Metrology of Free-Form Shape Parts

8/10/2019 Metrology of Free-Form Shape Parts

1/26Annals of the CIRP Vol. 56/2/2007 -810- doi:10.1016/j.cirp.2007.10.008

Metrology of freeform shaped parts

E. Savio1(2), L. De Chiffre

2(1), R. Schmitt

3

1Laboratory of Industrial and Geometrical Metrology, DIMEG - University of Padova, Padova, Italy

2Centre for Geometrical Metrology, IPL Technical University of Denmark, Lyngby, Denmark

3Chair of Metrology and Quality Management, RWTH - Aachen University, Aachen, Germany

Abstract

The scope of this keynote paper is to present the state of the art in the metrology of freeform shapes withfocus on the freeform capabilities of the most important measuring techniques and on related metrologicalissues.Some examples of products are presented, for which the metrology of freeform shapes is important toguarantee the desired functional performance of the product. A classification of freeform measuring tasks andthe corresponding metrological requirements are presented. A review of the most important measuringtechniques is presented along with their capabilities for freeform measuring tasks. Specification andverification of freeform surfaces, including data evaluation and comparison to specifications are discussed,along with the measurement uncertainty and traceability of freeform measurements.

Keywords: Freeform, Dimensional, Metrology.

Acknowledgments

The authors acknowledge the following persons whocontributed to the paper with suggestions, comments,references and corrections (* denotes CIRP member): E.Brinksmeier*, T. Bothe (BIAS, Germany), S. Carmignato(Univ. Padova, Italy), G. Goch*, C. Evans*, R. Fisker(3Shape A/S, Denmark), R. Henselmans (T.U.Eindhoven, The Nederlands), R. Hocken*, D. Imkamp(Carl Zeiss, Germany), J.-P. Kruth*, L. Monostori*, E. P.

Morse (UNC-Charlotte, USA), T. Pfeifer*, F. Puente*, H.Wang (Carl Zeiss, Germany), A. Weckenmann*, X. Jiang(Univ. Huddersfield, UK).

1 INTRODUCTION

Industrial manufacturing makes extensive use of simpleshapes for the production of goods, with many productshaving a geometry that is a combination of planes,cylinders, spheres and other simple shapes. These partsare fundamental for the functionality of most mechanicalproducts and, in general, they are easier and lessexpensive to manufacture than complex parts. However,in some applications they are not adequate, for instancewhen the functionality of the part is given by an interactionwith a fluid or a wave, as is the case of aerodynamics andoptics, for example.

In optics, a shape that is neither a portion of a sphere norof a cylinder is called asphere; the most commonly usedaspheric surfaces are axialsymmetric, since they areeasier to manufacture. In general, complex shapes withrotational symmetry are also easier to be measured, sincethe measurement may rely on the acquisition of fewprofiles.

Freeform surfaces, sometimes called sculptured orcurved surfaces, may be classified as complexgeometrical features. According to ISO 17450-1 [81],complex geometrical features have no invariance degree.The invariance degree of a geometrical feature is the

displacement of the ideal feature for which the feature iskept identical; it corresponds to the degree of freedomused in kinematics. In the following, the term freeform willbe used only for surfaces without rotational symmetry;

when relevant, aspheric geometry will be explicitlymentioned.

Freeform shaped parts are of great interest in manyapplications, either for functional or aesthetical reasons.Their relevance for industry is well-known in the designand manufacturing of products having complex functionalsurfaces [130] [150] [15] [16] [88] [93] [144] These partsare important components in industries such asautomotive, aerospace, household appliances and others.

Functional surfaces may have a great influence on theperformances of a product; in the design of aturbomachine, for example, freeform geometry for bothstatic and rotating components is of paramountimportance. Geometrical deviations in manufacturingcause inefficiencies that can cause waste of largequantities of energy. Design, engineering and testing ofparts having functional surfaces are key activities for thedevelopment of products with better performances [123].

In the modern design of many goods, aesthetic is alsobecoming more and more important for the market impactof a product. This is a critical success factor especially forconsumer items where shape and appearance may havemore impact on the customer than function. Examples are

very common: car bodies, mobile phones and consumerproducts in general.

Freeform shapes can be realised by many differentmanufacturing processes, whose capabilities have beenimproved for freeform geometry as reported by a numberof publications [15] [16] [88] [93] [14] [13] [29] [52] [168][98] [194] [125] [162] [11] [55]; most of these publicationsrefer to the metrology of freeform shapes as afundamental tool for both process troubleshooting andquality control.

For some demanding applications, it is advantageous tomeasure the workpiece during the production process; in-process metrology allows improvement of accuracy and areduction of machining time [124] [155] through the

elimination of repositioning and alignment operations.Another application in which the measurement of freeformshapes is highly relevant is the so-called reverseengineering, consisting in the creation of a computer-aided design (CAD) model starting from a physical part;


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the procedure is based on data points measured on thesurface of the physical part.

The scope of this keynote paper is to present the state ofthe art in the metrology of freeform shaped parts withfocus on the freeform capabilities of the most importantmeasuring techniques and on the related metrologicalissues. For details on the measuring principles the readeris referred to relevant previous CIRP keynote papers [55]

[41] [159] [66] [179] [31] [62] [28].The paper is organised as follows. Section 2 presentssome examples of products with functional freeformsurfaces, i.e. products for which the metrology of freeformshapes is important to guarantee the desired functionalperformances; the last part of the section presents aclassification of freeform measuring tasks and the relatedmetrological requirements. Section 3 is a review of themost important measuring techniques, with focus on theircapabilities for freeform measuring tasks. Section 4addresses the specification and verification of freeformsurfaces, while Section 5 deals with data evaluation andcomparison to specifications. Section 6 discussesmeasuring uncertainty and traceability of freeformmeasurements.

2 APPLICATIONS OF FREEFORM SHAPES

Freeform shapes are widely used in industrial production.They are important in key industries such as automotive(car bodies, 3D-cams, seals, gears, etc.), aerospace(turbine blades, impellers, fluid-dynamic ducts,aerodynamic parts, etc.), household appliances (waterpumps, fans, etc.), consumer products (mobile phones,cameras, etc.) and others. The aim of following examplesis to illustrate the wide field of challenges for measuringfreeform shapes. The description of each exampledelivers information about the industry it is applied in andthe function of the freeform shape, in order to address therequirements for an appropriate measurement of parts.

2.1 Airplane wings and fuselageNowadays there are two main goals in the aircraftindustry; to increase the amount of passengers and toreduce the consumption of fuel. To reach these goals, thefuselage and wing designs are improved by the usage ofnew materials like carbon fibre which reduce the overallaeroplane weight and enable the development of newgeometries reducing the aerodynamic resistance.Freeform measurement of the geometry is a key factor inthe development process as can be shown by two currentexamples, the development of the Airbus A380 and of theBoeing 787 Dreamliner.

Most important in the development of a new wing designsis the measurement of the aerodynamic resistance, the

ascending force and the rigidity in a wind tunnel (seeFigure 1). Measurement of the rigidity respectively thedeformation of the wing enables the online measurementof the coupled bending and twist spatial deflections [86].

Due to the dimensions, the tolerances during theassembly of the whole aircraft are quite challenging interms of resolution vs. the overall extension. To reducethe aerodynamic resistance, the structure, which consistsof several single parts, has to be even. So, the geometryof every single part has to be known to enable theassembly of the component. These subassemblies, likethe surface of airbrakes, the landing gear door or even awhole fuselage section, need to be measured.

A most challenging problem is the fabrication of thecarbon fuselage segments as one single part in anautoclave. One possible solution to rapidly gain a lot ofinformation about its surface and the geometry is themeasurement while the segment is fixed in a test bed.However, as long as the segments are not assembled to

a complete fuselage structure the gravitational forcesdistort the planned and produced geometry. So, onemeasurement task is the calculation of the as-builtgeometry of each sub-assembly out of the measurementdata [27].

Figure 1: Complete model of an A380 in theGerman-Dutch wind tunnel in Emmerloord (NL) [138].

2.2 Car body parts

One of todays challenges to the global automotiveindustry is owed to the fact that the success of the finalproduct not only depends on the engineering quality butincreasingly more on the emotional response it evokesfrom a potential buyer [92]. In addition to this, ergonomicand aerodynamic aspects are two other importantdemands on automotive products. With the background ofrising petrol prices, streamlined car bodies which result ineconomic cars are clear customers requirements.Freeform surfaces fulfil the demand of appealing design

in combination with aerodynamic shape and ergonomicproperties [188].

Figure 2: Part of a sheet metal forming tool withfreeform elements.

A commonly used manufacturing process to produce carbody parts with freeform elements is the stampingprocess. Fundamental components of stamping machinesare sheet metal forming tools (Figure 2), which makes the

tool and die making the key industry in the production offreeform geometries. In order to guarantee the correctshape of forming tools, it is necessary to digitize theirtopography and compare it to a CAD model. Thereby thechallenging task is to choose the optimal metrology


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system, the right measurement strategy and optimiseddata processing [69] [60]. Therefore the metrology systemshould provide high quality coordinates in terms of highdata density on structures with high spatial frequencies,and lower density on flat areas and the ability to detectsharp edges precisely [50].

2.3 Turbine blades

Turbine blades can be found in a wide variety of industrialproducts. Their field of application ranges from the massproduction of cooling fan solutions which can be found ine.g. personal computers and turbo charger systems as apart of combustion engines, up to the high performancerequirements of jet engine operation.

The use of blade shaped parts in turbojet engines can beseen as one of the most challenging areas regarding e.g.material selection and blade shape design. Over the last20-30 years, alloy improvement, directional and single-crystal solidification as well as the use of coating systemshas allowed an increase of gas temperature. This hasincreased system strength and thermodynamic efficiencyalong with reducing environmental pollutants.

The exposure to high pressure, temperatures up to

1400 C as well as the considerable stress caused byrotation at more than 10.000 rpm, gives an impression ofthe boundary conditions during a flight. Since the smallestfault during the design or manufacturing process of sucha turbine blade can lead to a catastrophic system failure,the implementation of all means of precautions availableis vital to ensure system reliability and blade integritywithout compromising economic aspects.

While FEM analysis and following optimization applyduring design, five axis milling is state of the art in theproduction of delicate parts like the axial flow bliskmanufactured from a single part (Figure 3). However, theneed for quality control is indispensable to ensure thesafe and efficient whole-life operation of such parts.

In addition to the manufacturing, quality control methodsvary widely depending on part size, weight, featureaccessability and measurement task. Since Bladed discsor Blisk, as a bunch of blades, are more complex then asingle blade, they also provide more challenges toacquire the necessary measurement data. Use of tactileprobing systems is time consuming because of spacerequirements for the movement between two adjacentblades, which also overlap. Overlapping surfaces alsohamper most optical systems since a clear line of sightand consistent illumination of the part needs to beestablished. Also strong curvatures at the leading andtrailing edge can oppose quite a challenge for optical dataacquisition.

Figure 3: Axial flow blisk

2.4 Optical parts

Aspheric and freeform optics offer significant advantagesover conventional flat and spherical surfaces. Examplesare found in many important applications, includingcomputational imaging, compact projection displays,document security, curing of polymer dental fillingmaterial, controlled diffusers for lithography, microscopy,and many others [26]. The use of freeform shapes in

optical design can significantly improve the performances,in terms of both system size reduction and improvedoptical functionality (i.e. lower wavefront error). Whenconventional optics are used, the optical system usuallydictates the mechanical design. By using freeforms, thenumber of components can be reduced, the componentscan be placed in mechanically favoured positions whilethe optical quality of the system can still be increased[122]. Their use is advantageous in many industries,ranging from mass production of consumer products tomanufacturing of single special components for largespace projects. In the following, some examples arebriefly described.

One of the first examples in mass production is the

Polaroid SX-70 folding Single Lens Reflex camera shownin Figure 4, which was on the market in 1972 [135].Because of its peculiar off-axis viewing optical system,two freeform optical components were used for distortioncorrection in its optical design. One large free-formsurface, deployed on the eye lens, corrected for field tiltand localized apparent power and astigmatic errorsacross the viewed scene, relayed from a textured Fresnelfield mirror focus screen. The other free-form surface, asmall corrector plate located just at the real aperture stop,mainly corrected the net coma and spherical aberrationcommon to all of the field. These two surfaces wereoptimized along with an off-axis aspheric concave mirrorto provide a well corrected system [136].

Another mass production example is the optics of laser

printers. Until recently, their optical scanning systemsutilized several optical elements to form a system. Byreplacing these with a single freeform mirror, the numberof components is reduced with corresponding benefits incost and size reduction. Further benefits include absenceof chromatic aberration and the ability to select anywavelength of laser. Since the shorter wavelength laserresults in spot size reduction, the preciseness of printingoutput is improved. Conversely, a long wavelength lasercan be used in a less expensive mass production printerwith the same freeform optical scanning system [29].

Figure 4: Cutaway view of the Polaroid SX-70 camera,showing its decentred reflective Fresnel focus screen and

the unusual Single Lens Reflex light path [136].


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An example of a special project using freeform shapes fora unique product is the NASA Infrared Multi-Objectimaging spectrometer IRMOS [192]. The optical designincludes four large, fast, off-axis, highly aspheric mirrorsand one off-axis concave biconic mirror [15] [14] [52][164]. The freeform mirror was introduced to help reducethe size of the system by an order of magnitude [51].Significant reduction in size can dramatically reduce theuse of exotic materials (such as beryllium) and theensuing mass reduction provide enhanced performancefor lightweight space systems. Freeform surfaces canalso be used to control astigmatism at multiple locationsin the field of view and thus reduce wavefront aberration.

In general, mirror freeform surfaces are more difficult tomanufacture and the inability of a designer to assess theirmanufacturability has limited their use in optical systems.Designers prefer to select rotationally symmetric surfaceshapes for which the manufacturability issues areunderstood, rather than specify a design based on theproperties of a freeform surface. For this reason, newoptical design tools [51] have been developed to improvethe manufacturability of these complex surfaces, in orderto keep costs under control.

Even with advanced design tools and ultra precisionmanufacturing technologies, the required accuracy ofoptical surfaces can often not be achieved in adeterministic manufacturing process. Besides thestatistical variation of the machining process the leastchanges in the boundary conditions lead to noticeableform deviations. Therefore the manufacturing isperformed iteratively. Hence the accurate measuring ofthe absolute shape of the surface is the missing keyfactor in the value chain of freeform optics manufacturing[122]. Even with efforts to establish machine integratedultra precise form testing, to enable an adaptivemanufacturing process [124] [155], the fundamentalmetrology specific challenges for optical surfaces remain,since the functional surfaces with their macroscopicdimensions demand a very high accuracy in thenanometre range. The dynamic range of the testequipment is the crucial point in measuring opticalfreeform shapes. Another important demand is that theoften delicate optical surface must not be damaged by theinspection.

2.5 Haptic sensor surfaces

The term haptic can be distinguished between tactileperception and proprioception. While tactile perceptionaddresses the sensitivity for surfaces, proprioceptionstands for perception of depths i.e. movements ormovement directions. Haptic perception, as the ability forsensory perception of mechanical excitations, strongly

depends on the forces perceived during a touch [5].A need for haptic sensors exists in many fields ofengineering. Intelligent haptic sensors can be used e. g.in robotic applications to manipulate 3D objects or forvirtual reality applications in sensor actuator systems inmedical tasks. In robotic assembly systems hapticsensors avoid damaging sensitive products by addingartificial sensing intelligence [126]. In medical tasks,haptic sensors can help to improve the work of surgeonsand also allow the design of intelligent prostheses [89].

A haptic sensor is not only used to evaluate freeforms butit also contains freeform elements itself. An exemplarysetup is shown in Figure 5. The sensor system consists ofa basic plane with integrated monomode light fibres and a

joint micro membrane of silicon. The quadratic micromembrane includes several freeform elements which aresimilar to pyramids (see Figure 5, right). The micromembrane and the basic plane have to be assembled insuch a way, that the light fibres in the basic plane lead

directly into the pyramid structures. Every fibre works as aminiature interferometric sensor. By applying a force F tothe membrane, the distance between the membrane andthe fibre tip changes. The change of distances in thewhole sensor array can be used to infer to a hapticbehaviour caused by the force F.

To allow for touching forces below 500 N, demands onthe accuracy of all parts of the sensor are high [165]. For

the setup shown, glass fibres with a diameter of 100 mhave to be positioned into a 400 m hole with a lateraltolerance of +/-50 m. To guarantee a working sensordevice, also the vertical distance between tip of the fibresand the bottom of the pyramid structure must be known .To solve this task, the freeform of the bottom of thepyramid structure must be evaluated accurately. Thisresults in a demand for metrology to offer suitablesolutions to measure the precise geometry tolerances.Furthermore, suitable sensors are necessary that canresolve the form of the highly sloped walls of the pyramidstructure while not deforming the soft silicon during themeasurement [36].

Micro membrane

Pyramid structure

Basic planeLight fibre

Figure 5: Setup of a tactile sensor system (left), photo ofthe micro membrane (right).

2.6 Classification of freeform shape applications

The above mentioned examples illustrate the range offreeform applications and show, depending on theindustry, the function, the material and the manufacturingprocess, the variety of measurement tasks and theirspecific challenges. These characteristics define therequirements for the technical properties of freeformsurface metrology.

Part dimension

mm cm damdm m

Profile

tolerance

mm

m

nm

10 -3

10 -5

optical parts

airplane

car bodieshapticinterfaces

turbineblades

10 -7

Part dimension

mm cm damdm m

Profile

tolerance

mm

m

nm

10 -3

10 -5

optical parts

airplane

car bodieshapticinterfaces

turbineblades

10 -7

Figure 6: Typical values of tolerances vs. dimensions forthe selected application examples.

A classification based on freeform part dimensions, shapecomplexity, material, surface and tolerances can bederived. Shape complexity is classified as low (e.g.nearly flat, aspheric, limited curvature change), medium


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(moderate to large curvature changes, multi-facetsurfaces) and high (with undercuts, access/visibilitylimitations, internal features). Concerning tolerances, theclassification is based on the relative profile tolerancedefined as the ratio:

tolerance / main part dimension.

The proposed criteria for classification are applied to theabove presented examples in Table 1, while Figure 6

shows a graphical representation of typical applicationranges for dimensions and related tolerances.

Airplane

fuselage/wings

Automotive

bodyparts

Turbineblades/

blisks

Opticalparts

Hapticsensor

surfaces

Part dimensions

large (100

- 102m)

medium (10-1 - 100m)

small (10-3

- 10-2

m)

micro (10-3

m)

Shape complexity

low

medium

high

Material, surface

hard, not sensitive

deformable

speculartransparent

opaque

Relative tolerance

medium (10-4

-10-3

)

fine (10-5

-10-4

)

ultra fine (< 10-5

)

Legend: typical

less frequent

Table 1: Summary of measurement requirements for thefreeform shapes presented in this section.

3 MEASURING SYSTEMS

Two general approaches can be identified for themeasurement of freeform shaped parts: direct andindirect comparison [71]. The basic principle of directcomparison is to check the degree of deviation betweenthe surface and master templates. In some cases, thesemasters represent two-dimensional cross sections of thesurface to be inspected. The measurement of the gapsbetween the surface and the template can be done with ameasuring microscope, or by bluing the surface to checkthe untouched area visually, or using a concentrated

source of light. Figure 7 illustrates this approach appliedto the inspection of a turbine blade. While the resultsachieved through direct inspection methods areacceptable for some applications, they are clearly notoptimal. The problems common to many types of direct

inspection are very basic: accuracy, speed and the needfor dedicated equipment.

With indirect comparison, the physical master template isreplaced by a computerised 3D geometrical model of thepart. The basic principle is to evaluate the degree ofdeviation between the measured surface and thecomputerised model, for instance the 3D CAD model ofthe part. In the following, only measuring techniques

based on the indirect comparison approach will be furtheranalysed.

Depending on the application requirements, the mostsuitable and cost effective measuring technique can beselected among several available. The measuringtechniques that have been identified for discussion areclassified in the following categories:

1. Systems for large scale metrology

2. Coordinate measuring machines

3. Stand still optical systems

4. Interferometric systems

5. Profilometry

6. Systems for micro/nano scale metrology

7. Other systems

The proposed classification is quite arbitrary but it isbelieved to be useful for an introduction to the subject.The measuring systems are presented with focus on theirfreeform capabilities.

Figure 7: The inspection process of a turbine blade basedon direct comparison with master templates

(Courtesy of Pietro Rosa TBM srl, Italy).


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3.1 Systems for large scale metrology

The measurement of large parts requires techniquescapable of recording point coordinates with measuringranges of several meters; the most common techniquesare laser tracking interferometry, photogrammetry andlaser radar. Measuring principles, capabilities, limitationsand application examples can be found in a CIRP keynotepaper dedicated to large scale metrology [41].

The laser tracker measures the position of anindependent portable target that is manually positioned onthe surface of interest by an operator. Measurements canbe performed at rates as high as 3000 points/s whilemoving the target, therefore high density profiles caneasily be scanned over large freeform surfaces. Thetarget is typically spherical with a diameter in the rangefrom 13 mm to 38 mm; special targets consisting ofportable probing systems, both contact and non-contact,are also available [100]. An example of laser trackersystem with an integrated handheld laser scanner isshown in Figure 8. The measuring process involves theuse of a high-speed camera, installed on the laser trackerin conjunction with a laser scanner, containing a diode

array and a reflector; while the laser tracker determinesthe exact position of the reflector, the camera determinesthe position of the diode array in the photogram and, fromthat, computes the spatial orientation of the laser scanner.

Photogrammetry is a technique in which the position ofpoints of interest is calculated via optical triangulationfrom two or more two-dimensional images taken fromdifferent locations. In industrial photogrammetry, themeasured surface is usually provided with physicalmarkers; the number of points measured on a freeformsurface is therefore intrinsically limited.

The laser radar is a measuring system in which abroadband frequency modulated infrared laser (100 GHzmodulation) provides a robust and eye-safe signal for anon-contact measurement capability up to 60 metersradius. The main advantage is the complete automationof the measurements process [41] [118].

Figure 8: Example of laser tracker system with anintegrated handheld laser scanner [100].

3.2 Coordinate Measuring Machines

The most important general-purpose instruments for theinspection of freeform shapes in industry are CoordinateMeasuring Machines (CMMs), which can be equippedwith both contact and non contact probing systems.

CMMs are well accepted in industry because they arevery flexible and allow the measurement of points inspace with high accuracy. They are today typicallyprovided with numeric control and scanning probes, both

contact and non-contact; automatic measurements ofthousands of points are easily performed, even oncomplex surfaces. When the geometrical accuracy offreeform surfaces is critical for the functionalperformance, or for the appearance of a product, theinspection process is typically conducted on CMMs.Figure 9 shows a classical example: a CMM equippedwith a contact measuring probe while scanning a freeformprofile on a mould.

In freeform metrology it is fundamental to measure a largenumber of points distributed on the surface to beinspected. Scanning probing systems are today preferredto touch trigger probes. Contact measuring probingsystems have been discussed in a recent keynote paper[179]; they can measure up to 200 points/s at speeds ashigh as 150 mm/s. Faster measurements are possiblewith non-contact probing systems based on opticalprinciples like autofocus, triangulation and conoscopicholograpy among others. Their advantages are, ingeneral, the non-contact nature of the measurementprinciple, fast acquisition of a large number of points andhigh degree of flexibility in measurement settings [22].Details of the measuring principles can be found in [159]

[66]. High-reflective surfaces may be measured on CMMsusing e.g. a special triangulation optical probe [196] thatuses three incident beams, passing through polarizingplates, so that specular light is eliminated to a largedegree and only the diffused light can reach the CCDcamera. Different practical solutions are available (see[194] for a list of commercial products), with an increasingmarket share of optical probing systems on CMMscompared to tactile ones. Optical probing systems aredistance sensors that can be classified into three basictypes, depending on the number and position of pointsthat are measured at the same time: point sensors, line(or stripe) sensors and area sensor. An example is shownin Figure 10.

Figure 9 : Example of a CMM equipped with a contactmeasuring probe while scanning a specular freeform

surface on a mould (Courtesy of Carl Zeiss IMT GmbH,Germany).

Articulated arms and other systems

The mechanical linkage between a fixed position and theprobing system can also be provided by a series of


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connected segments with encoders, called articulatedarm. Measuring volume can reach some meters inradius. The device is manually operated and is portable; itcan be useful for the measurement of freeform shapeswith accessibility difficulties on-site, such as die cavities,heavy equipment or simply parts that cannot be moved.Both contact and non-contact probing systems areavailable on commercial products [194].

Machine tools can also act as position systems for opticalsensors [146]. Industrial robots are another possible fully-automated solution for the positioning of opticalmeasuring systems, as an alternative to CMMs, forrepetitive inspection tasks on production lines where theneed of accuracy is limited, i.e. for sheet metalapplications.

Other systems are available on the market for specifictasks. Automated laser scanners with positioning androtating stages are common examples [194].

Figure 10: An example of optical probing system mountedon a CMM: multi-stripe laser sensor

(courtesy of Metris, Belgium).

3.3 Stand still optical systems

Other techniques do not require a controlled displacementwhile measuring and they can be classified as stand stilloptical systems, such as the two techniques based onstructured light presented in this section,photogrammetry introduced in Section 3.1 andinterferometric techniques discussed in Section 3.4.

Fringe projection

Fringe projection is a technique in which the position ofpoints of interest is calculated via optical triangulation;unlike industrial photogrammetry, in which the measuredsurface is usually provided with physical markers, fringeprojection techniques measure surfaces without physicalmarkers.

Two approaches are distinguished [159]: the first one is

based onto the projection of a pattern on the surface andat least two different camera views. In this case, thepattern only serves to generate homologous points fortriangulation within the camera images. The secondapproach evaluates the deformation of the pattern itself.

In this case, the projector of the pattern takes up the roleof one camera. See [159] [32] [149] [148] [20] [186] foradditional details. Fringe projection has been alsoimplemented by different authors in combination withother more accurate sensors; for example, the additionalsensor may provide a higher resolution measurement onsome region of interest [184]. Furthermore adaptive fringeprojection methods are currently researched in order towiden the range of measurable surfaces conditions andsurface angles [33].

A typical measurement volume of a fringe projectionsystem is in the range of side between 0.1 m and 1 m,with relative accuracy of up to 10

-4. An example of a fringe

projection system is shown in Figure 11.

Figure 11: Example of measuring device based on fringeprojection. Photo (a) and optical layout (b).

(Courtesy of University of Brescia, Italy)

Fringe reflection / Deflectometry

A general limitation of fringe projection systems is theirinability to measure non-cooperative surfaces, forexample specular ones. In the fringe reflection method,also called Deflectometry, a screen projects black andwhite fringes onto the reflective surface; the reflectedfringe pattern is viewed by a video camera and analysed

by a software routine. Thus, the reflection angles can bedetermined for every camera pixel and the local gradientscan be calculated with high lateral resolution. The surfacedata are obtained by integration, with a vertical resolutionmuch higher than in fringe projection. Details on the


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technique are available on [7] [9] [8] [10] [53]. Figure 12shows an example of horizontal setup while measuring afreeform mirror.

Deflectometric methods are suitable for the measurementof high precision optics, and fast inspection of largerparts, such as painted car bodies. Other methods underinvestigation by several research groups are Moirdeflectometry or the Reflection Grating Method [85] [134]

[176] [143] [127] [128] [137] [94]. Deflectometry is alsoadvantageous for the measurement of surface curvature,since only the first derivative has to be calculated. Asevery derivation increases the noise, the deflectometricmethods have an unbeatable advantage compared toother methods that capture shape data.

(a)

Screen

Projector

Specular surface

camera

(b)

Figure 12: Example of fringe reflection system: (a) picture

of an horizontal setup for small objects (see reflectedfringes in the freeform mirror) [7]; (b) a schematic optical

layout [94].

3.4 Interferometric techniques

Interferometric techniques are a well known solution forfast measurement of surfaces with sub-nm resolution inthe direction of beam propagation [159] [66]. Formtesting of simple shapes like flats and spheres is typicallyperformed by imaging interferometry. Aspheres can alsobe measured, using refractive or diffractive null-optics.This becomes necessary because the dynamic range ofinterferometers is generally insufficient to measureaspheres with large departures from the vertex sphere.The use of null-optics however does not only add cost butalso creates additional metrology problems and increasesmeasurement uncertainty. Perhaps the most problematicaspect of aspheric surface testing using interferometers

and null-lenses is the loss of universality in the metrologytool. A new surface geometry will generally require a newnull-lens [90]. Furthermore, null optics introducechallenging calibration issues, and the accuracy of theresulting measurements is notoriously difficult to validate.Interferometric tests are also relatively expensive for largeparts, especially for non-concave parts because thereference optics must then also be large [37].

The cost of null-lenses and the time needed to fabricateare therefore not acceptable for most applications. This iseven more evident for true freeform shapes, whichobviously cannot be measured when the departure from asphere is not within the dynamic range of theinterferometer. Alternative solutions are presented in thefollowing, which extend the measurement of complexsurfaces to the sub-millimetre vertical range.

Computer Generated Holograms (CGH) Interferometry

Complex shapes including aspheres and true freeformscan be measured using Computer Generated Holograms(CGH) [112] [129]. CGH interferometry is based on theuse of a surface specific diffractive element added to aninterferometer, which changes a spherical wavefront into

a more complex wavefront, as illustrated in Figure 13.The principle was first described by Lohmann and Paris in1967 [107] and then used for testing complex opticalgeometry [110] [17]. CGH masks consist of a planarsubstrate covered with a diffractive microstructureproduced by lithography. Due to the recent progress inlithography, it is possible to produce these diffractivemicrostructures with very high accuracy [159]. CGHmasks are today commercially available [35] [3].

Sphericalwavefront

Asphericalwavefront

ShapeunderTest

CGH

Referencesphere

Interferometer

Figure 13: The basic principle of CGH Interferometry.

A common arrangement for placing the CGH in theinterferometric setup is shown in Figure 14. An advantageof this setup is that it can be used with commercialinterferometers without any need for modifying theinterferometer. However, in this setup the quality of theCGH substrate (i.e. thickness variations in the hologram

plate) has greater influence on the measurement results.Alternative setups with the CGH element located in nearlycollimated light can reduce this and other error sources.Modern setups insensitive to vibrations have also beendescribed [17].

The most important error sources introduced by the CGHtechnique are due to the manufacturing of the diffractiveelement and to its alignment.

Manufacturing errors are related to the already mentionedthickness variation and to the accuracy with which thegrating curves are drawn, the latter typically being thesingle largest error source [139]. Alignment errors arealso important. Generally alignment marks, crosshairs,etc. are placed on the CGH to help in both aligning themask to the interferometer as well as aligning the sampleto the mask. Since traditional methods of finding fringesoften do not apply for freeform surfaces, such tools areindispensable for CGH metrology. Unfortunately, this also


9/26 -818-

means that fiducials must be applied to the sample to bemeasured [164].

Spatial filter

Image plane

Reference mirror

Test

shape

Laser

light

CGHBS

Figure 14: Example of a common interferometric setup forusing a CGH as a null lens (adapted from [17]).

While CGH metrology has been recently improved andextended for the measurement of true freeform shapes,

the techniques for doing so expediently and economicallyare not yet mature. CGH reference masks may takeweeks to design and fabricate; they are also specific for asingle shape, therefore the technique is quite expensive.Typical applications reported in literature aremeasurements of ultraprecision freeform mirrors forspace telescopes [192] [164] [17] [3].

Sub-aperture Interferometry

Another method that has been used for many years issub-aperture interferometry [169] [25] [106] [116] [43][42] [61] [47] [59]. Several commercially availableinstruments can automatically stitch flat surfaces,although stitching complex surfaces is inherently morecomplicated. The surface is measured in several

overlapping parts, or sub-apertures, that are within thedynamic range of the interferometer. Accurate positioningwith multiple axes is required. Subsequently, the sub-aperture measurements are combined, or stitchedtogether, to give a form error map for the entire surface ofthe part [90]. Stitching algorithms have been developed tocompensate for several types of error introduced by theinterferometer optics and stage mechanics. These includepositioning errors, viewing system distortion, and thesystem reference wave [47].

Figure 15: Example of measurement planning for sub-aperture stitching interferometry (adapted from [47]).

The technique has been successfully applied to differentmeasurement tasks, including local radius of curvature forhigh precision spheres [59], thickness of wafers [82] andaspherical surfaces [106] [116] [43] [42] [61] [47] [59][121]. As a result, it is possible to extend the measuringrange of aspheric departures from a few micrometres upto several tens of micrometres, with measurementuncertainty better than 0.2 m [42]. The generaldisadvantages of stitching are that measurements can betime consuming and data analysis is cumbersome anduncertainty may be difficult to estimate.

Interferometers as curvature sensors

Differential geometry offers curvature as a way tomeasure form or profile. Measuring systems based oninterferometers as curvature sensors are being developedat PTB [189] [40] [157] [158] and at NIST [90] [58] [111] tomeasure complex optical surfaces up to one meter indiameter, for which the form has to be known withnanometre uncertainty and lateral resolution smaller thanone millimetre.

Figure 16: Principle of the system developed at PTB,based on a interferometer (IF) as multiple distance sensorhead, scanned along the surface under test by the l inear

stage (S). A mirror (M) is mounted to the side of thesensor head and an autocollimator (AC) measures the tilt

angle of the sensor head [157].

Figure 17: NISTs prototype Geometry MeasuringMachine (GEMM) [111]

The prototype developed at PTB employs a commercialcompact Twyman-Green interferometer that is scannedalong the surface by a low-cost linear stage. Additionally,an autocollimator is utilized to account for angular


10/26


11/26 -820-

3D model can also be scaled according to the sonicspeed of material and the measuring point density.Depending on the used CT software various geometricmeasurements can be performed virtually within the partsvolume.

Delamination

Fluid-jet ultrasonic probe

B-scan

C-scan

Figure 18. Ultrasonic scan methods [156].

3.8 Classification of measuring equipment forfreeform geometry

Freeform surfaces are still representing challenging

measuring tasks, and the underlying manufacturingtechnology relies to a large extend on the capabilities ofthe metrological set-up that will deliver data from theproduct or process to stabilize production.

Measuring systems for freeform surfaces are using a widerange of physical phenomena to match the needs ofspecific measuring tasks. The above mentionedmeasuring examples may be compared to the

requirements which can be derived from the categories ofpart dimension, shape complexity, surface conditions andmaterial properties such as hardness and transparency(Table 2). Some techniques are better understood thanothers in terms of traceability and this reflects in the lastrow of Table 2; an indication of typical measuringuncertainty is given in Figure 19, even though a generalreference for all measuring tasks cannot be derived, dueto the different fields of application. In fact more importantthan just to understand the underlying physicalphenomena during the measuring process is tocomprehend the restrictions and boundary condition forevaluating a geometric feature and the interaction of thespecimens properties with the measuring instrument, e.g.bandwidth-limiting and thus resolution decreasing effects

whilst scanning a feature with a high shape complexity.

Lasertracker

DirectComparison

TactileCMM

OpticalCMM

X-raytomography

Fringeprojection

Fringereflection/

Deflectometry

Photogrammetry

Interferometry

TactileSurface

topography&Profilometry

OpticalSurface

topography&Profilometry

ConfocalMicroscopy

ScanningForce

Microscopy

Part dimensions

large

medium

small

micro

Shape complexity

low

medium

high

Material and surface

hard, not sensitive

deformable

specular

transparent

opaque

Traceability

Legend: full match:

little match:

Table 2: Evaluation of some measuring techniques with respect to the classification proposed in Table 1.


12/26 -821-

Measuringuncertainty

10 -3

10 -7

Interferometric techniques

Coordinate Measuring Machines

Stand still opticalsystems

Profilometry

Systems forlarge scalemetrology

Systems for micro/nano scale metrology

10 -5mm

m

nm

Part dimension

mm cm damdm m

Measuringuncertainty

10 -3

10 -7

Interferometric techniques

Coordinate Measuring Machines

Stand still opticalsystems

Profilometry

Systems forlarge scalemetrology

Systems for micro/nano scale metrology

10 -5mm

m

nm

Part dimension

mm cm damdm m

Figure 19: Typical range of measuring uncertainty vs. partdimension for different categories of measuring systems.

4 SPECIFICATION AND VERIFICATION OF

FREEFORM SURFACES

4.1 Definition of nominal shape

Curves and surface geometry is the underlying theory forthe description of complex shapes. Traditionally, curvesand surfaces have been represented by multipleorthographic projections. With the advent of computergraphics and the rapid developments in the area ofcomputational geometry, surface modelling has become acommon feature of computer-aided design (CAD)systems. Today, the nominal geometry of a freeformshape is typically defined by a CAD model.

Mathematical representation of surfaces

Freeform shapes are typically described using parametric

surface representations [44]. Commonly used parametricsurfaces are Bzier, B-spline and Non Uniform RationalB-spline (NURBS) surfaces. Parametric equations havemany advantages over non-parametric forms:

they are much more convenient to define and controlthe shape of a curve or a surface;

their mathematical description is easy to express interm of matrices and this allows the use of standardcomputation subroutines;

curves and surfaces descriptions are independent ofany coordinate system, therefore the choice of acoordinate system does not affect the shape ofsurfaces;

they are convenient for computer graphics softwareand hardware in terms of speed, since thecomputation of offset curves and surfaces can besimplified.

Modern CAD systems describe complex geometry usingNURBS that represent the industry standard for geometrydescription in CAD and Computer Graphics applications,even though B-spline surfaces are also popular incommercial modelling software due to their simplermathematics [190]. NURBS surfaces are a generalisationof Bzier and B-spline surfaces. They are widely used forthe representation of freeform surfaces due to theirinteresting properties such as the ability to handle largesurface patches, local controllability and the ability to

represent also simple analytical shapes such as planes,spheres, cylinders, cones, tori, etc. A NURBS surface isshown in Figure 20 and it is defined as [44] [190] [97]:

u v

u v

n

i

n

j

ijvjui

n

i

n

j

ijijvjui

wvBuB

cwvBuB

vup

1 1

1 1, (1)

where p is a point on the surface and u and v its location

parameters identifying the location of a pointpwithinthe surface

nu and nv are the number of control points in the uand vdirection

Bui(u) and Bvj(v)are the normalised B-spline functionsin the u andvdirection. Bui(u) is uniquely defined bythe order kuand knot sequence tuwith ku + tuu-knots.Similarly, Bvj(v) is uniquely defined by the order kvand knot sequence tvwith kv+ tvv-knots

cij are the control points controlling the shape of thesurface and wijtheir respective weights. When all theweights are set to 1, a NURBS surface becomes a B-spline surface.

Figure 20: Example of NURBS representation; uand vrepresent the location parameters of a point on the

surface [97].

4.2 Tolerancing

The specification of tolerances on a freeform shape maybe given on the basis of the profile tolerances defined forsurfaces, with or without reference to a datum, availablefrom ISO 1101 [72].

Form tolerances restrict the deviations of the real surfacefrom the nominal shape only and have no datum; thedefinition is as follows (see also Figure 21): Thetolerance zone is limited by two surfaces envelopingspheres of diameter t, the centres of which are situatedon a surface having the theoretically exact geometricalform. This definition of tolerance zone allows thespecification of form tolerances for any complex surface;it is the least restrictive type of tolerance, since there areno restrictions for location and orientation of the surface.

When the functionality of the part is dependent also onthe orientation and location of the surface, specificationsincluding datums are used; in this case the tolerancezone definition is more restrictive, since it controlssimultaneously location, orientation and form deviations.

Other international standards of relevance are ISO 1660(drawing indications) [74] and ISO/TR 5460 [78](verification principles and traditional inspection methods).However, modern specification and verification conceptsare not yet supported for freeform surfaces by a welldefined normative basis nor a universally acceptedcommon praxis [117].


13/26 -822-

Figure 21: Definition of profile tolerance not related to a

datum [70].

A general issue in tolerancing is understanding the link tofunctionality. This is particularly true in profile tolerancing,since both the designer and the fabricator are more orless working without knowing how much deviation fromthe desired surface can be tolerated. Poor understandingof the tolerances required for a given system, andespecially the inability to determine whether loosertolerances will still provide the desired performance, tendsto drive up costs [28].

4.3 Data exchange

The model of the part being inspected is typically used as

reference for the practical realisation of the measurementprogram on computer-controlled measuring devices. Thisprocess involves the exchange of geometric informationbetween CAD and measuring systems. Standardised dataexchange interfaces like STEP and IGES allow thetransfer of complex geometry including freeform surfaces.However, there are limitations on the automatic transfer ofinformation such as tolerances. To overcome thelimitations of actual transfer standards, both proprietaryand neutral standardised interfaces have been proposed;more information with a focus on coordinate metrology isavailable in [131].

4.4 Measurement strategy

The inspection of a general freeform shape is normally

based on a uniform distribution of points over thefreeform surface, even though the definition of uniformsampling is an open problem in mathematical terms evenfor a sphere [145]. Most measuring systems based onoptical principles provide a high number of measuredpoints, with relative point spacing adequate for theinspection of form. However other systems, especially theones relying on contact probing, have limitations inmeasuring speed and may require a more intelligentmeasuring strategy than uniform distribution of pointsover the whole surface. In the following, some criteria forthe optimisation of the measuring strategy will bediscussed.

Sampling strategy

Freeform shapes have continuous curvature changes; theoptimisation of the measurement strategy is typicallybased on the curvature of the part to be inspected. Areasor directions with smaller curvature have high density ofpoints, while in relatively flat areas a lower point density

may be acceptable for slow measuring techniques. Anexample is the measurement strategy for the inspection ofturbine blades on CMMs, based on high density contactscanning of a limited number of cross sections.

Figure 22: Example of sampling strategy based oncurvature, with a constraint on the maximum

distance between points.

Different criteria for more general surface sampling onfreeform shapes are discussed in [2] and summarised asfollows:

Uniform sampling in the u and v parametricdirections. The surface is broken down into arectangular grid, with even u-v spacing along thewhole of the surface. It is the simplest one, althoughits efficiency is limited since it is not sensitive tocurvature change.

Curvature based. For example, local samplingdensity may depend on a specified maximum chordaldeviation, this being the largest distance between aline connecting any two adjacent points and thesurface. This criterion is therefore dependent on thelocal curvature and allows a higher point density atlocations of high curvature.

Minimum sample density. It specifies the maximumallowed distance between any two neighbouringpoints. This criterion is useful in combination withcurvature based ones, to impose a minimum pointdensity for relatively flat surface portions.

Parametrisation-based sampling. The criterion is

based on the parametrisation of the surface; forexample, it specifies the number of points to besampled in each knot span.

Scanning probing systems may be optimised in terms ofmeasurement strategy. In [39] algorithms are proposedfor the identification of the optimal location of theisoparametric curves to be measured on the surface.

For applications in which the data processing involvesinterpolation of measured points, the optimised samplingstrategy is based on spectral sampling (e.g. Chebychevsampling) to prevent interpolation problems known as the"Runge phenomenon" [174] [12].

Adaptive measurements

An adaptive surface sampling strategy is presented in[38], suitable for probing systems delivering both pointposition and surface normal measurements. Using thisextended information and the nominal shape of the part, itshows that an estimate of the form error can be made


14/26 -823-

incrementally as sample points are acquired; therefore,with appropriate stopping criteria the number of points tobe probed is defined.

Adaptive techniques for scanning probing systems havealso been developed. An example of adaptive scanningdescribed in [46] is related to the use of a laser scannerfor fully automated measurement of complex anatomicshapes (like ear and dental impressions [1]); their

measurement is affected by the problem of uncoveredareas due to occlusion. Adaptive scanning is used toperform additional measurements that scan the missingareas until full coverage is obtained with no need forexpert knowledge for the creation of complex scansequences, tuning of scanning parameters or manualstitching/merging of different scans. The first step inadaptive scanning is to determine which areas are notproperly covered in the initial scan. Preferably this is doneby creating the surface model (e.g. by triangulation orfitting of parametric surfaces) or determined directly fromthe point cloud. Dependent on the application someholes, such as the bottom of the object, might be ignored.The full coverage stop criteria might be modified toexpress other priorities, e.g.: holes under a certain

threshold are ignored, only a certain number of iterationsare allowed, maximal scan time or a certain overallcoverage is reached.

Probe path planning

Probe path planning for the measurement of freeformsurfaces has also been investigated. In [67] [2] [34] andother papers the focus is on a general framework forprobe path planning for CMMs equipped with touchtrigger probes; the authors describe algorithms for theextension to freeform shape of typical functionalitiesavailable in computer-aided path planning systems,including measurement simulation, collision detection andoptimisation of the probing sequence.

4.5 Measurement executionWhen measurement strategy and planning have beendefined, the measurement of the freeform shaped partcan be performed using the selected equipment.

The basic result delivered by any measuring device is aset of points that represent the measured surface. Thesedata, in some applications, may be used for thereconstruction of a model of the part; in the next section asummary of this optional procedure is given.

4.6 Reverse engineering

The design and prototyping of freeform shaped parts mayinvolve the use of physical models at some stages of thedesign process. In the automotive industry, for example,the initial conceptual design of a car body is often done bystylists who formalise their ideas by making a clay orwooden model. Similar examples can also be found in theship-building industry, aircraft industry and industries formould and die making where freeform shapes areconcerned [109]. In order to start or continue theproduction process from these physical models, theshape information must be transferred to a CAD systemas a CAD model. Since this process aims at the creationof a CAD model from a physical part, it is called reverseengineering in mechanical engineering, as opposed toconventional engineering. While the latter transformsengineering concepts and models into real parts, inreverse engineering real parts are transformed intoengineering models [175].

Most of the proposed solutions for reverse engineeringare realised in two steps. During the first step, the surfaceof the physical model is measured and typically a relevantnumber of points, often called point cloud, is obtained. Inthe second step, a CAD model is reconstructed from the

measured points. The point cloud is first divided intoseveral subsets with a process called segmentation [175],consisting of the identification of the various componentsurfaces which meet along sharp or smooth edges; eachsubset is then used as reference for the fitting process.Fitting of a freeform surface to a point cloud can beformulated as the creation of a NURBS surface in Eq. (1)that approximates a cloud of mmeasured points within agiven tolerance [97]. The surface parameters to bedetermined from the points are the B-spline functions Buand Bv, uniquely defined by their order ku and kv andknots tuand tvrespectively, the n = nu nvcontrol points cijand their weights wij. In the case of grid distributed points,an interpolating surface can be obtained by choosing thenumber of control points n to be equal to the number ofmeasured points m. However, in most applications aninterpolating surface is not desired, since it may not bestable, due to the random errors contained in themeasured points. The number of control points will thusbe less than the number of measurement points and leastsquares techniques are used to minimise the resultingerror between the NURBS surface and the mmeasuredpoints.

When fitting NURBS surfaces, a large number ofunknown definition parameters are to be identified. Theidentification of all parameters at the same time wouldyield a non-linear least squares problem. The fittingprocedure usually includes the following steps [108]:

1. Parameterisation of measured points.

2. Parameterisation of knots.

3. Applying least-squares fitting to identify controlpoints.

For general least squares surface fitting withoutconstraints, the knots can be set uniformly or by using anaverage method [109]; automatic smoothing proceduresare also available [141].

Surface fitting, subject to boundary geometric constraints,is required when the created surface has to beincorporated into a CAD model consisting of severalsurface patches that join with positional, tangential orcurvature continuity. Positional continuity, sufficient forComputer Graphics, is clearly not enough for someengineering applications, due to functional or aestheticalreasons. Higher order continuity between adjacentpatches can be obtained by advanced modelling of localparameters, as described in [97] [140].

Many other efforts on the improvement of surfacemodelling have been reported, including integration ofsurface representations in more general productmodelling frameworks [91] [170] [84], advanced surfacefitting techniques [190], efficient computation of curves on

surfaces [142], techniques based on neural networks [96][49], interactive modelling [102], adaptive measurementprocedures and integrated modelling [23] andcombination of data from different measuring systems toimprove the efficiency of the reconstruction process [18].

5 DATA EVALUATION AND COMPARISON TONOMINAL GEOMETRY

In the following section, the focus is on the post-processing operations of the measured data and, finally,tolerance verification.

5.1 Registration of multiple measured data

Some measuring systems are not able to measure thecomplete freeform shape of interest in one single setup,due to intrinsic limitations of the measuring principle orpractical and economical reasons. Examples are fringeprojection systems and laser scanners with limited


15/26 -824-

positioning and orientation possibilities with respect to thepart being measured. When a single setup is notsufficient, multiple measurement views are taken with thepart placed in different orientations in front of themeasuring system, or vice versa. The relative alignmentof data points measured in multiple orientations is aprocess called registration [175].

Registration of multiple views is based on overlapping of

measured information and different approaches havebeen proposed. Hardware solutions based on multiplesensors or accurate rotating/ positioning systems is onepossibility, although the cost increase may not beacceptable for some applications. An alternative simplesolution is to add some reference objects (e.g. spheres orother targets) in such a way that at least some of themare measured in each view; the registration in then basedon the measured position of the added reference objects.

Many software solutions have been proposed for theregistration of the multiple views on the basis ofoverlapping measured data only, with and without use ofthe nominal geometry as a reference. This approach isvery popular in computer graphics literature, whereaccuracy is not critically restricted. The preferred

technique is based on the Iterative Closest Point (ICP)algorithm and its modifications [24] [197] [175] [103][178].

5.2 Filtering

Measured surface data require filtering operations for aproper separation of form from other surface geometricalfeatures and for the treatment of measurement noise.This applies also to freeform metrology, even though thelevel of implementation of filtering operations is limited ifcompared to other form measurements (i.e. roundness).Today, most software tools for the analysis of freeformmeasured data perform very basic filtering operations,mainly for the elimination of outliers based on threshold orstatistical detection, and for the reduction of the numberof points in case of too large data sets.

Four classes of filters have been proposed by the recentISO 16610 series of standards [80]:

Linear Filters replace every point of the measureddata with a weighted average of points in itsneighbourhoods. An example is the Gaussian filter,which provides a suppression of higher orderfrequencies exploiting a smoothing function. Anotherexample is the Spline filter, which overcomes somelimitations of the Gaussian filter when measuringcurved surfaces [99].

Morphological Filters are based on mathematicalMinkowski operations, in which a structuring element

(e.g. rolling ball or sliding straight-line segment)interacts in a given data set and modifies its shape.Dilation and erosion are two fundamentalmorphological operations. Dilation is the expansion ofthe input set by the structuring element, while erosionis obtained by shrinking the input set by thestructuring element [120].

Robust filters are tolerant to outliers, i.e. they areinsensitive to extreme points. These are effective inparticular when measurements are affected bylocalized disturbances, or when external featuresinterfere with measurement operations. Forthcomingstandards on robust filters are going to encompass inparticular Gaussian regression and Spline filters.

Segmentation filters are useful for dividing asurface into its constituent homogeneous sub-regions. The subdivision level depends on theproblem being solved: the segmenting operation

should stop when the objects of interest have beenisolated.

There are practical problems using many linear filters(e.g. Gaussian) on freeform surfaces, since the weightingfunction depends on the local geometry. Actually, thedefinition of those filters on a freeform surface is stillunclear; using filtering, only e.g. on the projection to aplane, will introduce distortions. The general use of linear

filters on a freeform surface is an unsolved problem andpart of current research. There are no similar problemsusing Morphological and Segmentation filters [83].

5.3 Alignment of measured data to nominalgeometry

The alignment procedure consists of finding thecorrespondence between the measurement coordinatesystem and the nominal coordinate system of the part,taking into account the type of tolerance to be verified.

In the case of profile tolerance related to datums withsimple geometry, the procedure is straightforward andsimilar to common tolerance verification. Datums aremeasured with an appropriate strategy and the alignmentis obtained using simple software operations for

coordinate transformation. In some applications, thealignment of a freeform shaped part is achieved at themeasurement stage by presenting the part at a desiredposition and orientation, using special tools, fixtures orother physical alignment devices, totally dedicated forspecific products, on which datums may be established.These devices are common in industry even though theyare not an optimal solution; they normally requireprecision manufacturing for both the mating surfaces andthe reference features used to align the part located by aprecision fixture. This procedure is usually costly, due totime and effort required to design and manufacture newfixtures [197] [103].

When no datums with simple geometry are specified for

the profile tolerance, the alignment procedure is based onthe freeform shape itself. Software procedures are thenrequired for mathematically aligning the measured data tothe nominal model.

In general, two alignment steps are needed and may beidentified as coarse and fine alignment.

Figure 23: An example of software based alignmentprocedure: initial stage, before coarse alignment [193].

Coarse alignment

A coarse alignment is necessary for the practicalfeasibility of those measurement processes based oncomputer-controlled measuring devices, and/or forproviding an initial good approximation for fine alignmentalgorithms. The coarse alignment is commonly provided


16/26 -825-

by fixturing devices with a certain degree of positioningrepeatability or by software operations assisted by anoperator, depending of the measuring principle andsoftware in use. Figure 23 shows an example of initialstate for a software based alignment procedure, in whicha point cloud originated by an optical system is aligned tothe CAD model.

A general review of the literature related to alignment

procedures is available in [103] [68]. Methods for thealignment of measured data to nominal geometry for bothcontact and non-contact measuring technology arediscussed. In most of the published approaches, thealignment is based on an iterative process; the ICPalgorithm is widely used also for this purpose.

A feature-based approach has been proposed [105] [104]as an automatic alignment procedure of measured datapresenting no initial coarse alignment to the nominalmodel. In the first step, both the measured data and thenominal geometry are analysed and surface features areextracted by an automatic segmentation process. Basedon curvature, points are classified in four basic shapecategories: concave, convex, saddle and flat. Extractedsurface features are defined as a group of neighbouring

points having the same shape classification. An exampleis shown in Figure 24. In the second step, the extractedfeatures are evaluated to find possible correspondingfeatures. After further classification and evaluation, fromidentified corresponding features, a set of feature centrepoints is calculated for both measured and nominal data.These two sets are then used to calculate thetransformation matrix for the coarse alignment of themeasured data to the nominal model.

Figure 24: An example of automatic surface featureextraction [105].

Fine alignment

After the completion of the initial coarse alignment usingan appropriate method, other mathematical algorithmsmay be used for the fine alignment. The nominal modelmay be represented, in principle, by a set of nominalpoints; the alignment process using measured and

nominal points instead of surfaces is faster, since themathematical representation is quite simple. However, forthe purpose of fine alignment, a nominal surface model isa better reference than a set of nominal points, as alreadydemonstrated decades ago when specifications werebased on nominal points and normals only [182] [185][181] [56] [54]. Today, accurate metrology of freeformsurfaces is based on the comparison of measured pointsto a nominal surface model.

The fine alignment can be calculated using differentmethods. The least-squares method is aligning themeasured data set in such a way that the sum of squaresof the deviations from the nominal surface model isminimised. This method is often the preferred solutionimplemented on analysis software. However it should beremembered that the correct evaluation of tolerances isbased on the minimum zone method, which bettermatches the tolerance definition.

Other alignment methods may be closer to applicationspecific requirements. For example, a weighted least-squares procedure may be more cost effective for thealignment of a freeform shaped part that has to berepaired; different weightings are given to the measuredpoints, depending on the cost needed to eventually fix orrepair the associated actual points [96]; for parts to bemachined, another criteria for alignment could be themaximisation of the minimum positive deviation, in orderto have enough material in all regions of the part.Another alignment method is based the use of virtualgauges for aligning specific points as required by theapplication. Figure 25 shows an example.

Figure 25: Example of virtual gauging through virtualcallipers. In the case here represented, virtual callipersare used to obtain the position of strategic points on the

edge of blades [70].

5.4 Evaluation of measurement

Common evaluation practice

The most common documentation of results is based ona 3D coloured map of deviations from the nominal model.As shown in the example of Figure 26, the CAD model ofthe part is displayed with different colours, each colourbeing associated to a range of deviations from thereference surface. Statistical data is often provided for thenumber of points out of tolerance, average deviations,deviation frequency distributions, etc.Visualisation of cross sections is also common for therepresentation of nominal profile, tolerance zone and

actual profile with magnification of deviations.Application specific parameters

Deviations and associated basic statistics may not beadequate for some applications, therefore specialpurpose evaluation parameters have been defined. Forexample, the measurement of turbine blade requires thecalculation of special parameters defined on crosssections of the aerofoil, as shown in Figure 27.

Curvature evaluation is also important. Principalcurvatures, Gaussian curvature and other curvature mapsare useful in some applications to detect specific surfacedefects [44] [163]. For example, in [87] the evaluation ofshape errors on freeform sheet metal parts wasinvestigated; the change in the principal curvatures was

used for local evaluation, while global evaluation wasbased on aggregate normal vectors for thecharacterisation of the representative directions of theportion of surfaces.


17/26 -826-

An approach based on the separation of topography intothree global form deviations in proposed in [132], where0th, 1st and 2nd order form deviations and relatedparameters have been derived mathematically.

An example focused on local thickness measurements onsheet metal parts is reported in [187]. The evaluation oflocal thickness may be inaccurate, due to incorrectcalculation of surface normals as represented in Figure

28. The proposed Medial Axis Transformation methodallows the identification of the medial surface, on whichnormals the local thickness may be calculated withreduced uncertainty. The calculation method is meetingthe definition of the extracted median surface given byISO 14660-1 and ISO 14660-2 [73].

Figure 26: Example of map of deviations [57].

Figure 27: Example of application-specific parameters forturbine blades, to be evaluated on sections: chord lengthand angle, radius of edges, maximum thickness, rotation

and translations [131].

Figure 28: Error introduced by thickness measurementsbased on surface normals [187].

6 MEASUREMENT UNCERTAINTY ANDTRACEABILITY

6.1 Error sources

The measurement of freeform shaped parts is a fully 3Dmeasurement, affected by the same error sources as for

other complex 3D geometrical features; therefore theiridentification and quantification may rely on the sameevaluation methods. However, some additional errorsmay be significant in freeform metrology and will bediscussed in the following.

Probe tip radius compensation

When using contact probing systems on freeform shapedparts, errors may be introduced by incorrect calculation ofthe correction for tip radius.

A common approach is illustrated in Figure 29 andcalculates the nominal point as a normal projection of themeasured point on the CAD surface. In most cases this isacceptable, although additional errors come into playwhen measuring surfaces with small curvatures in thepresence of form errors and misalignment between thepart and CAD model.

nominal point (target)

actual

point

probing direction

CMM probe

CAD surface

nominal point (recalculated)

Figure 29: Calculation of nominal and actual point.


18/26 -827-

An alternative approach is not to compensate proberadius during CMM measurements but only after therough data have been collected and the coordinatesystem has been well defined during data evaluation; inthis way the cosine-errors are minimized [95]. A possibleimplementation is to calculate offset surfaces in the CADmodel and use the probe centre points for fine alignmentand eventually data evaluation.

Error sources with optical systemsAdditional error sources may appear when using anoptical measuring system on a freeform object. Thesurface characteristic itself dominates the uncertainty ofmeasurement, therefore its variation in terms e.g. of localcurvature may add uncertainty. Other common errors arealso induced by: the slope of the surface (which mayproduce direct reflections to the detector), volumescattering (e.g. for plastic material), or an inhomogeneoussurface texture. Secondary reflections, specularreflections, volumetric scattering, colour transitions, orridges left by machining, may lead to gross systematicmeasuring errors [147] [21] [20].

Deformation effects

The effect of forces due to gravity, measuring principle orclamping operations on a deformable object may lead to asignificantly higher measuring uncertainty in freeformmetrology [183]. The deformation introduced by clampingis more significant for freeform shapes, due to theircomplexity.

Different approaches have been proposed for thereduction of these effects. A functional approach is to usea fixture that simulates the assembly of the part, asshown in Figure 30. The limitations of this approach aretime and costs for the preparation of the fixture. Analternative approach is to rely on rapid prototypingsystems for the preparation of a freeform dedicated fixturewith the same external geometry of the part being

inspected.

Figure 30: Example of functional fixture for the simulationof the assembly constraints on a flexible part [119].

A software based approach for the compensation ofdeformation has also been described [180]. The freeformworkpiece is placed in the measuring volume of theselected instrument with no special fixture; measurementdata is then corrected using a compensation methodbased on a FEM model in which the forces acting on thepart are added as boundary conditions. Figure 31 showsan example.

Figure 31: Example of virtual distortion compensationusing a FEM model with boundary conditions [180]

Software errors

The measuring and analysis software may be a significantsource of errors. The procedures for the registration of

multiple measured data, for example, quickly propagatethe measuring uncertainty found in a single measurementview; a practical example is reported in [147], where themeasurement uncertainty after the registration of a fewmultiple views is three times higher than the uncertaintyevaluated for a single view on a fringe projection system.Other errors may arise from the procedures for thealignment of measured data to nominal geometry, or inthe calculation of local or global evaluation parameters.

Data exchange is another potential error source.Geometric data are frequently exchanged betweendifferent modelling, manufacturing and inspectionsystems. For example a sphere can be represented as aquadratic NURBS surface, where in another system it

may be represented as a procedural type in terms of itscentre and radius. In a third system, the same spheremay be described as a surface of revolution, specifyingthe axis of revolution and a half circle as the crosssection. Some CMM measuring software systems are stillbased on Bezier mathematics and they might not be fullycompatible with the current CAD systems. Translationbetween Bezier and NURBS based systems can beproblematic and the continuity between the surfaces maynot be maintained. These error sources are particularlysignificant for ultraprecision manufacturing processes [11][48].

Therefore, the verification of measuring software is veryimportant and some organisations are concentrating theirefforts on this part of the measurement process. Some

National Metrology Institutes are active in providingreference data sets and services for the verification ofmeasuring and analysis software.

6.2 Performance verification of measuring systems

Performance verification is an important documentation ofthe measurement capability of any measuring device;standards and guidelines are available for the practicalimplementation of performance verification tests on somemeasuring instruments (e.g. the ISO 10360 series ofstandards for CMMs) or will be available in the future (e.g.national and international standards on optical coordinatemeasuring systems, laser trackers, articulated arms, etc.).

In general, performance verification is based on the use

of calibrated artefacts, for which geometry is usually verysimple and restricted to simple shapes, i.e. planes orspheres in different configurations. Such procedures aretherefore quite different from the typical use on themeasuring device when dealing with freeform shaped


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parts. Some examples of procedures and artefacts closerto application found in literature will be briefly describedin the following.

Investigations on contact scanning are described in [151],where calibrated cylinders and spheres have beenmeasured according to procedures called Cylinder testand Spiral test; measuring conditions were similar tothose of freeform measurements, in terms of measuring

speed, point density and complexity of scanning path.The use of cylindrical geometry has also been proposedin the micro range; in [115], the cylindrical surface of twooptical fibers has been used for the vertical calibration ofAFMs. Sinusoidal artefacts, shown in Figure 32, havealso been proposed to evaluate the scanningperformances of different CMMs as a function of speedand direction of the scan [45].

Figure 32: Example of artefacts proposed for tests oncontact scanning [45].

The proposed methods and artefacts may provideadditional information on the behaviour of instrumentswhen measuring actual complex parts. Their geometry isnot a true freeform; this allows low calibration uncertainty.

Freeform shaped artefacts are proposed in [160] for thepurpose of performance verification of optical measuringsystems. A general purpose sinusoidal shape, namedDoppelsinusflche, is shown in Figure 33, together withan example of an application specific artefact.

The relatively high calibration uncertainty andmanufacturing costs are the main reported issues ofcomplex and freeform shaped artefacts; in addition, fulltraceability of measurement results is not provided byperformance verification alone. The following section willtherefore focus on the available methods for uncertaintyevaluation and traceability establishment.

6.3 Methods to establish traceability

The evaluation of the measurement uncertainty, wheninspecting complex tolerances, is a difficult task andcommon praxis is to underestimate its amount [63].Traceability of measurements performed on coordinate

measuring systems (e.g. CMMs) may be guaranteed bytask-specific calibration methods [191], since no generalpurpose calibration of those devices is possible due to thecomplexity of the measuring process. When measuring

freeform surfaces, complexity is also increased by themeasurand itself.

The general model described by the Guide to theexpression of uncertainty in measurement [75] is difficultto apply to complex measurement processes, and morespecific evaluation procedures have been developed,especially for coordinate metrology [191]. Methods forindustrial level uncertainty estimation for almost any kind

of measurements using CMMs, and therefore for theestablishment of traceability of measurements in general,are suggested in a new series of standards currentlyunder preparation (ISO/TS 15530); the only onepublished, Part 3 [79], deals with the substitution bycalibrated workpieces. A new procedure, proposed asPart 2 [76], describes how to evaluate measurementuncertainty by performing measurements on the workpieces, involving different orientations and measurementstrategies. Part 4 [77] will specify common rules for theuncertainty estimation by computer simulation of themeasurement process [173] [171], while Part 5 willprovide guidance for the use of expertise in uncertaintystatements. In the following, a review of implementationsof these methods to the calibration of freeform shaped

parts is presented.

Figure 33: Above: example of general purpose freeformartefact: the Doppelsinusflche. Below: example ofapplication specific artefact: the Shoe Model [160].

Use of calibrated workpieces

The procedure is based on the substitution approach andrequires a calibrated artefact similar to the parts typicallymeasured. Some examples are described in literature,e.g. the already mentioned Doppelsinusflche andShoe Model [160], and the Bevel Gear Gauge,


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proposed for the calibration of CMMs measuring bevelgears [133], for which a synthetic gauge was built upwith basic geometric elements representing the freeformfunctional surfaces of tooth flanks. The declaredlimitations of the Substitution method are related to theavailability of artefacts with sufficient precision, stability,reasonable cost and sufficiently small calibrationuncertainty.

To overcome these limitations, the Modular FreeformGauge (MFG) concept has been proposed [152], in whichthe freeform surface is substituted by the surfaces ofsimple objects, assembled in such a way that the shapeof interest is simulated as closely as possible. The MFGconcept has practical limitations with respect to feasibleconfigurations and similarity requirements; however, itcan help in establishing traceability as shown in Figure34. An example of application to a turbine blade (seeFigure 35) [154], in which geometry has been substitutedby an assembly of a cylinder and two flat surfaces, wasused to evaluate freeform measuring uncertainty in theorder of 2-3 m for high accuracy CMM inspecting profiletolerances using contact scanning.

form and

dimensions of objects

Unit METRE

Single objects with

regular geometry

calibration of form

and dimensions

Length-measuring

capability of CMMs

calibration and

verification of CMMs

ModularFreeform Gauge

(MFG)

relative position

of objects

Actual freeform object

measured on the CMM

Experimental uncertainty

assessment procedure

(ISO TS 15530-3)

form and

dimensions of objects

Unit METRE

Single objects with

regular geometry

calibration of form

and dimensions

Length-measuring

capability of CMMs

calibration and

verification of CMMs

ModularFreeform Gauge

(MFG)

relative position

of objects

Actual freeform object

measured on the CMM

Experimental uncertainty

assessment procedure

(ISO TS 15530-3)

Figure 34: Traceability of CMM freeform measurements

using Modular Freeform Gauges [154].

Use of multiple measurement strategies

The draft ISO/TS 15530-2 [76] describes an experimentalprocedure for the evaluation of uncertainty; the principleis to randomly vary the uncertainty contributors frommeasurement to measurement, changing measurementpoint distribution and object position. The feasibility of themethod was also demonstrated for complex freeformgeometries, by calibration of local deviations from a CADmodel [172]. Experiments on various freeform partsevaluated an uncertainty of 2-3 m for a high accuracyCMM and 4-8 m for a medium accuracy CMM [153]

[154]. The evaluation procedure may be time consumingand deliver overestimated uncertainty, but it does notrequire freeform calibrated artefacts, special software orexpertise [172].

Figure 35: Example of MFG configuration (left) for theuncertainty assessment related to the measurement of a

turbine blade (right) [154].

Computer simulation

A modern solution for the assessment of uncertainty isrepresented by the Computer Simulation approach [173],but currently it has been implemented for simple featuresonly. In principle, the method may be extended tofreeform features, since the CAD model of the object andthe CNC measurement program well represent thegeometrical model of the measurement process. Theinteraction of tip geometry, form errors, positioningaccuracy and misalignment should be taken into accountfor extension of the method to freeform geometry.

6.4

Documents

Metrology of Free-Form Shape Parts