INSTITUTE OF PHYSICS PUBLISHING MEASUREMENT SCIENCE AND TECHNOLOGY

Meas Sci Technol 14 (2003) 1671ndash1682 PII S0957-0233(03)62385-4

Mathematical formulation and simulationof specular reflection based measurementsystem for gas tungsten arc weld poolsurfaceG Saeed and Y M Zhang

Center for Manufacturing and Department of Electrical and Computer EngineeringCollege of Engineering University of Kentucky Lexington KY 40506 USA

Received 17 April 2003 in final form 30 June 2003 accepted forpublication 16 July 2003Published 5 August 2003Online at stacksioporgMST141671

AbstractWeld pool surface can change dynamically during welding and is indicativeof information critical to controlling the process Research has picked up inthe field of observing the weld pool surface to understand the dynamics ofthe welding process This paper will help visualize and understand thephysics involved in observing the weld pool surface A study of laserproperties weld pool and camera optics was incorporated in developing amodel to describe the mechanism of observing the weld pool surface fromspecular reflection This observation method projects a laser beam on thepool surface through an optical grid with a frosted glass attached Thecorresponding specular reflection is calculated which is derived based onthe reflection law The reflected laser beams are then captured by the camerato form the image The model can be used to predict the outcome ofexperiments with grids placed in front of the laser and to determine theposition where the camera should be placed to acquire the best imagePreliminary results showed that the camera should be placed with the weldpool along the optical axis and the aperture should be as large as possible toallow as many rays into the camera as possible The model can be used tofind the optimal location of the laser and camera for materials of differentthickness by moving the electrode higher in the simulation and adjustingthe laser and camera location accordingly The paper will give some insightinto problems that might be encountered in observing the weld pool andsuggest the set-up of the laser and camera for obtaining the best image

Keywords specular surface image welding

(Some figures in this article are in colour only in the electronic version)

1 Introduction

Observation and the study of the weld pool surface canprovide information related to the welding process whichcan ultimately be used to develop a control procedure Weldpenetration control is the fundamental subject in automatedwelding Several methods have been studied includingpool oscillation [1ndash3] ultrasound infrared sensor [4 5]radiography etc Ultrasound based weld penetration sensing

has been extensively investigated at the Idaho NationalEngineering Laboratories [6] Despite significant progresspractical application is still restricted due to the requirementof a contact sensor At Georgia Tech a non-contact ultrasonicsensor is under development [7 8] The difficulty associatedwith this problem is to find a precise and reliable way tomeasure the weld penetration using only topside sensors thatmove with the torch

0957-023303091671+12$3000 copy 2003 IOP Publishing Ltd Printed in the UK 1671

G Saeed and Y M Zhang

Laserstripes

Camera

o y

x

Laser

Electrode

z

Image

Torch

Grid+frosted glass

Weld pool

Figure 1 Sensing mechanism of weld pool surface shape

Figure 2 Principle of virtual image of laser stripes without frostedglass

The three-dimensional surface of the weld pool canprovide abundant information about the welding processHowever measuring the three-dimensional surface of the weldpool is challenging First the bright arc light makes the weldpool difficult to see clearly Second the liquid surface of theweld pool is mirror-like which makes the regular structuredlight method for three-dimensional measurement ineffectiveTo measure the three-dimensional surface of the weld pool anovel measurement method has been proposed and studied atthe University of Kentucky [9 10] An algorithm has beendeveloped to process the image acquired [10] This methoduses a high-speed shutter camera assisted with a pulsed laserto view the weld pool through the bright arc light and usesa frosted-glass based grid to project laser stripes in order toimage the mirror-like liquid pool surface Initial experimentswith this sensing method at the University of Kentucky haveprovided successful results However due to the lack ofmathematical modelling of the system the study was limitedto image processing and qualitative explanation [9 10] Toquantitatively determine the three-dimensional surface of theweld pool from the image and to optimize the design of themeasurement system the measurement system must be fullyunderstood and be mathematically modelled

In this work a mathematical model for the three-dimensional surface measurement mechanism is establishedthat describes the relationship between the resultant imagethe three-dimensional surface of the weld pool in guided gastungsten arc welding (G-GTAW a novel arc welding processbeing developed under the grant which funds this presentstudy) and the image system parameters In order to help assistthe design of the system the optical model of the observationsystem has also been mathematically formulated The model

image

camera s (laser outlet)

grid

s3 (virtual image of s3)

s2 (virtual image of s2)

laser stripes

s1 (virtual image of s1)

s1

s2s3

frosted-glass

incident rays

mirror-like surface

reflectedrays

prime

prime

prime

Figure 3 Virtual image of laser stripes with frosted-glass

can be used to analyse the influence of system parameters suchas the position and angles of the imaging system It can alsobe used to maximize the accuracy and the robustness of thesystem

This paper is organized as follows In section 2 theprinciple of the 3D measurement system to be modelled isbriefly summarized Section 3 models the weld pool andlaser In section 4 the physics principle used to generatethe reflection and the algorithm used is explained Section 5is dedicated to the modelling of the optics principle of thecamera and the simulation of the measurement system usingthe developed models In sections 6 and 7 the simulation andthe results are discussed Section 8 summarizes the work doneand details future work

2 System

The system to be modelled is shown in figure 1 To acquire the3D weld pool surface information a special technique must beimplemented The common practice for determining the 3Dshape of a surface is to project a structured light onto the surfaceand sense the diffuse reflection of the structured light Theshape information can then be extracted from the deformationof the structured light However the weld pool surface ismirror-like smooth and no substantial amount of structuredlight can be reflected diffusely It appears that only specularreflection can be utilized In the case of specular reflection theobserved scene is the virtual image of the original object [11]The relationship between the original and its virtual imageis determined by the shape of the mirror surface Usuallystructured light is formed by projecting a laser through a gridThe sole function of the grid is to form the structured lightstripes As the light passes through the grid its directionof travel is unchanged This causes the virtual image of alaser point source to be a single spot despite the shape of themirror-like surface Thus a bright spot as shown in figure 2 issensed by the camera [9] and no shape information about thereflection surface is contained in the image To acquire shapeinformation a novel incident mechanism of structured lightas shown in figure 3 has been proposed [9]

The proposed incident mechanism of structured light isrealized through a specialized grid This grid consists of acommon grid and frosted glass When the laser is projectedonto the frosted glass the laser travel direction will be changedsee figure 3 From the viewpoint of light travel any point

1672

Mathematical formulation and simulation of specular reflection based measurement system

of the frosted glass can be considered as a new point lightsource which disperses light with a certain diffuse angle seefigure 3 The camera views the slits (grid openings) throughtheir reflection from the weld pool surface Their virtual imageconsists of bright stripes deformed by the weld pool surfacedeformation (figure 3) and is sensed by the camera The surfaceshape of the weld pool is contained in the acquired imageFurther the shutter of the camera is synchronized with a pulsedlaser so that the laser can be much brighter than the arc whenthe weld pool is imaged As a result the effect of the arcradiation is eliminated and a clear image of the weld pool isacquired [9]

3 Modelling weld pool surface and laser

31 Weld pool surface

Although the weld pool surface is not part of the measurementsystem simulation cannot be done to optimize themeasurement system without some assumed weld poolsurfaces It is known that the weld pool surface has twointerfaces the liquidndashvapour and solidndashliquid interface Theequation that governs the balance at the liquidndashvapour andsolidndashliquid interface under the quasi-steady state conditionis used to obtain the shapes of these interfaces [12 13] Theresultant liquidndashvapour interface Sl is

Sl(x y) = Kl

2Lb

(AP

πρv

) 23(

Lb + Lm + Cp(Tb minus T0)

3R0

) 13

times exp

(minus2(x + 2R0 Pel)

2

d2l

)exp

(minus2y2

L2l

)(1)

where dl = L l minus 2R0 if x minus2R0 Pel else dl = L l + 2R0Kl = Lb+Lm+Cp(TbminusT0)

Bb L l = ( 3AP R0

πρv[Lb+Lm+Cp (TbminusT0)])

13 the Peclet

parameter Pel = vR0al

Lm is the liquid coefficient Lb is theair coefficient T0 is the initial temperature of the work piecein kelvin (50 F) Tb is the boiling point in kelvin and Tm

is the melting point of the metal in kelvin Other variablesR0 A P ρ v and Cp are listed in table 1 The solidndashliquidinterface is expressed as

Ss(x y) = Ks

2Lm

(AP

πρv

) 23(

Lm + Cp(Tm minus T0)

3R0

) 13

times exp

(minus2(x + 2R0 Pes)

2

d2s

)exp

(minus2y2

L2s

)(2)

where ds = Ls minus 2R0 if x minus2R0 Pes else ds = Ls + 2R0Ks and Ls are given by Ks = [Lm+Cp(TmminusT0)]2

Lm[Lb+Lm+Cp (TbminusT0)] Ls =(

3AP R0πρv[Lm +Cp (TmminusT0)]

) 13

and Pes is defined as the Peclet

parameter and is given by Pes = vR0as

All variables and valuesare listed in table 1

Each state of the weld pool is defined by two equationsdepending on the value of the x-coordinate The equationswere reworked with the absolute function into one equationThe remodelled equation of the weld pool in liquidndashvapourstate is

Sl(x y) = Kl

2Lb

(AP

πρv

) 23(

Lb + Lm + Cp(Tb minus T0)

3R0

) 13

times exp

(minus2( x

2 + 2R0 Pe1)2

L2l + 4R2

0

minus 4L l R0s

|s|)

exp

(minus2y2

L2l

)

(3)

where s = x minus 2R0 Pel This facilitates the mathematicaldifferentiation of the shape

It should be pointed out that the above analytical modelsare derived from laser welding Although laser weld pools arein general different from those of gas tungsten arc shallowlaser weld pools generated by a low-power defocused (largediameter) beam should be similar to those of gas tungstenarc Further the focus of the present research is to measurethree-dimensional surfaces of weld pool and the measurementsystem is modelled to study the effect of the system parameterson measurement results The surface of the weld pool is givenonly as input parameters More importantly and practicallylimited work has been done to establish analytical models forgas tungsten arc weld pools Hence in this paper the analyticalmodels derived from laser welding are used to approximate gastungsten weld pool surfaces using an energy beam wider thantypical laser applications

32 Laser divergence

Light is a form of electromagnetic radiation It travelsthrough space as waves and occurs at different wavelengthsLaser light is different from other sources of light becauseit is monochromatic and its waves have a narrow rangeof wavelength Most lasers such as the Sealed NitrogenLaser LN300 manufactured by Laser Photonics with outputwavelength of 3371 nm used in this research emit light at onewavelength Laser light is also coherent This optical propertyof laser distinguishes it from other light sources Coherence issimply the measure of the degree of phase correlation that existsin the radiation field of a light source at different locations anddifferent times Laser light has a high degree of directionalityand the light is created with precise definition and minimumangular spread Laser light also has a high radiance (sourceintensity)

The beam spread angle φ and its characteristics outsidethe cavity are determined by solving electromagnetic wavesin an open cavity The beam divergence angle is given by therelationship

φ = 127λ

D(4)

where λ is the wavelength of the laser beam and D is thediameter of the laser beam at its beam waist [11] (figure 4)The pattern of the laser beam imaged on a screen consists ofa central bright circular spot the airy disc surrounded by aseries of bright rings as shown by figure 5 In this model theairy disc around the laser has been ignored for computationalsimplicity

Most lasers are fitted with mirrors and lenses to re-directthe beam The effect of these additions to the laser beam canbe observed and measured practically The fibre-optic throughwhich the laser is emitted can be placed at a predetermineddistance from a sheet of paper and the laser spot diameter

1673

G Saeed and Y M Zhang

Wavefronts

Laser cavity

Output couplingmirror

Highly reflectingmirror

Beam waist(diameter D)

External laser beam

= D

127λ

Beam divergenceangle

Figure 4 Beam divergence angle

Table 1 Values of parameters used in the weld pool equation

Name symbol and unit Value

Welding power P (W) Application dependentWelding speed v (m sminus1) Application dependent (002 or less)Absorptivity A 065Density ρ (kg mminus3) 7500Specific heat capacitance Cp (J kgminus1 Kminus1) 6785Diffusivity for the solid (liquid) state as(al) (m3 sminus1) 6435 times 10minus4

Boiling temperature Tb (K) 3342Melting temperature Tm (K) 1810Radius of welding heat source R0 (m) Application dependent (0002 or less)

Figure 5 The beam divergence angle φ is set by the edges of theAiry disc

measured The angle of divergence can be calculated usingtrigonometric triangles as

φ = tanminus1

(Diameter

Dis

)(5)

where lsquoDiameterrsquo is the measured diameter of the laser spotand lsquoDisrsquo is the distance between the fibre optic and the screen

4 Algorithm

The simulation represents lines as several points connectedtogether and surfaces are simply arrays or matrices with thecolumn and row being the x y coordinate respectively andwith the scalar at that location as the z-coordinate Theresolution of the figures can be significantly increased byincreasing the number of points used to represent each lineand increasing the size of arrays representing the surface

The surface of the weld pool is first plotted using surfaceequations as detailed in section 3 The torch is then introducedinto the same figure This will help visualize and find thelocation of the laser from where it will be targeted on the

weld pool The torch is cylindrical in shape and assumed tobe 14 mm end to end The torch and weld pool shape areillustrated in figure 6

Next the coordinates of the laser are entered and the laserbeams are drawn as emerging from that point onto the weldpool The divergence of the laser is considered in the modelIn order to display the divergence of the laser several lineswithin the area of divergence were drawn as shown in figure 7The diverging laser beams are calculated and drawn based onthe centre of the laser beam The distance between the laserand the weld pool should be large enough so that the beamdiverges and covers the shape of the weld pool completelyThe laser beam should also not pass through the torch Thiswill cause shadows to be formed on the image of weld pool

During welding the molten metal acts like a mirror andany incident light falling on the metal is reflected according tothe law of reflection which states that the angle of incidenceequals the angle of reflection In the simulation the reflectionof the laser beam was drawn by calculating the tangent planeof weld pool shape at the point of intersection of the incidentbeam and the pool shape The tangent equation is given as

Tangent =(

part

partxSl(x y)

)(x minus x0) +

(part

partySl(x y)

)(y minus y0)

(6)where part

partx Sl(x y) is the partial derivative of pool surface withrespect to x and part

party Sl(x y) is the partial derivative with respectto y [14] Next a normal vector was computed using twovectors on the tangent plane and by taking the cross productbetween the two vectors This normal vector would be usedto determine the angle of incidence Orthogonal projectionof the laser beam onto the normal vector was computed Theorthogonal projection of the incident ray onto the normal andreflected ray on the normal were matched The orthogonalprojection of the reflected ray onto the normal should be equalto the orthogonal projection of the incident ray onto the normal

1674

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 6 (a) Shape of the weld pool To approximate the gas tungsten arc weld pool surface a relatively low power (P = 2000 W) anddefocused beam (R0 = 0002 m) are used The travel speed is v = 002 m sminus1 (b) Torch placed directly above the weld pool Coordinateunits millimetres

because the normal the incident ray and the reflected rayform two similar triangles with equal angles between them(by the law of reflection) When treated as a vector in three-dimensional space the direction of the orthogonal projectionpoints towards the direction of the reflected ray Figure 8(a)details the use of orthogonal projection to find the reflected rayFigure 8(b) shows the laser rays being reflected off the weldpool in the simulation model

5 Modelling the camera

A camera can be modelled as a convex lens with a screen placedafter the lens where the image forms In this simulation thecamera was modelled using a simple aberration-free convex

lens with a finite focal length and a screen (figure 9) Using theoptics principle ie that rays converge to the focal length afterpassing through the convex length and the rays passing throughthe centre of optics will pass without refraction the behaviourof the rays passing through the camera can be modelled andused to predict the image formed on the screen Figure 10demonstrates the physics of the optics

The non-idealistic behaviour of the camera lens due to thecosine-fourth effect and the vignetting effect occurring becauseof the uneven exposure of the aperture were ignored as the weldpool will be approximately along the optical axis of the lensand a thin aperture will be used to observe the weld pool whichwill circumvent these two phenomena [11] Other parametersthat need to be controlled are the location of the camera andthe opening aperture of the camera

1675

G Saeed and Y M Zhang

(a)

(b)

Figure 7 Laser diverging from the source point onto the weld pool area (a) Two-dimensional view from the yndashz plane(b) Three-dimensional view Coordinate units millimetres

The simulation can be used to locate the position of thecamera where the highest quality of image can be acquiredThis location will be where most of the rays are reflected afterhitting the weld pool surface The goal should be to interceptas many rays as possible through the aperture of the cameraThe wider the opening of the aperture the more light entersinto the camera

In the simulation after the reflected rays are drawn thecamera is placed at the point where most rays seems toconverge and the screen is placed at the focal point on the lensThe rays after converging through the convex lens form an

image on the screen The distance of the object and the imageare related by the following equation

1

s+

1

s prime = 1

f(7)

where s = distance of the object from lens s prime = distance ofthe image from the lens and f is the focal length of the cameralens The camera captures the 3D world scene and transformsit into a 2D image Along with this transformation the sizeof the actual object is also changed The magnification of theimage is given by the equation m = sprime

s

1676

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 8 (a) The use of orthogonal projection to determine thereflected ray (b) Rays being reflected off the surface of the weldpool The concave shaped weld pool surface converges the reflectedrays this can be visually observed in the figure

Figure 9 Camera modelled as a convex lens and a screen for theformation of the image

A camera is a measurement device which uses theproperties of light and object reflectance to sense the world andimages can be defined as the record of energy (light) emittedfrom a particular scene in the world Surfaces can be thoughtof as sources of light which radiate energy The radiance ofthe surface is a function of the light falling on it and is given by

L = d2φ

dAf dw(8)

where φ is the power of light source illuminating the surfaceAf is the foreshortened area where radiance is being measured

Figure 10 Use of two rays one parallel to the optical axis and theother through the optical centre to find the location of the imageformed

(a)

(b)

(c)

Figure 11 (a) Laser rays reflected off the weld pool surface(b) The rays converge after passing through the convex lens(c) Image formed on the screen after converging through the lens

w is the angle at which the light is radiating from the surfaceand L is the power per unit foreshortened area emitted intoa solid angle The camera relates this radiance of a surfaceto the irradiance on the image plane Irradiance is defined asthe power per unit area falling on a surface or in other wordsthe amount of light falling on the plane If the surface is notradiating any light there will be no irradiance on to the imageplane in the camera and therefore no image will be capturedFor our application the reflection of the laser light can bethought of as the radiance of the weld pool surface If the lightreflected from the surface of the weld pool is not captured bythe camera then no image will be formed

1677

G Saeed and Y M Zhang

10

ndash10

8

0

6

4

2

0

ndash02

8 6 4 2 0 ndash2 ndash4 ndash6 ndash8

(a)

(b)

Figure 12 (a) Simulation run with the electrode in the scene and the laser placed such that no shadows are formed (b) Some laser isreflected into the electrode while some clear the electrode and are captured by the camera Coordinate units millimetres

In the simulation based on the judgment of the reflectedrays the camera and screen are placed on the desiredcoordinates and the image captured on the screen To findthe convergence of the reflected ray once it passes throughthe convex lens two more lines are imagined to come fromthe same point of reflection One ray is drawn parallel to theoptical axis of the convex lens which passes through the focalpoint after passing through the convex lens and the other rayis drawn to pass through the optical centre un-refracted Thepoint of intersection of these two rays behind the convex lenswill be the point where the reflected ray emerging from thesame point as the rays will converge as shown in figure 10

Figure 11(a) shows the reflection of the laser from the weldpool surface Figure 11(b) shows the placement of the cameraconvex lens in the simulation to capture the image Figure 11(c)shows the formation of image on the screen placed at 14 of thefocal length of the convex length The distance of the screenfrom the convex can be adjusted and hence the focusing ofthe camera changed A focused image is formed on a screen

placed at a distance of the focal length from the convex Thegoal of the simulation is to maximize the rays intercepted bythe camera

6 Simulation of measurement system

61 Determining the system parameters

The simulation can be run step by step to determine eachparameter individually For example first the model of laserweld pool surface and the electrode will be used to determinethe position of the laser Several trials can be made in theplacement of laser to determine the exact co-ordinates suchthat the laser does not hit the electrode and to ensure thatthe laser will cover the entire surface of the weld pool Theelectrode height can be adjusted to the desired value (typicallybetween 3 and 6 mm in G-GTAW) and the simulation run tofind the best location of the laser In figure 12(a) the simulation

1678

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 13 (a) Simulation run with a 3 times 3 grid placed before thelaser beam Each grid acts as a new source of the laser and the laserdiverges onto the surface and reflects (b) Simulation re-run withthe laser placed at a different location and angle notice that morerays are captured by the camera compared to figure 13(a)rsquosplacement of grid and camera

was run with the electrode at the height of 5 mm and the bestposition of the laser was determined to be (0 6 5) where 0will be the real world x-coordinate 6 mm is the y-coordinateand 5 mm is the z-coordinate

With the location of the laser determined the simulationcan be used to check the reflection of the laser rays at thatdesired location The reflected rays are drawn and traced to seeif the rays that pass the lens contain the weld pool informationFigure 12(b) shows a possibility where the rays reflecting offthe end of the weld pool surface are captured by the camerawhereas the rays in the centre of the weld pool and those closerto the laser hit the electrode after reflection and are not capturedby the camera This might suggest changing the laser locationso that the angle between the laser rays and the xndashy plane isreduced by lowering the height of the laser Decreasing theangle between the laser rays and the xndashy plane would make thereflection have a smaller angle between the xndashy plane (basedon the law of reflection) and hence make it possible for therays to clear the electrode

62 Simulation of measurement system

In the proposed measurement optical grids are used to formdifferent patterns The distortion of the patterns due to specular

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

(a)

(b)

(c)

Figure 14 (a) Image acquired of a flat shiny surface through thesimulation of the model developed (b) Image after joining the dots(c) Actual image of the flat shiny surface

reflection off the surface is used to determine the surface of theweld pool For example an opaque grid with ntimesn holes can beplaced between the laser and the weld pool The grid will blockmost of the laser allowing the laser to pass through the holesonly The laser which passes through the grid re-emits afterpassing through the frosted glass and diverges The diameterof the grid hole will be the new waist diameter D Equation (4)from section 22 can be used to calculate the divergence angle

Figure 13 shows the simulation with a 3 times 3 grid placedbetween the laser and weld piece The grid is an array of smallholes equi-distant from each other arranged in a rectangularpattern Figure 13(a) demonstrates that in some cases thereflected rays are scattered and only a few of them are captured

1679

G Saeed and Y M Zhang

05

04

03

02

01

0

ndash01

ndash02

ndash03ndash77 ndash76 ndash75 ndash74 ndash73 ndash72 ndash71 ndash7 ndash69 ndash68 ndash67

(a)

(b)

Figure 15 (a) Actual weld pool image (b) Image acquired throughthe simulation of the model developed

by the imaging system In this simulation run reflection forthe third row of the grid was not captured by the cameraFigure 13(b) shows a better placement of grid and camera Inthis simulation run the camera was placed a little further awayfrom the work piece to allow more rays to be captured by thecamera The image captured at the coordinates of the secondsimulation run will contain more information Sometimesthe rays undergo double reflections from the surface of theweld pool These lsquosecondrsquo reflections can be detected by thesimulation but they never make it to the imaging system dueto the angles they have after reflection

7 Results

To test the developed model the laser stripe was projected ontoa flat smooth surface and the camera used to observe the imageThe simulation result is displayed in figure 14(a) The surfacewas flat and reflecting and therefore the image is the sameas the grid pattern Since the simulation assumes a discretelaser source the dots on the simulated results are the reflectedlaser The result after joining the laser stripes and marking theboundary is shown in figure 14(b) and the actual picture froma camera is shown in figure 14(c) The simulation results and

-77 -76 -75 -74 -73 -72 -71 -7 -69 -68 -67-03

-02

-01

0

01

02

03

04

05

Figure 16 Simulation results with traced boundary and joined dotsto form stripes

the actual camera results show resemblance in that they bothcapture the continuous horizontal laser stripes as predicted

In an attempt to capture the weld pool image using thesystem described above the laser was projected onto the weldpool and a striped grid placed before the laser so that straightlines of laser fell upon the weld pool The camera was used toobserve the image The image with a striped grid yielded theresults shown in figure 15(a) The image was characterizedby clear boundaries and distorted laser stripe pattern Thesmall black projection on the top of the image is the shadowingcaused by the electrode tip

When the simulation was run with the striped grid placedbefore the laser to simulate the experimental set-up of the labpositive results were yielded which were exceptionally closeto the original image The simulation results are shown infigure 15(b) Since the simulation assumes a discrete lasersource the points on the simulated results are the reflected laserdots The result after joining the laser stripes and marking theboundary is shown in figure 16 The boundary of the weldpool is very prominent in the simulation result and the laserstripes are also distorted due to the shape of the weld poolas predicted from the real weld pool pictures However sincethe shape of the weld pool cannot be precisely controlled ordetermined during the experiments the simulation results willbe close to the experimental results but not exactly the samedue to the variation in the weld pool shape

In another simulation run the laser was projected at adifferent angle and the focusing of the camera changed Theset-up is shown in figure 17(a) The corresponding image onthe screen is shown in figure 17(b) After joining the dottedlaser points and marking the boundary the result is illustratedin figure 17(c) The image bears a close resemblance to theoriginal image with clear boundaries and laser stripes alongthe weld pool surface

8 Conclusions and future work

The models developed will assist future research in observingthe weld pool They can be used to predict the outcome ofexperiments with grids placed in front of the laser and to

1680

Mathematical formulation and simulation of specular reflection based measurement system

-03 -02 -01 0 01 02 03 04 -03 -02 -01 0 01 02 0325

3

35

4

45

5

25

3

35

4

45

5

(b)

(a)

(c)

Figure 17 (a) The set-up of a simulation run showing the laser stripes reflected laser camera and imaging plane (b) The result of thesimulation run (c) Simulation results after the boundary is traced and the points are joined to form stripes

determine the position where the camera should be placedto acquire the best image Preliminary results show that thecamera should be placed with the weld pool along the opticalaxis and the aperture should be as large as possible to allowas many rays into the camera as possible The model can beused to find the optimal location of the laser and camera formaterials of different thickness by moving the electrode higherin the simulation and adjusting the laser and camera locationaccordingly

This is a first step towards gaining an insight into theproposed measurement method for the three-dimensionalsurface of a weld pool The goal is to fully understandand mathematically model the proposed measurement system

The set of mathematical models obtained in this paper arecertainly subject to possible modification and revision Usingthe set of revised mathematical models the parameters of themeasurement system will be able to be further optimized andthe mathematical foundations will be able to be derived tocalculate the three-dimensional specular surface of the weldpool from the image captured using the proposed system

Acknowledgments

This work is funded by the National Science Foundation underGrant DMI-01144982 and the University of Kentucky Centerfor Manufacturing

1681

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682

Mathematical formulation and simulation of specular reflection based measurement system

of the frosted glass can be considered as a new point lightsource which disperses light with a certain diffuse angle seefigure 3 The camera views the slits (grid openings) throughtheir reflection from the weld pool surface Their virtual imageconsists of bright stripes deformed by the weld pool surfacedeformation (figure 3) and is sensed by the camera The surfaceshape of the weld pool is contained in the acquired imageFurther the shutter of the camera is synchronized with a pulsedlaser so that the laser can be much brighter than the arc whenthe weld pool is imaged As a result the effect of the arcradiation is eliminated and a clear image of the weld pool isacquired [9]

3 Modelling weld pool surface and laser

31 Weld pool surface

Although the weld pool surface is not part of the measurementsystem simulation cannot be done to optimize themeasurement system without some assumed weld poolsurfaces It is known that the weld pool surface has twointerfaces the liquidndashvapour and solidndashliquid interface Theequation that governs the balance at the liquidndashvapour andsolidndashliquid interface under the quasi-steady state conditionis used to obtain the shapes of these interfaces [12 13] Theresultant liquidndashvapour interface Sl is

Sl(x y) = Kl

2Lb

(AP

πρv

) 23(

Lb + Lm + Cp(Tb minus T0)

3R0

) 13

times exp

(minus2(x + 2R0 Pel)

2

d2l

)exp

(minus2y2

L2l

)(1)

where dl = L l minus 2R0 if x minus2R0 Pel else dl = L l + 2R0Kl = Lb+Lm+Cp(TbminusT0)

Bb L l = ( 3AP R0

πρv[Lb+Lm+Cp (TbminusT0)])

13 the Peclet

parameter Pel = vR0al

Lm is the liquid coefficient Lb is theair coefficient T0 is the initial temperature of the work piecein kelvin (50 F) Tb is the boiling point in kelvin and Tm

is the melting point of the metal in kelvin Other variablesR0 A P ρ v and Cp are listed in table 1 The solidndashliquidinterface is expressed as

Ss(x y) = Ks

2Lm

(AP

πρv

) 23(

Lm + Cp(Tm minus T0)

3R0

) 13

times exp

(minus2(x + 2R0 Pes)

2

d2s

)exp

(minus2y2

L2s

)(2)

where ds = Ls minus 2R0 if x minus2R0 Pes else ds = Ls + 2R0Ks and Ls are given by Ks = [Lm+Cp(TmminusT0)]2

Lm[Lb+Lm+Cp (TbminusT0)] Ls =(

3AP R0πρv[Lm +Cp (TmminusT0)]

) 13

and Pes is defined as the Peclet

parameter and is given by Pes = vR0as

All variables and valuesare listed in table 1

Each state of the weld pool is defined by two equationsdepending on the value of the x-coordinate The equationswere reworked with the absolute function into one equationThe remodelled equation of the weld pool in liquidndashvapourstate is

Sl(x y) = Kl

2Lb

(AP

πρv

) 23(

Lb + Lm + Cp(Tb minus T0)

3R0

) 13

times exp

(minus2( x

2 + 2R0 Pe1)2

L2l + 4R2

0

minus 4L l R0s

|s|)

exp

(minus2y2

L2l

)

(3)

where s = x minus 2R0 Pel This facilitates the mathematicaldifferentiation of the shape

It should be pointed out that the above analytical modelsare derived from laser welding Although laser weld pools arein general different from those of gas tungsten arc shallowlaser weld pools generated by a low-power defocused (largediameter) beam should be similar to those of gas tungstenarc Further the focus of the present research is to measurethree-dimensional surfaces of weld pool and the measurementsystem is modelled to study the effect of the system parameterson measurement results The surface of the weld pool is givenonly as input parameters More importantly and practicallylimited work has been done to establish analytical models forgas tungsten arc weld pools Hence in this paper the analyticalmodels derived from laser welding are used to approximate gastungsten weld pool surfaces using an energy beam wider thantypical laser applications

32 Laser divergence

Light is a form of electromagnetic radiation It travelsthrough space as waves and occurs at different wavelengthsLaser light is different from other sources of light becauseit is monochromatic and its waves have a narrow rangeof wavelength Most lasers such as the Sealed NitrogenLaser LN300 manufactured by Laser Photonics with outputwavelength of 3371 nm used in this research emit light at onewavelength Laser light is also coherent This optical propertyof laser distinguishes it from other light sources Coherence issimply the measure of the degree of phase correlation that existsin the radiation field of a light source at different locations anddifferent times Laser light has a high degree of directionalityand the light is created with precise definition and minimumangular spread Laser light also has a high radiance (sourceintensity)

The beam spread angle φ and its characteristics outsidethe cavity are determined by solving electromagnetic wavesin an open cavity The beam divergence angle is given by therelationship

φ = 127λ

D(4)

where λ is the wavelength of the laser beam and D is thediameter of the laser beam at its beam waist [11] (figure 4)The pattern of the laser beam imaged on a screen consists ofa central bright circular spot the airy disc surrounded by aseries of bright rings as shown by figure 5 In this model theairy disc around the laser has been ignored for computationalsimplicity

Most lasers are fitted with mirrors and lenses to re-directthe beam The effect of these additions to the laser beam canbe observed and measured practically The fibre-optic throughwhich the laser is emitted can be placed at a predetermineddistance from a sheet of paper and the laser spot diameter

1673

G Saeed and Y M Zhang

Wavefronts

Laser cavity

Output couplingmirror

Highly reflectingmirror

Beam waist(diameter D)

External laser beam

= D

127λ

Beam divergenceangle

Figure 4 Beam divergence angle

Table 1 Values of parameters used in the weld pool equation

Name symbol and unit Value

Welding power P (W) Application dependentWelding speed v (m sminus1) Application dependent (002 or less)Absorptivity A 065Density ρ (kg mminus3) 7500Specific heat capacitance Cp (J kgminus1 Kminus1) 6785Diffusivity for the solid (liquid) state as(al) (m3 sminus1) 6435 times 10minus4

Boiling temperature Tb (K) 3342Melting temperature Tm (K) 1810Radius of welding heat source R0 (m) Application dependent (0002 or less)

Figure 5 The beam divergence angle φ is set by the edges of theAiry disc

measured The angle of divergence can be calculated usingtrigonometric triangles as

φ = tanminus1

(Diameter

Dis

)(5)

where lsquoDiameterrsquo is the measured diameter of the laser spotand lsquoDisrsquo is the distance between the fibre optic and the screen

4 Algorithm

The simulation represents lines as several points connectedtogether and surfaces are simply arrays or matrices with thecolumn and row being the x y coordinate respectively andwith the scalar at that location as the z-coordinate Theresolution of the figures can be significantly increased byincreasing the number of points used to represent each lineand increasing the size of arrays representing the surface

The surface of the weld pool is first plotted using surfaceequations as detailed in section 3 The torch is then introducedinto the same figure This will help visualize and find thelocation of the laser from where it will be targeted on the

weld pool The torch is cylindrical in shape and assumed tobe 14 mm end to end The torch and weld pool shape areillustrated in figure 6

Next the coordinates of the laser are entered and the laserbeams are drawn as emerging from that point onto the weldpool The divergence of the laser is considered in the modelIn order to display the divergence of the laser several lineswithin the area of divergence were drawn as shown in figure 7The diverging laser beams are calculated and drawn based onthe centre of the laser beam The distance between the laserand the weld pool should be large enough so that the beamdiverges and covers the shape of the weld pool completelyThe laser beam should also not pass through the torch Thiswill cause shadows to be formed on the image of weld pool

During welding the molten metal acts like a mirror andany incident light falling on the metal is reflected according tothe law of reflection which states that the angle of incidenceequals the angle of reflection In the simulation the reflectionof the laser beam was drawn by calculating the tangent planeof weld pool shape at the point of intersection of the incidentbeam and the pool shape The tangent equation is given as

Tangent =(

part

partxSl(x y)

)(x minus x0) +

(part

partySl(x y)

)(y minus y0)

(6)where part

partx Sl(x y) is the partial derivative of pool surface withrespect to x and part

party Sl(x y) is the partial derivative with respectto y [14] Next a normal vector was computed using twovectors on the tangent plane and by taking the cross productbetween the two vectors This normal vector would be usedto determine the angle of incidence Orthogonal projectionof the laser beam onto the normal vector was computed Theorthogonal projection of the incident ray onto the normal andreflected ray on the normal were matched The orthogonalprojection of the reflected ray onto the normal should be equalto the orthogonal projection of the incident ray onto the normal

1674

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 6 (a) Shape of the weld pool To approximate the gas tungsten arc weld pool surface a relatively low power (P = 2000 W) anddefocused beam (R0 = 0002 m) are used The travel speed is v = 002 m sminus1 (b) Torch placed directly above the weld pool Coordinateunits millimetres

because the normal the incident ray and the reflected rayform two similar triangles with equal angles between them(by the law of reflection) When treated as a vector in three-dimensional space the direction of the orthogonal projectionpoints towards the direction of the reflected ray Figure 8(a)details the use of orthogonal projection to find the reflected rayFigure 8(b) shows the laser rays being reflected off the weldpool in the simulation model

5 Modelling the camera

A camera can be modelled as a convex lens with a screen placedafter the lens where the image forms In this simulation thecamera was modelled using a simple aberration-free convex

lens with a finite focal length and a screen (figure 9) Using theoptics principle ie that rays converge to the focal length afterpassing through the convex length and the rays passing throughthe centre of optics will pass without refraction the behaviourof the rays passing through the camera can be modelled andused to predict the image formed on the screen Figure 10demonstrates the physics of the optics

The non-idealistic behaviour of the camera lens due to thecosine-fourth effect and the vignetting effect occurring becauseof the uneven exposure of the aperture were ignored as the weldpool will be approximately along the optical axis of the lensand a thin aperture will be used to observe the weld pool whichwill circumvent these two phenomena [11] Other parametersthat need to be controlled are the location of the camera andthe opening aperture of the camera

1675

G Saeed and Y M Zhang

(a)

(b)

Figure 7 Laser diverging from the source point onto the weld pool area (a) Two-dimensional view from the yndashz plane(b) Three-dimensional view Coordinate units millimetres

The simulation can be used to locate the position of thecamera where the highest quality of image can be acquiredThis location will be where most of the rays are reflected afterhitting the weld pool surface The goal should be to interceptas many rays as possible through the aperture of the cameraThe wider the opening of the aperture the more light entersinto the camera

In the simulation after the reflected rays are drawn thecamera is placed at the point where most rays seems toconverge and the screen is placed at the focal point on the lensThe rays after converging through the convex lens form an

image on the screen The distance of the object and the imageare related by the following equation

1

s+

1

s prime = 1

f(7)

where s = distance of the object from lens s prime = distance ofthe image from the lens and f is the focal length of the cameralens The camera captures the 3D world scene and transformsit into a 2D image Along with this transformation the sizeof the actual object is also changed The magnification of theimage is given by the equation m = sprime

s

1676

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 8 (a) The use of orthogonal projection to determine thereflected ray (b) Rays being reflected off the surface of the weldpool The concave shaped weld pool surface converges the reflectedrays this can be visually observed in the figure

Figure 9 Camera modelled as a convex lens and a screen for theformation of the image

A camera is a measurement device which uses theproperties of light and object reflectance to sense the world andimages can be defined as the record of energy (light) emittedfrom a particular scene in the world Surfaces can be thoughtof as sources of light which radiate energy The radiance ofthe surface is a function of the light falling on it and is given by

L = d2φ

dAf dw(8)

where φ is the power of light source illuminating the surfaceAf is the foreshortened area where radiance is being measured

Figure 10 Use of two rays one parallel to the optical axis and theother through the optical centre to find the location of the imageformed

(a)

(b)

(c)

Figure 11 (a) Laser rays reflected off the weld pool surface(b) The rays converge after passing through the convex lens(c) Image formed on the screen after converging through the lens

w is the angle at which the light is radiating from the surfaceand L is the power per unit foreshortened area emitted intoa solid angle The camera relates this radiance of a surfaceto the irradiance on the image plane Irradiance is defined asthe power per unit area falling on a surface or in other wordsthe amount of light falling on the plane If the surface is notradiating any light there will be no irradiance on to the imageplane in the camera and therefore no image will be capturedFor our application the reflection of the laser light can bethought of as the radiance of the weld pool surface If the lightreflected from the surface of the weld pool is not captured bythe camera then no image will be formed

1677

G Saeed and Y M Zhang

10

ndash10

8

0

6

4

2

0

ndash02

8 6 4 2 0 ndash2 ndash4 ndash6 ndash8

(a)

(b)

Figure 12 (a) Simulation run with the electrode in the scene and the laser placed such that no shadows are formed (b) Some laser isreflected into the electrode while some clear the electrode and are captured by the camera Coordinate units millimetres

In the simulation based on the judgment of the reflectedrays the camera and screen are placed on the desiredcoordinates and the image captured on the screen To findthe convergence of the reflected ray once it passes throughthe convex lens two more lines are imagined to come fromthe same point of reflection One ray is drawn parallel to theoptical axis of the convex lens which passes through the focalpoint after passing through the convex lens and the other rayis drawn to pass through the optical centre un-refracted Thepoint of intersection of these two rays behind the convex lenswill be the point where the reflected ray emerging from thesame point as the rays will converge as shown in figure 10

Figure 11(a) shows the reflection of the laser from the weldpool surface Figure 11(b) shows the placement of the cameraconvex lens in the simulation to capture the image Figure 11(c)shows the formation of image on the screen placed at 14 of thefocal length of the convex length The distance of the screenfrom the convex can be adjusted and hence the focusing ofthe camera changed A focused image is formed on a screen

placed at a distance of the focal length from the convex Thegoal of the simulation is to maximize the rays intercepted bythe camera

6 Simulation of measurement system

61 Determining the system parameters

The simulation can be run step by step to determine eachparameter individually For example first the model of laserweld pool surface and the electrode will be used to determinethe position of the laser Several trials can be made in theplacement of laser to determine the exact co-ordinates suchthat the laser does not hit the electrode and to ensure thatthe laser will cover the entire surface of the weld pool Theelectrode height can be adjusted to the desired value (typicallybetween 3 and 6 mm in G-GTAW) and the simulation run tofind the best location of the laser In figure 12(a) the simulation

1678

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 13 (a) Simulation run with a 3 times 3 grid placed before thelaser beam Each grid acts as a new source of the laser and the laserdiverges onto the surface and reflects (b) Simulation re-run withthe laser placed at a different location and angle notice that morerays are captured by the camera compared to figure 13(a)rsquosplacement of grid and camera

was run with the electrode at the height of 5 mm and the bestposition of the laser was determined to be (0 6 5) where 0will be the real world x-coordinate 6 mm is the y-coordinateand 5 mm is the z-coordinate

With the location of the laser determined the simulationcan be used to check the reflection of the laser rays at thatdesired location The reflected rays are drawn and traced to seeif the rays that pass the lens contain the weld pool informationFigure 12(b) shows a possibility where the rays reflecting offthe end of the weld pool surface are captured by the camerawhereas the rays in the centre of the weld pool and those closerto the laser hit the electrode after reflection and are not capturedby the camera This might suggest changing the laser locationso that the angle between the laser rays and the xndashy plane isreduced by lowering the height of the laser Decreasing theangle between the laser rays and the xndashy plane would make thereflection have a smaller angle between the xndashy plane (basedon the law of reflection) and hence make it possible for therays to clear the electrode

62 Simulation of measurement system

In the proposed measurement optical grids are used to formdifferent patterns The distortion of the patterns due to specular

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

(a)

(b)

(c)

Figure 14 (a) Image acquired of a flat shiny surface through thesimulation of the model developed (b) Image after joining the dots(c) Actual image of the flat shiny surface

reflection off the surface is used to determine the surface of theweld pool For example an opaque grid with ntimesn holes can beplaced between the laser and the weld pool The grid will blockmost of the laser allowing the laser to pass through the holesonly The laser which passes through the grid re-emits afterpassing through the frosted glass and diverges The diameterof the grid hole will be the new waist diameter D Equation (4)from section 22 can be used to calculate the divergence angle

Figure 13 shows the simulation with a 3 times 3 grid placedbetween the laser and weld piece The grid is an array of smallholes equi-distant from each other arranged in a rectangularpattern Figure 13(a) demonstrates that in some cases thereflected rays are scattered and only a few of them are captured

1679

G Saeed and Y M Zhang

05

04

03

02

01

0

ndash01

ndash02

ndash03ndash77 ndash76 ndash75 ndash74 ndash73 ndash72 ndash71 ndash7 ndash69 ndash68 ndash67

(a)

(b)

Figure 15 (a) Actual weld pool image (b) Image acquired throughthe simulation of the model developed

by the imaging system In this simulation run reflection forthe third row of the grid was not captured by the cameraFigure 13(b) shows a better placement of grid and camera Inthis simulation run the camera was placed a little further awayfrom the work piece to allow more rays to be captured by thecamera The image captured at the coordinates of the secondsimulation run will contain more information Sometimesthe rays undergo double reflections from the surface of theweld pool These lsquosecondrsquo reflections can be detected by thesimulation but they never make it to the imaging system dueto the angles they have after reflection

7 Results

To test the developed model the laser stripe was projected ontoa flat smooth surface and the camera used to observe the imageThe simulation result is displayed in figure 14(a) The surfacewas flat and reflecting and therefore the image is the sameas the grid pattern Since the simulation assumes a discretelaser source the dots on the simulated results are the reflectedlaser The result after joining the laser stripes and marking theboundary is shown in figure 14(b) and the actual picture froma camera is shown in figure 14(c) The simulation results and

-77 -76 -75 -74 -73 -72 -71 -7 -69 -68 -67-03

-02

-01

0

01

02

03

04

05

Figure 16 Simulation results with traced boundary and joined dotsto form stripes

the actual camera results show resemblance in that they bothcapture the continuous horizontal laser stripes as predicted

In an attempt to capture the weld pool image using thesystem described above the laser was projected onto the weldpool and a striped grid placed before the laser so that straightlines of laser fell upon the weld pool The camera was used toobserve the image The image with a striped grid yielded theresults shown in figure 15(a) The image was characterizedby clear boundaries and distorted laser stripe pattern Thesmall black projection on the top of the image is the shadowingcaused by the electrode tip

When the simulation was run with the striped grid placedbefore the laser to simulate the experimental set-up of the labpositive results were yielded which were exceptionally closeto the original image The simulation results are shown infigure 15(b) Since the simulation assumes a discrete lasersource the points on the simulated results are the reflected laserdots The result after joining the laser stripes and marking theboundary is shown in figure 16 The boundary of the weldpool is very prominent in the simulation result and the laserstripes are also distorted due to the shape of the weld poolas predicted from the real weld pool pictures However sincethe shape of the weld pool cannot be precisely controlled ordetermined during the experiments the simulation results willbe close to the experimental results but not exactly the samedue to the variation in the weld pool shape

In another simulation run the laser was projected at adifferent angle and the focusing of the camera changed Theset-up is shown in figure 17(a) The corresponding image onthe screen is shown in figure 17(b) After joining the dottedlaser points and marking the boundary the result is illustratedin figure 17(c) The image bears a close resemblance to theoriginal image with clear boundaries and laser stripes alongthe weld pool surface

8 Conclusions and future work

The models developed will assist future research in observingthe weld pool They can be used to predict the outcome ofexperiments with grids placed in front of the laser and to

1680

Mathematical formulation and simulation of specular reflection based measurement system

-03 -02 -01 0 01 02 03 04 -03 -02 -01 0 01 02 0325

3

35

4

45

5

25

3

35

4

45

5

(b)

(a)

(c)

Figure 17 (a) The set-up of a simulation run showing the laser stripes reflected laser camera and imaging plane (b) The result of thesimulation run (c) Simulation results after the boundary is traced and the points are joined to form stripes

determine the position where the camera should be placedto acquire the best image Preliminary results show that thecamera should be placed with the weld pool along the opticalaxis and the aperture should be as large as possible to allowas many rays into the camera as possible The model can beused to find the optimal location of the laser and camera formaterials of different thickness by moving the electrode higherin the simulation and adjusting the laser and camera locationaccordingly

This is a first step towards gaining an insight into theproposed measurement method for the three-dimensionalsurface of a weld pool The goal is to fully understandand mathematically model the proposed measurement system

The set of mathematical models obtained in this paper arecertainly subject to possible modification and revision Usingthe set of revised mathematical models the parameters of themeasurement system will be able to be further optimized andthe mathematical foundations will be able to be derived tocalculate the three-dimensional specular surface of the weldpool from the image captured using the proposed system

Acknowledgments

This work is funded by the National Science Foundation underGrant DMI-01144982 and the University of Kentucky Centerfor Manufacturing

1681

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682

G Saeed and Y M Zhang

Wavefronts

Laser cavity

Output couplingmirror

Highly reflectingmirror

Beam waist(diameter D)

External laser beam

= D

127λ

Beam divergenceangle

Figure 4 Beam divergence angle

Table 1 Values of parameters used in the weld pool equation

Name symbol and unit Value

Welding power P (W) Application dependentWelding speed v (m sminus1) Application dependent (002 or less)Absorptivity A 065Density ρ (kg mminus3) 7500Specific heat capacitance Cp (J kgminus1 Kminus1) 6785Diffusivity for the solid (liquid) state as(al) (m3 sminus1) 6435 times 10minus4

Boiling temperature Tb (K) 3342Melting temperature Tm (K) 1810Radius of welding heat source R0 (m) Application dependent (0002 or less)

Figure 5 The beam divergence angle φ is set by the edges of theAiry disc

measured The angle of divergence can be calculated usingtrigonometric triangles as

φ = tanminus1

(Diameter

Dis

)(5)

where lsquoDiameterrsquo is the measured diameter of the laser spotand lsquoDisrsquo is the distance between the fibre optic and the screen

4 Algorithm

The simulation represents lines as several points connectedtogether and surfaces are simply arrays or matrices with thecolumn and row being the x y coordinate respectively andwith the scalar at that location as the z-coordinate Theresolution of the figures can be significantly increased byincreasing the number of points used to represent each lineand increasing the size of arrays representing the surface

The surface of the weld pool is first plotted using surfaceequations as detailed in section 3 The torch is then introducedinto the same figure This will help visualize and find thelocation of the laser from where it will be targeted on the

weld pool The torch is cylindrical in shape and assumed tobe 14 mm end to end The torch and weld pool shape areillustrated in figure 6

Next the coordinates of the laser are entered and the laserbeams are drawn as emerging from that point onto the weldpool The divergence of the laser is considered in the modelIn order to display the divergence of the laser several lineswithin the area of divergence were drawn as shown in figure 7The diverging laser beams are calculated and drawn based onthe centre of the laser beam The distance between the laserand the weld pool should be large enough so that the beamdiverges and covers the shape of the weld pool completelyThe laser beam should also not pass through the torch Thiswill cause shadows to be formed on the image of weld pool

During welding the molten metal acts like a mirror andany incident light falling on the metal is reflected according tothe law of reflection which states that the angle of incidenceequals the angle of reflection In the simulation the reflectionof the laser beam was drawn by calculating the tangent planeof weld pool shape at the point of intersection of the incidentbeam and the pool shape The tangent equation is given as

Tangent =(

part

partxSl(x y)

)(x minus x0) +

(part

partySl(x y)

)(y minus y0)

(6)where part

partx Sl(x y) is the partial derivative of pool surface withrespect to x and part

party Sl(x y) is the partial derivative with respectto y [14] Next a normal vector was computed using twovectors on the tangent plane and by taking the cross productbetween the two vectors This normal vector would be usedto determine the angle of incidence Orthogonal projectionof the laser beam onto the normal vector was computed Theorthogonal projection of the incident ray onto the normal andreflected ray on the normal were matched The orthogonalprojection of the reflected ray onto the normal should be equalto the orthogonal projection of the incident ray onto the normal

1674

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 6 (a) Shape of the weld pool To approximate the gas tungsten arc weld pool surface a relatively low power (P = 2000 W) anddefocused beam (R0 = 0002 m) are used The travel speed is v = 002 m sminus1 (b) Torch placed directly above the weld pool Coordinateunits millimetres

because the normal the incident ray and the reflected rayform two similar triangles with equal angles between them(by the law of reflection) When treated as a vector in three-dimensional space the direction of the orthogonal projectionpoints towards the direction of the reflected ray Figure 8(a)details the use of orthogonal projection to find the reflected rayFigure 8(b) shows the laser rays being reflected off the weldpool in the simulation model

5 Modelling the camera

A camera can be modelled as a convex lens with a screen placedafter the lens where the image forms In this simulation thecamera was modelled using a simple aberration-free convex

lens with a finite focal length and a screen (figure 9) Using theoptics principle ie that rays converge to the focal length afterpassing through the convex length and the rays passing throughthe centre of optics will pass without refraction the behaviourof the rays passing through the camera can be modelled andused to predict the image formed on the screen Figure 10demonstrates the physics of the optics

The non-idealistic behaviour of the camera lens due to thecosine-fourth effect and the vignetting effect occurring becauseof the uneven exposure of the aperture were ignored as the weldpool will be approximately along the optical axis of the lensand a thin aperture will be used to observe the weld pool whichwill circumvent these two phenomena [11] Other parametersthat need to be controlled are the location of the camera andthe opening aperture of the camera

1675

G Saeed and Y M Zhang

(a)

(b)

Figure 7 Laser diverging from the source point onto the weld pool area (a) Two-dimensional view from the yndashz plane(b) Three-dimensional view Coordinate units millimetres

The simulation can be used to locate the position of thecamera where the highest quality of image can be acquiredThis location will be where most of the rays are reflected afterhitting the weld pool surface The goal should be to interceptas many rays as possible through the aperture of the cameraThe wider the opening of the aperture the more light entersinto the camera

In the simulation after the reflected rays are drawn thecamera is placed at the point where most rays seems toconverge and the screen is placed at the focal point on the lensThe rays after converging through the convex lens form an

image on the screen The distance of the object and the imageare related by the following equation

1

s+

1

s prime = 1

f(7)

where s = distance of the object from lens s prime = distance ofthe image from the lens and f is the focal length of the cameralens The camera captures the 3D world scene and transformsit into a 2D image Along with this transformation the sizeof the actual object is also changed The magnification of theimage is given by the equation m = sprime

s

1676

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 8 (a) The use of orthogonal projection to determine thereflected ray (b) Rays being reflected off the surface of the weldpool The concave shaped weld pool surface converges the reflectedrays this can be visually observed in the figure

Figure 9 Camera modelled as a convex lens and a screen for theformation of the image

A camera is a measurement device which uses theproperties of light and object reflectance to sense the world andimages can be defined as the record of energy (light) emittedfrom a particular scene in the world Surfaces can be thoughtof as sources of light which radiate energy The radiance ofthe surface is a function of the light falling on it and is given by

L = d2φ

dAf dw(8)

where φ is the power of light source illuminating the surfaceAf is the foreshortened area where radiance is being measured

Figure 10 Use of two rays one parallel to the optical axis and theother through the optical centre to find the location of the imageformed

(a)

(b)

(c)

Figure 11 (a) Laser rays reflected off the weld pool surface(b) The rays converge after passing through the convex lens(c) Image formed on the screen after converging through the lens

w is the angle at which the light is radiating from the surfaceand L is the power per unit foreshortened area emitted intoa solid angle The camera relates this radiance of a surfaceto the irradiance on the image plane Irradiance is defined asthe power per unit area falling on a surface or in other wordsthe amount of light falling on the plane If the surface is notradiating any light there will be no irradiance on to the imageplane in the camera and therefore no image will be capturedFor our application the reflection of the laser light can bethought of as the radiance of the weld pool surface If the lightreflected from the surface of the weld pool is not captured bythe camera then no image will be formed

1677

G Saeed and Y M Zhang

10

ndash10

8

0

6

4

2

0

ndash02

8 6 4 2 0 ndash2 ndash4 ndash6 ndash8

(a)

(b)

Figure 12 (a) Simulation run with the electrode in the scene and the laser placed such that no shadows are formed (b) Some laser isreflected into the electrode while some clear the electrode and are captured by the camera Coordinate units millimetres

In the simulation based on the judgment of the reflectedrays the camera and screen are placed on the desiredcoordinates and the image captured on the screen To findthe convergence of the reflected ray once it passes throughthe convex lens two more lines are imagined to come fromthe same point of reflection One ray is drawn parallel to theoptical axis of the convex lens which passes through the focalpoint after passing through the convex lens and the other rayis drawn to pass through the optical centre un-refracted Thepoint of intersection of these two rays behind the convex lenswill be the point where the reflected ray emerging from thesame point as the rays will converge as shown in figure 10

Figure 11(a) shows the reflection of the laser from the weldpool surface Figure 11(b) shows the placement of the cameraconvex lens in the simulation to capture the image Figure 11(c)shows the formation of image on the screen placed at 14 of thefocal length of the convex length The distance of the screenfrom the convex can be adjusted and hence the focusing ofthe camera changed A focused image is formed on a screen

placed at a distance of the focal length from the convex Thegoal of the simulation is to maximize the rays intercepted bythe camera

6 Simulation of measurement system

61 Determining the system parameters

The simulation can be run step by step to determine eachparameter individually For example first the model of laserweld pool surface and the electrode will be used to determinethe position of the laser Several trials can be made in theplacement of laser to determine the exact co-ordinates suchthat the laser does not hit the electrode and to ensure thatthe laser will cover the entire surface of the weld pool Theelectrode height can be adjusted to the desired value (typicallybetween 3 and 6 mm in G-GTAW) and the simulation run tofind the best location of the laser In figure 12(a) the simulation

1678

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 13 (a) Simulation run with a 3 times 3 grid placed before thelaser beam Each grid acts as a new source of the laser and the laserdiverges onto the surface and reflects (b) Simulation re-run withthe laser placed at a different location and angle notice that morerays are captured by the camera compared to figure 13(a)rsquosplacement of grid and camera

was run with the electrode at the height of 5 mm and the bestposition of the laser was determined to be (0 6 5) where 0will be the real world x-coordinate 6 mm is the y-coordinateand 5 mm is the z-coordinate

With the location of the laser determined the simulationcan be used to check the reflection of the laser rays at thatdesired location The reflected rays are drawn and traced to seeif the rays that pass the lens contain the weld pool informationFigure 12(b) shows a possibility where the rays reflecting offthe end of the weld pool surface are captured by the camerawhereas the rays in the centre of the weld pool and those closerto the laser hit the electrode after reflection and are not capturedby the camera This might suggest changing the laser locationso that the angle between the laser rays and the xndashy plane isreduced by lowering the height of the laser Decreasing theangle between the laser rays and the xndashy plane would make thereflection have a smaller angle between the xndashy plane (basedon the law of reflection) and hence make it possible for therays to clear the electrode

62 Simulation of measurement system

In the proposed measurement optical grids are used to formdifferent patterns The distortion of the patterns due to specular

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

(a)

(b)

(c)

Figure 14 (a) Image acquired of a flat shiny surface through thesimulation of the model developed (b) Image after joining the dots(c) Actual image of the flat shiny surface

reflection off the surface is used to determine the surface of theweld pool For example an opaque grid with ntimesn holes can beplaced between the laser and the weld pool The grid will blockmost of the laser allowing the laser to pass through the holesonly The laser which passes through the grid re-emits afterpassing through the frosted glass and diverges The diameterof the grid hole will be the new waist diameter D Equation (4)from section 22 can be used to calculate the divergence angle

Figure 13 shows the simulation with a 3 times 3 grid placedbetween the laser and weld piece The grid is an array of smallholes equi-distant from each other arranged in a rectangularpattern Figure 13(a) demonstrates that in some cases thereflected rays are scattered and only a few of them are captured

1679

G Saeed and Y M Zhang

05

04

03

02

01

0

ndash01

ndash02

ndash03ndash77 ndash76 ndash75 ndash74 ndash73 ndash72 ndash71 ndash7 ndash69 ndash68 ndash67

(a)

(b)

Figure 15 (a) Actual weld pool image (b) Image acquired throughthe simulation of the model developed

by the imaging system In this simulation run reflection forthe third row of the grid was not captured by the cameraFigure 13(b) shows a better placement of grid and camera Inthis simulation run the camera was placed a little further awayfrom the work piece to allow more rays to be captured by thecamera The image captured at the coordinates of the secondsimulation run will contain more information Sometimesthe rays undergo double reflections from the surface of theweld pool These lsquosecondrsquo reflections can be detected by thesimulation but they never make it to the imaging system dueto the angles they have after reflection

7 Results

To test the developed model the laser stripe was projected ontoa flat smooth surface and the camera used to observe the imageThe simulation result is displayed in figure 14(a) The surfacewas flat and reflecting and therefore the image is the sameas the grid pattern Since the simulation assumes a discretelaser source the dots on the simulated results are the reflectedlaser The result after joining the laser stripes and marking theboundary is shown in figure 14(b) and the actual picture froma camera is shown in figure 14(c) The simulation results and

-77 -76 -75 -74 -73 -72 -71 -7 -69 -68 -67-03

-02

-01

0

01

02

03

04

05

Figure 16 Simulation results with traced boundary and joined dotsto form stripes

the actual camera results show resemblance in that they bothcapture the continuous horizontal laser stripes as predicted

In an attempt to capture the weld pool image using thesystem described above the laser was projected onto the weldpool and a striped grid placed before the laser so that straightlines of laser fell upon the weld pool The camera was used toobserve the image The image with a striped grid yielded theresults shown in figure 15(a) The image was characterizedby clear boundaries and distorted laser stripe pattern Thesmall black projection on the top of the image is the shadowingcaused by the electrode tip

When the simulation was run with the striped grid placedbefore the laser to simulate the experimental set-up of the labpositive results were yielded which were exceptionally closeto the original image The simulation results are shown infigure 15(b) Since the simulation assumes a discrete lasersource the points on the simulated results are the reflected laserdots The result after joining the laser stripes and marking theboundary is shown in figure 16 The boundary of the weldpool is very prominent in the simulation result and the laserstripes are also distorted due to the shape of the weld poolas predicted from the real weld pool pictures However sincethe shape of the weld pool cannot be precisely controlled ordetermined during the experiments the simulation results willbe close to the experimental results but not exactly the samedue to the variation in the weld pool shape

In another simulation run the laser was projected at adifferent angle and the focusing of the camera changed Theset-up is shown in figure 17(a) The corresponding image onthe screen is shown in figure 17(b) After joining the dottedlaser points and marking the boundary the result is illustratedin figure 17(c) The image bears a close resemblance to theoriginal image with clear boundaries and laser stripes alongthe weld pool surface

8 Conclusions and future work

The models developed will assist future research in observingthe weld pool They can be used to predict the outcome ofexperiments with grids placed in front of the laser and to

1680

Mathematical formulation and simulation of specular reflection based measurement system

-03 -02 -01 0 01 02 03 04 -03 -02 -01 0 01 02 0325

3

35

4

45

5

25

3

35

4

45

5

(b)

(a)

(c)

Figure 17 (a) The set-up of a simulation run showing the laser stripes reflected laser camera and imaging plane (b) The result of thesimulation run (c) Simulation results after the boundary is traced and the points are joined to form stripes

determine the position where the camera should be placedto acquire the best image Preliminary results show that thecamera should be placed with the weld pool along the opticalaxis and the aperture should be as large as possible to allowas many rays into the camera as possible The model can beused to find the optimal location of the laser and camera formaterials of different thickness by moving the electrode higherin the simulation and adjusting the laser and camera locationaccordingly

This is a first step towards gaining an insight into theproposed measurement method for the three-dimensionalsurface of a weld pool The goal is to fully understandand mathematically model the proposed measurement system

The set of mathematical models obtained in this paper arecertainly subject to possible modification and revision Usingthe set of revised mathematical models the parameters of themeasurement system will be able to be further optimized andthe mathematical foundations will be able to be derived tocalculate the three-dimensional specular surface of the weldpool from the image captured using the proposed system

Acknowledgments

This work is funded by the National Science Foundation underGrant DMI-01144982 and the University of Kentucky Centerfor Manufacturing

1681

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 6 (a) Shape of the weld pool To approximate the gas tungsten arc weld pool surface a relatively low power (P = 2000 W) anddefocused beam (R0 = 0002 m) are used The travel speed is v = 002 m sminus1 (b) Torch placed directly above the weld pool Coordinateunits millimetres

because the normal the incident ray and the reflected rayform two similar triangles with equal angles between them(by the law of reflection) When treated as a vector in three-dimensional space the direction of the orthogonal projectionpoints towards the direction of the reflected ray Figure 8(a)details the use of orthogonal projection to find the reflected rayFigure 8(b) shows the laser rays being reflected off the weldpool in the simulation model

5 Modelling the camera

A camera can be modelled as a convex lens with a screen placedafter the lens where the image forms In this simulation thecamera was modelled using a simple aberration-free convex

lens with a finite focal length and a screen (figure 9) Using theoptics principle ie that rays converge to the focal length afterpassing through the convex length and the rays passing throughthe centre of optics will pass without refraction the behaviourof the rays passing through the camera can be modelled andused to predict the image formed on the screen Figure 10demonstrates the physics of the optics

The non-idealistic behaviour of the camera lens due to thecosine-fourth effect and the vignetting effect occurring becauseof the uneven exposure of the aperture were ignored as the weldpool will be approximately along the optical axis of the lensand a thin aperture will be used to observe the weld pool whichwill circumvent these two phenomena [11] Other parametersthat need to be controlled are the location of the camera andthe opening aperture of the camera

1675

G Saeed and Y M Zhang

(a)

(b)

Figure 7 Laser diverging from the source point onto the weld pool area (a) Two-dimensional view from the yndashz plane(b) Three-dimensional view Coordinate units millimetres

The simulation can be used to locate the position of thecamera where the highest quality of image can be acquiredThis location will be where most of the rays are reflected afterhitting the weld pool surface The goal should be to interceptas many rays as possible through the aperture of the cameraThe wider the opening of the aperture the more light entersinto the camera

In the simulation after the reflected rays are drawn thecamera is placed at the point where most rays seems toconverge and the screen is placed at the focal point on the lensThe rays after converging through the convex lens form an

image on the screen The distance of the object and the imageare related by the following equation

1

s+

1

s prime = 1

f(7)

where s = distance of the object from lens s prime = distance ofthe image from the lens and f is the focal length of the cameralens The camera captures the 3D world scene and transformsit into a 2D image Along with this transformation the sizeof the actual object is also changed The magnification of theimage is given by the equation m = sprime

s

1676

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 8 (a) The use of orthogonal projection to determine thereflected ray (b) Rays being reflected off the surface of the weldpool The concave shaped weld pool surface converges the reflectedrays this can be visually observed in the figure

Figure 9 Camera modelled as a convex lens and a screen for theformation of the image

A camera is a measurement device which uses theproperties of light and object reflectance to sense the world andimages can be defined as the record of energy (light) emittedfrom a particular scene in the world Surfaces can be thoughtof as sources of light which radiate energy The radiance ofthe surface is a function of the light falling on it and is given by

L = d2φ

dAf dw(8)

where φ is the power of light source illuminating the surfaceAf is the foreshortened area where radiance is being measured

Figure 10 Use of two rays one parallel to the optical axis and theother through the optical centre to find the location of the imageformed

(a)

(b)

(c)

Figure 11 (a) Laser rays reflected off the weld pool surface(b) The rays converge after passing through the convex lens(c) Image formed on the screen after converging through the lens

w is the angle at which the light is radiating from the surfaceand L is the power per unit foreshortened area emitted intoa solid angle The camera relates this radiance of a surfaceto the irradiance on the image plane Irradiance is defined asthe power per unit area falling on a surface or in other wordsthe amount of light falling on the plane If the surface is notradiating any light there will be no irradiance on to the imageplane in the camera and therefore no image will be capturedFor our application the reflection of the laser light can bethought of as the radiance of the weld pool surface If the lightreflected from the surface of the weld pool is not captured bythe camera then no image will be formed

1677

G Saeed and Y M Zhang

10

ndash10

8

0

6

4

2

0

ndash02

8 6 4 2 0 ndash2 ndash4 ndash6 ndash8

(a)

(b)

Figure 12 (a) Simulation run with the electrode in the scene and the laser placed such that no shadows are formed (b) Some laser isreflected into the electrode while some clear the electrode and are captured by the camera Coordinate units millimetres

In the simulation based on the judgment of the reflectedrays the camera and screen are placed on the desiredcoordinates and the image captured on the screen To findthe convergence of the reflected ray once it passes throughthe convex lens two more lines are imagined to come fromthe same point of reflection One ray is drawn parallel to theoptical axis of the convex lens which passes through the focalpoint after passing through the convex lens and the other rayis drawn to pass through the optical centre un-refracted Thepoint of intersection of these two rays behind the convex lenswill be the point where the reflected ray emerging from thesame point as the rays will converge as shown in figure 10

Figure 11(a) shows the reflection of the laser from the weldpool surface Figure 11(b) shows the placement of the cameraconvex lens in the simulation to capture the image Figure 11(c)shows the formation of image on the screen placed at 14 of thefocal length of the convex length The distance of the screenfrom the convex can be adjusted and hence the focusing ofthe camera changed A focused image is formed on a screen

placed at a distance of the focal length from the convex Thegoal of the simulation is to maximize the rays intercepted bythe camera

6 Simulation of measurement system

61 Determining the system parameters

The simulation can be run step by step to determine eachparameter individually For example first the model of laserweld pool surface and the electrode will be used to determinethe position of the laser Several trials can be made in theplacement of laser to determine the exact co-ordinates suchthat the laser does not hit the electrode and to ensure thatthe laser will cover the entire surface of the weld pool Theelectrode height can be adjusted to the desired value (typicallybetween 3 and 6 mm in G-GTAW) and the simulation run tofind the best location of the laser In figure 12(a) the simulation

1678

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 13 (a) Simulation run with a 3 times 3 grid placed before thelaser beam Each grid acts as a new source of the laser and the laserdiverges onto the surface and reflects (b) Simulation re-run withthe laser placed at a different location and angle notice that morerays are captured by the camera compared to figure 13(a)rsquosplacement of grid and camera

was run with the electrode at the height of 5 mm and the bestposition of the laser was determined to be (0 6 5) where 0will be the real world x-coordinate 6 mm is the y-coordinateand 5 mm is the z-coordinate

With the location of the laser determined the simulationcan be used to check the reflection of the laser rays at thatdesired location The reflected rays are drawn and traced to seeif the rays that pass the lens contain the weld pool informationFigure 12(b) shows a possibility where the rays reflecting offthe end of the weld pool surface are captured by the camerawhereas the rays in the centre of the weld pool and those closerto the laser hit the electrode after reflection and are not capturedby the camera This might suggest changing the laser locationso that the angle between the laser rays and the xndashy plane isreduced by lowering the height of the laser Decreasing theangle between the laser rays and the xndashy plane would make thereflection have a smaller angle between the xndashy plane (basedon the law of reflection) and hence make it possible for therays to clear the electrode

62 Simulation of measurement system

In the proposed measurement optical grids are used to formdifferent patterns The distortion of the patterns due to specular

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

(a)

(b)

(c)

Figure 14 (a) Image acquired of a flat shiny surface through thesimulation of the model developed (b) Image after joining the dots(c) Actual image of the flat shiny surface

reflection off the surface is used to determine the surface of theweld pool For example an opaque grid with ntimesn holes can beplaced between the laser and the weld pool The grid will blockmost of the laser allowing the laser to pass through the holesonly The laser which passes through the grid re-emits afterpassing through the frosted glass and diverges The diameterof the grid hole will be the new waist diameter D Equation (4)from section 22 can be used to calculate the divergence angle

Figure 13 shows the simulation with a 3 times 3 grid placedbetween the laser and weld piece The grid is an array of smallholes equi-distant from each other arranged in a rectangularpattern Figure 13(a) demonstrates that in some cases thereflected rays are scattered and only a few of them are captured

1679

G Saeed and Y M Zhang

05

04

03

02

01

0

ndash01

ndash02

ndash03ndash77 ndash76 ndash75 ndash74 ndash73 ndash72 ndash71 ndash7 ndash69 ndash68 ndash67

(a)

(b)

Figure 15 (a) Actual weld pool image (b) Image acquired throughthe simulation of the model developed

by the imaging system In this simulation run reflection forthe third row of the grid was not captured by the cameraFigure 13(b) shows a better placement of grid and camera Inthis simulation run the camera was placed a little further awayfrom the work piece to allow more rays to be captured by thecamera The image captured at the coordinates of the secondsimulation run will contain more information Sometimesthe rays undergo double reflections from the surface of theweld pool These lsquosecondrsquo reflections can be detected by thesimulation but they never make it to the imaging system dueto the angles they have after reflection

7 Results

To test the developed model the laser stripe was projected ontoa flat smooth surface and the camera used to observe the imageThe simulation result is displayed in figure 14(a) The surfacewas flat and reflecting and therefore the image is the sameas the grid pattern Since the simulation assumes a discretelaser source the dots on the simulated results are the reflectedlaser The result after joining the laser stripes and marking theboundary is shown in figure 14(b) and the actual picture froma camera is shown in figure 14(c) The simulation results and

-77 -76 -75 -74 -73 -72 -71 -7 -69 -68 -67-03

-02

-01

0

01

02

03

04

05

Figure 16 Simulation results with traced boundary and joined dotsto form stripes

the actual camera results show resemblance in that they bothcapture the continuous horizontal laser stripes as predicted

In an attempt to capture the weld pool image using thesystem described above the laser was projected onto the weldpool and a striped grid placed before the laser so that straightlines of laser fell upon the weld pool The camera was used toobserve the image The image with a striped grid yielded theresults shown in figure 15(a) The image was characterizedby clear boundaries and distorted laser stripe pattern Thesmall black projection on the top of the image is the shadowingcaused by the electrode tip

When the simulation was run with the striped grid placedbefore the laser to simulate the experimental set-up of the labpositive results were yielded which were exceptionally closeto the original image The simulation results are shown infigure 15(b) Since the simulation assumes a discrete lasersource the points on the simulated results are the reflected laserdots The result after joining the laser stripes and marking theboundary is shown in figure 16 The boundary of the weldpool is very prominent in the simulation result and the laserstripes are also distorted due to the shape of the weld poolas predicted from the real weld pool pictures However sincethe shape of the weld pool cannot be precisely controlled ordetermined during the experiments the simulation results willbe close to the experimental results but not exactly the samedue to the variation in the weld pool shape

In another simulation run the laser was projected at adifferent angle and the focusing of the camera changed Theset-up is shown in figure 17(a) The corresponding image onthe screen is shown in figure 17(b) After joining the dottedlaser points and marking the boundary the result is illustratedin figure 17(c) The image bears a close resemblance to theoriginal image with clear boundaries and laser stripes alongthe weld pool surface

8 Conclusions and future work

The models developed will assist future research in observingthe weld pool They can be used to predict the outcome ofexperiments with grids placed in front of the laser and to

1680

Mathematical formulation and simulation of specular reflection based measurement system

-03 -02 -01 0 01 02 03 04 -03 -02 -01 0 01 02 0325

3

35

4

45

5

25

3

35

4

45

5

(b)

(a)

(c)

Figure 17 (a) The set-up of a simulation run showing the laser stripes reflected laser camera and imaging plane (b) The result of thesimulation run (c) Simulation results after the boundary is traced and the points are joined to form stripes

determine the position where the camera should be placedto acquire the best image Preliminary results show that thecamera should be placed with the weld pool along the opticalaxis and the aperture should be as large as possible to allowas many rays into the camera as possible The model can beused to find the optimal location of the laser and camera formaterials of different thickness by moving the electrode higherin the simulation and adjusting the laser and camera locationaccordingly

This is a first step towards gaining an insight into theproposed measurement method for the three-dimensionalsurface of a weld pool The goal is to fully understandand mathematically model the proposed measurement system

The set of mathematical models obtained in this paper arecertainly subject to possible modification and revision Usingthe set of revised mathematical models the parameters of themeasurement system will be able to be further optimized andthe mathematical foundations will be able to be derived tocalculate the three-dimensional specular surface of the weldpool from the image captured using the proposed system

Acknowledgments

This work is funded by the National Science Foundation underGrant DMI-01144982 and the University of Kentucky Centerfor Manufacturing

1681

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682

G Saeed and Y M Zhang

(a)

(b)

Figure 7 Laser diverging from the source point onto the weld pool area (a) Two-dimensional view from the yndashz plane(b) Three-dimensional view Coordinate units millimetres

The simulation can be used to locate the position of thecamera where the highest quality of image can be acquiredThis location will be where most of the rays are reflected afterhitting the weld pool surface The goal should be to interceptas many rays as possible through the aperture of the cameraThe wider the opening of the aperture the more light entersinto the camera

In the simulation after the reflected rays are drawn thecamera is placed at the point where most rays seems toconverge and the screen is placed at the focal point on the lensThe rays after converging through the convex lens form an

image on the screen The distance of the object and the imageare related by the following equation

1

s+

1

s prime = 1

f(7)

where s = distance of the object from lens s prime = distance ofthe image from the lens and f is the focal length of the cameralens The camera captures the 3D world scene and transformsit into a 2D image Along with this transformation the sizeof the actual object is also changed The magnification of theimage is given by the equation m = sprime

s

1676

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 8 (a) The use of orthogonal projection to determine thereflected ray (b) Rays being reflected off the surface of the weldpool The concave shaped weld pool surface converges the reflectedrays this can be visually observed in the figure

Figure 9 Camera modelled as a convex lens and a screen for theformation of the image

A camera is a measurement device which uses theproperties of light and object reflectance to sense the world andimages can be defined as the record of energy (light) emittedfrom a particular scene in the world Surfaces can be thoughtof as sources of light which radiate energy The radiance ofthe surface is a function of the light falling on it and is given by

L = d2φ

dAf dw(8)

where φ is the power of light source illuminating the surfaceAf is the foreshortened area where radiance is being measured

Figure 10 Use of two rays one parallel to the optical axis and theother through the optical centre to find the location of the imageformed

(a)

(b)

(c)

Figure 11 (a) Laser rays reflected off the weld pool surface(b) The rays converge after passing through the convex lens(c) Image formed on the screen after converging through the lens

w is the angle at which the light is radiating from the surfaceand L is the power per unit foreshortened area emitted intoa solid angle The camera relates this radiance of a surfaceto the irradiance on the image plane Irradiance is defined asthe power per unit area falling on a surface or in other wordsthe amount of light falling on the plane If the surface is notradiating any light there will be no irradiance on to the imageplane in the camera and therefore no image will be capturedFor our application the reflection of the laser light can bethought of as the radiance of the weld pool surface If the lightreflected from the surface of the weld pool is not captured bythe camera then no image will be formed

1677

G Saeed and Y M Zhang

10

ndash10

8

0

6

4

2

0

ndash02

8 6 4 2 0 ndash2 ndash4 ndash6 ndash8

(a)

(b)

Figure 12 (a) Simulation run with the electrode in the scene and the laser placed such that no shadows are formed (b) Some laser isreflected into the electrode while some clear the electrode and are captured by the camera Coordinate units millimetres

In the simulation based on the judgment of the reflectedrays the camera and screen are placed on the desiredcoordinates and the image captured on the screen To findthe convergence of the reflected ray once it passes throughthe convex lens two more lines are imagined to come fromthe same point of reflection One ray is drawn parallel to theoptical axis of the convex lens which passes through the focalpoint after passing through the convex lens and the other rayis drawn to pass through the optical centre un-refracted Thepoint of intersection of these two rays behind the convex lenswill be the point where the reflected ray emerging from thesame point as the rays will converge as shown in figure 10

Figure 11(a) shows the reflection of the laser from the weldpool surface Figure 11(b) shows the placement of the cameraconvex lens in the simulation to capture the image Figure 11(c)shows the formation of image on the screen placed at 14 of thefocal length of the convex length The distance of the screenfrom the convex can be adjusted and hence the focusing ofthe camera changed A focused image is formed on a screen

placed at a distance of the focal length from the convex Thegoal of the simulation is to maximize the rays intercepted bythe camera

6 Simulation of measurement system

61 Determining the system parameters

The simulation can be run step by step to determine eachparameter individually For example first the model of laserweld pool surface and the electrode will be used to determinethe position of the laser Several trials can be made in theplacement of laser to determine the exact co-ordinates suchthat the laser does not hit the electrode and to ensure thatthe laser will cover the entire surface of the weld pool Theelectrode height can be adjusted to the desired value (typicallybetween 3 and 6 mm in G-GTAW) and the simulation run tofind the best location of the laser In figure 12(a) the simulation

1678

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 13 (a) Simulation run with a 3 times 3 grid placed before thelaser beam Each grid acts as a new source of the laser and the laserdiverges onto the surface and reflects (b) Simulation re-run withthe laser placed at a different location and angle notice that morerays are captured by the camera compared to figure 13(a)rsquosplacement of grid and camera

was run with the electrode at the height of 5 mm and the bestposition of the laser was determined to be (0 6 5) where 0will be the real world x-coordinate 6 mm is the y-coordinateand 5 mm is the z-coordinate

With the location of the laser determined the simulationcan be used to check the reflection of the laser rays at thatdesired location The reflected rays are drawn and traced to seeif the rays that pass the lens contain the weld pool informationFigure 12(b) shows a possibility where the rays reflecting offthe end of the weld pool surface are captured by the camerawhereas the rays in the centre of the weld pool and those closerto the laser hit the electrode after reflection and are not capturedby the camera This might suggest changing the laser locationso that the angle between the laser rays and the xndashy plane isreduced by lowering the height of the laser Decreasing theangle between the laser rays and the xndashy plane would make thereflection have a smaller angle between the xndashy plane (basedon the law of reflection) and hence make it possible for therays to clear the electrode

62 Simulation of measurement system

In the proposed measurement optical grids are used to formdifferent patterns The distortion of the patterns due to specular

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

(a)

(b)

(c)

Figure 14 (a) Image acquired of a flat shiny surface through thesimulation of the model developed (b) Image after joining the dots(c) Actual image of the flat shiny surface

reflection off the surface is used to determine the surface of theweld pool For example an opaque grid with ntimesn holes can beplaced between the laser and the weld pool The grid will blockmost of the laser allowing the laser to pass through the holesonly The laser which passes through the grid re-emits afterpassing through the frosted glass and diverges The diameterof the grid hole will be the new waist diameter D Equation (4)from section 22 can be used to calculate the divergence angle

Figure 13 shows the simulation with a 3 times 3 grid placedbetween the laser and weld piece The grid is an array of smallholes equi-distant from each other arranged in a rectangularpattern Figure 13(a) demonstrates that in some cases thereflected rays are scattered and only a few of them are captured

1679

G Saeed and Y M Zhang

05

04

03

02

01

0

ndash01

ndash02

ndash03ndash77 ndash76 ndash75 ndash74 ndash73 ndash72 ndash71 ndash7 ndash69 ndash68 ndash67

(a)

(b)

Figure 15 (a) Actual weld pool image (b) Image acquired throughthe simulation of the model developed

by the imaging system In this simulation run reflection forthe third row of the grid was not captured by the cameraFigure 13(b) shows a better placement of grid and camera Inthis simulation run the camera was placed a little further awayfrom the work piece to allow more rays to be captured by thecamera The image captured at the coordinates of the secondsimulation run will contain more information Sometimesthe rays undergo double reflections from the surface of theweld pool These lsquosecondrsquo reflections can be detected by thesimulation but they never make it to the imaging system dueto the angles they have after reflection

7 Results

To test the developed model the laser stripe was projected ontoa flat smooth surface and the camera used to observe the imageThe simulation result is displayed in figure 14(a) The surfacewas flat and reflecting and therefore the image is the sameas the grid pattern Since the simulation assumes a discretelaser source the dots on the simulated results are the reflectedlaser The result after joining the laser stripes and marking theboundary is shown in figure 14(b) and the actual picture froma camera is shown in figure 14(c) The simulation results and

-77 -76 -75 -74 -73 -72 -71 -7 -69 -68 -67-03

-02

-01

0

01

02

03

04

05

Figure 16 Simulation results with traced boundary and joined dotsto form stripes

the actual camera results show resemblance in that they bothcapture the continuous horizontal laser stripes as predicted

In an attempt to capture the weld pool image using thesystem described above the laser was projected onto the weldpool and a striped grid placed before the laser so that straightlines of laser fell upon the weld pool The camera was used toobserve the image The image with a striped grid yielded theresults shown in figure 15(a) The image was characterizedby clear boundaries and distorted laser stripe pattern Thesmall black projection on the top of the image is the shadowingcaused by the electrode tip

When the simulation was run with the striped grid placedbefore the laser to simulate the experimental set-up of the labpositive results were yielded which were exceptionally closeto the original image The simulation results are shown infigure 15(b) Since the simulation assumes a discrete lasersource the points on the simulated results are the reflected laserdots The result after joining the laser stripes and marking theboundary is shown in figure 16 The boundary of the weldpool is very prominent in the simulation result and the laserstripes are also distorted due to the shape of the weld poolas predicted from the real weld pool pictures However sincethe shape of the weld pool cannot be precisely controlled ordetermined during the experiments the simulation results willbe close to the experimental results but not exactly the samedue to the variation in the weld pool shape

In another simulation run the laser was projected at adifferent angle and the focusing of the camera changed Theset-up is shown in figure 17(a) The corresponding image onthe screen is shown in figure 17(b) After joining the dottedlaser points and marking the boundary the result is illustratedin figure 17(c) The image bears a close resemblance to theoriginal image with clear boundaries and laser stripes alongthe weld pool surface

8 Conclusions and future work

The models developed will assist future research in observingthe weld pool They can be used to predict the outcome ofexperiments with grids placed in front of the laser and to

1680

Mathematical formulation and simulation of specular reflection based measurement system

-03 -02 -01 0 01 02 03 04 -03 -02 -01 0 01 02 0325

3

35

4

45

5

25

3

35

4

45

5

(b)

(a)

(c)

Figure 17 (a) The set-up of a simulation run showing the laser stripes reflected laser camera and imaging plane (b) The result of thesimulation run (c) Simulation results after the boundary is traced and the points are joined to form stripes

determine the position where the camera should be placedto acquire the best image Preliminary results show that thecamera should be placed with the weld pool along the opticalaxis and the aperture should be as large as possible to allowas many rays into the camera as possible The model can beused to find the optimal location of the laser and camera formaterials of different thickness by moving the electrode higherin the simulation and adjusting the laser and camera locationaccordingly

This is a first step towards gaining an insight into theproposed measurement method for the three-dimensionalsurface of a weld pool The goal is to fully understandand mathematically model the proposed measurement system

The set of mathematical models obtained in this paper arecertainly subject to possible modification and revision Usingthe set of revised mathematical models the parameters of themeasurement system will be able to be further optimized andthe mathematical foundations will be able to be derived tocalculate the three-dimensional specular surface of the weldpool from the image captured using the proposed system

Acknowledgments

This work is funded by the National Science Foundation underGrant DMI-01144982 and the University of Kentucky Centerfor Manufacturing

1681

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 8 (a) The use of orthogonal projection to determine thereflected ray (b) Rays being reflected off the surface of the weldpool The concave shaped weld pool surface converges the reflectedrays this can be visually observed in the figure

Figure 9 Camera modelled as a convex lens and a screen for theformation of the image

A camera is a measurement device which uses theproperties of light and object reflectance to sense the world andimages can be defined as the record of energy (light) emittedfrom a particular scene in the world Surfaces can be thoughtof as sources of light which radiate energy The radiance ofthe surface is a function of the light falling on it and is given by

L = d2φ

dAf dw(8)

where φ is the power of light source illuminating the surfaceAf is the foreshortened area where radiance is being measured

Figure 10 Use of two rays one parallel to the optical axis and theother through the optical centre to find the location of the imageformed

(a)

(b)

(c)

Figure 11 (a) Laser rays reflected off the weld pool surface(b) The rays converge after passing through the convex lens(c) Image formed on the screen after converging through the lens

w is the angle at which the light is radiating from the surfaceand L is the power per unit foreshortened area emitted intoa solid angle The camera relates this radiance of a surfaceto the irradiance on the image plane Irradiance is defined asthe power per unit area falling on a surface or in other wordsthe amount of light falling on the plane If the surface is notradiating any light there will be no irradiance on to the imageplane in the camera and therefore no image will be capturedFor our application the reflection of the laser light can bethought of as the radiance of the weld pool surface If the lightreflected from the surface of the weld pool is not captured bythe camera then no image will be formed

1677

G Saeed and Y M Zhang

10

ndash10

8

0

6

4

2

0

ndash02

8 6 4 2 0 ndash2 ndash4 ndash6 ndash8

(a)

(b)

Figure 12 (a) Simulation run with the electrode in the scene and the laser placed such that no shadows are formed (b) Some laser isreflected into the electrode while some clear the electrode and are captured by the camera Coordinate units millimetres

In the simulation based on the judgment of the reflectedrays the camera and screen are placed on the desiredcoordinates and the image captured on the screen To findthe convergence of the reflected ray once it passes throughthe convex lens two more lines are imagined to come fromthe same point of reflection One ray is drawn parallel to theoptical axis of the convex lens which passes through the focalpoint after passing through the convex lens and the other rayis drawn to pass through the optical centre un-refracted Thepoint of intersection of these two rays behind the convex lenswill be the point where the reflected ray emerging from thesame point as the rays will converge as shown in figure 10

Figure 11(a) shows the reflection of the laser from the weldpool surface Figure 11(b) shows the placement of the cameraconvex lens in the simulation to capture the image Figure 11(c)shows the formation of image on the screen placed at 14 of thefocal length of the convex length The distance of the screenfrom the convex can be adjusted and hence the focusing ofthe camera changed A focused image is formed on a screen

placed at a distance of the focal length from the convex Thegoal of the simulation is to maximize the rays intercepted bythe camera

6 Simulation of measurement system

61 Determining the system parameters

The simulation can be run step by step to determine eachparameter individually For example first the model of laserweld pool surface and the electrode will be used to determinethe position of the laser Several trials can be made in theplacement of laser to determine the exact co-ordinates suchthat the laser does not hit the electrode and to ensure thatthe laser will cover the entire surface of the weld pool Theelectrode height can be adjusted to the desired value (typicallybetween 3 and 6 mm in G-GTAW) and the simulation run tofind the best location of the laser In figure 12(a) the simulation

1678

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 13 (a) Simulation run with a 3 times 3 grid placed before thelaser beam Each grid acts as a new source of the laser and the laserdiverges onto the surface and reflects (b) Simulation re-run withthe laser placed at a different location and angle notice that morerays are captured by the camera compared to figure 13(a)rsquosplacement of grid and camera

was run with the electrode at the height of 5 mm and the bestposition of the laser was determined to be (0 6 5) where 0will be the real world x-coordinate 6 mm is the y-coordinateand 5 mm is the z-coordinate

With the location of the laser determined the simulationcan be used to check the reflection of the laser rays at thatdesired location The reflected rays are drawn and traced to seeif the rays that pass the lens contain the weld pool informationFigure 12(b) shows a possibility where the rays reflecting offthe end of the weld pool surface are captured by the camerawhereas the rays in the centre of the weld pool and those closerto the laser hit the electrode after reflection and are not capturedby the camera This might suggest changing the laser locationso that the angle between the laser rays and the xndashy plane isreduced by lowering the height of the laser Decreasing theangle between the laser rays and the xndashy plane would make thereflection have a smaller angle between the xndashy plane (basedon the law of reflection) and hence make it possible for therays to clear the electrode

62 Simulation of measurement system

In the proposed measurement optical grids are used to formdifferent patterns The distortion of the patterns due to specular

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

(a)

(b)

(c)

Figure 14 (a) Image acquired of a flat shiny surface through thesimulation of the model developed (b) Image after joining the dots(c) Actual image of the flat shiny surface

reflection off the surface is used to determine the surface of theweld pool For example an opaque grid with ntimesn holes can beplaced between the laser and the weld pool The grid will blockmost of the laser allowing the laser to pass through the holesonly The laser which passes through the grid re-emits afterpassing through the frosted glass and diverges The diameterof the grid hole will be the new waist diameter D Equation (4)from section 22 can be used to calculate the divergence angle

Figure 13 shows the simulation with a 3 times 3 grid placedbetween the laser and weld piece The grid is an array of smallholes equi-distant from each other arranged in a rectangularpattern Figure 13(a) demonstrates that in some cases thereflected rays are scattered and only a few of them are captured

1679

G Saeed and Y M Zhang

05

04

03

02

01

0

ndash01

ndash02

ndash03ndash77 ndash76 ndash75 ndash74 ndash73 ndash72 ndash71 ndash7 ndash69 ndash68 ndash67

(a)

(b)

Figure 15 (a) Actual weld pool image (b) Image acquired throughthe simulation of the model developed

by the imaging system In this simulation run reflection forthe third row of the grid was not captured by the cameraFigure 13(b) shows a better placement of grid and camera Inthis simulation run the camera was placed a little further awayfrom the work piece to allow more rays to be captured by thecamera The image captured at the coordinates of the secondsimulation run will contain more information Sometimesthe rays undergo double reflections from the surface of theweld pool These lsquosecondrsquo reflections can be detected by thesimulation but they never make it to the imaging system dueto the angles they have after reflection

7 Results

To test the developed model the laser stripe was projected ontoa flat smooth surface and the camera used to observe the imageThe simulation result is displayed in figure 14(a) The surfacewas flat and reflecting and therefore the image is the sameas the grid pattern Since the simulation assumes a discretelaser source the dots on the simulated results are the reflectedlaser The result after joining the laser stripes and marking theboundary is shown in figure 14(b) and the actual picture froma camera is shown in figure 14(c) The simulation results and

-77 -76 -75 -74 -73 -72 -71 -7 -69 -68 -67-03

-02

-01

0

01

02

03

04

05

Figure 16 Simulation results with traced boundary and joined dotsto form stripes

the actual camera results show resemblance in that they bothcapture the continuous horizontal laser stripes as predicted

In an attempt to capture the weld pool image using thesystem described above the laser was projected onto the weldpool and a striped grid placed before the laser so that straightlines of laser fell upon the weld pool The camera was used toobserve the image The image with a striped grid yielded theresults shown in figure 15(a) The image was characterizedby clear boundaries and distorted laser stripe pattern Thesmall black projection on the top of the image is the shadowingcaused by the electrode tip

When the simulation was run with the striped grid placedbefore the laser to simulate the experimental set-up of the labpositive results were yielded which were exceptionally closeto the original image The simulation results are shown infigure 15(b) Since the simulation assumes a discrete lasersource the points on the simulated results are the reflected laserdots The result after joining the laser stripes and marking theboundary is shown in figure 16 The boundary of the weldpool is very prominent in the simulation result and the laserstripes are also distorted due to the shape of the weld poolas predicted from the real weld pool pictures However sincethe shape of the weld pool cannot be precisely controlled ordetermined during the experiments the simulation results willbe close to the experimental results but not exactly the samedue to the variation in the weld pool shape

In another simulation run the laser was projected at adifferent angle and the focusing of the camera changed Theset-up is shown in figure 17(a) The corresponding image onthe screen is shown in figure 17(b) After joining the dottedlaser points and marking the boundary the result is illustratedin figure 17(c) The image bears a close resemblance to theoriginal image with clear boundaries and laser stripes alongthe weld pool surface

8 Conclusions and future work

The models developed will assist future research in observingthe weld pool They can be used to predict the outcome ofexperiments with grids placed in front of the laser and to

1680

Mathematical formulation and simulation of specular reflection based measurement system

-03 -02 -01 0 01 02 03 04 -03 -02 -01 0 01 02 0325

3

35

4

45

5

25

3

35

4

45

5

(b)

(a)

(c)

Figure 17 (a) The set-up of a simulation run showing the laser stripes reflected laser camera and imaging plane (b) The result of thesimulation run (c) Simulation results after the boundary is traced and the points are joined to form stripes

determine the position where the camera should be placedto acquire the best image Preliminary results show that thecamera should be placed with the weld pool along the opticalaxis and the aperture should be as large as possible to allowas many rays into the camera as possible The model can beused to find the optimal location of the laser and camera formaterials of different thickness by moving the electrode higherin the simulation and adjusting the laser and camera locationaccordingly

This is a first step towards gaining an insight into theproposed measurement method for the three-dimensionalsurface of a weld pool The goal is to fully understandand mathematically model the proposed measurement system

The set of mathematical models obtained in this paper arecertainly subject to possible modification and revision Usingthe set of revised mathematical models the parameters of themeasurement system will be able to be further optimized andthe mathematical foundations will be able to be derived tocalculate the three-dimensional specular surface of the weldpool from the image captured using the proposed system

Acknowledgments

This work is funded by the National Science Foundation underGrant DMI-01144982 and the University of Kentucky Centerfor Manufacturing

1681

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682

G Saeed and Y M Zhang

10

ndash10

8

0

6

4

2

0

ndash02

8 6 4 2 0 ndash2 ndash4 ndash6 ndash8

(a)

(b)

Figure 12 (a) Simulation run with the electrode in the scene and the laser placed such that no shadows are formed (b) Some laser isreflected into the electrode while some clear the electrode and are captured by the camera Coordinate units millimetres

In the simulation based on the judgment of the reflectedrays the camera and screen are placed on the desiredcoordinates and the image captured on the screen To findthe convergence of the reflected ray once it passes throughthe convex lens two more lines are imagined to come fromthe same point of reflection One ray is drawn parallel to theoptical axis of the convex lens which passes through the focalpoint after passing through the convex lens and the other rayis drawn to pass through the optical centre un-refracted Thepoint of intersection of these two rays behind the convex lenswill be the point where the reflected ray emerging from thesame point as the rays will converge as shown in figure 10

Figure 11(a) shows the reflection of the laser from the weldpool surface Figure 11(b) shows the placement of the cameraconvex lens in the simulation to capture the image Figure 11(c)shows the formation of image on the screen placed at 14 of thefocal length of the convex length The distance of the screenfrom the convex can be adjusted and hence the focusing ofthe camera changed A focused image is formed on a screen

placed at a distance of the focal length from the convex Thegoal of the simulation is to maximize the rays intercepted bythe camera

6 Simulation of measurement system

61 Determining the system parameters

The simulation can be run step by step to determine eachparameter individually For example first the model of laserweld pool surface and the electrode will be used to determinethe position of the laser Several trials can be made in theplacement of laser to determine the exact co-ordinates suchthat the laser does not hit the electrode and to ensure thatthe laser will cover the entire surface of the weld pool Theelectrode height can be adjusted to the desired value (typicallybetween 3 and 6 mm in G-GTAW) and the simulation run tofind the best location of the laser In figure 12(a) the simulation

1678

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 13 (a) Simulation run with a 3 times 3 grid placed before thelaser beam Each grid acts as a new source of the laser and the laserdiverges onto the surface and reflects (b) Simulation re-run withthe laser placed at a different location and angle notice that morerays are captured by the camera compared to figure 13(a)rsquosplacement of grid and camera

was run with the electrode at the height of 5 mm and the bestposition of the laser was determined to be (0 6 5) where 0will be the real world x-coordinate 6 mm is the y-coordinateand 5 mm is the z-coordinate

With the location of the laser determined the simulationcan be used to check the reflection of the laser rays at thatdesired location The reflected rays are drawn and traced to seeif the rays that pass the lens contain the weld pool informationFigure 12(b) shows a possibility where the rays reflecting offthe end of the weld pool surface are captured by the camerawhereas the rays in the centre of the weld pool and those closerto the laser hit the electrode after reflection and are not capturedby the camera This might suggest changing the laser locationso that the angle between the laser rays and the xndashy plane isreduced by lowering the height of the laser Decreasing theangle between the laser rays and the xndashy plane would make thereflection have a smaller angle between the xndashy plane (basedon the law of reflection) and hence make it possible for therays to clear the electrode

62 Simulation of measurement system

In the proposed measurement optical grids are used to formdifferent patterns The distortion of the patterns due to specular

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

(a)

(b)

(c)

Figure 14 (a) Image acquired of a flat shiny surface through thesimulation of the model developed (b) Image after joining the dots(c) Actual image of the flat shiny surface

reflection off the surface is used to determine the surface of theweld pool For example an opaque grid with ntimesn holes can beplaced between the laser and the weld pool The grid will blockmost of the laser allowing the laser to pass through the holesonly The laser which passes through the grid re-emits afterpassing through the frosted glass and diverges The diameterof the grid hole will be the new waist diameter D Equation (4)from section 22 can be used to calculate the divergence angle

Figure 13 shows the simulation with a 3 times 3 grid placedbetween the laser and weld piece The grid is an array of smallholes equi-distant from each other arranged in a rectangularpattern Figure 13(a) demonstrates that in some cases thereflected rays are scattered and only a few of them are captured

1679

G Saeed and Y M Zhang

05

04

03

02

01

0

ndash01

ndash02

ndash03ndash77 ndash76 ndash75 ndash74 ndash73 ndash72 ndash71 ndash7 ndash69 ndash68 ndash67

(a)

(b)

Figure 15 (a) Actual weld pool image (b) Image acquired throughthe simulation of the model developed

by the imaging system In this simulation run reflection forthe third row of the grid was not captured by the cameraFigure 13(b) shows a better placement of grid and camera Inthis simulation run the camera was placed a little further awayfrom the work piece to allow more rays to be captured by thecamera The image captured at the coordinates of the secondsimulation run will contain more information Sometimesthe rays undergo double reflections from the surface of theweld pool These lsquosecondrsquo reflections can be detected by thesimulation but they never make it to the imaging system dueto the angles they have after reflection

7 Results

To test the developed model the laser stripe was projected ontoa flat smooth surface and the camera used to observe the imageThe simulation result is displayed in figure 14(a) The surfacewas flat and reflecting and therefore the image is the sameas the grid pattern Since the simulation assumes a discretelaser source the dots on the simulated results are the reflectedlaser The result after joining the laser stripes and marking theboundary is shown in figure 14(b) and the actual picture froma camera is shown in figure 14(c) The simulation results and

-77 -76 -75 -74 -73 -72 -71 -7 -69 -68 -67-03

-02

-01

0

01

02

03

04

05

Figure 16 Simulation results with traced boundary and joined dotsto form stripes

the actual camera results show resemblance in that they bothcapture the continuous horizontal laser stripes as predicted

In an attempt to capture the weld pool image using thesystem described above the laser was projected onto the weldpool and a striped grid placed before the laser so that straightlines of laser fell upon the weld pool The camera was used toobserve the image The image with a striped grid yielded theresults shown in figure 15(a) The image was characterizedby clear boundaries and distorted laser stripe pattern Thesmall black projection on the top of the image is the shadowingcaused by the electrode tip

When the simulation was run with the striped grid placedbefore the laser to simulate the experimental set-up of the labpositive results were yielded which were exceptionally closeto the original image The simulation results are shown infigure 15(b) Since the simulation assumes a discrete lasersource the points on the simulated results are the reflected laserdots The result after joining the laser stripes and marking theboundary is shown in figure 16 The boundary of the weldpool is very prominent in the simulation result and the laserstripes are also distorted due to the shape of the weld poolas predicted from the real weld pool pictures However sincethe shape of the weld pool cannot be precisely controlled ordetermined during the experiments the simulation results willbe close to the experimental results but not exactly the samedue to the variation in the weld pool shape

In another simulation run the laser was projected at adifferent angle and the focusing of the camera changed Theset-up is shown in figure 17(a) The corresponding image onthe screen is shown in figure 17(b) After joining the dottedlaser points and marking the boundary the result is illustratedin figure 17(c) The image bears a close resemblance to theoriginal image with clear boundaries and laser stripes alongthe weld pool surface

8 Conclusions and future work

The models developed will assist future research in observingthe weld pool They can be used to predict the outcome ofexperiments with grids placed in front of the laser and to

1680

Mathematical formulation and simulation of specular reflection based measurement system

-03 -02 -01 0 01 02 03 04 -03 -02 -01 0 01 02 0325

3

35

4

45

5

25

3

35

4

45

5

(b)

(a)

(c)

Figure 17 (a) The set-up of a simulation run showing the laser stripes reflected laser camera and imaging plane (b) The result of thesimulation run (c) Simulation results after the boundary is traced and the points are joined to form stripes

determine the position where the camera should be placedto acquire the best image Preliminary results show that thecamera should be placed with the weld pool along the opticalaxis and the aperture should be as large as possible to allowas many rays into the camera as possible The model can beused to find the optimal location of the laser and camera formaterials of different thickness by moving the electrode higherin the simulation and adjusting the laser and camera locationaccordingly

This is a first step towards gaining an insight into theproposed measurement method for the three-dimensionalsurface of a weld pool The goal is to fully understandand mathematically model the proposed measurement system

The set of mathematical models obtained in this paper arecertainly subject to possible modification and revision Usingthe set of revised mathematical models the parameters of themeasurement system will be able to be further optimized andthe mathematical foundations will be able to be derived tocalculate the three-dimensional specular surface of the weldpool from the image captured using the proposed system

Acknowledgments

This work is funded by the National Science Foundation underGrant DMI-01144982 and the University of Kentucky Centerfor Manufacturing

1681

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682

Mathematical formulation and simulation of specular reflection based measurement system

(a)

(b)

Figure 13 (a) Simulation run with a 3 times 3 grid placed before thelaser beam Each grid acts as a new source of the laser and the laserdiverges onto the surface and reflects (b) Simulation re-run withthe laser placed at a different location and angle notice that morerays are captured by the camera compared to figure 13(a)rsquosplacement of grid and camera

was run with the electrode at the height of 5 mm and the bestposition of the laser was determined to be (0 6 5) where 0will be the real world x-coordinate 6 mm is the y-coordinateand 5 mm is the z-coordinate

With the location of the laser determined the simulationcan be used to check the reflection of the laser rays at thatdesired location The reflected rays are drawn and traced to seeif the rays that pass the lens contain the weld pool informationFigure 12(b) shows a possibility where the rays reflecting offthe end of the weld pool surface are captured by the camerawhereas the rays in the centre of the weld pool and those closerto the laser hit the electrode after reflection and are not capturedby the camera This might suggest changing the laser locationso that the angle between the laser rays and the xndashy plane isreduced by lowering the height of the laser Decreasing theangle between the laser rays and the xndashy plane would make thereflection have a smaller angle between the xndashy plane (basedon the law of reflection) and hence make it possible for therays to clear the electrode

62 Simulation of measurement system

In the proposed measurement optical grids are used to formdifferent patterns The distortion of the patterns due to specular

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

04 03 02 01 0 ndash01 ndash02 ndash03 ndash04

56

54

52

5

48

46

44

42

4

38

36

(a)

(b)

(c)

Figure 14 (a) Image acquired of a flat shiny surface through thesimulation of the model developed (b) Image after joining the dots(c) Actual image of the flat shiny surface

reflection off the surface is used to determine the surface of theweld pool For example an opaque grid with ntimesn holes can beplaced between the laser and the weld pool The grid will blockmost of the laser allowing the laser to pass through the holesonly The laser which passes through the grid re-emits afterpassing through the frosted glass and diverges The diameterof the grid hole will be the new waist diameter D Equation (4)from section 22 can be used to calculate the divergence angle

Figure 13 shows the simulation with a 3 times 3 grid placedbetween the laser and weld piece The grid is an array of smallholes equi-distant from each other arranged in a rectangularpattern Figure 13(a) demonstrates that in some cases thereflected rays are scattered and only a few of them are captured

1679

G Saeed and Y M Zhang

05

04

03

02

01

0

ndash01

ndash02

ndash03ndash77 ndash76 ndash75 ndash74 ndash73 ndash72 ndash71 ndash7 ndash69 ndash68 ndash67

(a)

(b)

Figure 15 (a) Actual weld pool image (b) Image acquired throughthe simulation of the model developed

by the imaging system In this simulation run reflection forthe third row of the grid was not captured by the cameraFigure 13(b) shows a better placement of grid and camera Inthis simulation run the camera was placed a little further awayfrom the work piece to allow more rays to be captured by thecamera The image captured at the coordinates of the secondsimulation run will contain more information Sometimesthe rays undergo double reflections from the surface of theweld pool These lsquosecondrsquo reflections can be detected by thesimulation but they never make it to the imaging system dueto the angles they have after reflection

7 Results

To test the developed model the laser stripe was projected ontoa flat smooth surface and the camera used to observe the imageThe simulation result is displayed in figure 14(a) The surfacewas flat and reflecting and therefore the image is the sameas the grid pattern Since the simulation assumes a discretelaser source the dots on the simulated results are the reflectedlaser The result after joining the laser stripes and marking theboundary is shown in figure 14(b) and the actual picture froma camera is shown in figure 14(c) The simulation results and

-77 -76 -75 -74 -73 -72 -71 -7 -69 -68 -67-03

-02

-01

0

01

02

03

04

05

Figure 16 Simulation results with traced boundary and joined dotsto form stripes

the actual camera results show resemblance in that they bothcapture the continuous horizontal laser stripes as predicted

In an attempt to capture the weld pool image using thesystem described above the laser was projected onto the weldpool and a striped grid placed before the laser so that straightlines of laser fell upon the weld pool The camera was used toobserve the image The image with a striped grid yielded theresults shown in figure 15(a) The image was characterizedby clear boundaries and distorted laser stripe pattern Thesmall black projection on the top of the image is the shadowingcaused by the electrode tip

When the simulation was run with the striped grid placedbefore the laser to simulate the experimental set-up of the labpositive results were yielded which were exceptionally closeto the original image The simulation results are shown infigure 15(b) Since the simulation assumes a discrete lasersource the points on the simulated results are the reflected laserdots The result after joining the laser stripes and marking theboundary is shown in figure 16 The boundary of the weldpool is very prominent in the simulation result and the laserstripes are also distorted due to the shape of the weld poolas predicted from the real weld pool pictures However sincethe shape of the weld pool cannot be precisely controlled ordetermined during the experiments the simulation results willbe close to the experimental results but not exactly the samedue to the variation in the weld pool shape

In another simulation run the laser was projected at adifferent angle and the focusing of the camera changed Theset-up is shown in figure 17(a) The corresponding image onthe screen is shown in figure 17(b) After joining the dottedlaser points and marking the boundary the result is illustratedin figure 17(c) The image bears a close resemblance to theoriginal image with clear boundaries and laser stripes alongthe weld pool surface

8 Conclusions and future work

The models developed will assist future research in observingthe weld pool They can be used to predict the outcome ofexperiments with grids placed in front of the laser and to

1680

Mathematical formulation and simulation of specular reflection based measurement system

-03 -02 -01 0 01 02 03 04 -03 -02 -01 0 01 02 0325

3

35

4

45

5

25

3

35

4

45

5

(b)

(a)

(c)

Figure 17 (a) The set-up of a simulation run showing the laser stripes reflected laser camera and imaging plane (b) The result of thesimulation run (c) Simulation results after the boundary is traced and the points are joined to form stripes

determine the position where the camera should be placedto acquire the best image Preliminary results show that thecamera should be placed with the weld pool along the opticalaxis and the aperture should be as large as possible to allowas many rays into the camera as possible The model can beused to find the optimal location of the laser and camera formaterials of different thickness by moving the electrode higherin the simulation and adjusting the laser and camera locationaccordingly

This is a first step towards gaining an insight into theproposed measurement method for the three-dimensionalsurface of a weld pool The goal is to fully understandand mathematically model the proposed measurement system

The set of mathematical models obtained in this paper arecertainly subject to possible modification and revision Usingthe set of revised mathematical models the parameters of themeasurement system will be able to be further optimized andthe mathematical foundations will be able to be derived tocalculate the three-dimensional specular surface of the weldpool from the image captured using the proposed system

Acknowledgments

This work is funded by the National Science Foundation underGrant DMI-01144982 and the University of Kentucky Centerfor Manufacturing

1681

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682

G Saeed and Y M Zhang

05

04

03

02

01

0

ndash01

ndash02

ndash03ndash77 ndash76 ndash75 ndash74 ndash73 ndash72 ndash71 ndash7 ndash69 ndash68 ndash67

(a)

(b)

Figure 15 (a) Actual weld pool image (b) Image acquired throughthe simulation of the model developed

by the imaging system In this simulation run reflection forthe third row of the grid was not captured by the cameraFigure 13(b) shows a better placement of grid and camera Inthis simulation run the camera was placed a little further awayfrom the work piece to allow more rays to be captured by thecamera The image captured at the coordinates of the secondsimulation run will contain more information Sometimesthe rays undergo double reflections from the surface of theweld pool These lsquosecondrsquo reflections can be detected by thesimulation but they never make it to the imaging system dueto the angles they have after reflection

7 Results

To test the developed model the laser stripe was projected ontoa flat smooth surface and the camera used to observe the imageThe simulation result is displayed in figure 14(a) The surfacewas flat and reflecting and therefore the image is the sameas the grid pattern Since the simulation assumes a discretelaser source the dots on the simulated results are the reflectedlaser The result after joining the laser stripes and marking theboundary is shown in figure 14(b) and the actual picture froma camera is shown in figure 14(c) The simulation results and

-77 -76 -75 -74 -73 -72 -71 -7 -69 -68 -67-03

-02

-01

0

01

02

03

04

05

Figure 16 Simulation results with traced boundary and joined dotsto form stripes

the actual camera results show resemblance in that they bothcapture the continuous horizontal laser stripes as predicted

In an attempt to capture the weld pool image using thesystem described above the laser was projected onto the weldpool and a striped grid placed before the laser so that straightlines of laser fell upon the weld pool The camera was used toobserve the image The image with a striped grid yielded theresults shown in figure 15(a) The image was characterizedby clear boundaries and distorted laser stripe pattern Thesmall black projection on the top of the image is the shadowingcaused by the electrode tip

When the simulation was run with the striped grid placedbefore the laser to simulate the experimental set-up of the labpositive results were yielded which were exceptionally closeto the original image The simulation results are shown infigure 15(b) Since the simulation assumes a discrete lasersource the points on the simulated results are the reflected laserdots The result after joining the laser stripes and marking theboundary is shown in figure 16 The boundary of the weldpool is very prominent in the simulation result and the laserstripes are also distorted due to the shape of the weld poolas predicted from the real weld pool pictures However sincethe shape of the weld pool cannot be precisely controlled ordetermined during the experiments the simulation results willbe close to the experimental results but not exactly the samedue to the variation in the weld pool shape

In another simulation run the laser was projected at adifferent angle and the focusing of the camera changed Theset-up is shown in figure 17(a) The corresponding image onthe screen is shown in figure 17(b) After joining the dottedlaser points and marking the boundary the result is illustratedin figure 17(c) The image bears a close resemblance to theoriginal image with clear boundaries and laser stripes alongthe weld pool surface

8 Conclusions and future work

The models developed will assist future research in observingthe weld pool They can be used to predict the outcome ofexperiments with grids placed in front of the laser and to

1680

Mathematical formulation and simulation of specular reflection based measurement system

-03 -02 -01 0 01 02 03 04 -03 -02 -01 0 01 02 0325

3

35

4

45

5

25

3

35

4

45

5

(b)

(a)

(c)

Figure 17 (a) The set-up of a simulation run showing the laser stripes reflected laser camera and imaging plane (b) The result of thesimulation run (c) Simulation results after the boundary is traced and the points are joined to form stripes

determine the position where the camera should be placedto acquire the best image Preliminary results show that thecamera should be placed with the weld pool along the opticalaxis and the aperture should be as large as possible to allowas many rays into the camera as possible The model can beused to find the optimal location of the laser and camera formaterials of different thickness by moving the electrode higherin the simulation and adjusting the laser and camera locationaccordingly

This is a first step towards gaining an insight into theproposed measurement method for the three-dimensionalsurface of a weld pool The goal is to fully understandand mathematically model the proposed measurement system

The set of mathematical models obtained in this paper arecertainly subject to possible modification and revision Usingthe set of revised mathematical models the parameters of themeasurement system will be able to be further optimized andthe mathematical foundations will be able to be derived tocalculate the three-dimensional specular surface of the weldpool from the image captured using the proposed system

Acknowledgments

This work is funded by the National Science Foundation underGrant DMI-01144982 and the University of Kentucky Centerfor Manufacturing

1681

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682

Mathematical formulation and simulation of specular reflection based measurement system

-03 -02 -01 0 01 02 03 04 -03 -02 -01 0 01 02 0325

3

35

4

45

5

25

3

35

4

45

5

(b)

(a)

(c)

Figure 17 (a) The set-up of a simulation run showing the laser stripes reflected laser camera and imaging plane (b) The result of thesimulation run (c) Simulation results after the boundary is traced and the points are joined to form stripes

determine the position where the camera should be placedto acquire the best image Preliminary results show that thecamera should be placed with the weld pool along the opticalaxis and the aperture should be as large as possible to allowas many rays into the camera as possible The model can beused to find the optimal location of the laser and camera formaterials of different thickness by moving the electrode higherin the simulation and adjusting the laser and camera locationaccordingly

This is a first step towards gaining an insight into theproposed measurement method for the three-dimensionalsurface of a weld pool The goal is to fully understandand mathematically model the proposed measurement system

The set of mathematical models obtained in this paper arecertainly subject to possible modification and revision Usingthe set of revised mathematical models the parameters of themeasurement system will be able to be further optimized andthe mathematical foundations will be able to be derived tocalculate the three-dimensional specular surface of the weldpool from the image captured using the proposed system

Acknowledgments

This work is funded by the National Science Foundation underGrant DMI-01144982 and the University of Kentucky Centerfor Manufacturing

1681

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682

G Saeed and Y M Zhang

References

[1] Renwick R J and Richardson R W 1983 Experimentalinvestigation of GTA weld pool oscillations Weld J 6229sndash35s

[2] Zacksenhouse M and Hardt D E 1983 Weld pool impedanceidentification for size measurement and control ASME JDyn Syst Meas Control 105 179ndash84

[3] Xiao Y H and den Ouden G 1993 Weld pool oscillation duringGTA welding of mild steel Weld J 72 428sndash34s

[4] Nagarajan S Bangerjee P Chen W and Chin B A 1992Control of the welding process using infrared sensors IEEETrans Robot Autom 8 86ndash93

[5] Beardsley H E Zhang Y M and Kovacevic R 1994 Infraredsensing of full penetration state in gas tungsten arc weldingInt J Machine Tool Manuf 34 1079ndash90

[6] Carlson N M et al 1992 Ultrasonic NDT methods for weldsensing Mater Eval 50 1338ndash43

[7] Graham G M and Ume I C 1997 Automated system for laserultrasonic sensing of weld penetration Mechatronics 7711ndash21

[8] Hopko S N and Ume I C 1999 Laser generated ultrasound bymaterial ablation using fiber optic delivery Ultrasonics 371ndash7

[9] Kovacevic R and Zhang Y M 1996 Sensing free surface of arcweld pool using specular reflection principle and analysisProc Inst Mech Eng Part B J Eng Manuf 210 553ndash64

[10] Kovacevic R and Zhang Y M 1997 Real-time imageprocessing for monitoring of free weld pool surface ASMEJ Manuf Sci Eng 119 161ndash9

[11] Pedrotti F L and Pedrotti L S 1993 Introduction to Optics 2ndedn (Englewood Cliffs NJ Prentice-Hall)

[12] Pecharapa W and Kar A 1997 Effects of phase changes onweld pool shape in laser welding J Phys D Appl Phys 303322

[13] Lankalapall K N Tu J F and Gartner M 1996 A model forestimating penetration depth of laser welding processesJ Phys D Appl Phys 29 1831

[14] Stewart J 1999 Calculus 4th edn (Pacific Grove CABrooksCole Publishing Company)

1682