M Kolluri

J P M Hoefnagels
email: j.p.m.hoefnagels@tue.nl

W Van Dommelen

G D Geers

J A W Van Dommelen

An improved miniature

published or not. The documents may come

Introduction

Interfacial delamination is a major concern in the micro-electronics industry (see, e.g. [START_REF] Van Der Sluis | Efficient damage sensitivity analysis of advanced Cu/low-k bond pad structures by means of the area release energy criterion[END_REF][START_REF] Van Driel | Prediction of delamination related IC and packaging reliability problems[END_REF] and Fig. 1(a,b)) due to the high density integration of dissimilar materials into a single so-called 'System in Package (SiP)', driven by ever-increasing demands for miniaturization and multi-functionality. Delamination occurs mainly due to significant thermal stresses generated at these interfaces during thermal cycling, triggered by the mismatch of the coefficient of thermal expansion (CTE) and the Poisson's ratio of the adherend layers. As a prime example, during encapsulation of SiPs with molding compound epoxy (MCE), interfaces between MCE and outer layers of the SiP are often susceptible to delamination (in particular lead frame-MCE and die-MCE interfaces, shown as dashed lines in Fig. 1(c)). Specifically, delamination of copper lead frame (CuLF)-MCE interfaces attracted significant attention from researchers [START_REF] Lee | Failure paths at copper-base leadframe/epoxy molding compound interfaces[END_REF][START_REF] Tanaka | Evaluating IC-package interface delamination by considering moisture-induced molding-compound swelling[END_REF][START_REF] Pan | Studies of the interface between the epoxy molding compound and the copper leadframe by X-ray photoelectron spectroscopy, auger electron spectroscopy, and secondary electron spectroscopy[END_REF][START_REF] Xu | Adhesion behavior between epoxy molding compound and different leadframes in plastic packaging[END_REF][START_REF] Xiao | Characterization and modeling of thin film interface strength considering mode mixity[END_REF][START_REF] Srikanth | Adhesion improvement of EMC-leadframe interface using brown oxide promoters[END_REF][START_REF] Thijsse | Characterization of semiconductor interfaces using a modified mixed mode bending apparatus[END_REF]. Yet the behavior of this interface over the full range of mode mixities and the underlying delamination mechanism are still not well understood. One of the reasons for this is that under certain loading configurations interface delamination occurs in concurrence with plasticity in the bulk layers, which prevents accurate determination of representative Critical Energy Release Rate (CERR) values. Accurate prediction of the interface behavior in SiP manufacturing and exploitation is necessary to minimize these delamination failures and to improve design rules accordingly. Successful prediction of interfacial failure using available interface models [START_REF] Van Hal | Cohesive zone modeling for structural integrity analysis of IC interconnects[END_REF][START_REF] Van Den Bosch | An improved description of the exponential Xu and Needleman cohesive zone law for mixed-mode decohesion[END_REF][START_REF] Borg | Simulating DCB, ENF and MMB experiments using shell elements and a cohesive zone model[END_REF], is only possible with the input of accurate and detailed measurements of the interface behavior obtained from well defined delamination experiments. To this end, this article presents a new improved Miniature Mixed Mode Bending (MMMB) delamination setup capable of measuring CERR values under simultaneous in-situ observation under the scanning electron microscope (SEM) or optical microscope. As an illustrative example, this setup is validated on measurements of the CERR of the CuLF-MCE interface for the full range of mode mixities. It is well documented in the literature that interface fracture toughness is not a unique material property but depends on the entire stress field ahead of the crack tip (i.e. loading mode) [START_REF] Hutchinson | Mixed mode cracking in layered materials[END_REF][START_REF] Suo | Interface crack between two elastic layers[END_REF][START_REF] Liechti | Asymmetric shielding in interfacial fracture under in-plane shear[END_REF][START_REF] Soboyejo | A modified mixed-mode bending specimen for the interfacial fracture testing of dissimilar materials[END_REF][START_REF] Banks-Sills | Interface fracture properties of a bimaterial ceramic composite[END_REF][START_REF] Shi | Determination of interface fracture toughness of the adhesive joint subjected to mixed-mode loading using finite element method[END_REF][START_REF] Rikards | Investigation of mixed mode I/II interlaminar fracture toughness of the laminated composites by using a CTS type specimen[END_REF]. A complete description of an interface necessitates its characterization over the complete range of mode angles from mode I to mode II. Figure 2 shows the loading configurations typically used for testing under mode I (double cantilever beam (DCB) test, Fig. 2(a)) and mode II (end notch flexure (ENF) test, Fig. 2(b)). In contrast, Figure 2(c) shows the mixed mode bending test, which is capable of testing the full range of mode mixities. Besides controlling interface delamination under a well-defined mode angle in a single loading configuration, it is highly beneficial if the experimental methodology is able to carry out accurate in-situ delamination tests of miniature samples under optical and scanning electron microscopes. This enables to precisely identify the crack tip position and to understand the active delamination mechanism in addition to the CERR measurements. Due to the small forces involved with miniature samples, such a setup should not entail undesirable and non-negligible dissipative contributions due to friction and clearance in the moving parts of the setup that prohibit accurate force measurement. Furthermore, non-linearities resulting from the geometry of the setup and from the self weight of the parts of the setup should be avoided. A number of experimental setups, which can carry out mixed mode delamination tests [START_REF] Xiao | Characterization and modeling of thin film interface strength considering mode mixity[END_REF][START_REF] Thijsse | Characterization of semiconductor interfaces using a modified mixed mode bending apparatus[END_REF][START_REF] Soboyejo | A modified mixed-mode bending specimen for the interfacial fracture testing of dissimilar materials[END_REF][START_REF] Reeder | Mixed mode bending method for delamination testing[END_REF][START_REF] Wang | Packaging effects on reliability of cu/low-k interconnects[END_REF][START_REF] Merrill | Effect of mode mixity and porosity on interface fracture of low-k dielectrics[END_REF][START_REF] Blanco | Analysis of the mixed-mode end load split delamination test[END_REF], have been proposed in the literature. The setup of Reeder and Crews [START_REF] Reeder | Mixed mode bending method for delamination testing[END_REF] is well known [START_REF] Thijsse | Characterization of semiconductor interfaces using a modified mixed mode bending apparatus[END_REF][START_REF] Soboyejo | A modified mixed-mode bending specimen for the interfacial fracture testing of dissimilar materials[END_REF][START_REF] Marannano | An analysis of interface delamination mechanisms in orthotropic and hybrid fiber-metal composite laminates[END_REF][START_REF] Tenchev | A correction to the analytical solution of the mixed-mode bending (mmb) problem[END_REF] and can access a large range of mode mixities, however, excluding pure mode I and mode II. Hence, separate DCB tests and ENF tests are required to fulfill the interface characterization over the complete range of mode mixities [START_REF] Thijsse | Characterization of semiconductor interfaces using a modified mixed mode bending apparatus[END_REF]. Despite being suitable for mixed mode testing, none of the above mentioned setups are directly suitable for in-situ testing of miniature samples due to (i) their size being too large to handle miniature samples and to fit in a SEM chamber, (ii) their inability to keep the direction of load application horizontal, which is required to enable the use of standard microscopes to trace the crack tip movement and to visualize the delamination mechanism during in-situ testing, and (iii) the presence of friction resulting from moving hinges and parasitic non-linearities (e.g. through the self weight of the setup acting on the sample), which is especially a problem when testing miniature samples. To overcome these constraints, a redesigned miniature mixed mode bending (MMMB) setup was recently introduced by Kolluri et al. [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF].

Proof of principle measurements on glued interfaces [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF] indeed demonstrated that the MMMB setup was successful in carrying out CERR measurements over the full range of mode mixities, under simultaneous in-situ microscopic visualization of the delamination mechanism. The remaining limitations of the setup proposed in [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF] are (i) the maximum applied load (too limited) and (ii) maximum applied displacement (too small), (iii) instrumental hysteresis that required an elaborate digital image correlation procedure to correct for, and (iv) high operator skill to compensate lack of setup robustness and ease-of-use. Particularly, the first two limitations prevent testing of the industrially-relevant CuLF-MCE interfaces. This paper presents an improved miniature mixed mode bending delamination setup that overcomes the abovementioned limitations, enabling a study on CuLF-MCE samples. First, a brief overview of the original MMMB design is given with its advantages and limitations. Next, the various improvements to the MMMB design are presented, and the design of the improved MMMB setup is confronted with the original MMMB setup. The functionality of the improved setup is demonstrated with tests on specially designed validation samples. Measurement accuracy and reproducibility are addressed by repeated tests on validation samples. Finally, the improved MMMB is used to study delamination in (preplated) CuLF-MCE samples over full range of mode mixities, and under in-situ SEM visualization of the delamination mechanism.

Brief review of the original MMMB setup

In this section, the main details of the loading configuration of the original MMMB design are reviewed [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF]. The mixed mode bending loading configuration shown in figure 2(c) was adopted because of its ability to apply different mixed mode loading conditions and the fact that it is standardized (ASTM D6671-01, 2001, [START_REF]Standard test method for mixed mode I-mode II interlaminar fracture toughness of unidirectional fiber reinforced polymer matrix composites[END_REF]), and widely accepted for interface delamination characterization. Figure 3 (left) shows the design of the original MMMB device to achieve the required mixed mode bending loading. The setup consists of several moving parts and special elastic hinges for frictionless rotation. The loads transferred to the sample, depicted in Fig. 3, can be written as:

P A = P MMMB (1 -ξ) ( 1 a) P B = P MMMB α β ξ (1b)
P C = P B 2 (1c)
P D = -P A + P B 2 , (1d)
where, ξ = H γ is a dimensionless shape parameter that represents the relative position of the applied load. H is the corresponding absolute position of the applied mixed mode load P MMMB and α, β, γ are the characteristic dimensions of the loading mechanism (indicated in Fig. 3). Variation in the mode mixities is obtained by changing ξ over 13 discrete positions from 0 (DCB) to 1 (ENF). The load on the sample can be decomposed in its pure mode I and pure mode II components, P I and P II , respectively, which are defined by the load configurations depicted in the top and bottom right hand side of figure 3. Pure mode II loading is defined as the case for which the separating parts of the cracked specimen deform with the same curvature, resulting in a zero mode I component at the interface. The corresponding expressions for these mode I and mode II components in a bilayer sample with the same materials and equal layer thickness were presented previously [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF]. Here, a more generalized form of these expressions for a bilayer sample with dissimilar materials and unequal layer thicknesses is presented. Details of the derivations are given in the appendix.

P I = P MMMB 1 -ξ - Kα 2β(1 + K) ξ , (2)
P II = P MMMB α β ξ, (3)
where K = P E P F is a constant determined by the geometry and material properties of the two adherend dissimilar layers, which is equal to K = EtIt E b I b under the assumption of linear beam theory, with E i and I i the Young's modulus and moment of inertia for the top and bottom layers respectively. It is clear from the Eqs. 2 and 3 that for ξ = 0, the applied loading corresponds to mode I (DCB) loading (P I = P A = P MMMB and P II = 0). For ξ = 1, the applied load corresponds to the conventional ENF loading (P II = P B and P I = -P MMMB Kα 2β(1+K) ), where the presence of a negative mode I component leads to complications in the measurement of the mode II fracture toughness because of the friction between the two delaminated crack surfaces. For instance, results published by Kolluri et al. [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF] showed a large difference in the CERR values obtained between pure mode II and ENF tests for glued interface samples. However, for any given bilayer sample, there is a position at which the mode I component is zero at the interface and consequently a pure mode II loading is obtained. This pure mode II loading position can be identified in this loading geometry by (see appendix)

ξ II = 2(1 + K)β Kα + 2(1 + K)β . (4)
The pure mode II position is ξ = 0.8 for the case of a homogeneous bilayer sample with equal thickness of the bulk layers and a setup with α = β.

Advantages of the MMMB setup

The following reported advantages of the original MMMB setup are recalled: (i) delamination can be tested over the full range of mode mixities, (ii) precise force measurement for small samples during delamination, (iii) in-situ testing under optical or electron microscope is possible which enables identification of the precise crack tip position and delamination mechanisms, and (iv) frictionless pure mode II test for accurate mode II fracture toughness determination. The functionality of the MMMB setup was demonstrated by in-situ delamination characterization of custom made glued interface structures [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF].

Limitations -Motivation for an improved design

Even though the original MMMB setup was successful in carrying out in-situ tests on miniature structures, some limitations persist. First of all, the setup is limited in maximum force (20N) and displacement (depending on the mode mixity) necessary to achieve delamination particularly in mode II and mode II dominant tests. As reported in [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF], the clearance in the dovetail connectors may still lead to hysteresis in loadingunloading cycles in the MMMB setup. Using image analysis of the displacements of the dovetail connectors, the influence of the hysteresis could be corrected (only) for mode I tests. Therefore, relatively high error margins may still be expected for mixed mode and mode II measurements. Additionally, some of the hinges in this design are unfavorably loaded in compression, which can lead to hinge failure due to buckling. Finally, the elastic hinges used in the design are fragile, and mounting and un-mounting of the sample demands special care. An improved version of the MMMB device is therefore designed and this design is analyzed, and validated on the CuLF-MCE specimens, as addressed in the next sections.

Improved Miniature Mixed Mode Bending setup

The design of the new improved MMMB focused on various fundamental improvements to the original MMMB setup in order to increase its maximum applicable load and stroke limits, to reduce undesirable clearances and to make the setup more robust. Improvements in the design are made by geometry optimization, performed at two structural levels to increase the maximum allowable displacement in the MMMB device. First, the total geometry of the improved MMMB was optimized for maximum global displacement using a finite element beam model of the setup. Next, the geometry of the elastic flexure hinge was optimized to provide more rotation in the critical hinges of the setup. The robustness of the setup was improved by several other modifications to the design including (i) a reversal of all compressive hinges (4 and 5) in the MLM to tensile hinges, (ii) increase in the thickness of the total device, and (iii) incorporation of additional tools for guiding, aligning and securing. and represented with black circles. A beam model was used because it allows for fast analysis and updates of the position of the hinges and insertion of new hinges.

Optimization of the elastic hinge positions

Simulations were performed at all loading positions from ξ = 0 to ξ = 1 to identify the maximum global displacement at each position and the corresponding critical hinge, i.e. the hinge that reaches its maximum rotation first and therefore limits the maximum global displacement. Hinge 1 appeared to be the critical one in most loading conditions, while mixed mode loading positions close to pure mode II were identified as the critical positions where the global displacements are minimal (Fig. 5). To improve the maximum displacements, two new hinges, one at the right bottom (hinge 9) and one at the left top corner (hinge 10) are inserted in the new design to reduce rotation of critical hinge 1. Figure 6 shows the schematic comparison between the old and new geometries at the same global displacement for a mixed mode test. It is clear from the figure that adding the extra hinges (9, 10) significantly reduces the rotation at hinge 1 (at the expense of increased rotation at hinge 6), which increases the maximum global displacement.

From the beam simulations, it became also obvious that more space between hinges 1 and 2 decreases the rotation at hinge 1. In addition, optimization (to maximize distance between hinge 1 and 2) was performed such that the setup uses all the space available (73 × 47 × 29 mm) in the micro-tensile stage (Kammrath & Weiss GmbH) which is sufficiently small to fit in a scanning electron microscope (SEM) vacuum chamber.

Optimization of the hinge geometry

After the above-mentioned optimization of the hinge positions, it was concluded that (depending on the loading position) the rotations in hinges 1 or 6 limit the maximum global displacement of the device. Hence, a special elliptic hinge design, which is optimized for an increased maximum rotation, is used to replace the previous circular hinges in these critical locations. The other hinges were kept circular because they more accurately preserve the center of rotation than elliptical hinges [START_REF] Chen | Geometrical profile optimization of elliptical flexure hinge using a modified particle swarm algorithm[END_REF][START_REF] Chen | A new generalized model for elliptical arc flexure hinges[END_REF]. Figure 7 shows the parameters representing the geometry of the hinge. The maximum rotation of a hinge is determined by the geometry parameters (b/a ratio) that define the profile of the hinge, width w and the thickness t at the thinnest region of the elastic hinge. The thinnest hinge region with thickness t carries the bending stress, σ Mz , and normal stress, σ Fy , that can be applied to the hinge. Optimization of this thickness is done using the closed form compliance equation derived by Smith et al., [START_REF] Smith | Elliptical flexure hinges[END_REF] for elliptic hinges:

M z = θ z Ewt 3 24aγ , ( 5
)
where γ is a lengthy function of b and t. The maximum total stress (σ T ), which is at the surface, is the sum of the stress contribution from the applied moment around the z-axis

(σ Mz = K t Mz.t
2Izz , where K t the stress concentration factor is a function of a and t, and where I zz is the moment of inertia at t) and the applied normal stress along the x-axis (σ Fy = Fy tw ). For a hinge in an elastic state, σ T should remain below the yield strength σ Y of the material, reduced with a safety factor ζ, i.e.

σ Mz + σ Fy ≤ σ Y 1 + ζ . ( 6
)
The maximum rotation of the hinge can be retrieved by rearranging equations ( 5) and ( 6) as,

θ z = σ Y 1 + ζ - F y tw 48aI zz K t Ewt 4 γ(b, t). (7)
θ z in equation ( 7) can be treated as the objective function to maximize, leading to an optimal geometry of the hinge (t, w, a and b). From Eq. 7, it is obvious that θ z increases monotonically with increasing hinge length, a, however, this is counteracted by a decrease in the precision of the rotation. Hence, the hinge profile ( b a = 0.8626) suggested by Chen et al., [START_REF] Chen | Geometrical profile optimization of elliptical flexure hinge using a modified particle swarm algorithm[END_REF] for optimal hinge performance with respect to maximum rotation at minimal stress concentration and maximum precision of rotation was used here. In addition, the width w was increased (from 2 mm in the original setup) to 6 mm in order to increase the allowable maximum axial force F y from 20 N to 50 N. The other constraint is b ≤ 3.25 mm which is limited by the available space in the tensile stage. Using b = 3.25 mm, θ z is plotted as a function of the thickness for an elliptic hinge in Fig. 8. A maximum rotation angle θ zmax of 8.62 • is found (without any safety factor) for t = 28 µm as shown in Fig. 8. With a safety factor of ζ = 0.2, the maximum rotation angle is reduced to θ zmax = 6.56 • for t = 34 µm. Based on the limited sensitivity to t around the maximum (Fig. 8), a value of t = 50 µm was adopted for the final design to avoid the region of high sensitivity of θ zmax to possible machining inaccuracies for small thicknesses. This slightly reduces the maximum rotation angle to 6.27 • . In all, the maximum rotation has improved with 32% compared to the hinges in the original design, using the same safety factor. Moreover, the extent of the profile of the hinge on one side was reduced to the minimum distance (y = x

2 ) without influencing the maximum rotation of the hinge [START_REF] Chen | A new generalized model for elliptical arc flexure hinges[END_REF], in order to maximize the distance between hinge 1 and 2 (Fig. 4).

Final improved MMMB design

Incorporating the above mentioned modifications, FE beam simulations were performed with the improved MMMB design to check its performance. The maximum global displacement was plotted as a function of the relative loading position in Fig. 5, which is improved by more than 60% for the critical loading positions close to pure mode II. Then, the improved MMMB device was manufactured by electrical discharge machining of the Ti-6Al-4V alloy plate. The final design of the improved MMMB device in comparison with the original MMMB device is shown in Fig. 9. All the important changes are highlighted and numbered. It is also noted that the improved design has 16 mode selector positions, instead of 13, in order to more accurately select the mode angle in the experiment. The main parts of the improved MMMB setup mounted in the micro-tensile stage are depicted in Fig. 10. Besides the above mentioned improvements in maximum load and displacement, also the remaining nonlinearities and robustness were further addressed, which is discussed next.

Clearance and Robustness of the setup

As was reported in [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF], the clearance in the dovetail connectors resulted in hysteresis in the load displacement curves during loading-unloading tests in the original MMMB setup. The hysteresis can only be corrected for in mode I tests by tracking the connector clearance using a digital image correlation technique [START_REF] Kolluri | Study of interface delamination by a newly designed miniature mixed mode bending setup[END_REF], a correction that is not possible in mixed mode and mode II measurements. To overcome this problem, new tapered dovetail connectors with wedge locking tools were manufactured as shown in Fig. 10b. With these tapered dovetails, mounting of the sample becomes easy and requires a relatively small force (in contrast to the original setup) which avoids any deformation of the hinges in the setup. After mounting the sample into the setup, the wedge tools are inserted to lock the dovetails without any clearance.

Since the elastic hinges used in the design are fragile, any damage of the hinge due to small forces during insertion and removal of the sample and the wedge locking tools and during actual loading should be avoided. Hence the following features have been adapted.

(i) The thickness of the device is increased from 2 to 6 mm (ii) The hinges at the mid loading position (hinge no. 4 and 5) are designed to be under tension to avoid any buckling of the hinges due to compressive loading in mode II dominated tests.

(iii) A separate bottom hinge shown in Fig. 10 (as 2b) is used when the load at this hinge is compressive (P D < 0) to avoid an unstable bottom hinge configuration.

(iv) Several components including guiding pins (to prevent in-plane motion) as well as a top and (movable) bottom supporting plate (to prevent out of plane motion) were introduced to provide mechanical isolation of the hinges and highly accurate alignment of the MLM during sample insertion and removal.

All these features significantly increase the robustness of the setup. Additionally, using these locking tools, the interface to be tested will also be protected from any external loads during the mounting process. In conclusion, the new robust setup increases the precision of the experimental load-displacement measurement by minimizing the risk for handling errors.

Analysis of the new improved MMMB setup

Validation and accuracy assessment

The performance of the improved MMMB setup is evaluated using specially designed validation samples to assess the influence from inaccuracies in the manufactured geometry, machine compliance, or other factors such as clearance at the connectors. These samples are homogeneous single layer brass samples (i.e. without an interface and hence no propagating crack) with a well-defined notch, having an opening width of 30 µm, representing an existing crack of fixed length. The thickness of these samples is 1 mm. The inset in Fig. 11a shows the extremity with the notch in one of the test samples. Test samples with 5 different notch lengths (3, 6, 9, 12 and 15 mm) were used to check the performance of the setup. Tests were performed at all loading positions from mode I to pure mode II (ξ = 0 -0.8). Figure 11a shows the results of these tests performed on the 12 mm notch sample. The hysteresis observed in these tests is negligibly small, confirming the efficacy of the new wedge-locking dovetails with minimal clearance at the connectors (Fig. 10b). The relative hysteresis, defined as the dissipated energy during a loading-unloading cycle, relative to the energy supplied during loading was calculated for all crack lengths and different loading positions. The maximum relative hysteresis was found to be 2% for pure mode II and the mean value is 1% for all loading positions. This is at least a factor of 2 smaller compared to the original design [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF], even without the need for an elaborate digital image correlation correction procedure. To assess the precision of the complete experimental procedure, repeated tests were performed at one loading position by unmounting the sample after each test and remounting it for the next test. Because of the increased robustness and the new alignment tools, consistent results can be obtained (Fig. 11b). Specifically, the error in the stiffness was less than 0.7% which demonstrates the achieved precision of the new design.

Finite Element Analysis

To determine the mode angle at the interface for different relative loading positions, finite element (FEM) simulations were performed for all loading positions using the beam model of the setup and an elastic material model of the CuLF-MCE sample. A fixed length of the precrack was modeled with a special rosette shape crack tip mesh. The mode angle, ψ, is defined by the orientation of the stress field. Because of the oscillating stress field in the dissimilar material interface (CuLF-MCE interface) [START_REF] Hutchinson | Mixed mode cracking in layered materials[END_REF], ψ is defined at a certain reference length, δ:

ψ = arctan σ 12 σ 22 δ , (8)
where σ 22 is the normal stress, σ 12 is the shear stress. The plastic process zone size is a common choice for the reference length δ [START_REF] Hutchinson | Mixed mode cracking in layered materials[END_REF][START_REF] Suo | Interface crack between two elastic layers[END_REF] as it is the only intrinsic length scale in the system. The size of this process zone can be defined by the point where the linear elastic stress field equals the yield stress of the weakest of the two materials [START_REF] Hutchinson | Mixed mode cracking in layered materials[END_REF], which is 5 µm in case of the CuLF-MCE samples. The mode angles calculated from the FEM simulations are plotted as a function of loading position in Fig. 12. The figure also shows this relation for a homogeneous bilayer sample with equal thickness of the bulk layers. These calculations demonstrate that the setup probes interface delamination over the complete range of mode angles for both homogeneous and dissimilar bilayer samples. Additionally, a strong feature of the MMMB concept is that the mode angle stays constant within a few degrees for different crack lengths and MLM displacements, as was shown previously [START_REF] Kolluri | In-situ characterization of interface delamination by a new miniature mixed mode bending setup[END_REF]. It is also worth mentioning that FEM simulations taking into account the actual geometry of the setup, showed that at a maximum load of 50N, the displacement resulting from the compliance of the MMMB setup can be neglected with respect to the measured global displacement.

Study of coated copper lead frame -molding compound epoxy (CuLF-MCE)

Experimental procedure

In-situ experiments with the improved MMMB setup were conducted in a scanning electron microscope and under an optical stereo microscope at high magnifications. A batch of bilayer samples (manufactured at Philips Applied Technologies), consisting of 0.2 mm thick preplated (Au-Pd-Ni) copper lead frame and 0.5 mm thick molding compound epoxy, was tested in these experiments. The samples were prepared by heating the lead frame to 180 • C for 2.5 minutes followed by high pressure hot injection molding of the molding compound epoxy. Afterwards, the samples were laser cut to the required dimensions (35 × 5 mm). After the molding, the samples received a post mold cure for 4 hours at 175 • C. At the end of each sample, a well defined 6 mm long pre-crack was created by sliding a thin knife edge between the two layers using a home-built device.

Prior to the experiment, samples were fine polished on one side (perpendicular to the plane of the interface) to visualize the interface at high magnifications. The dovetail connectors were attached on both sides of the samples with a stiff glue at the pre-cracked end, after which the samples were carefully mounted into the setup, by applying the robust sample insertion procedure with locked elastic hinges, as explained in section 3.4. Finally, the position selector was reinserted at the appropriate position to carry out a delamination experiment at the desired mode angle. During the test, the sample was unloaded and reloaded at regular intervals. At each load reversal, images of the crack tip were recorded at a magnification of 250x. These images were used to determine the crack length corresponding to the load reversals in the post processing analysis. On the basis of these experiments, the interface properties of the preplated CuLF-MCE interface structure were characterized over a complete range of mode mixities, thereby demonstrating the strengths of the setup.

Results and discussion

Figure 13 shows the load-displacement plots of the in-situ experiments conducted under a stereo microscope, at mode I (ξ = 0) and mixed mode loading positions (ξ = 0.4, 0.67 and 0.8). Initially, in all mode mixities, the load increases linearly with increasing displacement until the onset of crack initiation. Subsequently, a gradual drop in the load is observed with further displacement due to crack propagation. It is observed that the maximum at the onset of crack initiation in the first loading-unloading cycle is less pronounced compared to the sharp maximum observed in the subsequent cycles, as highlighted in red in Fig. 13 for the mode I curve. This smooth behavior in the first cycle is attributed to the gradual development of a sharp microscopic crack tip and stable crack front normal to the crack propagation direction. Hence, before the actual delamination experiment at certain mode mixity, each sample was first loaded in mode I (position ξ = 0), until the initiation of a sharp stable crack front was established. This first mode I cycle was ignored for the calculation of the CERR and is also not shown for the mixed mode experiments in Fig. 13. The critical energy release rate of the interface was calculated from the area between the successive loading-unloading stiffness lines (e.g. the hashed region in Fig. 13), divided by the delaminated area corresponding to an increase of the crack length from position a 1 to a 2 . A strong feature of the (improved) MMMB setup is that it allows highly accurate determination of the crack tip position under the SEM, therefore, there is no need to use approximate analytical formula or numerical tools for the crack length prediction to calculate the CERR values. The resulting CERR values are shown in Fig. 14 as a function of the mode angle (the latter is determined from FE simulations). Note that the CERR measured represents the macroscopic interfacial fracture energy, which may include contributions from other dissipative mechanisms like plasticity in the layers and/or at the interface. This is particularly the case when a ductile layer is present in the composite stack [START_REF] Lane | Plasticity contributions to interface adhesion in thin-film interconnect structures[END_REF][START_REF] Strohband | Interface separation in residually-stressed thin-film structures[END_REF]. For instance, the inset in Fig. 13 shows a zoom around the origin of the load-displacement plot for ξ = 0.8, revealing a residual opening after complete unloading indicating plasticity contributions to the measured CERR. Identification of the individual contributions of plasticity, e.g. to eliminate bulk plasticity from the calculation of CERR, requires a dedicated numerical-experimental methodology which is beyond the scope of this paper. For the present work, the maximum contribution to the measured CERR value due to bulk plasticity has been estimated from Fig. 13 by assuming a full damage unloading behavior to the origin. A maximum deviation of 6% about the mean CERR value was observed in mode II dominant test at ξ = 0.89, while a negligible influence was observed in mode I or mode I dominant mixed mode tests. Besides the energy absorbed between two successive loading-unloading cycles, other important details are visible in Fig. 13. The first important observation is that the hysteresis of the setup, which can be measured from the unloading-reloading curves is negligible in all of the tests (even better than Fig. 11(a)). It is also clear that the curves (during crack growth) are smooth for mode I dominant tests (at ξ = 0 and 0.4) compared to the wavy curves in mode II dominant tests (ξ = 0.67, 0.8). This difference in behavior can be attributed to the roughness of the bonded surfaces (which is ∼1 µm RMS for the copper layer before molding) at the interface. Locally, the micro-scale roughness can lead to local shearing of and/or sliding between roughness asperities causing additional frictional dissipation which leads to an increased macroscopic interface fracture energy. In both cases, apart from the bulk layers that need to store more energy, a local crack opening displacement on the order of the roughness or size of the asperities is necessary for the crack to grow. In shear dominant tests (e.g. pure mode II), where the crack opening in normal direction is constrained, individual asperities will need to shear for the crack to grow. Therefore, either frictional sliding or local shearing can explain the discrete crack growth with its resulting characteristic wavy behavior for mode II dominant curves. To substantiate this hypothesis, in-situ delamination tests under SEM at high magnifications were carried out, confirming that the crack grows smoothly in mode I dominant tests (ξ = 0-0.53), compared to more discrete or jerky crack growth in mode II dominant tests (ξ = 0.53-1). This is also consistent with the trend in the CERR in Fig. 14, where a CERR of ∼ 4.5 J/m 2 that is constant within experimental uncertainty was observed until ψ ≈ 15

• (i.e. ξ = 0-0.53), followed by a sharp increase with increasing mode angle towards position ξ = 1. In addition to the CERR measurements, real-time microscopic visualization allowed extraction of additional information at the delamination front to trace the formation and propagation of the crack in detail, providing valuable input for understanding of the fracture mechanism and for predictive simulations of the delamination in these interfaces. Specifically, the microscopic images allow for the following measurements.

(i) The precise crack tip position can be determined within an accuracy of 5 µm.

(ii) The delamination mechanism can be visualized at the interface. Figure 15 shows the evolution of the crack in the process zone. A brittle cleavage type of failure was observed as the dominant mechanism indicating the brittle nature of the interface. At high magnifications, some bridges were observed as in Fig. 16.

The displacement field during delamination can be mapped a digital image correlation (DIC) technique. a first of resolving local strain fields, a DIC generated y-displacement field around the crack tip is shown in Fig. 17. By measuring the evolution of the full profile of the crack opening during initiation of the delamination crack, important local information such as the elongation of the interface (glue) layer at the crack tip, and the rotation of the loading point can be extracted, which can be used in combination with the measured applied load and the sample width to calculate the full interface traction-separation law using the approach proposed by Andersson et al. [START_REF] Andersson | The stress-elongation relation for an adhesive layer loaded in peel using quilibrium of energetic forces[END_REF].

Conclusions

An improved miniature mixed mode bending setup capable of in-situ delamination testing in advanced microscope systems (e.g. scanning electron microscope) has been proposed, designed, manufactured and tested. Improvements in the present design are: (i) optimal number and location of the hinges, (ii) optimal elastic hinge geometries for increased maximum rotation and (iii) minimized clearance at the connectors and increasing robustness of the setup with additional guiding and alignment tools. It was demonstrated that the improved setup exhibits (i) a wider application range to more interface systems because of its significant increase of the maximum accessible load and stroke in all mode mixities, (ii) good accuracy in measuring the load-displacement response, as demonstrated from experiments on validation samples, confirming the significantly reduced hysteresis, (iii) excellent experimental reproducibility characteristics due to the newly added alignment tools.

The improved setup was used for testing industrially relevant CuLF-MCE samples in the full range of mode mixities. Unloading-reloading tests performed at different mode mixities were used to calculate the CERR. In addition to the CERR measurements, realtime microscopic visualization allowed the identification of the difference in the crack growth behavior between different mode mixity tests. At small mode angles (ψ < 15 • ) relatively smooth crack growth behavior was observed resulting in a constant CERR value of ∼ 4.5 J/m 2 . At mode angles greater than 15 • , however, a jerky crack growth was observed with a serrated load-displacement response, which suggest its direct correlation with the observed increase in CERR with increasing mode angle from 15 • to 90 • . Insitu visualization also allowed to determine the precise position of the crack tip, and the crack opening displacement using DIC technique, as well as a qualitative assessment of the active delamination mechanism at the interface. A brittle cleavage failure type was observed for the investigated CuLF-MCE samples, while at high magnifications, some crack bridges were also observed. which is the equation for mode I component (Eq. 2). For pure mode II loading, substituting P I = 0 in Eq. 2 and solving for ξ yields Eq. 4:

ξ II = 2(1 + K)β Kα + 2(1 + K)β . (4)

Figure 1 .

 1 Figure 1. (a,b) SEM micrographs (arrows) showing delamination between copper metal lines and dielectric material in back end structures (Courtesy of NXP Semiconductors). (c) Schematic of the lead frame package molded in a molding compound epoxy. All the interfaces which are prone to delamination are marked as dashed lines.

Figure 2 .

 2 Figure 2. Sketch of different loading configurations for interface delamination; (a) double cantilever beam (DCB) test; (b) end notch flexure (ENF) test; and (c) mixed mode bending test.

Figure 3 .

 3 Figure 3. Design of the original MMMB device with: (1) the main loading mechanism (MLM) with (2) elastic hinges, (3) the bottom support hinge, (4) the left support, (5) the bilayer sample with (6) precrack, (7) the dovetail connector, and (8) loading tip. Also shown is the schematic representation of the load distribution under mixed mode bending, with the loads on the sample (shown on the left) and the decomposition of these loads into mode I and pure mode II components (shown on the right).

Figure 4 .

 4 Figure 4. FE beam model of the MMMB setup with new additional hinges. An asterisk '*' indicates the newly added elastic hinges (9 and 10). Details of the nodal ties and hinges are also shown.

AFigure 5 .

 5 Figure 5. Comparison of the maximum global displacement between the original and improved MMMB configurations, obtained from FE beam simulations performed on a homogeneous bilayer sample with a fixed crack length of 6 mm.

Figure 6 .

 6 Figure 6. Schematic comparison between the old (red) and new (green) geometries loaded to the same global displacement for a mixed mode test, showing how the introduction of new hinges (9, 10) significantly reduces rotation (ω < η) at the critical hinge (no. 1).

Figure 7 .

 7 Figure 7. Elliptical flexure hinge with geometrical parameters. The stress distributions across the thickness at t due to the applied moment and the axial force are also shown.

Figure 8 .

 8 Figure 8. Maximum rotational angle θ z as a function of the thickness t of the hinge for w = 6 mm, b = 3.25 mm and b a = 0.8626. Inset: (a) a conventional elastic hinge, (b) the circular hinge used in the original MMMB setup and (c) the new elliptic hinge design with a reduced profile on one side. The safety element closes the safety groove, g, once the hinge rotates to the maximum allowable rotational angle.

Figure 9 .

 9 Figure 9. Design of (a) the original and (b) the improved MMMB setup. Improvements are highlighted and numbered: (1) introduction of new hinges, (2) replacement of the critical hinges with elliptical hinges for increased rotation, (3) optimization of hinge geometry and increased robustness, (4) redesign of compressive hinges to tensile loading, and (5) increased length of the MLM.

Figure 10 .

 10 Figure 10. (a) the improved MMMB setup mounted in the micro-tensile stage: (1) main loading mechanism, (2a,2b) bottom hinges, (3) sample, (4) position selector, (5) handling plate, (6) micro-tensile stage, (7) position bar, (8) guiding rods, (9) set screw, (10) screw lock, (11) alignment pins, (12) sample height adjuster, (13) alignment screw, (14) bottom plate. (b): (15) new tapered dovetail connectors and, (16) wedge locking tools that remove any clearance in the connectors. (c): (17) top supporting plate (to lock the device during insertion and removal of the wedges) and (d): improved MMMB setup mounted in (18) SEM chamber.

Figure 11 .

 11 Figure 11. (a) Load-displacement results for loading-unloading cycles with the 12 mm validation sample at different loading positions (indicated by ξ). The inset shows the notched extremity of a validation sample used for mode I to pure mode II tests ξ = 0 -0.8. (b) Results of complete experimental repetition of 3 validation tests performed at ξ = 0. An error of less than 0.7% in stiffness was observed.

Figure 12 .

 12 Figure 12. Mode angle obtained by FEM analysis as a function of relative loading position (ξ) for (i) a homogeneous equal thickness bilayer sample and (ii) the CuLF-MCE sample used in this study, with a crack length of 6 mm at 200 µm MLM displacement.

Figure 13 .

 13 Figure 13. Comparison of load-displacement plots of a mode I (ξ = 0) and mixed mode (ξ = 0.4, 0.67, 0.8) experiments conducted on CuLF-MCE samples. The inset shows a magnification around the origin of the ξ = 0.8 curve illustrating the residual opening after unloading to zero load.

Figure 14 .

 14 Figure 14. Critical energy release rate with standard deviation bandwidth as a function of mode angle.

Figure 15 .

 15 Figure 15. SEM micrographs showing the evolution of crack (from a-c) in the process zone during delamination. The crack is growing from the right to left.

Figure 16 .

 16 Figure 16. SEM micrograph showing small cracks and a limited number of bridges at the interfaces.

Figure 17 .

 17 Figure 17. SEM micrograph with a DIC overlay of the y-displacement at a certain opening. Opening displacement as a function of section length is shown in the graph below.

Acknowledgements This research was carried out under the project number MC2.05235 in the framework of the research program of the Materials Innovation Institute M2i (www.m2i.nl). The authors are grateful to Marc van Maris for his assistance in conducting some of the reported experiments and Lisanne Dautzenberg for her contributions to the FE simulations. Peter Timmermans and Olaf van der Sluis from Philips are acknowledged for their help in manufacturing and supply of samples.

Appendix

Expressions for the mode I (Eq. 2), mode II (Eq. 3) and relative loading position for pure mode II (Eq. 4) are derived here. From Fig. 3, the force and moment balances directly yields Eq. 3,

Other balance equations are, P A = P I + P E (A.1)

Combining K = P E P F with Eq. A.2 gives,

3)

Substituting Eqs. 1a and A.4 in Eq. A.1, results in,

Furthermore, substituting Eq. 3 in Eq. A.5 yields,


Luca Di Angelo

Paolo Di Stefano

Anna Eva Morabito

Di Angelo

L Di

P Di

Automatic evaluation of form errors in high-density acquired surfaces

Keywords: TOLERANCING, INSPECTION user): Geometric Product Specification, Form Errors GPS tolerancing, automated inspection, form error evaluation

In this paper the authors present an original methodology aiming at the automation of the geometric inspection, starting from a high-density acquired surface. The concept of intrinsic nominal reference is herein introduced in order to evaluate geometric errors. Starting from these concepts, a new specification language, which is based on recognizable geometric entities, is defined. This work also proposes some surface differential properties, such as the intrinsic nominal references, from which new categories of form errors can be introduced. Well-defined rules are then necessary for the unambiguous identification of these intrinsic nominal references. These rules are an integral part of the tolerance specification. This new approach requires that a recognition process be performed on the acquired model so as to automatically identify the already-mentioned intrinsic nominal references. The assessable errors refer to recognizable geometric entities and their evaluation leaves the nominal reference specification aside since they can be intrinsically associated with a recognized geometric shape. Tolerance specification is defined based on the error categories which can be automatically evaluated and which are an integral part of the specification language.

Introduction

The verification process is the manufacturing step at which a metrologist determines whether the real surface of a workpiece conforms to the range of permissible deviations that have been specified during the design process. These specifications express the functional requirements that have to be satisfied so that the manufactured parts work as designed. In the last few years, geometric inspection has become critical due, on the one hand, to the increase of geometrical complexity and product variety, and on the other hand to the ever-growing demands for geometrical accuracy. Consequently, a reliable, effective and automated inspection system will definitely improve the industrial manufacturing responsiveness and increase quality and product competition.

In order to harmonize and fill in the gaps of what has already been standardized in tolerancing and metrology by the ISO, the technical committee ISO/TC 213 has recently proposed a language, supported by mathematical principles, which is referred to as Geometrical Product Specification (GPS).

The measuring equipments, with their specific way to acquire the geometric properties of the object under verification, affect error evaluation and specification; GPS standards include this concept. With the advent of CMM (Coordinate Measuring Machines) and high-resolution optical digitizers, alternative approaches for 2 dimensional and geometrical error definition and evaluation can be identified. In particular, nowadays, the approach to be used cannot ignore the fact that a large amount of point data can be extracted from the workpiece itself, by means of optical 3D scanners characterized by an increasingly high accuracy. The measurement carried out by these devices is not a set of single measures (as it is the case when using a CMM device); rather, it consists in a 3D acquisition of the real object. The acquired points identify the surfaces of the object so it is possible to recognize from them the geometric properties from which to detect references for error evaluation. The acquired surface can be compared to the recognized nominal entities which assume the role of virtual references because of their mathematical definition. For this reason, new categories of nominal reference can be conceived which may improve on the signs in the traditional language for tolerance specification. Error evaluation always requires the knowledge of the corresponding nominal reference. In this work nominal references are classified into two main categories: explicit and intrinsic. The explicit reference can be given by a specification, also by using a CAD model [START_REF] Prieto | An Automated Inspection System[END_REF], Gao et al. 2006). Intrinsic references are nominal entities that can be recognized in the scanned object, independently of the tolerance specification.

This paper then presents an original methodology aiming at the automation of the inspection, starting from a high-density acquired surface. This new approach requires that a recognition process be performed on the acquired model so as to automatically identify the intrinsic nominal references. The identification of intrinsic nominal references avoids the registration process between CAD model and data points. What is here carried out is an original classification of the form error categories for the recognized surfaces based on the above-mentioned concepts. Specific form tolerances are defined for any recognizable surface. These categories, together with the traditional ones, have been implemented in an original software and tested in real cases. An example is hereafter presented.

Related works

Recent efforts have been made to develop methods which apply non-contact digitizing techniques to the geometric inspection. A great number of the methodologies proposed in literature require the knowledge of the CAD model of the workpiece under inspection, which provides the nominal references in the form of the analytical surface describing the geometric model [START_REF] Prieto | An Automated Inspection System[END_REF][START_REF] Li | Free-form surface inspection techniques -state of the art review[END_REF], Li and Gu 2005, Li and Gu 2005, Gao et al. 2006). Since tolerance specifications usually refer to some features of the workpiece, the mapping between one surface (or feature) of the CAD model and the corresponding scanned point sub -cloud has to be performed. Tolerance specifications can be either included in the CAD model as textual information, or interactively defined by the user. Nowadays no standard language able to specify tolerances and suited to automatic verification has yet been defined. [START_REF] Prieto | An Automated Inspection System[END_REF] propose and implement a methodology for the automated inspection of manufactured parts. This methodology is able to verify both dimensional and geometric tolerances. Firstly, the high-density point cloud is registered with the corresponding CAD model of the workpiece by means of the Iterative Closest Point (ICP) algorithm. The 3D point cloud is then segmented by associating the points matching the same local geometric properties with the nearest CAD surface. Since the registration process minimizes the sum of the squared distances between the point cloud and the CAD model, the authors state that the distances, between a generic CAD surface and the corresponding sub -cloud, will have a Gaussian distribution. The specified errors are estimated based on the average value and the standard deviation of the point location distribution. Particularly, the form error is evaluated as four times the standard deviation of the distances between the points and the analogous CAD surface which approximates them. A similar approach is proposed by Li and Gu [START_REF] Li | Free-form surface inspection techniques -state of the art review[END_REF], Li and Gu 2005, Li and Gu 2005). Gao in (Gao et al. 2006) develops an automated geometric inspection system within a commercial RE software. The authors define a Nominal Inspection Frame (NIF) for a CAD model or a reference digitized model (Master Model) in which every dimensional and geometrical tolerance specification may be defined and interactively specified by the user. A similar approach is proposed by Campana et al (Campana et al. 2006), which automatically performs the mapping between the tolerance specification in the CAD model and the corresponding acquired sub -cloud.

All these approaches evaluate the form errors in the same way as the "profile tolerance of a surface" does (ISO 1101(ISO : 2004) (the tolerance zone is limited by two CAD surfaces placed up and down the CAD reference surface at a distance t/2 from it), without taking into account specific geometric properties of the surface (for example, axially -symmetric surface, extruded surface, etc.). Some specific properties of the surface can actually play an important functional role in the object. For this purpose, ISO 1101 and ASME 14.5Y consider form tolerances, such as straightness, circularity, etc., which can be applied to derived or extracted features from the surface (axis, planar section of the surface, etc.). The approaches presented in literature, which use a CAD model as an analytical reference, show limitations if compared to the traditional approach to tolerance inspection. In other words, the specification "language" based on these methods is generally characterized by fewer terms than in the traditional approaches of tolerance specification. Furthermore, these approaches do not take advantage of the specific way to inspect the real object and the numerical devices that can be used to evaluate the acquired point cloud. Based on these specific characteristics, new categories of form error can be introduced in accordance with the duality principle reported in GPS standards [START_REF] Nielsen | New Concepts in Specifications, Operators and Uncertainties and Their Impact on Measurement and Instrumentation[END_REF]. Finally, another drawback of the tolerance inspection methods based on CAD systems is the lack of standards for geometric data exchange.

Automatic error evaluation

The methodology proposed in this paper is suited to automatically evaluate the form errors of high-density acquired surfaces. The flowchart of the methodology is shown in figure 1. It can be decomposed into the following four steps:

(1) Non-ideal feature recognition;

(2) Nominal reference association;

(3) Error evaluation; (4) Form tolerance evaluation.

During the first phase, the GPS feature operation, called partition, is carried out. In the GPS language this operation aims at "obtaining, from the non-ideal surface model (skin model) or the real surface, the non-ideal features corresponding to the nominal features" (ISO/TS 17450-1: 2005).

In this work the non-ideal feature is defined as a set of adjacent mesh vertices that are recognized as smooth and of the same type (flat, umbilical, ruled and generic). The concept of non-ideal feature then involves the concept of point smoothness. In a continuous surface a point is smooth if its neighbourhood can be represented by a smooth function. In other words, in a non -smooth point the differential geometrical properties cannot be defined. Typically, non -smooth points are those pertaining to the edges of the geometrical model where two different surfaces meet. The concept of smooth point can be extended from a point of a continuous surface to a vertex of a tessellated surface. In a tessellated surface the point smoothness property must be recognized, by using some criteria (Di [START_REF] Angelo | A New Method For Automatic Detection Of Non-Regular Points In Triangular Meshes[END_REF].

Error evaluation always requires the knowledge of a nominal reference from which the error can be measured. The nominal reference can either be given explicitly (for example, by a GPS specification or a CAD model) or can be implicitly or intrinsically present in the acquired object. In the related literature several researchers have used a CAD model as a nominal reference [START_REF] Prieto | An Automated Inspection System[END_REF], Gao et al. 2006). In this work the nominal references are recognized as intrinsic references from the acquired workpiece. The categories of valuable errors can be deduced from these references. Error evaluation involves the quantitative identification of the reference element from which the deviation can be evaluated. In figure 1 this phase is referred to as Intrinsic Nominal Reference association. For this purpose, some specific rules, referred to as association rules, need to be introduced. They play an important role in quantitatively identifying the references; in fact, different rules can produce different estimations of the references. These rules then assume the role of identifiers of nominal references just as the reference surfaces of a gauge do in the traditional approach to error measurement.

In the next phase, and for each identified intrinsic reference, the deviations from the corresponding associated non-ideal feature are evaluated and assumed as form errors.

The procedure ends with the comparison between the estimated form errors and the GPS specifications. Thus, from all the errors associated with all the intrinsic references identified, some are extracted based on the tolerance specifications.

Non-ideal feature recognition

To carry out a reliable recognition of the non-ideal feature types it is necessary to segment the acquired surface into regular continuous patches. Segmentation is a complex procedure that makes it possible to capture high-level geometric information that is essential for the following Intrinsic Nominal Reference association. The non-ideal feature recognition is carried out based on the procedure outlined in figure 2, where two main segmentation levels are identified.

The first level aims at identifying regular surfaces or, equivalently, at detecting surface discontinuities. The second level is intended for recognizing the geometric shape by investigating some intrinsic local and global differential properties of the tessellated model. The evaluation of these differential properties is a complex task since the triangulated model represents only a discrete approximation of the smooth geometry. In literature, two are the main approaches used for data segmentation; namely, the edge-based, the region-based methods [START_REF] Liu | Automatic segmentation of unorganized noisy point clouds based on the Gaussian map[END_REF]. In order to overcome the lack of robustness, non-uniqueness or complexity deriving from the use of one of these two strategies, some hybrid approaches, based on edge-based and region-based methods, have been developed. This work makes use (3) segmentation of the triangles into groups, corresponding to surface features characterized by some definite geometrical properties; (4) identification of the non-ideal features by analyzing the global properties of the segmented triangles.

At the end of this process, the triangles are segmented in accordance with the following analytical features: plane, sphere, cylinder and cone. The non-ideal feature types, which cannot be included among the analytical features, are classified, based on the recurrence of specific differential geometric properties among the points pertaining to the non-ideal feature, as: generic extruded, generic cone, generic axiallysymmetric and free form.

Intrinsic nominal reference

In order to perform an automatic error evaluation, we introduce the original concept of Intrinsic Nominal Reference. The Intrinsic Nominal Reference is associated with the concept of recognisability. There exists an Intrinsic Nominal Reference when a property is recognized as common to a set of adjacent points of the point cloud of the acquired object. For example, planarity is an intrinsic property since it can be recognized in a set of points that are evaluated as sufficiently close to a plane surface. In this paper three different types of intrinsic nominal references are being considered:

-Intrinsic Shape Reference (ISR); -Intrinsic Derived Reference (IDR); -Intrinsic Local Reference (ILR).

Depending on the recognized non-ideal feature type, one or more Intrinsic Nominal References can be identified. Table 1 shows the intrinsic references related to each recognized non-ideal feature.

Intrinsic Shape Reference

The Intrinsic Shape Reference is recognized in those points of the acquired workpiece which can be considered as lying on an analytical surface (for example, plane, sphere, cylinder, cone, torus, etc.). The recognition of this type of reference requires the identification of an analytical surface, which approximates the points belonging to the tessellated surface. The concept of non-ideal feature is more general than that of Intrinsic Shape Reference since it can also include the smooth free-form surfaces. For these surfaces it is not possible to identify an Intrinsic Shape Reference. This nominal reference can be provided by a CAD model, as it is the case in other approaches described in literature [START_REF] Prieto | An Automated Inspection System[END_REF], Gao et al. 2006). The Intrinsic Shape References are recognized by means of some rules which play a leading role in the definition of the domain of the recognizable intrinsic references and of the assessable errors.

Intrinsic Derived Reference

The Intrinsic Derived References are the analytically known geometric entities deriving from the recognition of some geometric properties. These entities are not physical and, for this reason, are not directly measurable from the surface, but can nevertheless be derived from the surface. In many cases these references are the situation features of the non-ideal feature (for example, the axis for a generic axiallysymmetric or helicoidal surface). In other cases they are special sections of the surface (for instance, the geometric shape of the directrix curve for a generic extruded surface, the generatrices of a ruled surface or the circular cross-section for a generic axially-symmetric surface).

Intrinsic Local Reference

The Intrinsic Local Reference is not considered in the current tolerancing standards. It deals with the uniformity of some intrinsic differential geometric properties, such as: regularity, curvature recurrences, and so on. These references do not pertain to the global analytical properties of a surface, but rather to properties which locally characterize it. In what follows, two Intrinsic Local References are introduced.

Regularity

One of the most important Intrinsic Local References is the regularity of the points pertaining to a continuous surface or, more properly, to a non-ideal feature. A continuous surface is regular if the following two conditions are satisfied (Do Carmo 1974):

(1) At each point, derivatives of any order exist in any direction(which are automatically continuous). A surface which meets this condition is referred to as differentiable surface;

(2) At each point, there exists a tangent plane (i.e. there are no singular points).

A regular surface is, roughly speaking, assimilable to a differentiable surface and any imperfection and/or error is associated with a deviation from it. A tessellated surface is a non-regular reconstruction of an acquired surface, which can be assimilated to a nominally regular surface. For this reason, surface regularity is not a characteristic of the acquired point cloud, but it should be recognized from the cloud as an intrinsic reference. The surface regularity measurement is useful for the evaluation of the surface imperfections. For example, surface imperfections, such as scratches, cracks and acquisition errors, can produce surface regularity errors. This kind of intrinsic reference can be recognized also for surfaces that are not analytical.

In the case of generic extruded or axially-symmetric surfaces it could also be useful to distinguish the geometrical property of the surface transverse section, which has a nominal analytical reference (straight line or circle), from the longitudinal section which, generally speaking, cannot be assimilated to an analytical reference. In order to evaluate this type of imperfections, the concept of profile regularity can be introduced. For instance, for a turned generic axially-symmetric surface, any surface imperfection, which is associated with an undesired circumferential groove caused by the tool vibration, will not give rise to axis straightness error or to circularity error; the imperfection will nonetheless be identified because of a loss of profile regularity.

Ruledness

Another Intrinsic Local Reference, introduced in this work, concerns ruled surfaces. This reference has to do with the confirmation of the ruled property of the acquired surface. It assumes an important role in generic ruled surfaces for which a reference of the type ISR cannot be identified. For this type of surfaces an Intrinsic Derived Reference is typically used. It is a straight line representing a surface generatrix and the related tolerance is the straightness tolerance (ISO/TS 12780-1:2003). This tolerance is adequate to the nominal concept of ruled surface, but is difficult to verify practically. The method here proposed to verify the property of a surface to be ruled evaluates the uniformity of differential geometric properties along a straight line (a principal curvature is always null). In other words, a ruled surface in the neighbourhood of a generatrix can be approximated by a ruled paraboloid (figure 3).

With this approach the conformity of the generatrix to a straight line is not directly verified, but the uniformity of some local properties related to curvatures is (the neighborhood of the generatrix can be approximated by a unique ruled paraboloid). In what follows, the deviation from this kind of intrinsic reference is referred to as ruledness error. This approach to evaluate errors in ruled surfaces is possible thanks to the specific way to acquire the object under verification (which guarantees a high-density point cloud). It solves some problems dealing with the nature of the measurement process, which provides points that, in general, are not aligned with the surface generatrices. Furthermore, the generatrix direction is not preventively known and must be sought by investigating surface differential geometric properties. Hence, any straightness tolerance evaluation can be largely affected by mesh sizes. The high density of point clouds makes it possible to implement easily methods which perform local surface approximations. This method, when applied to tessellated surfaces, is a more rigorous way to evaluate ruledness and is intrinsically more robust than generatrices straightness tolerance verification.

Intrinsic nominal reference association

In the GPS language the association corresponds to the feature operation: "fit an ideal feature to a non ideal feature according to a criterion" (ISO/TS 17450-1: 2005). In this work the criterion is referred to as rule and the association is carried out in order to quantitatively identify the nominal reference from which the error can be evaluated. The recognition step just performs a qualitative identification of the reference (analytical ideal feature), which must be evaluated quantitatively by approximating the non-ideal feature. For this purpose, several rules can be used, each of which identifies a different Intrinsic Shape Reference, which is affected by the rule under examination. At present, GPS standards do not define univocally the rules to identify ideal features. In literature, in order to carry out this operation, several rules are presented. Roughly speaking, the rules perform the median reference feature identification around which the tolerance zone is constructed. From a mathematical point of view, a constrained optimization problem needs to be solved. In order to determine the nominal reference S (the ideal feature) that best fits the related point cloud p i (non-ideal feature), the most used association rule is based on the L α -norm. It is defined as follows:

( ) α = α α       = ∑ 1 N 1 i , S d N 1 L i p (1)
where 0<α<∞, N is the total number of data points and d(S, p i ) is the shortest distance (or equivalently, the residual error) between p i and S. Based on the α value, several association rules are possible: the most used are summarized in table 2.

From a practical point of view, the L 1 -norm and the L 2 -norm rules do not yield very different results. The L 1 -norm rule is less sensitive to outliers but can nevertheless offer multiple solutions. The L 2 -norm rule, on its part, can overestimate the error (especially, if we compare it to the error evaluated by the min -max approach), but it undoubtedly gives the best estimation of the nominal reference. With the L 2 -norm rule, the error can also be evaluated by following a statistical approach, based on the standard deviation of the points' distance from the estimated reference.

( ) ∑ = N 1 i , S d i p α α α α=2 L2-norm rule or least squares approach ( ) ∑ = N 1 i 2 , S d i p α α α α=∞ ∞ ∞ ∞ L∞ ∞ ∞ ∞-norm rule or min -max approach ( ) { } i N ,.... 1 i p d Max S, =
The L ∞ -norm rule (min -max approach) minimizes the maximum distance between the point p i and the ideal feature. In other words, the L ∞ -norm rule minimizes the following function: where d i is the signed distance between the intrinsic nominal reference and the i th point p i of the related point cloud. The L ∞ -norm rule performs an ideal feature association whose maximal distance f from the points conforms to the error evaluated according to the ISO standards. Generally speaking, the L ∞ -norm rule does not carry out the best approximation of a non-ideal feature, but it identifies the ideal feature from which the distance between the most external and internal points is minimized. In other words, the L ∞ -norm rule is strongly affected by particularities in single point locations. Since different association rules give rise to different values for the error, the rule needs to be clearly stated in the tolerance specification. Hence, in this work both the L 2 -norm and the L ∞ -norm rules are considered. Some approaches are described in literature [START_REF] Traband | Evaluation of straightness and flatness tolerances using the minimum zone[END_REF][START_REF] Novaksy | Utilization of Voronoi Diagrams for circularity algorithms[END_REF][START_REF] Lagarias | Convergence properties of the Nelder-Mead Simplex Method in low dimensions[END_REF][START_REF] Huang | An exact solution for the roundness evaluation problems[END_REF][START_REF] Moroni | Geometric Tolerance Evaluation: a Discussion on Minimum Zone Fitting Algorithms[END_REF] which evaluate the minimum tolerance zone. In particular, in this work, minimization is carried out by means of the Nelder-Mead iterative technique used by [START_REF] Lagarias | Convergence properties of the Nelder-Mead Simplex Method in low dimensions[END_REF]).

f = max{d i } -min{d i } (2)

The Intrinsic Shape Reference association

The quantitative definition of the Intrinsic Shape Reference involves, respectively, the evaluation and the identification of the corresponding intrinsic characteristics and/or situation features. In table 3 the intrinsic characteristics and the situation features for each ISR are reported. The intrinsic references, with the value of their characteristics and the location of the situation features, strictly depend on the rules used to evaluate them. For each type of ISR, the L 2 -norm and the L ∞ -norm rules have been implemented.

The Intrinsic Derived Reference association: the case of the axiallysymmetric features

Not every Intrinsic Derived Reference can be directly deduced from the point cloud. Some of them need a characteristic situation reference to be preventively estimated. This is also the case of the axially-symmetric non-ideal feature.

The quality of its estimation affects significantly the subsequent error evaluation. In this work, in order to estimate the axis, two methods are implemented: the normal intersections method [START_REF] Pottmann | An introduction to line geometry with applications[END_REF]) (henceforth denoted as Π 1 ) and the curvature centres method [START_REF] Cao | Geometric Structure Estimation of Axially Symmetric Pots from Small Fragments[END_REF] (from now on denoted as Π 2 ). These rules must be stated in the tolerance specification. The axis itself is an Intrinsic Derived Reference, but it is also used to identify the set of cutting planes (perpendicular to the axis), from which the circularity and the location of the centre of the section ("associated derived centres" (UNI EN ISO 14660-2: 2002)) can be evaluated. The non-ideal circumferential line is extracted by projecting the points in the neighbourhood of the cutting plane onto it. Based on the optimization criterion, which is defined by a specific rule, for each plane section a reference ideal circle is associated with the non-ideal circumferential line. In this work the ideal circle is obtained by using the L 2 -norm and the L ∞ -norm rules.

The Intrinsic Local Reference association

In order to evaluate profile regularity, surface regularity and surface ruledness, Intrinsic Local References need to be conveniently defined and associated. The rules here proposed to identify these nominal references can be considered as examples. Other rules could be defined to evaluate new error categories that require Intrinsic Local References. Since these rules play an important role in reference identification, error specification must also define the way to evaluate these intrinsic references.

Profile regularity

Roughly speaking, a regular profile is assimilable to a differentiable curve and any imperfection is associated with a deviation from it. This work takes the best local approximation of the data point with a regular curve as the nominal reference for profile regularity evaluation. The rules to identify the nominal reference include: the type of regular curve (polynomial curve, exponential curve, etc.) and the method to locally approximate the profile. In the approach herein proposed, the reference curve is evaluated, at each point of the profile, as the approximating curve of its neighbourhood. It is a quadric or a cubic polynomial, calculated by the weighted L 2norm rule. The weights of the approximation rule are assumed as the values of a Gaussian function having the mean located at the analyzed point and a properly selected value for the standard deviation σ (figure 4). This weighting approach aims at defining a regular profile by filtering the local irregularities. The width of this filter is conventionally assumed as λ = 6σ of the Gaussian function. The λ value is assumed as the maximum value between the expected maximum size of the profile imperfection and the mesh dimension. The profile regularity error, at each point analyzed, is defined as the distance between the point and the related approximating curve.

Surface regularity

Surface regularity is a generalization (three -dimensional extension) of the profile regularity concept. A non-ideal feature consists of adjacent mesh vertices having some uniform differential geometric properties. Surface regularity measures locally the membership of each point to a regular surface which locally approximates its neighbourhood. Similarly to profile regularity, the rules to identify the nominal reference include: the type of regular surface and the method to locally approximate the surface. For each recognized non-ideal feature, the reference surface is evaluated at each vertex as the approximating surface by means of the weighted L 2 -norm rule. Surface regularity is not related to a specific shape of surface; it refers to the differentiability, which is a local property. Thus, in this work, the quadric paraboloid is used as a reference; it is the typical analytical regular surface used to evaluate the differential geometric properties. The weights of the approximation rule are assumed as the values of a two -dimensional symmetric Gaussian function having the mean located at the analyzed point and a properly selected value for the standard deviation (figure 5). In this case, the λ value is conventionally assumed as equal to three times the maximum dimension of the mesh. The surface regularity error, at each point analyzed, is defined as the distance between the point and the related approximating surface.

Ruledness

Another important concept introduced in this paper is ruledness. The rule to evaluate the related intrinsic reference, herein proposed, is based on a typical growth algorithm. The intrinsic reference is an analytical ruled paraboloid whose form is expressed in the coordinate system (ξ, ψ, ζ) as follows:

ζ=c•ψ 2 +d•ξ+e•ψ+f (4)
Its parameters (c, d, e and f) are obtained by best fitting the point cloud around its vertex generatrix. For this purpose, a weighted approximation rule is used; the weight factors are the values of a Gaussian function having the mean located in the vertex generatrix and a value for the standard deviation, chosen according to the mesh dimensions (σ = max mesh dimension). Since the generatrix direction is unknown, it must be sought by a growing algorithm from a seed point. Figure 6 shows the flow chart of the algorithm for ruledness evaluation.

The process starts with the identification of the seed point p s (x, y, z) belonging to the ruled surface under examination. A local coordinate system (ξ, ψ, ζ) is defined in such a way that the axis ζ overlaps the normal n at p s ; the axis ξ is parallel to the estimated principal direction (x 2 ) related to the null curvature at p s (figure 7 a). In this local coordinate system, the equation of the ruled paraboloid (Γ ruled ) locally approximating the mesh is expressed by the equation ( 4). In the first step of the algorithm, the surface approximation is performed by analyzing the 1-ring neighbourhood around the seed point (in figure 6 it is denoted as NGH s ) (figure 7 b).

In order to identify the generatrix direction which best fits the surface, the growing algorithm explores the nearest point (denoted as p 1 in figure 7c) along the x 2,1 direction. At the j-th step of the growing algorithm, the nearest point in the estimated generatrix direction is found (denoted as p j in figure 7d); the set of points in the neighborhood of the points explored are approximated by the paraboloid (4) (points in red shown in figure 7e) and a new principal direction x 2,j is reevaluated. The set of points to approximate ( j ∑ ) is defined as follows:

{ } { } U U j i NGH j i NGH i i s j ,.... 1 , ,.... 1 , = = = ∑ p (5)
The process continues exploring step-by-step the nearest points in this direction until a specified limited length (L 0 ) is reached. The value of L 0 must be included in the tolerance specification. Seed points are chosen so as to lie on the unique directrix curve of the surface. The ruledness error is evaluated as the distance between the last estimation of Γ ruled and p s and the set of nearest points p j .

Error evaluation

The final phase of the methodology herein proposed is error evaluation. For each type of non-ideal feature (see table 4), specific categories of errors can be identified. These are the categories that can be automatically evaluated as deviations from the recognizable intrinsic references. Table 4 does not show the curve and the surface profile tolerances, which rather require that a nominal reference be specified.

Total Roundness -- -- X X X -- -- -- Cylindricity -- -- X -- -- -- -- -- Conicity -- -- -- X -- -- -- -- Form error Sphericity X -- -- -- -- -- -- -- Profile X X X X X X -- -- Regularity error Surface -- -- X X X X X X Ruledness error -- -- X X -- X X --
All the form errors can be expressed both as the maximum (Max) and as the standard deviation (σ) of the point cloud distances from the corresponding intrinsic reference. A probabilistic evaluation of the error can be more significant than the maximum deviation measure. The point cloud acquisition process is affected by singular errors which can be ascribed to the typical measuring errors affecting the optical scanner devices. The σ value performs a probabilistic estimation of the acquired point location with respect to the intrinsic nominal reference. This way to specify error is possible thanks to the large set of points acquired for each non-ideal feature.

Error Evaluation: the case of the axially -symmetric features

The typical form errors considered in the ISO standards for axiallysymmetric surfaces are roundness and axis straightness. For all these tolerances the Intrinsic Derived Reference is the axis of the feature, which is estimated by using specific rules such as those presented in section 6.2. Roundness error evaluation requires that, in any cross-section of the corresponding non-ideal feature (cylinder, cone and generic axially-symmetric), the non-ideal circumferential line be extracted (figure 8b). This line is compared to the circularity ideal reference and the error is calculated as the maximum radial distance between them. Axis straightness error evaluation requires the extraction of the median line of axially -symmetric features (cylindrical, conical and generic). The median line is the locus of the centres of the ideal circles associated with the non-ideal circumferential lines; in the GPS standards it is referred to as "associated derived centres" (UNI EN ISO 14660-2: 2002) (figure 8c). The median line is compared to the reference axis and the error is calculated as two times its maximum distance from the centres.

Thanks to the new techniques for geometric property measurement, which allow a large amount of point data to be automatically computed and virtual references to be simulated, new categories of tolerances can be introduced. This paper introduces the tolerance category of total roundness, suited to specify the maximum deviation from a nominal axially -symmetric surface. Traditional roundness performs single evaluations of the error, section by section (figure 8d). The total roundness error is similar to the run-out, but it is measured by virtually rotating the surface around its own estimated axis. Whereas the run-out requires that an external datum (axis) be assigned, total roundness is estimated from its intrinsic datum: its axis (its revolution axis). In this case, the intrinsic nominal references consist of a set of circles, all having their centre on the estimated axis (figure 8e). The algorithm used in this work for the automatic evaluation of the total roundness error is as follows:

(1) identification of the reference axis (rules Π 1 and Π 2 );

(2) definition of a cylindrical coordinate system (ρ, θ, ζ) with ζ -axis coincident with the reference axis; (3) circular projection (along the θ direction) of the point cloud onto a plane passing through the reference axis; (4) evaluation of the local radial thickness of the projected point cloud.

Application example

The methodology, described in the previous sections, has been implemented in an original software, coded in C++, by using a library dedicated to the processing of tessellated geometric models, which has been developed in the University of L'Aquila.

In order to verify the reliability of the proposed methodology, a specific test case, including different types of non-ideal features, has been analyzed. This test case (figure 9) refers to a real object whose acquisition has been carried out by an optical 3D scanner (www.scansystems.it). This test case consists of five non-ideal features, four of which are analytical features. Figure 10 shows the identification of features with their respective labels. The areas recognized as non-regular are depicted in yellow and therefore automatically excluded from the error evaluation. The results of the error analysis are quoted in table 5.

The rules used to estimate the intrinsic references play an important role in error evaluation. This is confirmed by the results reported in table 5, where it is possible to see how the different rules used give rise to differences in the estimated errors. For example, in the analysis of generic_axially-symmetric_1, different values for profile regularity errors are obtained by varying the rules for axis estimation (Π 1 and Π 2 ) and for curve approximation (quadric and cubic polynomial). The difference between the results obtained by using the L 2 rule and those generated by means of the L ∞ rule is due to the different concept of reference that is associated with them. This specific aspect has already been discussed in section 6.

The errors are evaluated as the maximum or standard deviation of the acquired points from the intrinsic reference. The errors evaluated at a confidence level of 3σ are, in all the cases analyzed, close to the maximum deviation (table 5). The comparison between these two methods, especially when they show large differences in the error estimation, gives out a warning signal of a probable systematic factor affecting the measuring process.

In figure 11 a screen capture of our developed system is depicted. Figure 12 presents an example of tolerances' specification for the test case under consideration. In this case, the tolerance categories are denoted by a series of symbols whose meaning are explained in table 4. The rules to identify the intrinsic references are specified in braces. Each rule is separated by a comma. The way to measure the error by using the probabilistic approach is specified by means of the symbol σ.

Conclusion

This paper proposes a general approach to form error evaluation and specification. This approach is founded on the concepts of non-ideal feature and intrinsic nominal reference. The object to be inspected is segmented into a set of nonideal features and for each of them one or more intrinsic nominal references are identified. The recognizability of non-ideal features and the rules to identify the intrinsic references from a point cloud are the key elements of this approach to tolerance specification. The tolerance categories that can be specified are those referring to recognizable non-ideal features for which intrinsic nominal references can be identified. Recognizable non-ideal features, with their intrinsic references, define the signifiers of the signs in a language specifically oriented to form error specification. These signs are suited to be automatically recognized in the inspected object. Based on the above-mentioned considerations, new error categories have been introduced. Some of these errors have to do with the intrinsic references deduced through the recognized differential geometric properties. The tolerances of the ISO specification standards are herein implemented with other tolerances similar to the traditional types, such as conicity, sphericity, total roundness, regularity and ruledness. These new tolerances are supported by the specific nature of the measuring process, which allows a large amount of point data to be automatically acquired and a virtual reference to be simulated. Since the surface sampling interval affects error evaluation, it must be selected according with tolerance specification.

An important directive of GPS standards, which deals with the concept of imperfection, finds a concrete application in the regularity error. At present, the concept of imperfection still has not found a formal and rigorous way to be defined and measured. The measurement of the surface regularity error, which is here proposed, is useful for the evaluation of all surface imperfections (scratches, cracks), and it can also be carried out for surfaces that are not analytical.

Another important concept, which is also introduced herein, is the ruledness which could be an effective substitute for the traditional generatrix straightness tolerance.

The identification of intrinsic nominal references avoids the registration process between CAD model and data points. However, this approach does not rule out the possibility of supplementing tolerance verification by means of the comparison with a nominal representation of the inspected object performed by a CAD system. This integration could be useful to control the free-form features which cannot be automatically associated with analytical surfaces. Further work is addressed at extending the concepts of Intrinsic Nominal Reference also in the case of orientation and localization tolerances.

  hybrid segmentation approach[START_REF] Angelo | Fuzzy sets for geometric shape recognition in triangular meshes[END_REF], based on fuzzy logic. The non-ideal feature recognition process is performed in the following steps:(1) smooth point identification and elimination from the mesh of the non -smooth points (Di Angelo and Di Stefano 2009); (2) smooth point classification;

Table 4 .

 4 Types of non-ideal feature and related errors

Figure 1 .Figure 2 .Figure 3 .Figure 6 .

 1236 Figure 1. An approach to the automatic geometric inspection based on the recognition of the intrinsic references

Figure 7 .

 7 Figure 7. Explanation of the steps in the ruledness evaluation process

Table 1 .

 1 Non-ideal feature types and the related intrinsic references

	Intrinsic Reference Types

Table 2 .

 2 The most used association rules for median reference feature evaluation

		Rule name	RULE EXPRESSION
	α α α α=1	L1-norm rule

Table 3 .

 3 The intrinsic characteristics and the situation features of the Intrinsic Shape References

	Intrinsic Shape Reference	Intrinsic characteristics	Situation features
	Plane	none	• •	a point x0 ≡ (x0, y0, z0) located on the plane; the direction a ≡ (a, b, c) of the plane;
	Sphere	diameter	•	its centre x0 ≡ (x0, y0, z0)
	Cylinder	diameter	•	its axis a ≡ (a, b, c);
			•	a point lying on its axis x0 ≡ (x0, y0, z0);
	Cone	apex angle	•	its axis a ≡ (a, b, c);
			•	its apex x0 ≡ (x0, y0, z0);

Table 5 .

 5 The results of the error evaluation for the test case under examination

														generic axially-
					sphere_1	cylinder_1	cylinder_2	cone_1	symmetric_1
					L2			L∞	L2	L∞	L2	L∞	L2	L∞	Π1	Π2
			Straightness		--			--	0.011	0.018	0.004	0.049	0.079	0.079	0.057	0.035
	Form error	Flatness Roundness Total Roundness Cylindricity Conicity	Max	----------			----------	--0.055 0.019 0.058 --	--0.055 0.028 0.057 --	--0.033 0.020 0.036 --	--0.034 0.054 0.036 --	--0.009 0.082 --0.082	0.009 0.049 0.032	--0.022 0.057 ----	--0.017 0.036 ----
			Sphericity		0.043	0.037	--	--	--	--	--	--	--
	Regularity error	Profile	Quadratic polynomial Cubic polynomial Surface	Max Std Max Std Max Std	--------	0.012 0.003	--------	0.004 0.001 0.002 0.001 0.018 0.006 0.002 0.006 0.002 0.004	0.002 0.001 0.002 0.001 0.036 0.015 0.005 0.016 0.005 0.004	0.032 0.009 0.032 0.010 0.011 0.027 0.007 0.025 0.008 0.002	0.077 0.023 0.071 0.023 0.024 0.127 0.041 0.113 0.040 0.003
		Ruledness error	Max Std		----			0.017 0.0034	0.0064 0.0016	0.0103 0.0026	----

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