometer triad. Hence we have two
optionshere:
1) Use the form of equations for the AIV that hasdifferent
separation distances given in (5)—(7) andcompute the corresponding
error covariance to beused later in the estimation.2) Instead of
using different separation distances,
we can adjust the distance manually for everydistributed triad
so that each is unique and validateit experimentally. Having a
unique separation distancefrom every distributed tri-axial to the
central tri-axialsimplifies computations. For this reason, we
considerthe equally spaced triads configuration in the rest ofthis
paper.
IV. THE THREE-STATE MODEL EKF
Based on the previously derived AIV (8)—(10),we can formulate
the EKF setup. The quadraticterms do not give a unique angular
velocity vectorsolution. Instead we get two solutions. In our
setup,we consider only a fixed accelerometers configuration.For the
determination of the algebraic sign in acompletely GF-IMU, there
exist solutions usingaccelerometers that vibrate in a known matter
to bodyas proposed initially by Merhav [5] and recentlyby Costello
[10] who gave a solution that does notrequire integration. The
conventional low-cost gyroscan be used to insure a correct sign
convergencein the GF-IMU as shown in [20]. Other possible
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integration schemes include GPS [9, 12] and
tri-axialmagnetometers [7] to have a nondrifting
orientationestimate. We are interested in estimating the
angularvelocity component along each body axis in 3D.In reality the
continuous angular velocity vector isreplaced with the angular
change vector as given in(16) because the computerized
implementation isdiscrete. The angular change is the sampled
angularvelocity multiplied by the sampling time and is givenas
x= [x1 x2 x3]T = [¾x ¾y ¾z]
T: (16)
A. Initialization
The initial state vector can be set as
x̂+0 = Efx0g= [¾x0 ¾y0 ¾z0 ]T: (17)
The initial estimation error covariance is given as
P+0 = Ef(x0¡ x̂+0 )(x0¡ x̂+0 )Tg: (18)
B. Prediction
In discrete time the actual output of eachaccelerometer is the
velocity change; hence the outputof (8) is the angular velocity
change vector ®. Theprocess model based on Euler integration is
¾xk = ¾xk¡1 +®xk¡1¢t+wxk¡1
¾yk = ¾yk¡1 +®yk¡1¢t+wyk¡1
¾zk = ¾zk¡1 +®zk¡1¢t+wzk¡1 :
(19)
We define the following vector as the processinput
u= ®¢t= [®x¢t ®y¢t ®z¢t]T: (20)
Using (20), the process given in (19) has the linearform of
xk = Fk¡1xk¡1 +Gk¡1uk¡1 +wk¡1
Fk¡1 =Gk¡1 = I3£3:(21)
We assume that the uncertainty in the processis mainly due to
the uncertainty in the angularvelocity change. In this work, we
consider the errorof each accelerometer as white Gaussian noise
forsimplicity. For an accelerometer error accountingfor the
remaining bias, we developed a solutionutilizing the dynamic models
to estimate the biasparameters in the nine angular information
termsgiven in [21]. In that solution, the bias parametersand the
angular acceleration vector are augmentedwithin the state-space
model to form a 15-state model.For that process update, we used the
Wiener processor simply the nearly constant acceleration model.When
using such a model, all the bias parameters inthe AIV become
observable under the condition thatthe angular acceleration has a
non-zero magnitude.
The three-state model has the advantage of simplecalculations
because only three states need to beestimated. Moreover there is no
need to make anassumption about the dynamics of the motion,and
hence such a solution fits most scenarios.Each accelerometer
discrete time measurementis composed of true value plus a white
noisecomponent.
ameas = atrue +wacc=p¢t: (22)
The white noise wacc has the unit of g=pHz, where
g is the gravity or its equivalent derivatives. Thediscrete
white noise depends on the square root of thesampling time
p¢t. The measured velocity change of
each accelerometer is expressed as
¢ºmeas =¢ºtrue +waccp¢t: (23)
The variance Ra of each accelerometer measurementof velocity
change is
Ra = Ef(¢ºmeas¡¢ºtrue)2g= Efw2accg¢t: (24)All accelerometers are
modeled with a commonupper bound of the noise variance, as they
wouldbe in reality. The error corrupting the angularacceleration
vector given in (8) is inherited from theaccelerometers’ errors, as
it is a linear combinationof accelerometers. This combination
results ina correlated process noise, and its covariance iscomputed
as
Qk¡1 = (¢t)2μRad2
¶264 1 ¡14 ¡ 14
¡ 14 1 ¡ 14¡ 14 ¡ 14 1
375 : (25)The predicted or a priori estimation error covariance
isupdated as
P¡k = Fk¡1P+k¡1F
Tk¡1 +Qk¡1: (26)
The predicted or a priori state estimate is updated as
x̂¡k = Fk¡1x̂
+k¡1 +Gk¡1uk¡1: (27)
C. Measurement Update
We plug the measured velocity changes of theaccelerometers in
the six quadratic terms in to (9)and (10) and multiply the
resulting sum by thesampling time to derive the measurement of
thestate vector. Considering the existence of whiteGaussian noise
in each accelerometer measurement,the observation inherits also a
white Gaussian noisev vector
yk= hk(x,v) = [x1x2 x1x3 x2x3 x
21 x
22 x
23]Tk
+[v1 ¢ ¢ ¢v6]Tk : (28)
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Fig. 3. Angular change profile.
The Jacobian of the measurement vector is computedas
Hk =@hk(x,v)@x
¯̄̄̄x=x̂¡k
=
26666666664
x2 x1 0
x3 0 x10 x3 x22x1 0 0
0 2x2 0
0 0 2x3
37777777775x=x̂¡k
:
(29)The measurement is correlated, and its covariance iscomputed
as
Rk = (¢t)2μRad2
¶266666666664
1 1414 0 0 ¡ 12
14 1
14 0 ¡ 12 0
14
14 1 ¡ 12 0 0
0 0 ¡ 12 32 ¡ 12 ¡ 120 ¡ 12 0 ¡ 12 32 ¡ 12¡ 12 0 0 ¡ 12 ¡ 12
32
377777777775:
(30)
The process noise is correlated with the measurementnoise, and
its cross-covariance is computed as
EfwkvTj g=Mk±k¡j+1
Mk = (¢t)2μRad2
¶(31)
£
264¡14
14 0 0 ¡ 12 12
14 0 ¡ 14 12 0 ¡ 120 ¡ 14 14 ¡ 12 12 0
375 :The Kalman gain is updated since we consider
across-covariance to have a better performance as
shown in the literature covering the Kalman filtering,e.g.,
[22].
Kk = (P¡k H
Tk +Mk)(HkP
¡k H
Tk +HkMk +M
Tk H
Tk +Rk)
¡1:
(32)The corrected or a posteriori estimation errorcovariance is
updated as
P+k = P¡k ¡Kk(HkP¡k +MTk ): (33)
The corrected or a posteriori state estimate is updatedas
x̂+k = x̂
¡k +Kk(yk ¡ hk(x̂
¡k ,0)): (34)
V. THREE-STATE MODEL SIMULATION RESULTS
A. The Trajectory Profile
In simulations we consider the trajectory ofstabilizing an
object subject to parasitic vibrations inone dimension [23]
combined with a constant angularrotation in the other dimension and
no motion in thethird dimension. The mathematical description of
thisangular motion is given as
!(t) = [!x 0 !z sin(2¼ft)]T
¾k =¢t!(tk):(35)
The sampling time ¢t is chosen to be 0.01 s, andthe frequency of
vibration f is set to 0.1 Hz. Themagnitudes of angular velocity
components areset to !x = 3:29 rad/s and !z = 0:658 rad/s. Thetotal
simulation time is 20 s, and the separationdistance is set to 0.1
m. We seeded a white noiseerror of 50 ¹g=
pHz (velocity random walk) in each
accelerometer measurement and extracted the AIVbased on
(8)—(10). Fig. 3 shows a plot of the angularchange profile.
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Fig. 4. Angular change estimation errors.
Fig. 5. Angular change estimation errors with bias in
accelerometers.
We plotted the estimation errors for the trajectoryas shown in
Fig. 4.
B. Effect of Noncalibrated Bias
To test the effect of noncalibrated bias, we addeda random
constant bias generated from a Gaussiandistribution with one
standard deviation of 50 ¹gto each noisy accelerometer measurement
describedpreviously. We repeated the same setup as before
andcreated a plot of the estimation errors shown in Fig. 5.One
important note here: at a relatively high
sampling rate (e.g., 0.01 s or more), the magnitudeof the error
due to white noise is about 10 times themagnitude of error due to
the remaining bias for this
example of accelerometer specifications. Consequentlyfor this
sampling rate, the noise error dominates thebias error in this
accelerometer, and for this scenariothe EKF model works without a
big difference fromthe one without bias. Hence the use of a white
noiseerror model can be justified considering that the Kalmanfilter
will tolerate such a small remaining bias error.
C. Effect of Improper Initialization
Here we investigated the effect of improperinitialization on the
filter performance. We simulatedthe previously described setup
under initializationerrors. The initialization error is set to be
equal to0.01 rad in each of three angular change components.
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Fig. 6. Angular change estimation errors for improper
initialization.
In general a proper sign initialization is important toavoid
convergence to the opposite sign. The filterconverges to the
reference angular change profilebecause of the availability of
measurements ofquadratic angular change as shown in Fig. 6.
VI. ATTITUDE UPDATE USING ROTATION ANGLEVECTOR
The rotation angle vector is used to update theattitude
direction cosing matrix (DCM) as advocatedby Bortz [24] or to
update the quaternion vector[25]. In this section, we show how to
update theDCM using a rotation angle vector; however, thesame
principle applies to quaternion’s update. Theattitude DCM, denoted
next as C, is the attitude ofthe body with respect to the fixed
inertial frame. Thematrix Ak is another DCM that transforms from
bodycoordinates at the (k+1)th computer cycle to bodycoordinates at
the kth computer cycle using the anglerotation vector as shown in
[24]—[26]. The continuousform of the DCM differential equation and
its discretetime solution are given as
_C = C!bib£
Ck+1 = CkAk, Ak = eR tk+1tk
!bib£dt:
(36)
Next we consider the case where the direction of theangular rate
vector !bib remains fixed in space, i.e.,nonconing motion. Assuming
that the angular velocityvector has little change during the update
interval, wecan approximate the angle rotation by the
previouslyestimated angular change ¾.Z tk+1
tk
!bibdt ¼ ¾: (37)
This transformation matrix Ak is computed as
Ak = I+sin¾¾[¾£] + 1¡ cos¾
¾2[¾£]2
¾ =q¾2x +¾2y +¾2z :
(38)
Equation (38) is composed of either linear orquadratic terms of
the angular rotation vector. It canbe approximated considering the
first three terms asgiven next:
Ak ¼
266666641¡ (¾
2y +¾
2z )
2
¾x¾y2¡¾z
¾x¾z2+¾y
¾x¾y2
+¾z 1¡(¾2x +¾
2z )
2
¾y¾z2¡¾x
¾x¾z2¡¾y
¾y¾z2
+¾x 1¡(¾2x +¾
2y )
2
37777775 :
(39)
VII. CONSTRAINTS DERIVATION
In this section we derive the equality constraintsthat exist in
the model because of orthogonalaccelerations. For a nonconing
motion, the centripetalacceleration vector !bib£!bib £pb and the
tangentialacceleration vector _!bib£pb in (1) are orthogonal.
Thetwo vectors are represented as
ac =
264¡(!2y +!
2z )px+!x!ypy +!x!zpz
!x!ypx¡ (!2z +!2x )py +!y!zpz!x!zpx+!y!zpy ¡ (!2x +!2y )pz
375 (40)
at =
264¡ _!zpy + _!ypz_!zpx¡ _!xpz¡ _!ypx+ _!xpy
375 : (41)EDWAN ET AL.: CONSTRAINED ANGULAR MOTION ESTIMATION IN
A GYRO-FREE IMU 603
-
We use three vectors as a basis set that spans allpossible
distributed positions264pxpy
pz
375 28>:264100
375 ,264010
375 ,264001
3759>=>; : (42)
Applying the inner product of the two vectorsgiven in (40) and
(41) with three different values of[px py pz]
T as given in (42) produces the followingset of
equations:264¡(!
2y +!
2z )
!x!y
!x!z
375¯264 0_!z¡ _!y
375= 0 (43)264 !x!y¡(!2z +!2x )
!y!z
375¯264¡ _!z0_!x
375= 0 (44)264 !x!z!y!z¡(!2x +!2y )
375¯264 _!y¡ _!x0
375= 0: (45)After computing the inner product in the
aboveequations, we get
!x!y _!z ¡!x!z _!y = 0!y!z _!x¡!x!y _!z = 0!x!z _!y ¡!y!z _!x =
0:
(46)
The last constraint is dependent on the first two.Therefore, the
remaining two independent constraintsare
!x!y _!z ¡!x!z _!y = 0!y!z _!x¡!x!y _!z = 0:
(47)
VIII. HANDLING OF CONSTRAINTS WITHIN THEEKF
We derived two constraints that need to beintegrated within the
EKF algorithm. The constraintequations shown in (47) contain
angular accelerationcomponents. The derivation of the
angularacceleration vector from the angular velocity vector
bydifferentiation gives a noisy value; hence we augmentthe angular
acceleration in the state vector. Moreoverthe derivation of the
Jacobian of the constraints willbe much easier. Using the angular
change vectoraugmented with the angular velocity change vectoras a
state vector results in the following state vector:
x= [x1 ¢ ¢ ¢x6]T = [¾x ¾y ¾z ®x ®y ®z]T:(48)
A. Initialization
We initialize the state vector and the estimationerror
covariance in a similar way to (17) and (18).
B. Prediction
We use a nearly constant angular velocitychange state-space
model, which is also known asWiener-sequence model, for the process
update. Itassumes that the angular velocity change incrementis an
independent (white noise) process. Othersophisticated models for
target dynamics which canbe used for the process update are
surveyed in [27].As discussed previously, using this model, we
canestimate the bias parameters in the AIV when they areaugmented
to the state vector. This model is expressedin discrete time as
xk = F0k¡1xk¡1 +G
0k¡1wk¡1
F 0k¡1 =·I3£3 ¢tI3£303£3 I3£3
¸G0k¡1 =
·¢tI3£3I3£3
¸:
(49)
This process noise covariance is computed as
Q0k¡1 =G0k¡1cov(wk¡1)G
0k¡1
T: (50)
The a priori estimation error covariance is updated as
P¡k = F0k¡1P
+k¡1F
0k¡1
T+Q0k¡1: (51)
The a priori state estimate is updated as
x̂¡k = F
0k¡1x̂
+k¡1: (52)
C. Measurement Update
The whole AIV given in (8)—(10) is used to derivethe measurement
of the new state vector. Consideringthe existence of a white
Gaussian noise in eachaccelerometer measurement, the observation
inheritsalso white Gaussian noise v vector
y0k= h0k(x,v)
= [x4 x5 x6 x1x2 x1x3 x2x3 x21 x
22 x
23]Tk
+[v1 ¢ ¢ ¢v9]Tk : (53)The Jacobian of the new measurement model
H 0kis expressed in terms of the Jacobian of three-statemodel H,
given in (29)
H 0k =@h0k(x,v)@x
¯̄̄̄x=x̂¡k
=·03£3 I3£3H 06£3
¸x=x̂¡k
: (54)
The measurement covariance R0k of the new modelis computed based
on the previously derivedmeasurement covariance Rk,
cross-correlatedcovariance Mk, and covariance Qk¡1 of the
three-statemodel
R0k =·Qk¡1=(¢t)
2 Mk=¢t
MTk =¢t Rk
¸: (55)
The Kalman gain is updated as
Kk = P¡k H
0kT(H 0kP
¡k H
0kT+R0k)
¡1: (56)
604 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.
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The a posteriori estimation error covariance can beupdated
as
P+k = P¡k ¡KkH 0kP¡k : (57)
The a posteriori state estimate is updated as
x̂+k = x̂
¡k +Kk(y
0k¡h0k(x̂¡k ,0)): (58)
D. Handling of the Nonlinear Equality Constraints
There are many approaches for handling equalityconstraints in
the Kalman filtering literature. Commonapproaches include
pseudomeasurement, projection,and model reduction approaches. Due
to thenonlinearity of the constraints, model reduction is
notapplicable in our system. In the pseudomeasurementapproach, also
known as the pseudoobservationapproach, each nonlinear constraint
is modeledas an observation with additive white Gaussiannoise to
relax the rigidity of the constraints. Anexample of that approach
for the use of the kinematicconstraint is described in [28] with
the decreasingcovariance rationale which was introduced to relaxthe
rigidity of the constraints. In the projectionapproach, the
unconstrained estimate is projectedonto the constraints surface as
developed by Simon[29] using its second-order nonlinear extension
[30].The UKF, which has proven to be superior to theEKF in the case
of significant nonlinearities, canalso be applied to incorporate
nonlinear constraintsto achieve a better estimation performance
eitherusing the pseudomeasurement approach or projectionfunction
[31]. However, we choose an algorithm,which might be
computationally less demandingcompared with other algorithms. In
our simulationswe use the smoothly constrained Kalman filter(SCKF)
developed by Geeter for applying nonlinearconstraints because
nonlinear constraints are differentfrom linear constraints due to
truncation andbase-point errors resulting from EKF
linearization[31, 32]. Essentially the SCKF algorithm is
classifiedas a pseudomeasurement approach combined withthe iterated
EKF (IEKF), which iteratively appliesnonlinear observations to
reduce truncation errors.In every time step, each constraint is
linearized withthe last available estimate and applied iteratively
withexponentially decreasing uncertainty. The smoothapplication of
constraints is motivated by the fact thatapplying a measurement of
variance ³ is equivalentto applying the same measurement k times
each withvariance k³. The equality constraints function and
itsJacobian matrix, considering the new state vector, arederived
from (47) as
ec(x) =·x1x2x6¡ x1x3x5x2x3x4¡ x1x2x6
¸= 0
Ec =@ec(x)@x
=·x2x6¡ x3x5 x1x6 ¡x1x5 0 ¡x1x3 x1x2¡x2x6 x3x4¡ x1x6 x2x4 x2x3 0
¡x1x2
¸: (59)
The following algorithm summarizes how toincorporate the
constraints after measurement updateusing the SCKF
x̂= x̂+k , P = P+k
for c= 1,2
fi = 0, e(x) = rowc(ec(x)), E = rowc(Ec)jx̂³0 = °EPE
T, s=maxj(EjPjjEj)=EPET
while (s < sth)
f³c = ³0e¡i, E = rowc(Ec)jx̂K = (PET)(EPET+ ³c)
¡1
x̂= x̂¡Ke(x̂)P = (I6£6¡KE)Pi= i+1
s=maxj(EjPjjEj)=EPET
g end while g end forx̂+k = x̂, P
+k = P:
(60)
The function rowc finds the cth row of ec(x) andEc to apply
constraints sequentially. The value ofthe fraction factor ° for the
initial variance needs tobe verified by simulations. The relative
strength scompares the variance in the denominator, which isa sum
of terms, to one of the largest terms formingthat variance. The
symbol sth is a termination criterionfor the application of
constraints, which Geeter setsto be sth = 100 as a practical value.
To avoid anyconvergence problems, we apply those constraintsafter
sufficiently applying the observations. For therapidly changing
angular velocity profile, it is difficultto estimate the angular
velocity change; consequentlyin such situations the constraints are
not applied.
IX. CONSTRAINED MODEL SIMULATION RESULTSAND ANALYSIS
A similar setup to the one described in Section Vis repeated
here for the constrained model. In thefollowing plots, we compare
the performance of theEKF constrained model with the EKF
unconstrainedmodel. We choose a constant angular velocitytrajectory
and we set the separation distance d to 1 m.
!(t) = ![1 1 1]T: (61)
EDWAN ET AL.: CONSTRAINED ANGULAR MOTION ESTIMATION IN A
GYRO-FREE IMU 605
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Fig. 7. RMSE of angular change vector.
Fig. 8. RMSE of ¾x.
TABLE IAccelerometers Categories
PerformanceParameter Consumer Automotive Tactical
Noise Floor VRW(¹g=
pHz)
2000 1000 500
The angular velocity along each component is setto ! = 3:29
rad/s. Three levels of white noise errorare considered in the
simulation, namely, tactical,automotive, and consumer grade, as
categorized
by Gebre-Egziabher [33] with a doubling of theconsumer-grade
noise level to distinguish it from theautomotive grade as shown in
Table I. We evaluatethe improvement in the estimation performanceby
computing the root mean square error (RMSE)over the Monte Carlo
runs. Clearly there is animprovement due to the use of the dynamic
modelover the three-state model.The RMSE is computed over 500 Monte
Carlo
runs. Due to space limitations, we show here thesimulation
results in Figs. 7—10 using the tacticalaccelerometers category
only. We applied theconstraints directly after 10 s had passed to
distinguish
606 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.
47, NO. 1 JANUARY 2011
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Fig. 9. RMSE of ¾y .
Fig. 10. RMSE of ¾z .
TABLE IIComparison of Steady State RMSE for Different
Accelerometers Categories
d = 1 m Tactical Grade Automotive Grade Consumer Grade
EKF Model
Term Unconstrained Constrained Unconstrained Constrained
Unconstrained Constrained
¾ [rad] 4:09£ 10¡5 8:54£ 10¡6 8:44£ 10¡5 1:47£ 10¡5 1:66£ 10¡4
2:64£ 10¡5¾x [rad] 2:38£ 10¡5 4:97£ 10¡6 4:74£ 10¡5 8:17£ 10¡6
9:94£ 10¡5 1:55£ 10¡5¾y [rad] 2:29£ 10¡5 4:87£ 10¡6 4:84£ 10¡5
8:53£ 10¡6 9:32£ 10¡5 1:50£ 10¡5¾z [rad] 2:41£ 10¡5 4:95£ 10¡6
5:04£ 10¡5 8:74£ 10¡6 9:42£ 10¡5 1:52£ 10¡5
between the improvement due to use of the dynamicmodel and the
improvement due to the application ofconstraints. The RMSE of the
angular change
components and the angular change vector aretabulated in Table
II for the three categories ofaccelerometers.
EDWAN ET AL.: CONSTRAINED ANGULAR MOTION ESTIMATION IN A
GYRO-FREE IMU 607
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For all categories of accelerometers, there was aclear
superiority of the constrained model over thethree-state model in
the steady-state RMSE. Typicallywe should have a similar
improvement reflected in theRMSE of each estimated angular change
componentdue to the symmetry of the GF-IMU configuration,equal
signal input in the 3D components, andsymmetry of the constraints.
The simulation resultsare in agreement with this.
X. CONCLUSION
We proposed two EKF models for the estimationof the angular
motion from a multiple distributedtri-axial accelerometers
measurements in a GF-IMU.The performance of the constrained EKF
wasremarkably better than the unconstrained EKF. Afuture work is to
incorporate rotating accelerometersin the GF-IMU and to find the
corresponding properfilter scheme.
ACKNOWLEDGMENTS
We greatly appreciate the support of the GermanAcademic Exchange
Service (DAAD) for the doctoralwork of Ezzaldeen Edwan within the
InternationalPostgraduate Programme (IPP) Multi Sensorics.
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Ezzaldeen Edwan (M’01) received the B.Sc. degree in electrical
engineeringfrom Birzeit University, West Bank, Palestine in 1997
and the M.Sc. degree inelectrical engineering from Oklahoma State
University, OK, in 2003.Since June 2006, he has been pursuing his
doctoral degree in the IPP
Multi Sensorics at the Center for Sensorsystems (ZESS),
University of Siegen,Germany. His main research interests are
low-cost INS, gyro-free INS, andestimation theory applied in the
integrated navigation systems. He has experiencein the academic
field lecturing for electrical engineering courses.Mr. Edwan is a
recipient of the Fulbright and the DAAD scholarships,
respectively.
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[33] Gebre-Egziabher, D.Design and performance analysis of a
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Stefan Knedlik (M’04) received the Diploma degree in electrical
engineering andthe Dr. Eng. degree from the University of Siegen,
Siegen, Germany in 1998 and2003, respectively.Since 1998, he has
been a member of the Center for Sensorsystems (ZESS),
University of Siegen, and since 2003 he has also been a
researcher/assistantprofessor at the Institute of Signal Processing
and Communication Theory atthe Department of Electrical Engineering
and Computer Science. In severalresearch projects, e.g., in
cooperation with the German Aerospace Agency (DLR),he developed
state and parameter estimation strategies, often with respect
tosynthetic aperture radar (SAR) interferometry. A few years ago,
he founded aresearch group on navigation. He is also the executive
director of the InternationalPostgraduate Program (IPP) Multi
Sensorics and of the NRW Research Schoolon Multi Modal Sensor
Systems for Environmental Exploration and Safety(MOSES). His
current research interests include signal processing and
appliedestimation theory in navigation.
Otmar Loffeld (M’05–SM’06) received the Diploma degree in
electricalengineering from the Technical University of Aachen,
Aachen, Germany in1982 and the Eng.Dr. and the “Habilitation”
degrees in the field of digital signalprocessing and estimation
theory from the University of Siegen, Siegen, Germanyin 1986 and
1989, respectively.Since 1991, he has been a Professor of Digital
Signal Processing and
Estimation Theory with the University of Siegen. In 1995, he
became aMember of the Center for Sensorsystems (ZESS), which is a
central scientificresearch establishment at the University of
Siegen, where since 2005 he hasbeen the chairman. He is author of
two textbooks on estimation theory. He iscurrently PI for
interferometric techniques in the German TerraSAR-X mission,and
together with Prof. Ender from Forschungsgesellschaft für
AngewandteNaturwissenschaften (FGAN), he is one the PIs for a
bistatic spaceborne-airborneexperiment, where TerraSAR-X serves as
the bistatic illuminator while FGAN’sPAMIR system mounted on a
Transall airplane is used as a bistatic receiver. In2002, he
founded the International Postgraduate Program “Multi
Sensorics,”and based on that program, he established the “NRW
Research School onMulti Modal Sensor Systems for Environmental
Exploration and Safety” at theUniversity of Siegen as an upgrade of
excellence in 2008. He is the speaker andcoordinator of both
doctoral degree programs, hosted by ZESS. Furthermore,he is the
university’s Scientific Coordinator for Multidimensional and
ImagingSystems.His current research interests include multisensor
data fusion, Kalman filtering
techniques for data fusion, optimal filtering and process
identification, SARprocessing and simulation, SAR interferometry,
phase unwrapping, and baselineestimation. A recent field of
interest of his is bistatic SAR processing.Professor Loffeld is a
Member of the ITG/VDE and Senior Member of
the IEEE/GRSS. He was the recipient of the Scientific Research
Award ofNorthrhine-Westphalia (“Bennigsen-Foerder Preis”) for his
works on applyingKalman filters to phase-estimation problems such
as Doppler centroid estimationin SAR, phase, and frequency
demodulation.
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