Folie 1

High precision image-based tracking of a rigid body moving within a fluid

Stuart Laurence, Jan Martinez Schramm

German Aerospace Center (DLR), Göttingen, Germany

APS/DFD, 23 November 2010

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Motivation

Visualization-based techniques an attractive option for measuring displacements (and derived quantities) of rigid bodies in fluids, as they are completely non-intrusive

Particularly attractive for force-measurement in short-duration hypersonic facilities, as few other options available

However, measurement precision critical – in past (film-based analog techniques) displacement measurements limited to ~50 μm

Focus here on edge-detection-based techniques combined with least-squares fitting (suitable for silhouette images from schlieren, etc.)

Assumptions: no changes to body profile; motion two dimensional + one axis of rotation

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Analytic-fitting technique

Edge detection Model edge

tracing and sub-pixel detection

Least-squares fitting

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Free-flight measurements with analytic-fitting technique

Image-based measurements show reasonable agreement with accelerometer measurements

Response time for 14 kfps estimated to be ~0.5 ms

1 1.5 2 2.5 3 3.5 4

x 10-3

0

1000

2000

3000

4000

5000

t [s]

Dra

g ac

cele

ratio

n [m

/s2 ]

Accelerometers

Images

1 1.5 2 2.5 3 3.5 4

x 10-3

0

500

1000

1500

t [s]

Lift

acc

eler

atio

n [m

/s2 ]

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Problems with analytic-fitting technique

Model cross-sectional profile must be expressible analytically (can be avoided by using, e.g., splines)

For all but simplest geometries, fitting procedure is iterative (slow!)

Reasonably complete profile required for convergence

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Edge-tracking technique

Based on matching closest edge-points in reference and displaced images

Edge angle assumed to be the same for each edge-point pair

cos)(sin)(cossin eeee yyxxyx

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Edge-tracking technique

Based on matching closest edge-points in reference and displaced images

Edge angle assumed to be the same for each edge-point pair

cos)(sin)(cossin eeee yyxxyx

linear least-squares problem for Δx and Δy

a) no errors b) with errors

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Application of edge-tracking technique

2 4 6 8 10 12

x 10-3

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

-3

Time [s]

Dis

plac

emen

ts [

m]

x

y

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Error estimation through artificial image analysis

Errors introduced by pixellation/edge-detection (can be reduced through more precise algorithms) and CCD noise (unavoidable at given light conditions)

Such errors can be estimated through analysis of artificially constructed images

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Error determination from calibrated sphere measurements

Precision-machined 40-mm diameter sphere controlled by linear displacement stepper

Magnification ~300 μm/pixelPosition determination from tracking techniques

compared with inputted displacementsStandard error ~1.3 μm

(A) Shimadzu HPV-1; (B) Telephoto lens; (C) Precision-machined sphere; (D) Linear displacement stage; (E) Light source; (F) Light-diffusing material

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Errors in constant acceleration measurements

10-3

10-2

10-5

10-4

10-3

10-2

10-1

100

101

Measurement time [s]

Err

or in

mea

sure

d ac

cele

ratio

n: S

(a)/

a

20 kfps,a=1000 m/s2,s=1m

20 kfps,a=1000 m/s2,s=5m

20 kfps,a=1000 m/s2,s=50m

20 kfps,a=100 m/s2,s=1m

80 kfps,a=1000 m/s2,s=1m

Error in measured constant acceleration, a, can be determined from assumed displacement error (δ):

(n = number of measurement points)

For micron-level precision in displacement, accurate (~1%) acceleration measurements possible even for millisecond test times

2/1

56)(2ta

s

na

aS

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Errors in constant acceleration measurements

10-3

10-2

10-5

10-4

10-3

10-2

10-1

100

101

Measurement time [s]

Err

or in

mea

sure

d ac

cele

ratio

n: S

(a)/

a

20 kfps,a=1000 m/s2,s=1m

20 kfps,a=1000 m/s2,s=5m

20 kfps,a=1000 m/s2,s=50m

20 kfps,a=100 m/s2,s=1m

80 kfps,a=1000 m/s2,s=1m

Error in measured constant acceleration, a, can be determined from assumed displacement error (δ):

(n = number of measurement points)

For micron-level precision in displacement, accurate (~1%) acceleration measurements possible even for millisecond test times

2/1

56)(2ta

s

na

aS

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Conclusions

Technique originally developed for bodies with analytically expressible cross-sections

Generalized to arbitrary body geometries

Displacement measurements to micron level for wind-tunnel scale models – allows acceleration measurements to <1% under typical conditions

Generalization to three-dimensional motions?

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Application of edge-tracking technique

2 4 6 8 10 12

x 10-3

0

0.1

0.2

0.3

0.4

0.5

t [s]

Vel

ocity

[m

/s]

vx

vy

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Shock-wave surfing

-200 -150 -100 -50 0 50 100 150 200-0.015

-0.01

-0.005

0

0.005

0.01

0.015

(degrees)

(r -

r0)/

r 0

Corrected

Uncorrected

Optical distortions can become problematic for large fields-of-view

Can be corrected for using reference images

Error in edge-point locations

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Shock-wave surfing

0 2 4 6 8 10 120

20

40

60

80

100

t [ms]

x [m

m]

Corrected

Uncorrected

0 2 4 6 8 10 12

-25

-20

-15

-10

-5

0

t [ms]

y [m

m]

Displacements

Optical distortions can become problematic for large fields-of-view

Can be corrected for using reference images

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0 2 4 6 8 10 12

x 10-3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t [s]

CD

Corrected

Uncorrected

0 2 4 6 8 10 12

x 10-3

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

t [s]

CL

Shock-wave surfing

Force coefficients

Optical distortions can become problematic for large fields-of-view

Can be corrected for using reference images