© NMISA 2010

Multilateration Laser Tracker Systems

Speaker : Pieter Greeffe-mail: pgreeff@nmisa.org

© NMISA 2010

Contents

1. Introduction

2. The Project

3. Multilateration

4. Conclusion

http://www.metronics.com/

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Lower Accuracy CMMLower Accuracy CMM

Reproducible standard as prescribe in Metrology: Laser Radiation (Iodine 127)

SI unit for length

(metre definition)

Step GaugeStep

GaugeNMISA

activities inside border

National Standard for length as in Government Gazette: Laser Interferometer

(CSIR 4)

National Standard for length as in Government Gazette: Laser Interferometer

(CSIR 4)

End Standards: CMM Standard(R10 000)

End Standards: CMM Standard(R10 000)

Length Bar SystemLength Bar System

Standard CMM (R 15 000)

Standard CMM (R 15 000)

SquaresSquares

Laser InterferometerLaser Interferometer

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CMM Measurements (R 50 mil estimation)CMM Measurements (R 50 mil estimation)

Lower Accuracy CMMLower Accuracy CMM

Reproducible standard as prescribe in Metrology: Laser Radiation (Iodine 127)

SI unit for length

(metre definition)

Step GaugeStep

Gauge

CMM Calibrations(350 CMMs/year: R3 ,5mil)

CMM Calibrations(350 CMMs/year: R3 ,5mil)

CMM Measurements(R 120 000)

CMM Measurements(R 120 000)

Automotive Exports(R 65 bill estimation)Automotive Exports(R 65 bill estimation)

NMISA activities

inside border

National Standard for length as in Government Gazette: Laser Interferometer

(CSIR 4)

National Standard for length as in Government Gazette: Laser Interferometer

(CSIR 4)

End Standards: CMM Standard(R10 000)

End Standards: CMM Standard(R10 000)

Length Bar SystemLength Bar System

Standard CMM (R 15 000)

Standard CMM (R 15 000)

SquaresSquares

Laser InterferometerLaser Interferometer

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IntroductionApplication and Technology

• Applications of three dimensional metrology:– measure parts and assemblies: ship building, aeroplane construction, rotor

blades, satellite dish antennas, turbines, cars…

• Instruments for three dimensional metrology:– photogrammetry systems, bridge type CMM (Coordinate Measuring System),

portable measurement arms, total stations, GPS, indoor GPS systems, laser

trackers http://www.gom.com/

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IntroductionLaser Tracker Selection Motivation

• Accuracy: – A laser tracker is the most accurate type of device, for its measurement volume,

on the market

• Volume restriction:– not have the size restriction of the bridge type CMM or portable measuring arm.

• Traceability – A multilateration system is directly traceable to the metre, reducing uncertainties

caused by intermediate calibration steps.

• The level of the required accuracy of dimensional metrology in manufacturing industries are increasing

http://www.leica-geosystems.com/

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Laser Tracker

http://www.fieldcmm.com/Laser_Tracker.htm

Laser InterferometerLaser Interferometer

CMM MeasurementsCMM MeasurementsCMM CalibrationsCMM Calibrations

Laser

Laser Tracker

Home Position

Retroreflective Target

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Laser Tracker

http://www.fieldcmm.com/Laser_Tracker.htm

Laser Tracker

Target

Relative

Distance

L r

θ

α

(L,θ,α) (X,Y,Z)L = Li + Lr

(X,Y,Z)

(0,0,0)

Initial Distance

Li

© NMISA 2010Beam Offset δx

Measurement Beam

Beam Steering Mechanism

Optical Tracking Sensor

Beam Splitter

Control System

Beam Offset δy

Movement of Retroreflective Target

The Laser TrackerWorking Principle: Tracking

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The Project

http://www.metronics.com/

1. What is it about?

a) Traceability

b) Resource Development

c) Technical Development

2. What did we obtain?

a) Better System Understanding

b) Multilateration Algorithm

c) Multilateration Simulations

http://www.faro.com/

http://www.primemachine.com/files/inspsvc.html

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Understand System BetterMain Kinematic Error Source

Deadpath

Beam Steering Mechanism

Interferometer

Deadpath Error

Target

Deadpath

Change in Target Position

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Understand System Better Kinematic ModellingModel based on [10], described gimbal type mirror with 10 parameters

Source Beam

bi

Mirror Front Surface

bo

bc

Target Position

P

I

Om

Gimbal Axis 2

xb, x1

zm

θ2, z1

θ1

Gimbal Axis 1

Ob O1

xm

zb

bi

bo

bc

Om

zm

xm

ym

cx

cy

I

Mirror Frame

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Z

X

Y

Mirror Centre

Covariance Ellipsoid

Understand System Better Kinematic Modelling

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Understand System Better Build a Laser Tracker PrototypeDesign Scope

• Design scope

– select type of beam steering mechanism, design and build it

– sensor signal conditioning

– control of the system

CAD model of manufactured prototype tracker

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Understand System Better Laser Tracker PrototypeComponent Integration

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Multilateration: Solve target point coordinates, with only the distance between the target points and the station points precisely known

•Similar to triangulation or trilateration

•At least 4 tracker Stations Points (SP)

•At least 10 Target Points (TP)

Z

X

Y

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𝑒𝑖𝑗 = ට൫𝑥𝑖 −𝑋𝑗൯2 +൫𝑦𝑖 − 𝑌𝑗൯2 +൫𝑧𝑖 − 𝑍𝑗൯2 −൫𝑙𝑖𝑗 − 𝐿𝑗൯

Multilateration Concept

Cost Function: minimise residual

Residual (eij); (ith target point, jth station position):

Initial distances

Measured distancesAssumed TP location Assumed SP location

TP: Target PointSP: Station Point

Optimisation algorithm seeks local minimum for E:

Receives: •initial TP ((xyz)i) and SP ((XYZ)j)•and initial length (lj)•Measured Distances (Lij)

𝐸= 𝑒𝑖𝑗2𝑛𝑖=0

4𝑗=0

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Multilateration Test Sequential Multilateration Test Setup

•20 target points

•Use only one tracker

•Sequentially at 6 different positions

Z

X

Y

TP1

TP3

TP4

TP5

TP6

(700)

(450)

(700)

(416)

(Distances in mm)

TP2

(200)

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Multilateration Test Result Sequential Multilateration: Optimisation History

Cos

t F

unct

ion,

E (

mm

)

Iteration Number

𝑒𝑖𝑗 = ට൫𝑥𝑖 −𝑋𝑗൯2 +൫𝑦𝑖 − 𝑌𝑗൯2 +൫𝑧𝑖 − 𝑍𝑗൯2 −൫𝑙𝑖𝑗 − 𝐿𝑗൯

𝐸= 𝑒𝑖𝑗2𝑛𝑖=0

4𝑗=0

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Multilateration Test Result Sequential Multilateration Tests Results: 3D Distances: 200 mm: (-1,5 μm < Error < 2,0 μm)

mm

Distance Number

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Multilateration Test Result Sequential Multilateration Tests Results: 3D Distances: 450 mm: (140 μm < Error < 160 μm)

mm

Distance Number

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Multilateration and Uncertainty Estimation

Worst Case Error

CMM

(2,4 + 3L) µm, Max 1 m/ axis

Maximum TP (CMM): 5,4 µm

TP1 TP25,4 µm

SP1

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SMR Repeatability (Spherically Mounted Retroreflector)

10,67 15,33 18,33 10,67Average of Differences (µm)

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Multilateration and Uncertainty

SMR and Tracker Repeatability

From table = 10,67 µm

CMM: 5,4 µm

SMR: 10,67 µm

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Multilateration and Uncertainty

CMM: 5,4 µm

SMR: 10,67 µm

Laser Tracker (L in metres)

Radial accuracy (1 + 1L) µm, (Max radial distance 4 m)

Transverse accuracy (3 + 1L) µm. (Max transverse displacement 1 m)

Worst case maximum:

Sqrt(5^2 + 4^2*2) = 7,55 µm

TRACKER: 7,55 µm

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Multilateration and Uncertainty

5,4 µm

10,67 µm

7,55 µm

Combined worst case TP uncertainty:

Sqrt(5,4^2 + 10,67^2 + 7,55^2) = 14,2 µm

Worst case 3D distance uncertainty:

14,14x 2 = 28,28 µm

16,07 µm

14,14 µm

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Multilateration and Uncertainty: Tracker Measurement Result

Tracker Average Error per 3D Distance (Rounded to 5 μm)

Axes Distance Min

(μm) Max(μm)

Z 450 -5 10X,Z 492 -5 10Y,Z 832 0 25

X,Y 728 0 30X,Y,Z 855 5 25

Y 700 5 30X,Y,Z 461 -20 30Y,Z 416 -20 30X 400 -5 5X 200 -5 5

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Multilateration and Uncertainty: Fit Measurement Result

Total: 10 μmVariable: 140 nm

Max and Min for 3D Distances(With Initial Length Suppressed,

Rounded to 5 μm)

Components Distance

(mm)Min

(μm) Max(μm)

Z 450 135 160X,Z 492 120 150Y,Z 832 5 35X,Y 728 -80 -55

X,Y,Z 855 5 35Y 700 -85 -60

X,Y,Z 461 -15 25Y,Z 416 -5 30X 400 -15 15X 200 0 5

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Multilateration and UncertaintyAssumed SP Position

(cm precision)

Distance

(1 + 1L) µ

m

Distance (1 + 1L) µm

FIT SP Position (µm precision)

TP Repeatability 14 µm

Residual Error

1. If TP, SP1 or SP2 moves relative to each other in the X-axis, it will have a 1 to 1

effect on residual.

2. However, if SP3 moves in X-axis, it will only have a cosine effect on residual.

3. Since there are more SP in the X-axis, over the full range of it, more information

is available to solve it, while for the Y axis less data is available.

SP1

SP2

SP3

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Multilateration: Effect of SP Positions: Setup 1

TPSP

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Multilateration: Effect of SP Positions : Setup 2

TPSP

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Multilateration: Effect of SP Positions : Setup 3

TPSP

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Result Summary of SP Effect Analysis For a 100 iterations for each setup

Max at 450 and 492

Min at 200 and 400

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Result Summary of SP Effect Analysis For a 100 iterations for each setup

Max at 450 and 492

Min at 200 and 400

Setup 1

Setup 2

Setup 3 Measured with Trackers

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Reduction in Measurement Error with MultilaterationFor a 100 iterations for each setup

Measurement with Fit (Setup 3)

Simulated Measurement with Trackers

Average Error: -300 nm Max Deviation: 800 nm

Average Error: 600nm Max Deviation: 1100 nm

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Conclusion

• Laser Tracker

– A laser tracker is a highly accurate measuring instrument, with many

applications in the coordinate measurement field.

• Multilateration

– The concept of multilateration should be able to even further improve

the obtainable accuracy, due to direct traceability.

– However constraints are the beam steering mechanisms dead path

error contribution and the fit algorithm’s dependence on SP coordinates.

• Future Work

– Investigate

• the absolute accuracy of the fit and effect of the kinematic errors

– Further development of the beam steering mechanism

© NMISA 2010

Acknowledgements

• OA Kruger and the NMISA for their sponsorship for this project

• Mechatronic Engineering Department of the University of Stellenbosch

• Prof. K Schreve, for supervising this project.

Photo of prototype tracker

© NMISA 2010

References

[1] Benjamin B. Gallagher, "Optical shop applications for laser tracking metrology systems," Department of Optical Sciences, University of Arizona, 2003.

[2] W. Cuypers et al., "Optical measurement techniques for mobile and large-scale dimensional metrology," Optics and Lasers in Engineering, vol. 47, pp. 292-300, 2009, Optical Measurements.

[3] Kam C Lau and Robert J Hocken, "Three and five axis laser tracking systems," Patent 1987.

[4] T. Takatsuji, M. Goto, T. Kurosawa, and Y. Tanimura, "The first measurement of a three-dimensional coordinate by use of a laser tracking interferometer system based on trilateration," Measurement Science and Technology, vol. 9, pp. 38-41, 1998.

[5] P.D. Lin, C.H. Lu, and others, "Modeling and sensitivity analysis of laser tracking systems by skew-ray tracing method," Journal of Manufacturing Science and Engineering, vol. 127, p. 654, 2005.

[6] EB Hughes, A Wilson, and GN Peggs, "Design of a High-Accuracy CMM Based on Multi-Lateration Techniques," CIRP Annals - Manufacturing Technology, vol. 49, pp. 391-394, 2000.

[7] G.X. Zhang et al., "A Study on the Optimal Design of Laser-based Multi-lateration Systems," CIRP Annals - Manufacturing Technology, vol. 52, pp. 427-430, 2003.

[8] M. Vincze, JP Prenninger, and H. Gander, "A Laser Tracking System to Measure Position and Orientation of Robot End Effectors Under Motion," The International Journal of Robotics Research, vol. 13, p. 305, 1994.

[9] Y. Bai, H. Zhuang, and ZS Roth, "Fuzzy logic control to suppress noises and coupling effects in a laser tracking system," IEEE Transactions on Control Systems Technology, vol. 13, pp. 113-121, 2005.

[10] H. Zhuang and Z.S. Roth, "Modeling Gimbal Axis Misalignments and Mirror Center Offset in a Single-Beam Laser Tracking Measurement System," The International Journal of Robotics Research, vol. 14, p. 211, 1995.

[11] H. Zhuang, S.H. Motaghedi, Z.S. Roth, and Y. Bai, "Calibration of multi-beam laser tracking systems," Robotics and Computer-Integrated Manufacturing, vol. 19, p. 301, 2003.

[12] P.L. Teoh, B. Shirinzadeh, C.W. Foong, and G. Alici, "The measurement uncertainties in the laser interferometry-based sensing and tracking technique," Measurement, vol. 32, pp. 135-150, 2002.

[13] H. Schwenke, M. Franke, and J. Hannaford, "Error mapping of CMMs and machine tools by a single tracking interferometer," CIRP Annals - Manufacturing Technology, vol. 54, p. 475, 2005.

© NMISA 2010

Thank You!

Any questions?

Covariance Ellipsoid Plot, for system kinematic parameters