research article quantification of trunk postural...

Research ArticleQuantification of Trunk Postural Stability Using ConvexPolyhedron of the Time-Series Accelerometer Data

Roman Melecky1 Vladimir Socha1 Patrik Kutilek1 Lenka Hanakova1

Peter Takac2 Jakub Schlenker1 and Zdenek Svoboda3

1Faculty of Biomedical Engineering Czech Technical University in Prague 272 01 Kladno Czech Republic2Department of Rehabilitation and Spa Medicine Faculty of Medicine P J Safarik University in Kosice 040 01 Kosice Slovakia3Palacky University of Olomouc Faculty of Physical Culture 771 11 Olomouc Czech Republic

Correspondence should be addressed to Lenka Hanakova lenkahanakovafbmicvutcz

Received 1 May 2015 Revised 1 February 2016 Accepted 29 March 2016

Academic Editor Jiann-Shing Shieh

Copyright copy 2016 Roman Melecky et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Techniques to quantify postural stability usually rely on the evaluation of only two variables that is two coordinates of COPHowever by using three variables that is three components of acceleration vector it is possible to describe human movementmore precisely For this purpose a single three-axis accelerometer was used making it possible to evaluate 3D movement by use ofa novel method convex polyhedron (CP) together with a traditional method based on area of the confidence ellipse (ACE) Tenpatients (Pts) with cerebellar ataxia and eleven healthy individuals of control group (CG) participated in the studyThe results showa significant increase of volume of the CP (CPV) in Pts or CG standing on foam surface with eyes open (EO) and eyes closed (EC)after the EC phase Significant difference between Pts and CGwas found in all cases as well Correlation coefficient indicates strongcorrelation between the CPV and ACE in most cases of patient examinations thus confirming the possibility of quantification ofpostural instability by the introduced method of CPV

1 Introduction

Postural stability of the body segments during standing espe-cially trunk stability can be negatively influenced by manydiseases of the nervous ormusculoskeletal system [1] Patientswith these deficits often show instability during stance tasks[2] Thus the trunk accelerations during stance can be quan-titative indicators of impaired balance control in individualswith neurological disorders [3] Hence the evaluation ofaccelerations suits the needs of clinical practice since theyreflect the changes in position as well as the intensity andmagnitude of tremorThe biomedical community is currentlystarting to use the triaxial inertial measurement units (IMU)for high-accuracy measurement of human body segmentmovements (accelerations and orientations) instead of com-monly used force (posturography) platforms used to studythe center of pressure (CoP) movements of whole body [4]Assessment of trunkmovements using IMUmay yield clearer

insights into balance deficits and provide a considerablycheaper and better diagnostic tool than more traditionalpreviously documented measures The IMUs were placed onspinous processes of T1 andor S1 for measuring the motionof trunk and pelvis during quiet standing Although theIMU can measure three angles and three accelerations thetechniques to quantify segmentmovements using only one ortwo measured quantities were introduced in clinical practice[4] Thus in clinical practice one of the greatest advantagesof the IMUcompared to traditional posturography platformswhich allow for measuring only the 2D movements intransversal plane is not used The reason for this is that theapplication of the IMU for trunk acceleration measurementduring stance is relatively new and the IMU has not beenpreviously used to study the range postural balance problemsand patients with specific types of diseases Thus the postur-ography platforms are always the main tool for the study ofbody movement of the patients for instance suffering from

Hindawi Publishing CorporationJournal of Healthcare EngineeringVolume 2016 Article ID 1621562 9 pageshttpdxdoiorg10115520161621562

2 Journal of Healthcare Engineering

cerebellar diseases [5] Therefore the first objective of thispaper is to test the application of the IMU in an area whereIMUs have not been used before and where IMUs will replaceconventional posturography platforms

Since the design is intended to identify connectionsbetween trunk movement in space and neurological disor-ders it is used to diagnose cerebellar disease characterizedby tremor or sway Cerebellar ataxia can arise as a result ofmany diseases and present itself with symptoms of coordi-nation balance gait and extremity movements difficultiesLesions to the cerebellum can cause dyssynergia dysmetriadysdiadochokinesia dysarthria ataxia of stance and gait andso forth see [6 7] Deficits can be observed on movementson the same side of the body as the lesion [6] Cliniciansoften use motor tasks in order to verify the signs of ataxia[7] Patients with a cerebellar ataxia diagnosis are interestingnot only because of their impaired postural stability butalso because there is no effective causal pharmacotherapyTherefore it is necessary to look for methods which canenhance the accuracy of evaluation procedures of patientsrsquopostural stability during the treatment

The second objective of the paper is to design and testa new method of quantitative evaluation of 2D data setfor the 3D trunk movement measured by IMU Traditionalmore complexmethods for processing themeasured data andassessing the postural instability using at least two measuredvariables aremethods based on the 2D convex hull 2D confi-dence ellipse or length of trajectory obtained by plotting twovariables versus each other [8ndash10] Usually thesemethods areused to evaluate 2D data set from posturography platforms[6] However there are limitations of these traditional solu-tions to quantify postural stability which were originally usedto evaluate 2D data set from the posturography platforms andwhich are now being used to evaluate data from the IMUsAs mentioned before the limitation is that these methodsare based solely on the evaluation of only two variables eachin one of the two human body planesaxes This can leadto a loss of important information about physical activityspecifically the third physical quantity (ie acceleration) of3D movement However we can also model the distributionof the measured 3D data (ie three orthogonal accelerations)of human body segment instead of the analysis of 2D data

Therefore this study is aimed at introducing a novelmethod used in the identification of pathological balancecontrol using IMU to measure accelerations as well as theconvex polyhedron of plotting three accelerations versus eachother (the evaluation of 3D data set of superior-inferior accel-eration mediolateral acceleration and anterior-posterioracceleration of body segments) It follows practice consistingin the assessment of the convex hull of only two variables(specifically angles) measured by IMU [11] The convex hullarea has already been used in clinical practice to study postu-ral balance problems but the concept of convex polyhedronvolume (CPV) has never been used before in clinical practiceto study postural balance problems by three accelerations[8] The choice for this novel design came from the abilityof a single variable which defines the shape of the convexpolyhedron used to describe changes in three accelerationsconsidering wide availability of new cheaper triaxial IMUs

(ordinary cell phones or watches) [12 13]The applicability ofthe convex polyhedron variable describing the body segmentmovement foreshadows the immense potential of a simpleand inexpensive IMU capable of direct evaluation of complex3D movement (ie 3D acceleration) as a whole Moreoverthe combinations of superior-inferior mediolateral andanterior-posterior accelerationmeasurements during specificbalance tasks could identify new and specific differences inbalance control of patients compared to healthy subjectsThefinal significant reason for themeasurement and evaluation ofall three accelerations instead of only two is that the calibra-tion of the cheap triaxial IMU may not be accurate and thusthe measurement of all three accelerations minimalizes theloss of information about the 3D movement as a whole Theinclusion of all three accelerations allows us tominimalize theinfluence of mispositioning the IMU on a body segmentTheclaim follows the general assumption that acceleration vectorin 3D space is defined by the three segments of the vectorwhile these segments are typically considered in accordancewith the Earthrsquos coordinate system axes or anatomical axesof a particular body part If the sensor is misplaced on thebody part or miscalibrated as for the coordinate system themeasured segments of acceleration in their respective axeswill be inaccurate If we choose to examine 3D movement byusing only two segments of acceleration a piece of informa-tion may be lost which is never the case if all three segmentsare observed The inclusion of all three accelerations allowsus to minimalize the influence of mispositioning of the IMUon a body segment

The method of recording and processing 3D data thusoffers the possibility to measure stability in smaller medicalcenters or at home by using IMU implemented in forexample a mobile phone instead of the traditional and spa-cious posturography platforms There is also a possibility forpatients to perform the therapy examination or training athome The method also eliminates the risk of mispositioningthe sensor and loss of vertical movement data and at thesame time presents patients and medical staff with an easy-to-interpret 3D data processing methodTherefore the focusof this work is to identify suitability of the convex polyhedronand IMU data for clinical application

2 Methods and Materials

21 Participants In order to test the new methods it isnecessary to compare healthy subjects without any posturalbalance problems to participants who have postural balanceproblems Ten volunteer patients (Pts) (six women and fourmen age of 522 (SD 117) years) with degenerative cerebellarataxia participated [7] The patients were recruited from theFaculty Hospital Motol Prague Czech Republic A board-certified neurologist had previously diagnosed progressivecerebellar disease Diagnostic evaluation included a neuro-logic examination laboratory blood tests and a brain MRIThe patients were measured in the initial phase of the clinicrsquostwo-week rehabilitation program Eleven healthy individualsof control group (CG) (five women and six men age of 260(SD 64) years) were also recruited for comparable analysisHealthy subjects were recruited from the studentsvolunteers

Journal of Healthcare Engineering 3

A-P axis

M-L

axis

S-I a

xis

3DoF orientation trackerSy

naps

ys P

ostu

rogr

aphy

Sys

tem

(MTx unit)

Figure 1 The MTx unit used to measure angles and accelerationsof the trunk and the Synapsys Posturography System used tomeasure the COP displacements S-I superior-inferior axis M-Lmediolateral axis and A-P anterior-posterior axis

at Charles University in Prague In the case of the CG thediagnostic evaluation included a detailed disease history aneurological examination and a routine laboratory testingThe study was performed in accordance with the HelsinkiDeclaration The study protocol was approved by the localEthical Committee and the University Hospital Motol andan informed consent was obtained from each subject Thesubjects were chosen for measurement randomly and ondifferent days

22 Measurement Equipment and Test Procedure

221 Measurement Equipment Xbus Master a lightweight(330 g) and portable device using Motion Tracker Xsens(product abbreviation MTx) units for orientation and accel-eration measurement of body segments (see Figure 1) wasused for measurements of trunk movements MTx unit withan embedded accelerometer and gyroscope is accurate IMUmeasuring drift-free 3D orientation and 3D accelerationTheMTx unit was calibrated before each clinical examination bycalibration measurement The MTx unit was set up in a waythat the one axis of the coordinate system of the MTx unitwas parallel to the anterior-posterior axis that is symmetryaxis of the fixed stationary platform of the Synapsys Pos-turography System on which the participants stood and theother two axes were perpendicular to the anterior-posterioraxis (ie symmetry axis of the platform) with respect to thedirection of Earthrsquos gravity that is superior-inferior axis wascolinear with the direction of gravity After calibration theMTx unit was placed on patientrsquos trunk according to [14 15]at the level of the lower back (lumbar 2-3) see Figure 1

The data sets comprised of the three Euler angles (roll(Φ) yaw (Ψ) and pitch (Θ)) as well as three orthogonalaccelerations (119886119878119909 119886119878119910 119886119878119911) in the accelerometer coordinatesystem (ie three orthogonal components of acceleration

direction corresponding to the three principal axes of theMTx unit accelerometer) were measured using an MTxunit placed on the subjects trunks while Pts and CG wereperforming a quiet stance on a fixed stationary platform ofthe Synapsys Posturography System [16 17] Conventions ofEuler angles are described in [18] The three accelerationsmeasured by the accelerometer of MTx unit are described indetail in [19]

Also measurement of the human body center of pressure(COP) displacement (ie postural sway) by force platformSynapsys Posturography System was performed to comparethe data obtained by the traditional method with the dataobtained by the IMU The Synapsys Posturography Systemprovides information about the area of the 95 confidenceellipse of COP excursions [20] Comparison between COPcharacteristics and accelerometer data took place due tothe fact that COP movement is given by COM (center ofmass) movement position of which follows the position ofindividual segments A significant body segment is the trunkon which the accelerometer (ie IMU) is placed Change inthe position of the trunk which is measured as accelerationusing IMU thus directly affects the position of COM andtherefore also COP These data can of course differ withrespect to the influence of other segments on the COP posi-tion it is however assumed that the trunk position (or itsmovement) has the most significant influence on the COPposition change It would be examined and verified whetherthere is indeed a correlation between the data from the IMUand data from the posturography platform data which wouldsuggest that the platform is fully replaceable by the IMU

222 Test Procedure The body sway of each participant wasmeasured by the Xsens system (Xsens Technologies BV) andSynapsys Posturography System (Synapsys Inc) during quietstance on a firm surface (FiS) and soft foam surface (FoS)witheyes open (EO) and eyes closed (EC) [21]

The sequence of the four measurement settings for eachsubject was as follows EO FiS EC FiS EO FoS and EC FoSThe order of the four settings was set and followed in all par-ticipants The subjectrsquos bare feet were positioned next to eachother splayed at the angle of 30∘ and arms were always inhanging positionThe tasks included standing on both feet forat least 60 seconds [22] Both systems recorded body activityat the same time that is data were recorded simultaneouslyby both systems Time synchronization of the measured data(ie data from both systems) was achieved by controllingboth systems and processing data on the same computerThemeasurements usually lasted a few seconds longer and theinitial data have been cut off so that all data sets have a recordlength of 60 seconds The data were recorded with a samplefrequency of 100Hz (for both systems) Kalman filter wasimplemented in the Xbus Kit system and the MT Managersoftware of the Xbus Kit system was used for data storage

23 Data ProcessingMethod Thethree Euler angles and threeaccelerations in the accelerometer coordinate system are usedto calculate the accelerations in the global reference systemand then in the anatomical coordinate frameThe calculationis based on the rotational matrices The first rotation matrix


119877119866119878

rotates an acceleration vector 119878= (119886119878119909 119886119878119910 119886119878119911)

119879 in thesensor coordinate system (119878) to the global reference system(119866)

119866= 119877119866119878sdot 119878 (1)

where the matrix 119877119866119878

is interpreted in terms of Euler angles[23]

119877119866119878 = 119877119885

Ψsdot 119877119884

Θsdot 119877119883

Φ (2)

where

119877119885

Ψ=[[

[

cosΨ minus sinΨ 0sinΨ cosΨ 00 0 1

]]

]

119877119884

Θ=[[

[

cosΘ 0 sinΘ0 1 0

minus sinΘ 0 cosΘ

]]

]

119877119883

Φ=[[

[

1 0 0

0 cosΦ minus sinΦ0 sinΦ cosΦ

]]

]

(3)

The acceleration vector 119866 = (119886119866119909 119886119866119910 119886119866119911)

119879 in theglobal reference system is then rotated to the anatomicalcoordinate frame (119860)

119886= 119877119860119866sdot 119866 (4)

where second rotation matrix 119877119860119866

is

119877119860119866= 119877119885

Ψ0=[[

[

cosΨ0minus sinΨ

00

sinΨ0 cosΨ0 00 0 1

]]

]

(5)

The angle sinΨ0is obtained during the calibration process

of the MTx unit The calculated acceleration vector 119860=

(119886AP 119886ML 119886SI)119879 represents the superior-inferior acceleration

(119886SI) mediolateral acceleration (119886ML) and anterior-posterioracceleration (119886AP) The acceleration vectors or in otherwords time dependent data (119886SI 119886ML 119886AP) are plotted as a 3DplotThe set of points is obtained by plotting the accelerationsversus each other see Figure 2 The time of measurementthat is record length of the data set (60 s) and the samplefrequency (100Hz) affect the number of points in the setRecording frequency must be sufficiently high to recordalso a short time and random displacements of the bodysegment and no information about the range of motionduringmaintaining postural stability of stance is lost Using agreater number of registered points of higher frequency it ispossible to record movement more accurately without losinginformation on certain phases of rapid movement that is reg-istered through low frequency data collection For instanceinformation loss could occur on tremor of segments causedby higher frequencies Therefore the data was collected atthe frequency of 100Hz using MT Manager Version 170

19 195 2 205 21 215 2224

25

2692

925

93

935

aSI(m

sminus2)

aAP

(ms minus2)

aML (msminus2)

Figure 2 Example of a convex polyhedron obtained by plottingsuperior-inferior (SI) mediolateral (ML) and anterior-posterior(AP) accelerations versus each other

adjusted for human movements as recommended by themanufacturer [24] The choice of the sampling frequency of100Hz was made also by following previous studies whichused the same system for motion tracking (see [25ndash29])

The novel method for identification of pathologicalbalance control is based on mathematical tools for staticposturography [20 30] We can model the distribution of themeasured data by 2D convex hull or 3D convex polyhedron(CP) [31] A variable which can be used to describe the shapeof the 2D convex hull or 3D convex polyhedron can be thearea or volume We used a method based on the descriptionof the distribution of the measured data (ie 119886SI 119886ML and119886AP) by CP In mathematics the convex hull of a set of points(SP) in the Euclidean space is the smallest convex set thatcontains SP [32] The CP may be defined as the intersectionof all convex sets containing SP or as the set of all convexcombinations of points in SP [33]The set of points is obtainedby plotting three accelerations versus each other see Figure 2The CP of a set of points in 3D space is the smallest convexregion enclosing all points in the set see Figure 2Thenumberof points is determined by the time ofmeasurement (60 s) andthe sample frequency (100Hz) A custom-designedMATLABprogram based on the functions of the MATLAB software(MATLAB R2010b Mathworks Inc Natick MA USA) wasused to calculate the CP of the 3D plot The convex hullcomputation in MATLAB uses the Delaunay triangulation[34] Since there is no knownmethod of calculating the CPVwe can use the equations used to calculate the volume of anypolyhedron (PV) [35ndash37]

PV = 13

10038161003816100381610038161003816100381610038161003816100381610038161003816

119895

sum

119894=1

1198751119894sdot 119873119894sdot 119860119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

(6)

where 119860119894 is the surface area of a planar polygonal face 119878119894

119860119894=1

2

1003816100381610038161003816100381610038161003816100381610038161003816

119873119894sdot

119897

sum

119896=1

(119875119896119894times 119875(119896+1)119894)

1003816100381610038161003816100381610038161003816100381610038161003816

(7)


Thus

PV = 16

10038161003816100381610038161003816100381610038161003816100381610038161003816

119895

sum

119894=1

(1198751119894sdot 119873119894) sdot

1003816100381610038161003816100381610038161003816100381610038161003816

119873119894sdot

119897

sum

119896=1

(119875119896119894times 119875(119896+1)119894)

1003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816

(8)

where 1198751119894 119875119897119894 are the vertices of a planar polygonal face

119878119894 oriented counterclockwise with respect to the outwardpointing normal of planar polygonal face and 119873119894 is a unitoutward pointing vector normal to specific planar polygonalface 119878

119894

119873119894=(1198752119894minus 1198751119894) times (119875

3119894minus 1198751119894)

1003816100381610038161003816(1198752119894 minus 1198751119894) times (1198753119894 minus 1198751119894)1003816100381610038161003816

(9)

To calculate the PV the custom-designed MATLABprogram was also used Because the volume corresponds tothe volume of CP of 3D plot obtained by plotting 119886SI 119886MLand 119886AP versus each other the physical unit of the volume ism3sdotsminus6 Although the MTx unit also senses the gravitationalacceleration it is not necessary to subtract the gravitationalacceleration because the method of calculating the PV usesonly changes in the accelerations and the gravitational accel-eration is constant and perpendicular to the horizontal planeof the Earthrsquos surface

The area of the 95 confidence ellipse (ACE) of COPexcursions was used to compare the data obtained by theposturography system with data obtained by the IMU TheSynapsys Posturography System directly calculates the areasThus it is not necessary to convert the measured data Thephysical unit of the area is mm2

24 Statistical Analysis After calculating the CPV of eachpatient and healthy subject standing on a FiS and FoSwith EO and EC the Jarque-Bera test was used to test thenormal distribution of calculated CPVs The median (Mdn)minimum (Min) maximum (Max) first quartile (Q1) andthird quartile (Q3) of the CPV were then used to comparethe results Also theWilcoxon signed rank test andWilcoxonrank sum test were used to assess the significance of thedifferences between the measurements results that is tocompare the different stance conditions and CG with PtsThe significance level was set at 119901 lt 005 Also Spearmanrsquosrank correlation coefficient between the volume of convexpolyhedron and the area of the confidence ellipse of COPexcursions were calculated to study the differences betweenthe data from IMU and the center of pressure data Thestatistical analysis was performed using MATLAB software

A comparison of the same age groups is not necessarysince studies show that parameters of the body sway of thehealthy subjects within the age range between 20 and 60years vary only slightly [38 39] Aoki et al [38] found thatthere are insignificant differences in 10ndash60-year-old subjectsin COP sway parameters (ie Romberg quotients) Also adetailed analysis of age-related increase of CoP parametersby the polynomial type of regression showed that the gradualincrease of body sway that is significant degradation ofstability characterized by increase of CoP oscillations startsafter the age of 60 [39]

Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

Volu

me o

f con

vex

poly

hedr

on(m

3sminus

6)

p lt 005

p lt 005

p lt 005

70 times 100

70 times 10minus1

70 times 10minus2

70 times 10minus3

70 times 10minus4

Figure 3 Comparison of the volume of convex polyhedron of con-trol group (CG) and patients (Pts) standing on a firm surface (FiS)with eyes open (EO) and eyes closed (EC)

CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts

Volu

me o

f con

vex

poly

hedr

on(m

3sminus

6)

p lt 005

45 times 101

45 times 100

45 times 10minus1

45 times 10minus2

45 times 10minus3

Figure 4 Comparison of the volume of convex polyhedron of con-trol group (CG) and patients (Pts) standing on a foam surface (FoS)with eyes open (EO) and eyes closed (EC)

3 Results

The statistical data are used to illustrate the differencesbetween the CPVs of Pts and CG see Figures 3 and 4 Resultsobtained from the posturography platform are listed as well(see Figures 5 and 6) to provide for the evaluation of thedata from IMU by comparing them with the data from theposturography platformThe following plots display theMinMax Mdn Q1 and Q3 for the calculated CPVs and ACEsSince some calculated values were not distributed normallythe Wilcoxon test was used to compare and analyze thedata sets In all cases of comparisons between the groups of


Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

p lt 005

p lt 005

p lt 005

38 times 103

96 times 102

24 times 102

60 times 101

Are

a of c

onfid

ence

ellip

se(m

m2)

Figure 5 Comparison of the area of confidence ellipse of controlgroup (CG) and patients (Pts) standing on a firm surface (FiS) witheyes open (EO) and eyes closed (EC)

p lt 005

13 times 104

32 times 103

80 times 102

20 times 102

Are

a of c

onfid

ence

ellip

se(m

m2)

CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts

Figure 6 Comparison of the area of confidence ellipse of controlgroup (CG) and patients (Pts) standing on a foam surface (FoS) witheyes open (EO) and eyes closed (EC)

data the effect sizes ranged from moderate to large that iscalculated values were greater than 05 GlowastPower software(GlowastPower 319 Universitat Kiel Germany) was used for thecalculations

31 ComparingQuiet Stance Trials Thecomparison ofCGonFiS with EO and CG on FiS with EC (119901 = 0206) did not showany differences Differences were found when comparing Ptson FiS with EO and Pts on FiS with EC (119901 = 0014) CG onFoS with EO and CG on FoS with EC (119901 = 0003) and Ptson FoS with EO and Pts on FoS with EC (119901 = 0002) In thecase of the CG or Pts with EO and EC standing on the FiS

Table 1 Spearmanrsquos rank correlation coefficient between the volumeof convex polyhedron and the area of the confidence ellipse of COPexcursions of control group (CG) and patients (Pts) standing on afirm surface (FiS) and a foam surface (FoS) with eyes open (EO) andeyes closed (EC)

CG EO Pts EO CG EC Pts ECFiS 003 053 037 093FoS 003 083 018 070

the measured data show the slight increase of the median ofthe CPVs after the eyes closed phase (Figure 3) In the caseof the CG or Pts with EO and EC standing on the FoS themeasured data show a significant increase of the median ofthe CPVs after the EC phase (Figure 4)

32 Comparing Patients and Healthy Subjects Significantdifferences were found when comparing CG on FiS with EOand Pts on FiS with EO (119901 = 0010) CG on FiS with EC andPts on FiS with EC (119901 = 0005) CG on FoS with EO andPts on FoS with EO (119901 = 0001) and CG on FoS with ECand Pts on FoS with EC (119901 = 0001) In all cases significantdifferences between the data for CG and Pts were observed

The median of the CPVs in Pts standing on the FiS withEO is 40 times larger than the median of the CPVs in theCG standing on the FiS with EOThe median of the CPVs inPts standing on the FiS with EC is 97 times larger than themedian of the CPVs in the CG standing on the FiS with ECThe median of the CPVs in Pts standing on the FoS with EOis 372 times larger than the median of the CPVs in the CGstanding on the FoS with EOThe median of the CPVs in Ptsstanding on the FoS with EC is 2224 times larger than themedian of the CPVs in the CG standing on the FoS with EC

33 Correlation between the Data from IMU and the Centerof Pressure Data In the case of the CG with EO standing onthe FiS and FoS the Spearman rank correlation coefficientindicates a negligible correlation between the CPV and ACEIn the case of the CGwith EC standing on the FiS and FoS thecorrelation between the CPV andACE is weak Inmost casesthe patient examinations show strong correlation between theCPV and ACE see Table 1 A moderate positive relationshipbetween the data from IMU and the data from force platformwas found in all cases

4 Discussion

This study tested and verified a novel method utilizing thevolume of convex polyhedron obtained by plotting threeaccelerations versus each other It also proved the importanceof incorporating the phase of the stance task on a foam surfacewith EC during the measurement process since the resultsbetween patients with cerebellar ataxia and CG differed themost significantly in the CPVs observed during the FoSstance task both in the EC and EO phases [40 41] Thisconcludes that complicating stance tasks by reducing themechanoreceptor perception highlights the differences intrunk movements between the CG and Pts Also the method


identified significant differences between patients and CG inall instances of measurement The CPV of the CG and Ptsdiffered even in the hundreds of measured units of the CPV

Although the method yielded results corresponding tothose obtained by traditional methods in the case of Ptsand CG standing on the FoS with EO and EC the novelmethod revealed significant differences in the balance controlbetween Pts and CG [8] The above findings point to somecomplexity in the relation between the loss of perception andcompensation for this loss by moving the trunk In the caseof the patients the correlation coefficients indicate moderateand strong positive correlation between the CPV and ACEThe important information for the description of the situationis that very significant changes in the trunk positionwere seenonly in the PtsThe reason for this is that the largemovementsonly in the trunk which is primarily used to improve thestability of the patientrsquos body have a great impact on changingthe COM of the whole body and which corresponds to theposition of the COP [42] Conversely in the case of theCG the correlation coefficients indicate a weak or very weakcorrelation between the CPV and ACEThe reason for this isthat theCOPposition is a result of a complex kinematic chainSmall movements of the COMof the trunk are overshadowedby movements of other body segments Thus in the case ofthe CG small changes in trunk position may be differentfrom the changes in position of the center of mass (COM)of the whole body which differs from the COM of the trunkTherefore the lowest correlations correspond to the smallestmovements of the torso when CG is standing with eyes openThefinal reasonwhy the standard parameter (eg ACE) is notcorrelated to CPV is because the CPV describes the complex3D trunk accelerations (ie 3D movement) and the ACEdescribes distribution of only 2D data in transversal plane(2D space) and neglects movement in the third direction (ievertical direction) [10] Even a very small vertical movementwhich the posturography platform may fail to record canresult in a significant change in theCPV If we assume the areaof CPV reflected into the horizontal plane this may correlatewith ACE However if the reflected CP area is multiplied by arelatively small value of the vertical movement which mightbe small compared to the horizontal movement the volumeof the resulting pattern varies according to the number ofmultiples in the value of vertical movement

Significant correlations revealed between ACE and CPVduring stance on FiS with EC suggest future use of IMU in theevaluation of postural stability in patients with for examplecerebellar disease at smaller medical centers or even at homePosturography platforms are spacious and less convenientfor use in small medical centers or at home and IMUsmight therefore replace the platforms in the therapy processwhile enabling the patient to perform postural stabilitytraining individually with the IMU implemented in forexample a mobile phone Interpretation of the results fromthe measured data is identical to the interpretation of theresults obtained by posturography platforms and thus theiruse might be identical as well when it comes to therapy orevaluation of postural stability following surgeries Based onthe above it is clear that the determination and assessmentof postural instability may be now additionally carried out

by using the CPV for example in the form of examining3D trunk accelerationThemajor difference between utilizingtriaxial IMU and the traditional method using 2D trajectoryis the ability to describe the trunk movement in all threehuman body axesplanes The novel method thus clearlyfound differences in postural control between Pts and CGand even greater differences in postural stability between Ptsand CG Clinical trials which are generally used to assess animpaired postural stability have large variability (eg ClinicalTest of Sensory Interaction for Balance and others) This dis-advantage could be eliminated using the proposed systemandmethodwhich can be used to evaluatemore subtle changes inpostural stability in an individual patient course of treatmentConcluding from what is above mentioned the potential ofCPV may prove instrumental in gaining clearer insight intothe postural stability and postural balance problems which iscrucial in medical examination and rehabilitation medicine

This study also adopted a few limitations The major oneis that the size of the recruited subjects was rather smaller andpossibly not fully applicable to the larger scope of populationeven though it yielded statistically relevant results and it stillmight prove the results in a larger study Another possiblelimitation is the chance of the existence of difference incomparisons which did not yield a statistically relevant resultHowever it is safe to say that the sample of 10 Pts and 11CG was relevant to design a preliminary study aimed on thestudy of degenerative cerebellar disorder sufferers using thebasic features of the proposed techniques Moreover possiblelimitation might be also the nonhomogenous age groups thatwere compared Even though previous works cited that agedifferences should not pose a significant influence on thepresented results [38 39] it would be interesting to use thepresented method to compare age groups and thus verify orrefute the influence of age on the relevance of the results

5 Conclusion

All previous applications based on convex hull consider onlytwo variablesmdashtwo coordinates or two anglesmdashnot acceler-ations [22 32] However the three-dimensional version ofthe convex hull can also be used to study 3D movements ofpatients measured by triaxial IMU used in clinical practiceThere are two main reasons for this design The first reasonis that one variable defining the shape of the 3D convex poly-hedron allows us to study the change in the 3D movementas a whole by new and cheaper triaxial IMUs Moreoverthe combinations of the measurement of three accelerationsduring specific balance tasks could identify new and spe-cific differences in balance control of patients compared tohealthy subjects The second reason for the measurementand evaluation of all three accelerations instead of only twoangles is that the calibration of the cheap triaxial IMU maynot be accurate and thus the measurement of accelerationsin all three directions minimalizing the loss of informationabout the 3D movement as a whole Particularly cheap IMUsin the contemporary mobile phones or watches price ofwhich is constantly falling and compared to expensive profes-sional motion capture systems and posturography platformsis already considerably favorable may find use in smaller


medical centers as well as in long distancemedicine thanks tothe designedmethodAlso inclusion of all three accelerationsallows forminimalizing the influence ofmispositioning of thesensor on a body segmentThuswith the proposedmethod itis capable of providing postural examination in everyday lifeusing a cheap 3D IMU and it is also possible to use the newmethod in a wide field of medicine including the rehabilita-tion with consideration of trunk coordination for physicallychallenged people

The designed method and the 3D IMU not only arecapable of replacing the expensive posturography platformsbut also they can become their complementary part Postur-ography platforms allow for evaluation of body movement asa whole and the introduced method provides for the evalu-ation of postural stability of a given segment The proposedmethod of the evaluation of IMU data follows the traditionalmethod used with posturography platforms and therefore isalready familiar to the medical staff and easily interpretedThe use of parallel measurements of 3D data using IMUand posturography platforms for 2D data would requirefurther research into the applicability of such solution andits contribution for medical examinations for example inrehabilitation process For the future clinical use it wouldalso be appropriate to add dynamic tests of postural stabilityto the static ones because of the larger complexity of theexamination and also because performing the static tests dur-ing treatment does not correlate with the improvements ofdynamic tests (which are also important for patient life quality)

Competing Interests

The authors state no conflict of interests

Acknowledgments

This work was done in the Joint Department of the Facultyof Biomedical Engineering CTU and Charles Universityin Prague in the framework of Research Program noVG20102015002 (2010ndash2015 MV0VG) sponsored by theMinistry of the Interior of the Czech Republic and ProjectSGS16109OHK41T17 of Czech Technical University inPrague The authors would also like to thank Andrej Mado-ran BA for the translation of this paper

References

[1] M Mancini F B Horak C Zampieri P Carlson-Kuhta JG Nutt and L Chiari ldquoTrunk accelerometry reveals posturalinstability in untreated Parkinsonrsquos diseaserdquo Parkinsonism andRelated Disorders vol 17 no 7 pp 557ndash562 2011

[2] R Moe-Nilssen and J L Helbostad ldquoTrunk accelerometry as ameasure of balance control during quiet standingrdquo Gait amp Pos-ture vol 16 no 1 pp 60ndash68 2002

[3] W Maetzler M Mancini I Liepelt-Scarfone et al ldquoImpairedtrunk stability in individuals at high risk for Parkinsonrsquos dis-easerdquo PLoS ONE vol 7 no 3 Article ID e32240 2012

[4] B W Fling G G Dutta H Schlueter M H Cameron and FB Horak ldquoAssociations between proprioceptive neural pathwaystructural connectivity and balance in people with multiplesclerosisrdquo Frontiers in Human Neuroscience vol 8 article 8142014

[5] S Kammermeier J F Kleine T Eggert S Krafczyk and UButtner ldquoDisturbed vestibular-neck interaction in cerebellardiseaserdquo Journal of Neurology vol 260 no 3 pp 794ndash804 2013

[6] M U Manto Cerebellar Disorders A Practical Approach toDiagnosis andManagement Cambridge University Press Cam-bridge UK 2010

[7] O Cakrt M Vyhnalek K Slaby et al ldquoBalance rehabilitationtherapy by tongue electrotactile biofeedback in patients withdegenerative cerebellar diseaserdquo NeuroRehabilitation vol 31no 4 pp 429ndash434 2012

[8] B P C Van de Warrenburg M Bakker B P H Kremer B RBloem and J H J Allum ldquoTrunk sway in patients with spino-cerebellar ataxiardquoMovement Disorders vol 20 no 8 pp 1006ndash1013 2005

[9] L F Oliveira D M Simpson and J Nadal ldquoCalculation of areaof stabilometric signals using principal component analysisrdquoPhysiological Measurement vol 17 no 4 pp 305ndash312 1996

[10] J A RaymakersMM Samson andH J J Verhaar ldquoThe assess-ment of body sway and the choice of the stability parameter(s)rdquoGait amp Posture vol 21 no 1 pp 48ndash58 2005

[11] J Gill J H J Allum M G Carpenter et al ldquoTrunk sway mea-sures of postural stability during clinical balance tests effectsof agerdquoThe Journals of Gerontology Series A Biological Sciencesand Medical Sciences vol 56 no 7 pp M438ndashM447 2001

[12] CUManohar S KMcCrady Y Fujiki and I T Pavlidis ldquoEval-uation of the accuracy of a triaxial accelerometer embedded intoa cell phone platform formeasuring physical activityrdquo Journal ofObesity amp Weight loss Therapy vol 1 no 106 p 3309 2011

[13] V Socha P Kutılek A Stefek et al ldquoEvaluation of relationshipbetween the activity of upper limb and the piloting preci-sionrdquo in Proceedings of the 16th International Conference onMechatronics-Mechatronika (ME rsquo14) T Brezina D Maga andA Stefek Eds pp 405ndash410 IEEE Brno Czech RepublicDecember 2014

[14] A L Adkin B R Bloem and J H J Allum ldquoTrunk sway mea-surements during stance and gait tasks in Parkinsonrsquos diseaserdquoGait amp Posture vol 22 no 3 pp 240ndash249 2005

[15] F Ochi K Abe S Ishigami K Otsu and H Tomita ldquoTrunkmotion analysis inwalking using gyro sensorsrdquo inProceedings ofthe 19th Annual International Conference of the IEEE Engineer-ing in Medicine and Biology Society R J Jaeger Ed pp 1824ndash1825 Chicago Ill USA November 1997

[16] S T AwGMHalmagyi R A Black I S Curthoys R A Yavorand M J Todd ldquoHead impulses reveal loss of individual semi-circular canal functionrdquo Journal of Vestibular Research vol 9no 3 pp 173ndash180 1999

[17] T J M Kennie and G Petrie Engineering Surveying TechnologyThomson Science and Profesional Glasgow UK 1990

[18] O Findling J Sellner NMeier et al ldquoTrunk sway inmildly dis-abled multiple sclerosis patients with and without balanceimpairmentrdquo Experimental Brain Research vol 213 no 4 pp363ndash370 2011

[19] A Gil-Agudo A de los Reyes-Guzman I Dimbwadyo-Terrer etal ldquoA novelmotion tracking system for evaluation of functionalrehabilitation of the upper limbsrdquoNeural RegenerationResearchvol 8 no 19 pp 1773ndash1782 2013

[20] P Schubert M Kirchner D Schmidtbleicher and C T HaasldquoAbout the structure of posturography sampling durationparametrization focus of attention (part I)rdquo Journal of Biomed-ical Science amp Engineering vol 5 no 9 pp 496ndash507 2012


[21] F Honegger G J van Spijker and J H J Allum ldquoCoordinationof the head with respect to the trunk and pelvis in the roll andpitch planes during quiet stancerdquoNeuroscience vol 213 pp 62ndash71 2012

[22] M Zadnikar and D Rugelj ldquoPostural stability after hippother-apy in an adolescent with cerebral palsyrdquo Journal of NovelPhysiotherapies vol 1 no 1 article 106 2011

[23] N Ying and W Kim ldquoUse of dual Euler angles to quantify thethree-dimensional joint motion and its application to the anklejoint complexrdquo Journal of Biomechanics vol 35 no 12 pp 1647ndash1657 2002

[24] Xsens MTi and MTx User Manual and Technical Documenta-tion 2010 Xsens Technoloties EnschedeTheNetherlands 2010

[25] K Lebel P Boissy M Hamel and C Duval ldquoInertial measuresof motion for clinical biomechanics comparative assessment ofaccuracy under controlled conditionsmdasheffect of velocityrdquo PLoSONE vol 8 no 11 Article ID e79945 2013

[26] A Martınez-Ramırez P Lecumberri M Gomez L Rodriguez-Manas F J Garcıa andM Izquierdo ldquoFrailty assessment basedon wavelet analysis during quiet standing balance testrdquo Journalof Biomechanics vol 44 no 12 pp 2213ndash2220 2011

[27] H M Schepers D Roetenberg and P H Veltink ldquoAmbulatoryhumanmotion tracking by fusion of inertial andmagnetic sens-ing with adaptive actuationrdquoMedical and Biological Engineeringamp Computing vol 48 no 1 pp 27ndash37 2010

[28] J Hejda O Cakrt V Socha J Schlenker and P Kutilek ldquo3-D trajectory of body sway angles a technique for quantifyingpostural stabilityrdquo Biocybernetics and Biomedical Engineeringvol 35 no 3 pp 185ndash191 2015

[29] H J Luinge Inertial Sensing of Human Movement TwenteUniversity Press Enschede Nederlands 2002

[30] L Ferrufino B Bril G Dietrich T Nonaka andO A CoubardldquoPractice of contemporary dance promotes stochastic posturalcontrol in agingrdquoFrontiers inHumanNeuroscience vol 5 article169 2011

[31] M de Berg M van Kreveld M Overmars and O Schwarz-kopf Computational Geometry Algorithms and ApplicationsSpringer Berlin Germany 2nd edition 2000

[32] B Chazelle ldquoAn optimal convex hull algorithm in any fixeddimensionrdquo Discrete amp Computational Geometry vol 10 no 1pp 377ndash409 1993

[33] D E KnuthAxioms andHulls Springer Berlin Germany 1992[34] F P Preparata and S J Hong ldquoConvex hulls of finite sets of

points in two and three dimensionsrdquo Communications of theACM vol 20 no 2 pp 87ndash93 1977

[35] B Grunbaum and V Klee ldquoLectures by Branko Grunbaum andVictor Kleerdquo in CUPM (Committee on the Undergraduate Pro-gram in Mathematics) Geometry Conference Proceedings PartI Convexity and Applications L K Durst Ed MathematicalAssociation of America 1967

[36] C S Ogilvy Excursions in Geometry Dover Books on Mathe-matics Dover New York NY USA 1990

[37] R Goldman ldquoArea of planar polygons and volume of polyhe-drardquo in Graphics Gems II J Arvo Ed pp 170ndash171 AcademicPress Boston Mass USA 1991

[38] H Aoki S Demura H Kawabata et al ldquoEvaluating the effectsof openclosed eyes and age-related differences on center of footpressure sway during stepping at a set tempordquoAdvances in AgingResearch vol 1 no 3 pp 72ndash77 2012

[39] D Abrahamova and F Hlavacka ldquoAge-related changes ofhuman balance during quiet stancerdquo Physiological Research vol57 no 6 pp 1ndash17 2008

[40] J T Blackburn B L Riemann J B Myers and S M LephartldquoKinematic analysis of the hip and trunk during bilateralstance on firm foam and multiaxial support surfacesrdquo ClinicalBiomechanics vol 18 no 7 pp 655ndash661 2003

[41] M Patel P A Fransson D Lush et al ldquoThe effects of foamsurface properties on standing body movementrdquo Acta Oto-Laryngologica vol 128 no 9 pp 952ndash960 2008

[42] T Teranishi I Kondo S Sonoda et al ldquoValidity study of thestanding test for imbalance and disequilibrium (SIDE) is theamount of body sway in adopted postures consistent with itemorderrdquo Gait amp Posture vol 34 no 3 pp 295ndash299 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



cerebellar diseases [5] Therefore the first objective of thispaper is to test the application of the IMU in an area whereIMUs have not been used before and where IMUs will replaceconventional posturography platforms

Since the design is intended to identify connectionsbetween trunk movement in space and neurological disor-ders it is used to diagnose cerebellar disease characterizedby tremor or sway Cerebellar ataxia can arise as a result ofmany diseases and present itself with symptoms of coordi-nation balance gait and extremity movements difficultiesLesions to the cerebellum can cause dyssynergia dysmetriadysdiadochokinesia dysarthria ataxia of stance and gait andso forth see [6 7] Deficits can be observed on movementson the same side of the body as the lesion [6] Cliniciansoften use motor tasks in order to verify the signs of ataxia[7] Patients with a cerebellar ataxia diagnosis are interestingnot only because of their impaired postural stability butalso because there is no effective causal pharmacotherapyTherefore it is necessary to look for methods which canenhance the accuracy of evaluation procedures of patientsrsquopostural stability during the treatment

The second objective of the paper is to design and testa new method of quantitative evaluation of 2D data setfor the 3D trunk movement measured by IMU Traditionalmore complexmethods for processing themeasured data andassessing the postural instability using at least two measuredvariables aremethods based on the 2D convex hull 2D confi-dence ellipse or length of trajectory obtained by plotting twovariables versus each other [8ndash10] Usually thesemethods areused to evaluate 2D data set from posturography platforms[6] However there are limitations of these traditional solu-tions to quantify postural stability which were originally usedto evaluate 2D data set from the posturography platforms andwhich are now being used to evaluate data from the IMUsAs mentioned before the limitation is that these methodsare based solely on the evaluation of only two variables eachin one of the two human body planesaxes This can leadto a loss of important information about physical activityspecifically the third physical quantity (ie acceleration) of3D movement However we can also model the distributionof the measured 3D data (ie three orthogonal accelerations)of human body segment instead of the analysis of 2D data

Therefore this study is aimed at introducing a novelmethod used in the identification of pathological balancecontrol using IMU to measure accelerations as well as theconvex polyhedron of plotting three accelerations versus eachother (the evaluation of 3D data set of superior-inferior accel-eration mediolateral acceleration and anterior-posterioracceleration of body segments) It follows practice consistingin the assessment of the convex hull of only two variables(specifically angles) measured by IMU [11] The convex hullarea has already been used in clinical practice to study postu-ral balance problems but the concept of convex polyhedronvolume (CPV) has never been used before in clinical practiceto study postural balance problems by three accelerations[8] The choice for this novel design came from the abilityof a single variable which defines the shape of the convexpolyhedron used to describe changes in three accelerationsconsidering wide availability of new cheaper triaxial IMUs

(ordinary cell phones or watches) [12 13]The applicability ofthe convex polyhedron variable describing the body segmentmovement foreshadows the immense potential of a simpleand inexpensive IMU capable of direct evaluation of complex3D movement (ie 3D acceleration) as a whole Moreoverthe combinations of superior-inferior mediolateral andanterior-posterior accelerationmeasurements during specificbalance tasks could identify new and specific differences inbalance control of patients compared to healthy subjectsThefinal significant reason for themeasurement and evaluation ofall three accelerations instead of only two is that the calibra-tion of the cheap triaxial IMU may not be accurate and thusthe measurement of all three accelerations minimalizes theloss of information about the 3D movement as a whole Theinclusion of all three accelerations allows us tominimalize theinfluence of mispositioning the IMU on a body segmentTheclaim follows the general assumption that acceleration vectorin 3D space is defined by the three segments of the vectorwhile these segments are typically considered in accordancewith the Earthrsquos coordinate system axes or anatomical axesof a particular body part If the sensor is misplaced on thebody part or miscalibrated as for the coordinate system themeasured segments of acceleration in their respective axeswill be inaccurate If we choose to examine 3D movement byusing only two segments of acceleration a piece of informa-tion may be lost which is never the case if all three segmentsare observed The inclusion of all three accelerations allowsus to minimalize the influence of mispositioning of the IMUon a body segment

The method of recording and processing 3D data thusoffers the possibility to measure stability in smaller medicalcenters or at home by using IMU implemented in forexample a mobile phone instead of the traditional and spa-cious posturography platforms There is also a possibility forpatients to perform the therapy examination or training athome The method also eliminates the risk of mispositioningthe sensor and loss of vertical movement data and at thesame time presents patients and medical staff with an easy-to-interpret 3D data processing methodTherefore the focusof this work is to identify suitability of the convex polyhedronand IMU data for clinical application

2 Methods and Materials

21 Participants In order to test the new methods it isnecessary to compare healthy subjects without any posturalbalance problems to participants who have postural balanceproblems Ten volunteer patients (Pts) (six women and fourmen age of 522 (SD 117) years) with degenerative cerebellarataxia participated [7] The patients were recruited from theFaculty Hospital Motol Prague Czech Republic A board-certified neurologist had previously diagnosed progressivecerebellar disease Diagnostic evaluation included a neuro-logic examination laboratory blood tests and a brain MRIThe patients were measured in the initial phase of the clinicrsquostwo-week rehabilitation program Eleven healthy individualsof control group (CG) (five women and six men age of 260(SD 64) years) were also recruited for comparable analysisHealthy subjects were recruited from the studentsvolunteers


A-P axis

M-L

axis

S-I a

xis


naps

ys P

ostu

rogr

aphy

Sys

tem

(MTx unit)












119877119866119878



119866= 119877119866119878sdot 119878 (1)



119877119866119878 = 119877119885

Ψsdot 119877119884

Θsdot 119877119883

Φ (2)

where

119877119885

Ψ=[[

[


]]

]

119877119884

Θ=[[

[

cosΘ 0 sinΘ0 1 0

minus sinΘ 0 cosΘ

]]

]

119877119883

Φ=[[

[

1 0 0


]]

]

(3)



119886= 119877119860119866sdot 119866 (4)


is

119877119860119866= 119877119885

Ψ0=[[

[

cosΨ0minus sinΨ

00


]]

]

(5)





19 195 2 205 21 215 2224

25

2692

925

93

935

aSI(m

sminus2)

aAP

(ms minus2)

aML (msminus2)




PV = 13

10038161003816100381610038161003816100381610038161003816100381610038161003816

119895

sum

119894=1

1198751119894sdot 119873119894sdot 119860119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

(6)


119860119894=1

2

1003816100381610038161003816100381610038161003816100381610038161003816

119873119894sdot

119897

sum

119896=1

(119875119896119894times 119875(119896+1)119894)

1003816100381610038161003816100381610038161003816100381610038161003816

(7)


Thus

PV = 16

10038161003816100381610038161003816100381610038161003816100381610038161003816

119895

sum

119894=1

(1198751119894sdot 119873119894) sdot

1003816100381610038161003816100381610038161003816100381610038161003816

119873119894sdot

119897

sum

119896=1

(119875119896119894times 119875(119896+1)119894)

1003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816

(8)



119894

119873119894=(1198752119894minus 1198751119894) times (119875

3119894minus 1198751119894)


(9)





Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

Volu

me o

f con

vex

poly

hedr

on(m

3sminus

6)

p lt 005

p lt 005

p lt 005

70 times 100

70 times 10minus1

70 times 10minus2

70 times 10minus3

70 times 10minus4


CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts

Volu

me o

f con

vex

poly

hedr

on(m

3sminus

6)

p lt 005

45 times 101

45 times 100

45 times 10minus1

45 times 10minus2

45 times 10minus3


3 Results



Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

p lt 005

p lt 005

p lt 005

38 times 103

96 times 102

24 times 102

60 times 101

Are

a of c

onfid

ence

ellip

se(m

m2)


p lt 005

13 times 104

32 times 103

80 times 102

20 times 102

Are

a of c

onfid

ence

ellip

se(m

m2)

CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts










4 Discussion








5 Conclusion





Competing Interests


Acknowledgments


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










A-P axis

M-L

axis

S-I a

xis


naps

ys P

ostu

rogr

aphy

Sys

tem

(MTx unit)












119877119866119878



119866= 119877119866119878sdot 119878 (1)



119877119866119878 = 119877119885

Ψsdot 119877119884

Θsdot 119877119883

Φ (2)

where

119877119885

Ψ=[[

[


]]

]

119877119884

Θ=[[

[

cosΘ 0 sinΘ0 1 0

minus sinΘ 0 cosΘ

]]

]

119877119883

Φ=[[

[

1 0 0


]]

]

(3)



119886= 119877119860119866sdot 119866 (4)


is

119877119860119866= 119877119885

Ψ0=[[

[

cosΨ0minus sinΨ

00


]]

]

(5)





19 195 2 205 21 215 2224

25

2692

925

93

935

aSI(m

sminus2)

aAP

(ms minus2)

aML (msminus2)




PV = 13

10038161003816100381610038161003816100381610038161003816100381610038161003816

119895

sum

119894=1

1198751119894sdot 119873119894sdot 119860119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

(6)


119860119894=1

2

1003816100381610038161003816100381610038161003816100381610038161003816

119873119894sdot

119897

sum

119896=1

(119875119896119894times 119875(119896+1)119894)

1003816100381610038161003816100381610038161003816100381610038161003816

(7)


Thus

PV = 16

10038161003816100381610038161003816100381610038161003816100381610038161003816

119895

sum

119894=1

(1198751119894sdot 119873119894) sdot

1003816100381610038161003816100381610038161003816100381610038161003816

119873119894sdot

119897

sum

119896=1

(119875119896119894times 119875(119896+1)119894)

1003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816

(8)



119894

119873119894=(1198752119894minus 1198751119894) times (119875

3119894minus 1198751119894)


(9)





Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

Volu

me o

f con

vex

poly

hedr

on(m

3sminus

6)

p lt 005

p lt 005

p lt 005

70 times 100

70 times 10minus1

70 times 10minus2

70 times 10minus3

70 times 10minus4


CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts

Volu

me o

f con

vex

poly

hedr

on(m

3sminus

6)

p lt 005

45 times 101

45 times 100

45 times 10minus1

45 times 10minus2

45 times 10minus3


3 Results



Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

p lt 005

p lt 005

p lt 005

38 times 103

96 times 102

24 times 102

60 times 101

Are

a of c

onfid

ence

ellip

se(m

m2)


p lt 005

13 times 104

32 times 103

80 times 102

20 times 102

Are

a of c

onfid

ence

ellip

se(m

m2)

CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts










4 Discussion








5 Conclusion





Competing Interests


Acknowledgments


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119877119866119878



119866= 119877119866119878sdot 119878 (1)



119877119866119878 = 119877119885

Ψsdot 119877119884

Θsdot 119877119883

Φ (2)

where

119877119885

Ψ=[[

[


]]

]

119877119884

Θ=[[

[

cosΘ 0 sinΘ0 1 0

minus sinΘ 0 cosΘ

]]

]

119877119883

Φ=[[

[

1 0 0


]]

]

(3)



119886= 119877119860119866sdot 119866 (4)


is

119877119860119866= 119877119885

Ψ0=[[

[

cosΨ0minus sinΨ

00


]]

]

(5)





19 195 2 205 21 215 2224

25

2692

925

93

935

aSI(m

sminus2)

aAP

(ms minus2)

aML (msminus2)




PV = 13

10038161003816100381610038161003816100381610038161003816100381610038161003816

119895

sum

119894=1

1198751119894sdot 119873119894sdot 119860119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

(6)


119860119894=1

2

1003816100381610038161003816100381610038161003816100381610038161003816

119873119894sdot

119897

sum

119896=1

(119875119896119894times 119875(119896+1)119894)

1003816100381610038161003816100381610038161003816100381610038161003816

(7)


Thus

PV = 16

10038161003816100381610038161003816100381610038161003816100381610038161003816

119895

sum

119894=1

(1198751119894sdot 119873119894) sdot

1003816100381610038161003816100381610038161003816100381610038161003816

119873119894sdot

119897

sum

119896=1

(119875119896119894times 119875(119896+1)119894)

1003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816

(8)



119894

119873119894=(1198752119894minus 1198751119894) times (119875

3119894minus 1198751119894)


(9)





Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

Volu

me o

f con

vex

poly

hedr

on(m

3sminus

6)

p lt 005

p lt 005

p lt 005

70 times 100

70 times 10minus1

70 times 10minus2

70 times 10minus3

70 times 10minus4


CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts

Volu

me o

f con

vex

poly

hedr

on(m

3sminus

6)

p lt 005

45 times 101

45 times 100

45 times 10minus1

45 times 10minus2

45 times 10minus3


3 Results



Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

p lt 005

p lt 005

p lt 005

38 times 103

96 times 102

24 times 102

60 times 101

Are

a of c

onfid

ence

ellip

se(m

m2)


p lt 005

13 times 104

32 times 103

80 times 102

20 times 102

Are

a of c

onfid

ence

ellip

se(m

m2)

CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts










4 Discussion








5 Conclusion





Competing Interests


Acknowledgments


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Thus

PV = 16

10038161003816100381610038161003816100381610038161003816100381610038161003816

119895

sum

119894=1

(1198751119894sdot 119873119894) sdot

1003816100381610038161003816100381610038161003816100381610038161003816

119873119894sdot

119897

sum

119896=1

(119875119896119894times 119875(119896+1)119894)

1003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816

(8)



119894

119873119894=(1198752119894minus 1198751119894) times (119875

3119894minus 1198751119894)


(9)





Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

Volu

me o

f con

vex

poly

hedr

on(m

3sminus

6)

p lt 005

p lt 005

p lt 005

70 times 100

70 times 10minus1

70 times 10minus2

70 times 10minus3

70 times 10minus4


CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts

Volu

me o

f con

vex

poly

hedr

on(m

3sminus

6)

p lt 005

45 times 101

45 times 100

45 times 10minus1

45 times 10minus2

45 times 10minus3


3 Results



Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

p lt 005

p lt 005

p lt 005

38 times 103

96 times 102

24 times 102

60 times 101

Are

a of c

onfid

ence

ellip

se(m

m2)


p lt 005

13 times 104

32 times 103

80 times 102

20 times 102

Are

a of c

onfid

ence

ellip

se(m

m2)

CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts










4 Discussion








5 Conclusion





Competing Interests


Acknowledgments


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Nonsignificant

CGEO FiS EC FiS

CGEC FiS

PtsEO FiS

Pts

p lt 005

p lt 005

p lt 005

38 times 103

96 times 102

24 times 102

60 times 101

Are

a of c

onfid

ence

ellip

se(m

m2)


p lt 005

13 times 104

32 times 103

80 times 102

20 times 102

Are

a of c

onfid

ence

ellip

se(m

m2)

CGEO FoS EC FoS

CGEC FoS

PtsEO FoS

Pts










4 Discussion








5 Conclusion





Competing Interests


Acknowledgments


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















5 Conclusion





Competing Interests


Acknowledgments


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Competing Interests


Acknowledgments


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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