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Correction of rigid body motion in deformation

measurement of rotating objects

Pedro J Sousaablowast Joatildeo Manuel R S Tavaresab Paulo J S Tavaresa PedroM G P Moreiraa

aInstituto de Ciecircncia e Inovaccedilatildeo em Engenharia Mecacircnica e Engenharia IndustrialUniversidade do Porto Rua Dr Roberto Frias sn 4200-465 Porto Portugal

bFaculdade de Engenharia Universidade do Porto Rua Dr Roberto Frias 400 4200-465Porto Portugal

Abstract

When using image-based measurement solutions for monitoring the displace-ment of rotating objects it is necessary to trigger the image acquisition systemat precise positions Triggering errors lead to articial rigid motions in the out-put that are hard to remove independently Thus a methodology was developedto enable the removal of these errors while maintaining the true rotations suchas those relating to the angle of attackThe proposed methodology involves detecting an objects axis and centre ofrotation and then using that knowledge to correct acquired deformation dataAdditionally it also enables the representation of displacements in a more rep-resentative coordinate system ie one that includes the rotation axisWhile common solutions for this problem use either manually positioned refer-ences or track a single-points rotation the proposed methodology is based ona least-squares approach This results in a more accurate measurement of therotation axis and thus improves the quality of the resultsThe methodology was experimentally validated and it was shown that its imple-mentation is accurate with errors below 3 Additionally it was applied to anactual experimental situation and the results were compared to the uncorrecteddata while highlighting the most relevant improvements

Keywords Digital Image Correlation Helicopter blade deformation Imageprocessing Point cloud processing

1 Introduction

Rotating structures are commonly used for a wide range of applicationswhich makes their motion study important for the improvement of current so-lutions [1] particularly in aeronautics where most rotating objects are long andslender exhibiting considerable deformation [2] There are several methodsto study the deformation of such objects either with or without contact forexample using strain gauges or digital image correlation [3 4] Image-basedmethods have a number of advantages over traditional methods such as being

lowastCorresponding authorEmail address psousafeuppt (Pedro J Sousa)

non-contact or able to acquire full displacement or deformation elds as op-posed to point-to-point techniques [5] As such their use does not inuence themovementdeformation of the object under study which is important in manyapplications [3 6]

In recent years image methods have been used to perform measurements inrotating objects ranging from shafts [7] to propellers [6 8] and wind turbines[9] Many of these works such as [6 3 10 11 12 13 9 14 15 8 16 17] involvethe acquisition of shape and displacements using 3D digital image correlation

The alignment of the world coordinates with the rotation axis is crucial fordisplacement measurements [13 18 17] in that if the coordinate systems aremisaligned the comparatively large deformations along the rotation axis alsocontribute towards measurements along other axes As such this problem isoften tackled by dening the rotation axis [18] and creating a new coordinatesystem intrinsic to the motion Sicard and Sirohi approached this by carefullyaligning a calibration plate with the rotor hub plane and dening the coordinatesystem from this plate [13] On the other hand Winstroth et al proposed amethod where one data point from the tip of the rotor is extracted for an entirerotation which would then be used to dene a 3D circle using the least-squaresmethod dening the centre of rotation [18] Other works involving rotatingobjects often do not approach this problem because the target of their researchcan be obtained even with misaligned axis For example Rizo-Patron and Sirohiwere able to perform modal analysis [15 8] and obtain the natural frequenciesand mode shapes without considering misalignments as it had no eect onthese results

Another important factor when performing measurements in rotating ob-jects is that the camera should be precisely synchronized with the rotation torecord the target object always in the same position [6] However in [16 17] itwas noted that when using a static reference its positioning is also importantas misalignments between the acquired images in stationary and the rotatingsituations will cause rigid-body rotations that may hide the sought-after defor-mations As such they should be avoided or removed prior to the analysisbecause they only represent a dierent angular position and not an actual dis-placement

Commercial Digital Image Correlation (DIC) software usually feature rigidbody motion removal features that can remove rotations [19 20] in order tond only the deformation components of displacement [19] For example usingVIC3D from Correlated Solutions there are three rigid body motion removalapproaches [19] average transformation keeping three xed points or just onexed point While the latter will not aect rotations maintaining the originalvalues with the other two all displacements and rotations will be relative toeither the average transformation or the three selected points This meansthat to remove rotation it is necessary to do it along all axes when using thatsoftware

Winstroth and Seume suggest as a nal remark in [18] that one way toremove these rotations could be to realign the deformed point clouds with thereference one by minimizing the distance between them close to the blade hubAdditionally this could also be approached using a relative technique wherethe reference image is also acquired during rotation as suggested by Stasickiand Boden [6] but with a dierent goal which was that of reducing motion blureects

2

The proposed methodology is aimed at tackling simultaneously the afore-mentioned two problems by dening the rotation axis using two reference sit-uations in dierent angular positions and using this knowledge to align everydynamically loaded situations to one of these references where a projection ofthe blade perpendicular to the rotation axis is the matching target This en-ables the removal of rigid rotations that do not impact the systems functionwhile maintaining the others of which the attack angle is the most importantexample Afterwards that same information is used to represent the results inthe new coordinate system

In this article a novel methodology is proposed Then two implementedtest procedures are described The rst one deals with synthetic data andattempts to recover the parameters that were used to generate it The other isan experimental validation procedure where known displacements were imposedand the proposed methodology was used in order to accurately measure themFinally an example application is presented and the results obtained from theproposed methodology are compared with the uncorrected ones and a criticaldiscussion is presented

2 Proposed Methodology

A novel methodology was developed to detect an objects axis and centre ofrotation and use that information to correct acquired deformation data from arotating object

For this an experimental setup that is similar to the one used in [17] andshown schematically in Figure 1 was considered It is used to measure thedisplacements of rotating objects such as helicopter blades and requires thatthe region of interest is painted with a random speckle pattern as the blade inthe right side of the schematic

GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

Figure 1 Schematic of the experimental setup

In order to measure deformation using 3D Digital Image Correlation (DIC)it is necessary to rst calibrate the stereo camera system using images of aknown pattern and then acquire a pair of stereo images to dene the referencesituation [21] Afterwards while target is rotating the blades will periodically

3

interrupt a laser beam which generates a signal on the photodetectors outputThis is then processed by the controller to trigger the high-speed cameras andsimultaneously acquire stereo images of the rotating object in a particular posi-tion in space The displacements are then obtained by comparing the referenceand deformed situations [21]

The proposed methodology aims at removing any rigid rotations that arecreated by misalignments between the reference and deformed situations andaligning the world coordinates with the rotation axis Its main steps include

Acquisition of reference images for calibration of the rotation axis in twodierent angular positions

Acquisition of images under dynamic loading conditions

Image processing using Digital Image Correlation [21 22] that includesthe removal of outliers that are created in regions with poor or no specklepattern or with other correlation issues such as motion blurred or out-of-focus regions

Calculation of the rotation axis from the reference images data

Projection of the data from the situations with loading in a plane normalto the rotation axis

Calculation of the best-t rotation angles between these situations and areference one

Correct the point clouds using the calculated rotations

Calculate displacements and other interest parameters

These steps are shown schematically in Figure 2The images are initially acquired and then processed with the DIC software

package VIC-3D 2012 from USAs Correlated Solutions The resulting pointclouds are exported and corrected according to the following workow

21 Calculation of the centre and axis of rotation

The rst step of the methodology is to calculate the best-t rotation axisusing the Least-Squares method developed by Spoor and Veldpaus [23 24] andtaking into account Rose and Richards suggestions [25]

Two point clouds at dierent angular positions of the object are used Inorder to reduce errors a large angle between them is advantageous and thereference images should have only undergone rotation along the target axisDening the points in the rst point cloud as ai and the ones in the secondpoint cloud as pi it is possible to dene one matrix for each point cloud

[a] =[[a1] [an]

] [p] =

[[p1] [pn]

](1)

The centroids of both sets a and p can be calculated simply as an averageof each coordinate

An auxiliary matrix M can then be calculated as

M =1

n[a] middot [p]T minus a middot pT (2)

4

Figure 2 Proposed acquisition and measurement methodology

5

MatrixM can be used to calculate a symmetric matrix of Lagrangian multi-pliers as S2 = MT middotM [23] From the denition of eigenvalues and eigenvectorsit is seen that MTM = V D2V T where the eigenvectors Vi are the columns ofmatrix V and the eigenvalues di are the positive square roots of the diagonalof D2

D2 =

d21 0 00 d22 00 0 d23

(3)

The parameters in (2) are calculated according to Spoor and Veldpaus analy-sis [23] towards the minimization of an overall measure for the dierence betweenthe two point clouds

The rotation matrix is calculated as in [25] by following the appropriatebranch of

If V2 times V3 = +V1 then R =[m2timesm3

d2middotd3

m2

d2

m3

d3

]middot V T

If V2 times V3 = minusV1 then R =[minusm2timesm3

d2middotd3

m2

d2

m3

d3

]middot V T

(4)

where mi are the columns of

m =[m1 m2 m3

]= M middot V (5)

After the application of Spoor and Veldpaus approach it is then necessaryto calculate the centre of rotation

There are two possible representations of a rotation matrix [26 27] follow-ing the logic of object or axis rotation whose relationship is Robject = RT

axisTherefore the result from Eq 4 can be transposed to match the intendedrepresentation

From matrix R it is trivial to obtain the axis of rotation minusrarru However inorder to fully characterize the rotation a point contained in this axis is alsonecessary Thus the centre of rotation x0 is calculated by considering that therelationship between a and p is

p = R middot (aminus x0) + x0 (6)

What means that the centre of rotation can then be calculated as

x0 = (Rminus I)(R middot aminus p) (7)

where I is the R3 identity matrixSince any point that belongs to the axis will verify Eq 6 the system of

equations will have multiple solutions and the matrix (RminusI) will usually be closeto singular which can lead to incorrect results with usual linear solvers As suchpossible alternative solving methodologies include the use of a minimum normleast-squares solution through Complete Orthogonal Decomposition [28 29] orthe use of the MoorePenrose pseudoinverse (RminusI)+ which uses Singular ValueDecomposition [29 30] It is important to note that both of these options shouldprovide accurate results but not exactly the same point The rst approach isalso generally regarded as more ecient [28]

Having obtained one point in the axis it is possible to obtain any other Forthe problem under study a good choice of centre of rotation can be one thathas z(x0) = 0 which can be calculated as

6

x0ref = x0 minusminusrarru middotz(x0)

z(minusrarru )(8)

where z(x0) is the Z coordinate of x0 and z(minusrarru ) is the component of minusrarru along Z

22 Correction of the Point Clouds

In order to fully characterize the necessary movement to correct the pointclouds there are three essential parameters the axis the centre of rotation andthe angle The rst two are previously obtained from the calibration procedureAs such using these parameters the rotation angle is then calculated from eachmeasurement through a least-squares t in a projection plane normal to the axisof rotation This allows to get the best-t angle of rotation even with signicantdeformations in other axes

When dening that projection plane it was intended to maintain an axis asclosely aligned with the original X axis as possible As such the axes minusrarrv1 and minusrarrv2are calculated as

minusrarrv2 = minusrarru times

100

(9)

minusrarrv1 = minusrarrv2 timesminusrarru (10)

After dividing each new vector by its norm the new orthonormal base(minusrarrv1minusrarrv2 minusrarru ) is dened

To calculate the angle of rotation bi are dened as the reference situationpoints and qi as the ones with dynamic load Here the reference set of pointsbi could be the same as one of the sets used in Section 21 ai or pi

It should be noted that it is not necessary to use the total amount of pointsfor this registration It is also possible to select one edge or other notable featureas the registration target

Dening minusrarrx1i as the smaller norm vector from the rotation axis to a particularreference point bi yields

minusrarrx1i = bi minus ei (11)

where ei is the rotation axis point closest to bi It can be calculated as [31]

ei = x0 + ti middot minusrarru (12)

where x0 is the previously calculated rotation centre and ti is

ti = minus (x0 minus bi) middot minusrarru||minusrarru ||2

(13)

Next the projection of minusrarrx1i along both minusrarrv1 and minusrarrv2 [32] is obtained by

minusrarrx1i|v1 =minusrarrv1 middot minusrarrx1iminusrarrv1

(14)


(15)

7

If the base is orthonormal the norm of minusrarrv1 and minusrarrv2 will be 1 (one) and assuch do not have to be considered

Finally the point projected in this plane will be

bi|v1v2 =

minusrarrx1i|v1minusrarrx1i|v2

0

(16)

With the due considerations this can be extended to the remaining pointsqi in order to obtain the projection

qi|v1v2=


(17)

where minusrarrx2i is the smallest norm vector from the rotation axis to a particularpoint qi

Following the notation used in Eq 1 the projected points can be denedas

[b|v1v2

]=[[b1|v1v2 ] [bn|v1v2 ]

][q|v1v2

]=[[q1|v1v2 ] [qn|v1v2 ]

](18)

The angle is then calculated by applying Spoor and Valdepaus algorithm

until Eq 4 replacing the matrices [a] and [p] with[b|v1v2

]and

[q|v1v2

] This

again returns a rotation matrix from where an angle α and a new axis can beextracted

If the orthogonal base was not created using Eqs 9 and 10 it may benecessary to correct the angles signal α = s middot α where

s = minusrarru middot (minusrarrv1 timesminusrarrv2) (19)

Additionally the axis extracted from the last rotation matrix can be either[0 0 1

]or[0 0 minus1

] In the rst case the angle has to be corrected as

α = minusα (20)

From this angle α and the minusrarru axis a new rotation matrix is created RC and is then used to correct the dynamic load situation points [q]

[Q] = RC middot ([q]minus [X0]) + [X0] (21)

where [X0] is a matrix of the same size as [q] where in every column are thecoordinates of the centre of rotation

[X0] =[[x0] [x0]

](22)

The application of this approach is schematically shown in Figure 3

8

Figure 3 Approach developed for point cloud correction

9

23 Calculation of the Displacements

Finally it is possible to calculate the displacements in the (minusrarrv1minusrarrv2 minusrarru ) coor-dinate system enabling a more intuitive presentation of the acquired data Forthis it is necessary to dene each point in the new coordinate system This canbe achieved by applying Eqs 11 to 17 and considering the result of Eq 13 asthe third component of Eqs 16 and 17

bi|minusrarrv1minusrarrv2minusrarru =

minusrarrx1i|v1minusrarrx1i|v2ti

qi|minusrarrv1minusrarrv2minusrarru =


(23)

Then displacements in the new coordinate system can be directly calculatedfrom the coordinates of matching points in both sets

3 Tests and discussion

In order to ensure that the proposed methodology gives good results forpractical application it is necessary to test its behaviour As such two com-plementary approaches were used numerical and experimental The rst oneinvolves the use of synthetic data to test the applicability of the methodologyin identifying movement parameters The second one was performed by impos-ing known displacements on a polymer blade and measuring them using theproposed methodology The obtained displacements were then compared withthe imposed ones in order to validate the results For each approach a briefdiscussion regarding the obtained errors is included

31 Synthetic data

A set of 5 (ve) points with coordinates in the [0 10] range was denedthrough a pseudo-random function which led to

[a] =

90579 12699 91338 63236 0975427850 54688 95751 96489 1576197059 95717 48538 80028 14189

(24)

Several test scenarios were then dened by combining a rotation axis anangle and a centre of rotation as indicated in Table 1

Table 1 Parameters that were combined to dene each test scenario

minusrarru α x0

1

0

0

1

40deg

09

09

09

2

minus03333

06667

06667

1518341deg

65574

03571

84913

Every possible combination of the three parameters in Table 1 (minusrarru i αj and

x0k) was tested As such a rotation of αj degrees around axisminusrarru i and centre x0k

10

(input values) was applied to the set of random points [a] Then the developedalgorithm was applied to recover an angle and an axis dened by a normal vectorand a centre of rotation (output values) from the generated points

Without forcing the third component of x0 to 0 (zero) the calculation of theoutput centre of rotation is underdetermined and as such it can be any pointin the axis as long as it satises

x0output= x0input

+ t middot minusrarru (25)

where t isin RThe tested combinations as well as the obtained results including the value

of t required to verify Eq 25 are indicated in Table 2

Table 2 Obtained results after application of the algorithm to the synthetic data sets

Input Output

(minusrarru α x0) Normal vector Angle Rotation centre t

minusrarru 1 α1 x01

00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131

While it is easily perceived from Table 2 that the obtained normal vectorsand angles are all accurate at least to the fourth decimal place the output

11

centres of rotation are related to the input values through Eq 25 A moredetailed presentation of the obtained errors is presented in Table 3 Here thecentre of rotation error was calculated after application of Eq 25 and any errorvalues of 0 (zero) should be understood as being inferior to the machine eplison(eps) ie the minimum dierence between two oating-point double-precisionnumbers

Table 3 Errors obtained from the application of the algorithm to the synthetic data sets

Input Errors

(minusrarru α x0) Normal vector Angle Rotation centre


minus57348times 10minus17


18874times 10minus15















2125times 10minus16


0

















9992times 10minus16










0





0











0

Since all of the calculated errors are below 10minus13 it is appropriate to consider

that the proposed methodology is capable of obtaining accurate results

32 Experimental Validation

To complement the previous analysis an experimental validation procedurewas devised A manual linear stage rigidly connected to a 12mm diametermetallic rod was used to impose and measure displacements Due to the size and

12

material of the rod it is expected that its deformation is neglectable comparedto the polymeric blade

To dene the reference point from which the displacements applied to themanual stage are also applied to the target an external methodology based inProjection Moireacute [33] was used This involved the use of a fringe projector and acamera to compare the position of the projected grid in dierent points in timeby subtracting the images and analysing the generated fringe pattern Thesealong with the remaining components of the setup are shown in Figures 4 and5 Even though the experimental validation was performed statically this setupwas designed for dynamic usage [16 17] Here the laser and photodetectorwere used to generate a signal from the rotation of the blades that was thenprocessed by the controller to trigger both high-speed cameras simultaneouslyat a particular location Besides this the controller is also capable of generatingtrigger signals independently of the input enabling manual simultaneous imageacquisition

GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage

Figure 4 Schematic of the experimental setup used for validation

The required images were then obtained Bundle calibration was performedby placing a pattern in multiple orientations inside the work area and the cali-bration images were acquired

Reference images were then obtained by manually positioning the RC heli-copters blade in two angular positions with a metallic part keeping it rigidlyconnected to the rotation axis

Before imposing displacements to the blade the Projection Moireacute systemwas used to position the linear manual stage at the point where fringes startappearing

From previous works the displacements that were measured for rotationsof around 680 rpm along the Z-axis were close to 6 mm [17] Considering thatthis is smaller than the real value a set of four displacements 20 40 50 and70 mm were imposed and images were acquired for each Here the maximumdisplacement matches the expected real value at 680 rpm and there are threeadditional not evenly spaced points at integer values The acquired imageswere processed by Correlated Solutions VIC-3D and exported to MATLAB forprocessing Applying the developed algorithm the axis and centre of rotation

13

Figure 5 Experimental setup used for validation

were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)

The angle between the two references was also calculated as αref = 108424degconsidering the top edge as the target This consideration was used consistentlyas well for the remaining reported situations

From the calculated value for minusrarru it is possible to compare the orientationof both coordinate systems Figure 6 The position of minusrarru between the Z andY axis helps to explain the errors that were highlighted in previous work [17]where the displacements along the rotation axis would contribute to these twoaxes representations

Figure 6 Comparison of the orientation of the original (XY Z) and the new (minusrarrv1minusrarrv2minusrarru )coordinate systems

For each of the imposed displacements the best-t angle was calculated by

14

registration of the top edges obtaining a correction angle for each

α2mm = minus71568deg (27)




Finally applying the Algorithm in Sections 22 and 23 the points werecorrected and the displacements in (minusrarrv1minusrarrv2 minusrarru ) were calculated

Figure 7 shows the resulting displacements for the 5 mm imposed displace-ment case For each of the cases a value in a constant position close to theapplication point was extracted (red cross in Figure 7c) having obtained

∆minusrarru 2mm = minus20512




(a) Displacements along minusrarrv1 (b) Displacements along minusrarrv2

(c) Displacements along minusrarru

Figure 7 Displacements in the (minusrarrv1minusrarrv2minusrarru ) coordinate system for an imposed displacement of5 mm

15

As such the errors for each situation were

Error(∆minusrarru 2mm) = +00512 mm = +256

Error(∆minusrarru 4mm) = minus00450 mm = minus112



Thus the errors were less than 01 mm which is below 3 Consideringthat the movement was performed with resolution of a tenth of a millimetrethe obtained results closely match the expected result and as such help validatethe developed methodology

4 Application Example

As a nal analysis the developed methodology was applied to a dynamicsituation It was performed using the setup shown in Figure 1 and excludingProjection Moireacute and the manual linear stage in Figure 5

Before the actual dynamic measurements the preparation phase again in-volved the acquisition of reference and calibration images Afterwards thetrigger controller was congured for the situation at hand the helicopter wasstarted and one acquisition per rotation was performed This generated 115pairs of images at a speed of approximately 680 rpm An exposure of 1 micros wasused during which it is expected that the blades tip moves around 0013 mm

The used cameras allow for high framerates 2000 fps at full resolutionHowever in this experiment the framerate is not predened as the acquisitionof each image is performed asynchronously with just one frame acquired foreach rotation If other slower cameras were used or if the rotation speed wastoo high it would also be possible for example to acquire just one frame everytwo or more rotations

The images were processed using VIC-3D 2012 from USAs CorrelatedSolutions and the data was exported to MATLAB for processing First theaxis and centre of rotation were calculated having obtained

minusrarru =

000990227409738

x0 =

minus1035792minus68325

0

αref = 176933deg (31)

It should be noted that since this experiment was not performed simul-taneously with the experimental validation small dierences in the axis andcentre of rotation are expected due to slight orientation changes in the cameraspositioning and other factors

Afterwards for each of the 115 revolutions the point clouds can be correctedand the displacements in the (minusrarrv1minusrarrv2 minusrarru ) coordinate system can be calculatedSince the results are very similar among these frames a single ones the 15thare presented

In this case the calculated best-t angle was small but still signicant

α = 08106deg (32)

16

(a) Displacement along minusrarrv1 (b) Displacement along minusrarrv2

(c) Displacement along minusrarru

Figure 8 Displacements in the (minusrarrv1minusrarrv2minusrarru ) coordinate system using the developed algorithm

After correcting the point clouds using the calculated values it was nallypossible to calculate the displacements in the (minusrarrv1minusrarrv2minusrarru ) coordinate system

The obtained displacements are shown in Figure 8 while the uncorrectedones ie the output from VIC3D are shown in Figure 9

By comparing both sets of data it is possible to see that the main improve-ments are

For the displacements along minusrarrv1 and X it is possible to see slight changesin the pattern mainly due to the rotation correction since the new co-ordinate system was created to maintain minusrarrv1 and X close and thereforetheir dierence is small

The displacements along minusrarrv2 and Y are noticeably dierent mainly becausethe displacements along the minusrarru rotation axis contributed to both Y andZ displacements and since their magnitude is larger than the ones alongminusrarrv2 they mask their presence

Finally the displacements along minusrarru and Z are very similar showing mostlychanges in magnitude due to the new coordinate system with the rsthaving larger displacements due to a better alignment For example theminimum values are minus67490mm along Z and minus71205mm along minusrarru

17

(a) Displacement along X (b) Displacement along Y

(c) Displacement along Z

Figure 9 Displacements in the (XY Z) coordinate system calculated by VIC3D software

It was also possible to notice that the average displacements for minusrarrv1 and minusrarrv2were 02443 and -08482 This dierence drew attention to the existence of gapsbetween the helicopters blade and axis assembly of this order of magnitude

5 Conclusions

A methodology for data correction through detection of the rotation axiswas developed and validated This enables the correction of two factors mis-placement of the reference image causing over-rotation in the images and themisalignment of the coordinate system and the rotation axis which resulted inunintuitive representations

Through the proposed methodology by comparing two dierent referencesituations it was possible to obtain a centre and an axis of rotation for anRC helicopters blade that were close to the expected values Afterwards it waspossible to use this knowledge to calculate the best-t rotation angle between thedynamic load situation and one of the references This enabled both the removalof the eects of an over-rotation between these situations and the calculationof the displacements in a more appropriate coordinate system ie one thatcontains the rotation axis and two perpendicular axes Combining these twofactors it was possible to obtain an accurate measurement of the deformationstate of an RC helicopters blade under rotation

18

The new methodology was also experimentally validated by statically im-posing known displacements and measuring the results having obtained valuesthat are very close to the imposed ones

Acknowledgements

Pedro J Sousa gratefully acknowledges the FCT (Fundaccedilatildeo para a Ciecircncia ea Tecnologia) for the funding of the PhD scholarship SFRHBD1293982017

Dr Moreira acknowledges POPH QREN-Tipologia 42 Promotion ofscientic employment funded by the ESF and MCTES

The authors also gratefully acknowledge the funding of Project NORTE-01-0145-FEDER-000022 - SciTech - Science and Technology for Competitiveand Sustainable Industries co-nanced by Programa Operacional Regional doNorte (NORTE2020) through Fundo Europeu de Desenvolvimento Regional(FEDER)

References

[1] I Bucher D J Ewins Modal analysis and testing of rotating struc-tures Philosophical Transactions of the Royal Society of London SeriesA Mathematical Physical and Engineering Sciences 359 (1778) (2001) 6196 doi101098rsta20000714

[2] J Winstroth J R Seume Wind Turbine Rotor Blade Monitoring usingDigital Image Correlation Assessment on a Scaled Model in 32nd ASMEWind Energy Symposium American Institute of Aeronautics and Astro-nautics 2014 doidoi10251462014-139610251462014-1396URL httpdxdoiorg10251462014-1396

[3] F Boden K Bodensiek B Stasicki Application of image pattern corre-lation for non-intrusive deformation measurements of fast rotating objectson aircrafts Vol 7522 2009 pp 75222S75222S10URL httpdxdoiorg10111712852703

[4] M Ozbek D J Rixen Operational modal analysis of a 25 MW windturbine using optical measurement techniques and strain gauges WindEnergy 16 (3) (2013) 367381 doi101002we1493URL httphttpsdoiorg101002we1493

[5] F Hild S Roux Digital Image Correlation from Displacement Measure-ment to Identication of Elastic Properties a Review Strain 42 (2) (2006)6980 doi101111j1475-1305200600258xURL httphttpsdoiorg101111j1475-1305200600258x

[6] B Stasicki F Boden Application of high-speed videography for in-ightdeformation measurements of aircraft propellers Vol 7126 2008 pp712604712612URL httpdxdoiorg10111712822046

[7] X Zi S Geng S Zhao F Shu Measurement of short shaft power based ona digital speckle correlation method Measurement Science and Technology26 (4) (2015) 45001

19

[8] S Rizo-Patron J Sirohi Operational Modal Analysis of a Helicopter RotorBlade Using Digital Image Correlation Experimental Mechanics 57 (3)(2017) 367375 doi101007s11340-016-0230-6

[9] J Winstroth L Schoen B Ernst J R Seume Wind turbine rotor blademonitoring using digital image correlation a comparison to aeroelastic sim-ulations of a multi-megawatt wind turbine Journal of Physics ConferenceSeries 524 (1) (2014) 12064URL httpstacksioporg1742-6596524i=1a=012064

[10] M N Helfrick P Pingle C Niezrecki P Avitabile Optical Non-contactingVibration Measurement of Rotating Turbine Blades (2009)

[11] C Warren C Niezrecki P Avitabile Optical Non-contacting VibrationMeasurement of Rotating Turbine Blades II in Proceedings of theIMAC-XXVIII 2010 doi101007978-1-4419-9716-6URL httplinkspringercom101007978-1-4419-9716-

65Cnhttpwwwspringerlinkcomindex101007978-1-4419-

9716-6

[12] J Sirohi M S Lawson Measurement of helicopter rotor blade deformationusing digital image correlation Optical Engineering 51 (4) (2012) 4360343608 doi1011171OE514043603URL httpdxdoiorg1011171OE514043603

[13] J Sicard J Sirohi Measurement of the deformation of an extremely ex-ible rotor blade using digital image correlation Measurement Science andTechnology 24 (6) (2013) 65203URL httpstacksioporg0957-023324i=6a=065203

[14] S Rizo-Patron J Sirohi Operational Modal Analysis of a Helicopter RotorBlade Using Digital Image Correlation Experimental Mechanics (2016) 19doi101007s11340-016-0230-6URL httpdxdoiorg101007s11340-016-0230-6

[15] S S Rizo-Patron J Sirohi Operational Modal Analysis of a RotatingCantilever Beam Using High-Speed Digital Image Correlation in 57thAIAAASCEAHSASC Structures Structural Dynamics and MaterialsConference American Institute of Aeronautics and Astronautics 2016doidoi10251462016-195710251462016-1957URL httpdxdoiorg10251462016-1957

[16] P J Sousa F Barros P J Tavares P M Moreira Displacement mea-surement and shape acquisition of an RC helicopter blade using DigitalImage Correlation Procedia Structural Integrity 5 (2017) 12531259doi101016jprostr201707097URL httplinkinghubelseviercomretrievepii

S2452321617302093

[17] P J Sousa F Barros P J Tavares P M Moreira Digital im-age correlation displacement measurement of a rotating RC heli-copter blade Engineering Failure Analysis 90 (March) (2018) 371379doi101016jengfailanal201804005

20

URL httplinkinghubelseviercomretrievepii

S1350630717313328

[18] J Winstroth J R Seume Wind turbine rotor blade monitoring using dig-ital image correlation 3d simulation of the experimental setup in Euro-pean Wind Energy Association - EWEA 2014 no March 2014 Barcelona2014

[19] Correlated Solutions Vic-3D 2010 Reference Manual (2010)

[20] G mbH GOM Correlate Professional V8 SR1 Manual Basic Inspection 3D Testing (2015)

[21] M A Sutton J J Orteu H Schreier Image correlation for shape motionand deformation measurements basic concepts theory and applicationsSpringer Science amp Business Media 2009

[22] Correlated Solutions Digital Image CorrelationURL httpcorrelatedsolutionscomdigital-image-correlation

[23] C W Spoor F E Veldpaus Rigid body motion calculated from spatialco-ordinates of markers Journal of Biomechanics 13 (4) (1980) 391393doi1010160021-9290(80)90020-2

[24] F E Veldpaus H J Woltring L J M G Dortmans A least-squares al-gorithm for the equiform transformation from spatial marker co-ordinatesJournal of Biomechanics 21 (1) (1988) 4554 doi1010160021-

9290(88)90190-X

[25] W C Rose J G Richards Estimating Body Segment Motion byTracking Markers Journal of Chemical Information and Modeling53 (9) (2012) 16891699 arXivarXiv10111669v3 doi101017

CBO9781107415324004

[26] E W Weisstein Rotation MatrixURL httpmathworldwolframcomRotationMatrixhtml

[27] MathWorks Rotation matrix for rotations around z-axis - MATLAB rotzURL httpswwwmathworkscomhelpphasedrefrotzhtml

[28] MathWorks Minimum norm least-squares solution to linear equation -MATLAB lsqminnorm (2017)URL httpswwwmathworkscomhelpmatlabreflsqminnormhtml

[29] The GSL Team Linear Algebra - GSL 24 documentation (2017)URL httpswwwgnuorgsoftwaregsldochtmllinalghtml

[30] MathWorks Moore-Penrose pseudoinverse - MATLAB pinvURL httpswwwmathworkscomhelpmatlabrefpinvhtml

[31] E W Weisstein Point-Line Distance3-DimensionalURL httpmathworldwolframcomPoint-LineDistance3-

Dimensionalhtml

21

[32] E W Weisstein ProjectionURL httpmathworldwolframcomProjectionhtml

[33] G Cloud Optical Methods in Experimental Mechanics Part 22 Projec-tion Moireacute Experimental Techniques 30 (4) (2006) 1518 doi101111

j1747-1567200600056x

22

Introduction

Proposed Methodology

Calculation of the centre and axis of rotation

Correction of the Point Clouds

Calculation of the Displacements

Tests and discussion

Synthetic data

Experimental Validation

Application Example

Conclusions

non-contact or able to acquire full displacement or deformation elds as op-posed to point-to-point techniques [5] As such their use does not inuence themovementdeformation of the object under study which is important in manyapplications [3 6]

In recent years image methods have been used to perform measurements inrotating objects ranging from shafts [7] to propellers [6 8] and wind turbines[9] Many of these works such as [6 3 10 11 12 13 9 14 15 8 16 17] involvethe acquisition of shape and displacements using 3D digital image correlation

The alignment of the world coordinates with the rotation axis is crucial fordisplacement measurements [13 18 17] in that if the coordinate systems aremisaligned the comparatively large deformations along the rotation axis alsocontribute towards measurements along other axes As such this problem isoften tackled by dening the rotation axis [18] and creating a new coordinatesystem intrinsic to the motion Sicard and Sirohi approached this by carefullyaligning a calibration plate with the rotor hub plane and dening the coordinatesystem from this plate [13] On the other hand Winstroth et al proposed amethod where one data point from the tip of the rotor is extracted for an entirerotation which would then be used to dene a 3D circle using the least-squaresmethod dening the centre of rotation [18] Other works involving rotatingobjects often do not approach this problem because the target of their researchcan be obtained even with misaligned axis For example Rizo-Patron and Sirohiwere able to perform modal analysis [15 8] and obtain the natural frequenciesand mode shapes without considering misalignments as it had no eect onthese results

Another important factor when performing measurements in rotating ob-jects is that the camera should be precisely synchronized with the rotation torecord the target object always in the same position [6] However in [16 17] itwas noted that when using a static reference its positioning is also importantas misalignments between the acquired images in stationary and the rotatingsituations will cause rigid-body rotations that may hide the sought-after defor-mations As such they should be avoided or removed prior to the analysisbecause they only represent a dierent angular position and not an actual dis-placement

Commercial Digital Image Correlation (DIC) software usually feature rigidbody motion removal features that can remove rotations [19 20] in order tond only the deformation components of displacement [19] For example usingVIC3D from Correlated Solutions there are three rigid body motion removalapproaches [19] average transformation keeping three xed points or just onexed point While the latter will not aect rotations maintaining the originalvalues with the other two all displacements and rotations will be relative toeither the average transformation or the three selected points This meansthat to remove rotation it is necessary to do it along all axes when using thatsoftware

Winstroth and Seume suggest as a nal remark in [18] that one way toremove these rotations could be to realign the deformed point clouds with thereference one by minimizing the distance between them close to the blade hubAdditionally this could also be approached using a relative technique wherethe reference image is also acquired during rotation as suggested by Stasickiand Boden [6] but with a dierent goal which was that of reducing motion blureects

2






GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades



3
















[a] =[[a1] [an]

] [p] =

[[p1] [pn]

](1)



M =1


4


5


D2 =

d21 0 00 d22 00 0 d23

(3)




d2middotd3

m2

d2

m3

d3

]middot V T


d2middotd3

m2

d2

m3

d3

]middot V T

(4)


m =[m1 m2 m3

]= M middot V (5)











6


z(minusrarru )(8)






100

(9)











(13)



(14)


(15)

7



bi|v1v2 =


0

(16)


qi|v1v2=


(17)



[b|v1v2

]=[[b1|v1v2 ] [bn|v1v2 ]

][q|v1v2

]=[[q1|v1v2 ] [qn|v1v2 ]

](18)



]and

[q|v1v2

] This





]or[0 0 minus1


α = minusα (20)




[X0] =[[x0] [x0]

](22)


8


9







(23)




31 Synthetic data


[a] =

90579 12699 91338 63236 0975427850 54688 95751 96489 1576197059 95717 48538 80028 14189

(24)



minusrarru α x0

1

0

0

1

40deg

09

09

09

2

minus03333

06667

06667

1518341deg

65574

03571

84913



10



x0output= x0input





Input Output



00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131


11



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions






GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades



3
















[a] =[[a1] [an]

] [p] =

[[p1] [pn]

](1)



M =1


4


5


D2 =

d21 0 00 d22 00 0 d23

(3)




d2middotd3

m2

d2

m3

d3

]middot V T


d2middotd3

m2

d2

m3

d3

]middot V T

(4)


m =[m1 m2 m3

]= M middot V (5)











6


z(minusrarru )(8)






100

(9)











(13)



(14)


(15)

7



bi|v1v2 =


0

(16)


qi|v1v2=


(17)



[b|v1v2

]=[[b1|v1v2 ] [bn|v1v2 ]

][q|v1v2

]=[[q1|v1v2 ] [qn|v1v2 ]

](18)



]and

[q|v1v2

] This





]or[0 0 minus1


α = minusα (20)




[X0] =[[x0] [x0]

](22)


8


9







(23)




31 Synthetic data


[a] =

90579 12699 91338 63236 0975427850 54688 95751 96489 1576197059 95717 48538 80028 14189

(24)



minusrarru α x0

1

0

0

1

40deg

09

09

09

2

minus03333

06667

06667

1518341deg

65574

03571

84913



10



x0output= x0input





Input Output



00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131


11



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions
















[a] =[[a1] [an]

] [p] =

[[p1] [pn]

](1)



M =1


4


5


D2 =

d21 0 00 d22 00 0 d23

(3)




d2middotd3

m2

d2

m3

d3

]middot V T


d2middotd3

m2

d2

m3

d3

]middot V T

(4)


m =[m1 m2 m3

]= M middot V (5)











6


z(minusrarru )(8)






100

(9)











(13)



(14)


(15)

7



bi|v1v2 =


0

(16)


qi|v1v2=


(17)



[b|v1v2

]=[[b1|v1v2 ] [bn|v1v2 ]

][q|v1v2

]=[[q1|v1v2 ] [qn|v1v2 ]

](18)



]and

[q|v1v2

] This





]or[0 0 minus1


α = minusα (20)




[X0] =[[x0] [x0]

](22)


8


9







(23)




31 Synthetic data


[a] =

90579 12699 91338 63236 0975427850 54688 95751 96489 1576197059 95717 48538 80028 14189

(24)



minusrarru α x0

1

0

0

1

40deg

09

09

09

2

minus03333

06667

06667

1518341deg

65574

03571

84913



10



x0output= x0input





Input Output



00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131


11



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions


5


D2 =

d21 0 00 d22 00 0 d23

(3)




d2middotd3

m2

d2

m3

d3

]middot V T


d2middotd3

m2

d2

m3

d3

]middot V T

(4)


m =[m1 m2 m3

]= M middot V (5)











6


z(minusrarru )(8)






100

(9)











(13)



(14)


(15)

7



bi|v1v2 =


0

(16)


qi|v1v2=


(17)



[b|v1v2

]=[[b1|v1v2 ] [bn|v1v2 ]

][q|v1v2

]=[[q1|v1v2 ] [qn|v1v2 ]

](18)



]and

[q|v1v2

] This





]or[0 0 minus1


α = minusα (20)




[X0] =[[x0] [x0]

](22)


8


9







(23)




31 Synthetic data


[a] =

90579 12699 91338 63236 0975427850 54688 95751 96489 1576197059 95717 48538 80028 14189

(24)



minusrarru α x0

1

0

0

1

40deg

09

09

09

2

minus03333

06667

06667

1518341deg

65574

03571

84913



10



x0output= x0input





Input Output



00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131


11



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions


D2 =

d21 0 00 d22 00 0 d23

(3)




d2middotd3

m2

d2

m3

d3

]middot V T


d2middotd3

m2

d2

m3

d3

]middot V T

(4)


m =[m1 m2 m3

]= M middot V (5)











6


z(minusrarru )(8)






100

(9)











(13)



(14)


(15)

7



bi|v1v2 =


0

(16)


qi|v1v2=


(17)



[b|v1v2

]=[[b1|v1v2 ] [bn|v1v2 ]

][q|v1v2

]=[[q1|v1v2 ] [qn|v1v2 ]

](18)



]and

[q|v1v2

] This





]or[0 0 minus1


α = minusα (20)




[X0] =[[x0] [x0]

](22)


8


9







(23)




31 Synthetic data


[a] =

90579 12699 91338 63236 0975427850 54688 95751 96489 1576197059 95717 48538 80028 14189

(24)



minusrarru α x0

1

0

0

1

40deg

09

09

09

2

minus03333

06667

06667

1518341deg

65574

03571

84913



10



x0output= x0input





Input Output



00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131


11



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions


z(minusrarru )(8)






100

(9)











(13)



(14)


(15)

7



bi|v1v2 =


0

(16)


qi|v1v2=


(17)



[b|v1v2

]=[[b1|v1v2 ] [bn|v1v2 ]

][q|v1v2

]=[[q1|v1v2 ] [qn|v1v2 ]

](18)



]and

[q|v1v2

] This





]or[0 0 minus1


α = minusα (20)




[X0] =[[x0] [x0]

](22)


8


9







(23)




31 Synthetic data


[a] =

90579 12699 91338 63236 0975427850 54688 95751 96489 1576197059 95717 48538 80028 14189

(24)



minusrarru α x0

1

0

0

1

40deg

09

09

09

2

minus03333

06667

06667

1518341deg

65574

03571

84913



10



x0output= x0input





Input Output



00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131


11



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions



bi|v1v2 =


0

(16)


qi|v1v2=


(17)



[b|v1v2

]=[[b1|v1v2 ] [bn|v1v2 ]

][q|v1v2

]=[[q1|v1v2 ] [qn|v1v2 ]

](18)



]and

[q|v1v2

] This





]or[0 0 minus1


α = minusα (20)




[X0] =[[x0] [x0]

](22)


8


9







(23)




31 Synthetic data


[a] =

90579 12699 91338 63236 0975427850 54688 95751 96489 1576197059 95717 48538 80028 14189

(24)



minusrarru α x0

1

0

0

1

40deg

09

09

09

2

minus03333

06667

06667

1518341deg

65574

03571

84913



10



x0output= x0input





Input Output



00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131


11



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions


9







(23)




31 Synthetic data


[a] =

90579 12699 91338 63236 0975427850 54688 95751 96489 1576197059 95717 48538 80028 14189

(24)



minusrarru α x0

1

0

0

1

40deg

09

09

09

2

minus03333

06667

06667

1518341deg

65574

03571

84913



10



x0output= x0input





Input Output



00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131


11



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions







(23)




31 Synthetic data


[a] =

90579 12699 91338 63236 0975427850 54688 95751 96489 1576197059 95717 48538 80028 14189

(24)



minusrarru α x0

1

0

0

1

40deg

09

09

09

2

minus03333

06667

06667

1518341deg

65574

03571

84913



10



x0output= x0input





Input Output



00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131


11



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions



x0output= x0input





Input Output



00000

00000

10000

400000

09000

09000

minus00000

-09000


00000

00000

10000

400000

65574

03571

minus00000

-84913


00000

minus00000

10000

1518341

09000

09000

00000

-09000


minus00000

00000

10000

1518341

65574

03571

00000

-84913


minus03333

06667

06667

400000

12000

03000

03000

-09000


minus03333

06667

06667

400000

59396

15926

97268

18533


minus03333

06667

06667

1518341

12000

03000

03000

-09000


minus03333

06667

06667

1518341

77951

minus21183

60159

-37131


11



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions



Input Errors




















2125times 10minus16


0

















9992times 10minus16










0





0











0





12



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions



GigabitSwitch

TriggerController

Laser

Photodetector

High-speedCameras

Computer

Helicopter Blades

MoireCamera Fringe

Projector

LinearManualStage






13


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions


were calculated as

minusrarru =

003050258609655

x0 =

minus103018109453

0

(26)





14















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions















15












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions












minusrarru =

000990227409738

x0 =


0

αref = 176933deg (31)




α = 08106deg (32)

16










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions










17





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions





5 Conclusions



18


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions


Acknowledgements




References








19






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions






9716-6






S2452321617302093


20


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions


S1350630717313328








9290(88)90190-X


CBO9781107415324004







Dimensionalhtml

21



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions



j1747-1567200600056x

22

Introduction






Synthetic data


Application Example

Conclusions

Correction of rigid body motion in deformation measurement ...tavares/downloads/... · Correction...

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Transcript of Correction of rigid body motion in deformation measurement ...tavares/downloads/... · Correction...