A Novel Approach for the Inspection of Flexible Parts

Gad N. Abenhaim

1

Department of Mechanical Engineering, Université de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada e-mail: [email protected]

Antoine S. Tahan Department of Mechanical Engineering, École de Technologie Supérieure (ÉTS), Montreal, QC, H3C 1K3, Canada e-mail: [email protected]

Alain Desrochers Department of Mechanical Engineering, Université de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada e-mail: [email protected]

Roland Maranzana Department of Mechanical Engineering, École de Technologie Supérieure (ÉTS), Montreal, QC, H3C 1K3, Canada e-mail: [email protected]

1

A Novel Approach for the Inspection of Flexible Parts Without the Use of Special Fixtures In a free state, flexible parts may have different shapes compared to their computer-aided design (CAD) model. Such parts may likewise undergo large deformations depending on their space orientation. These conditions severely restrict the feasibility of inspecting flexible parts without restricting the deformations of the part and therefore require dedicated and expensive tools such as a conformation jig or a fixture to maintain the integrity of the part. To address these challenges, this paper proposes a new inspection method, the iterative displacement inspection (IDI) algorithm, that evaluates profile variations without the need for specialized fixtures. This study examines 32 models of simulated manufactured parts to show that the IDI algorithm can iteratively deform the meshed CAD model until it resembles the scanned manufactured part, which enables their comparison. The method deforms the mesh in such a manner so as to ensure its smoothness. This way, neither surface defects nor the measurement noise of the scanned parts are concealed during the matching process. As a result, the case studies illustrate that the method’s error essentially only represents the scanned part’s measurement noise. The inspection results, therefore, solely reflect the effect of variations from the manufacturing process itself and not the deformation of the part. 关DOI: 10.1115/1.4003335兴

Introduction 2

Manufactured parts naturally deviate from their nominal geometry due to process variations. Standards such as ASME Y14.5M assume that inspections of these parts are carried out in a free state. Such an assumption is clearly inappropriate when dealing with flexible parts because as a result of dimensional variation, gravity loads, residual stress induced distortion, and/or assembly force, flexible parts take on different shapes in a free state compared to their nominal CAD model. For example, a skin panel of an aircraft can be lightly warped, making it unacceptable in a free state. This same panel may nevertheless be assembled on the airframe. This example therefore highlights the delicacy of inspection if insufficient fixing constraints are applied to ensure measurement repeatability, an endeavor that involves the use of expensive and dedicated fixtures. Furthermore, the cost of dedicated fixtures such as conformation jigs is amplified since two jigs are usually required—one for the client and the other for the supplier to ensure that the client can independently assess the part’s quality. This paper seeks to address these problems by outlining a method that allows for inspection without such fixtures. Specifically, the proposed method evaluates the surface profile deviations of a manufactured part through a comparison of its point cloud obtained via a noncontact measurement approach and its meshed CAD model. The shapes of the scanned part and the CAD will inevitably differ, and thus a simple comparison is not sufficient to evaluate the profile deviations. A comparison of this sort would merely identify large deviations while masking the actual profile deviation of the part. The inspection algorithm this paper proposes rectifies this in that it deforms the meshed CAD model until it 1

Corresponding author. The term deviations is preferred to defects since the latter is dependent on the assigned tolerance. Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received January 7, 2010; final manuscript received November 30, 2010; published online February 1, 2011. Assoc. Editor: Prof. Shreyes N. Melkote. 2

matches the scanned part. The two point sets are compared at each iteration of the algorithm, followed by an identification method, which separates the profile deviation from the part’s deformation. Finally, after each iteration, the meshed CAD model is smoothly deformed in a way so as not to mask the profile deviations. Before any substantive discussion of the proposed method can transpire, it is necessary to take a moment to explain the way in which this paper will unfold. Section 1 introduces the challenges involved in inspecting flexible parts. Having established these challenges, Sec. 2 provides an outline of existing research to highlight the extent to which it has not adequately addressed these inspection difficulties. Section 3 then elaborates on the specific problems of inspecting flexible parts without the use of specialized fixtures. Section 4 details each module of the proposed iterative displacement inspection 共IDI兲 algorithm. Section 5 presents 22 case studies using three types of shapes to evaluate the algorithm’s performance. Finally, Sec. 6 describes the directions of future research.

2

Research Background

2.1 Inspection by a Scanning-Based Approach. Freeform surface inspection is a field in constant evolution due to the growing complexity of parts and the rising demand for high precision tools for their inspection. Two types of measurement method are used for such surfaces: contact measurements—traditionally using a touch probe mounted on a coordinate measuring machine 共CMM兲—and noncontact measurements—such as laser and optical scanning. A comprehensive review of these types of measuring systems was presented by Savio et al. 关1兴. Noncontact scanning approaches are gaining much popularity; for instance, they have been used for inspection in numerous papers such as those of Gu and Huang 关2兴, Li and Gu 关3,4兴, Yao 关5兴, Gao et al. 关6兴, Shi and Xi 关7兴, and, most recently, Ravishankar et al. 关8兴. Throughout these inspections, the manufactured part is assumed to have a shape similar to that of the CAD model, enabling their comparison. Noncontact scanning systems are additionally used in defect detection applications, such as those presented by Karbacher et al.

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关9兴, Leopold et al. 关10兴, Eichhorn et al. 关11兴, Dorïng et al. 关12兴, and, lately, Megahed and Camelio 关13兴. This type of algorithm flags surface deviations, but it does not quantify them. Therefore, the method proposed in this paper differs from both applications in that it makes possible the quantification of the surface deviation, even though the manufactured part has a significantly different shape compared to the CAD model due to the absence of a master jig. 2.2 Registration Issues. A CMM mounted with a laser/ scanner is a prevalent method for scanning a manufactured part for its inspection. The point cloud generated from such a scan must then be compared with the CAD model in order to evaluate the deviation of the part with respect to its tolerances. When designed, the CAD model exists in the design coordinate system 共DCS兲, while the scan generates a point cloud in the measurement coordinate system 共MCS兲. Localization, registration, and alignment are the processes used to unify these two coordinate systems. Mathematically, registration refers to finding an optimal transformation matrix between the DCS and the MCS. An extensive review of the surface representation methods and associated registration techniques was provided by Li and Gu 关3兴. Since this paper develops an inspection methodology without pretreatment of the point cloud and also independent of reengineering software, the next paragraph addresses registration algorithms that do not require a parametrical representation of the surfaces. Besl and McKay 关14兴 introduced in 1992 the iterative closest point 共ICP兲 algorithm. The ICP method uses the quaternion’s proprieties to estimate, at each iteration, the optimal transformation matrix minimizing the Euclidean distance between two point clouds. To improve the computing time of ICP while simultaneously conserving the method’s robustness, Masuda and Yokoya 关15兴 proposed the adoption of random sampling of the point cloud. Similarly, with the aim of improving the computing time of ICP, Refs. 关16,17兴 introduced alternative metrics to a pointEuclidean distance to find the matching point between two point clouds. Rusinkiewicz and Levoy 关18兴 classified and compared several ICP variants to then introduce a normal-space-directed sampling based on the distribution of the normal vector associated with each point. Finally, Gelfand et al. 关19兴 proposed a sampling method that minimizes the instability of the ICP algorithm when using distance metrics, such as point-plan rather than point-point. The aforementioned approaches all begin with the assumption that the shapes of two rigid parts are similar. This assumption’s extension to cover flexible parts, however, is not viable. The registration problem therefore is not only limited to finding a rigid transformation but also requires the introduction of nonrigid registration. The difference between rigid and nonrigid registrations is that rigid registration can only align two parallel lines, for example, whereas nonrigid registration can align a line with a curve. Medical imaging has significantly contributed to the evolution of nonrigid registration techniques. Dawant 关20兴 and Holden 关21兴 presented literature reviews of the registration algorithms used in this field. Holden 关21兴 regrouped the methods by their founding theory in terms of those that are constrained by the physical proprieties of their subjects and those that rely on an interpolation method. Using the physical proprieties of their subjects, Ferrant et al. 关22兴 proposed a nonrigid registration algorithm that minimizes a two-term energy function. The first term in Ferrant’s approach limits the deformation of the physical properties of the subject, while the second term deals with the distance between the two subjects. On the other hand, using the interpolation method, Feldmar and Ayache 关23,24兴 introduced the locally affine deformation method. Instead of using one rigid transformation to align the point clouds, they suggested using one affine transformation metric for each point while maintaining a similar affine transformation between those applied to a point and those of its neighbor points within spherical regions of space. Animation has likewise contributed to the literature on nonrigid 011009-2 / Vol. 133, FEBRUARY 2011

registration. Allen et al. 关25兴 created a parametric representation of a human body by transforming a high resolution template mesh of a body until it replicates the scan of the real body. Drawing from the work of Feldmar and Ayache 关23,24兴, an affine transformation is applied at each iteration to each node of the template mesh. The smoothness of the mesh is preserved throughout this process. In order to find the affine transformations, the method minimizes a three-term error function: distance error, smoothness error, and marker error. This method, despite originating from an outside discipline, is applicable to the one proposed in this paper because it remains functional even when sections of the scanned body are incomplete. Amberg et al. 关26兴 improved this method by reformulating the error function of Allen et al. 关25兴 into a quadratic function that is more easily solved. 2.3 Tolerance of Compliant Assemblies. Tolerance allocation for flexible parts may also be a challenging task since tolerance analysis methods presuppose that the part is rigid and thus fail to take into consideration permissible deformations during assembly. These methods overestimate the tolerance allocated to flexible parts. Three research groups have tried to remove this flaw by developing a tolerance analysis method that integrates the flexibility of the parts. The first group, the Laboratory for Manufacturing System Realization and Synthesis 共MA/RS兲 of the University of Wisconsin in collaboration with the Collaborative Research Laboratory in Advanced Vehicle Manufacturing 共GM CRL-AVM兲, with Hu, Ceglarek, and their students 共Liu, Camelio, etc.兲 created the foundation for tolerance analysis for compliant assemblies. Their work primarily concerns managing variations in sheet metal assemblies for autobodies. Liu et al. 关27兴 highlighted the importance of the assembly sequence of sheet metals on their final assembly result. Camelio et al. 关28兴 proposed an algorithm optimizing the fixture’s positions while minimizing assembly variations as a function of the parts’ and tools’ variations. Liu and Hu 关29兴 also presented two methods for predicting the variation of sheet metal assembly using the finite element method, the direct Monte Carlo simulation, and the method of influence coefficients. The key in the method of influence coefficients is the establishment of a linear relation, called the mechanistic variation model, between the induced variation of parts and the variation of their final assembly. The second group, the Association for the Development of Computer-Aided Tolerancing Systems 共ADCATS兲, founded by Kenneth W. Chase of Brigham Young University 共BYU兲, extended their research on the tolerance analysis of rigid parts to also treat cases of compliant assemblies. Rather than subjecting the meshes’ nodes to random variations to simulate surface variations, such as in the work of Liu and Hu 关29兴, Merkley 关30兴 developed a random Bezier curve to describe these variations. Bihlmaier 关31兴 evaluated the average and the geometrical covariance matrix of the space between two surfaces to be assembled. This information is then used in a finite element analysis to predict the distribution of the assembly force. The methods previously outlined, however, are not designed for complex assemblies, such as assemblies of parts with variable dimensions when in operation. A prime example of such an assembly is an injector, which the aforementioned tolerance analysis methods cannot accommodate. To address this shortcoming, the third group composed by Markvoort et al. 关32–34兴 proposed a finite element analysis of the assembly for each combination of simulated tolerance allocations of its parts. The methods in this section are designed to predict the variations of an assembly, while likewise considering potential changes in the nature of the part’s assembly. The utility of tolerance analysis is restricted to prediction in that surface variations and external forces are imposed, meaning that as it stands, tolerance analysis cannot be used for inspection. Transactions of the ASME


CAD model is simulated without profile deviation, with the same deformation as that of the scanned part. The outcome of this method consists of two models that are geometrically close, enabling the evaluation of the profile deviations.

4 Fig. 1 Illustration of a dent shaped deviation characteristics; area AD and peak deviation HD

3

The Problem

Despite a number of well-documented efforts to incorporate a part’s flexibility into tolerance analysis with the aim of reducing the quality cost, this literature review nonetheless exposes a void in the research into the inspection of such parts. The purpose of this paper, as Sec. 1 makes clear, is to take a step in rectifying this gap by proposing an alternative to the use of inspection fixtures to constraint a flexible part to its assembled configuration. In the absence of the latter, the effects of gravity and the deformations generated by the internal constraint induced by the manufacturing process necessarily create a situation wherein the manufactured part is significantly different from its CAD geometry. Consequently, the proposed method treats and isolates this factor so that the inspection results reflect only the deviation due to the manufacturing process. The remainder of this paper will focus on explaining IDI as a new algorithm to locate and quantify profile deviations of skin parts. The inputs of IDI consist of the point cloud of the manufactured part when supported by simple fixtures and its meshed CAD model. The method assumes the following. •

The component is a skin part. It is defined and is made available by math data 共STEP, IGES, etc.兲 file. • The mounting method used during the scanning allows the part to deform elastically in an amount close to or greater than the profile tolerance specified. • The manufactured part is completely scanned and represented as a point cloud 共x , y , z兲. • Profile deviations are dent shapes 共see Fig. 1兲. • The inspection is restrained to the profile deviation as required by the ASME Y14.5M standard.

Despite the deformation induced by a part’s flexibility, IDI essentially works by comparing a scanned part’s geometry with its meshed CAD model. Since the two shapes are different, it is necessary to deform the meshed CAD model so that it more accurately reflects the scanned part. In this case, the registration problem is not limited just to finding a transformation matrix minimizing the distance between the two sets, like it is with rigid registration methods. It is necessary, however, to add a displacement field 共see Fig. 2兲 estimating the deformation required by the meshed CAD model to reflect the scanned part. This field must represent only the deformation induced to the scanned part by the mounting method, the gravity, and the deformation resulting from the manufacturing process; deformations attributed to the profile deviations ought to be excluded. Therefore, by adding a displacement field that deforms the meshed CAD model, a new meshed

Fig. 2 Deformation required by the CAD model to reflect a scanned part without profile deviation

Journal of Manufacturing Science and Engineering

Algorithm: IDI

The proposed IDI algorithm combines rigid with nonrigid registration methods, as well as a newly developed identification method, to distinguish deviation from deformation of a given part. The proceeding explanation of this algorithm begins by briefly presenting the registration problem mathematically. Then, the five major modules of the algorithm are presented, which are as follows: pre-alignment, rigid registration, nonrigid registration, correction of the point-point distance, and the identification method. The pre-alignment module aligns the meshed CAD model and the scanned point cloud using landmarks, enabling the rigid registration using the ICP algorithm. Next, the nonrigid registration finds an affine transformation for each node in the meshed CAD, deforming the mesh into a shape closer to that of the scanned point cloud. Throughout the process, the projection of the point-point distance metric onto the normal vector of the node is used to correct the inaccuracies of the point-point distance metric. Furthermore, an identification method is used between the two registration modules to tag each node of the meshed CAD as having 共or not having兲 a corresponding point in a zone with a profile deviation in the scanned part. Finally, this section combines the five modules to present the IDI algorithm in a step-based process, shown in the flow chart in Fig. 8. 4.1 Spatial Alignment. This subsection briefly formulates the registration problem mathematically. Let P = 兵p1 , p2 , p3 , . . . , pNP 兩 p j 苸 R3其 be a set of N P points representing the scanned part and S = 兵s1 , s2 , s3 , . . . , sNS 兩 si 苸 R3其 be a set of NS 兩 NS Ⰶ N P nodes of the meshed CAD model. Since the two point clouds are in distinct coordinate systems, their alignment is achieved by finding the three translation qT 苸 R3⫻1 and three rotation R 苸 R3⫻3 vectors that minimize the Euclidean distance dsi between the point clouds. The registration problem is therefore represented by the objective function f, NS

f=

兺储d 储 i=1

si

共1兲

2

NS

f共R,qT兲 =

兺储c − 关R · s + q 兴储 i

i

T

2

共2兲

i=1

where ci 苸 C and C is a set of points in P closest to the set of nodes S. 4.2 Pre-Alignment. Given that the nodes of the meshed CAD and the scanned point clouds are in distinct coordinate systems and that the ICP algorithm 关14兴 demands that the two be positioned relatively closely, this module estimates the transformation matrix by optimizing the registration objective function 共Eq. 共1兲兲 with the Nelder–Mead simplex method. This is done by using identifiable landmark points, which may be manually selected in each point set. Alternatively, these may be selected only once for a type of part based on knowledge of the mounting fixture contact points in the MCS used during scanning, as well as their nominal position in the DCS. 4.3 Rigid Registration. As previously mentioned, this rigid registration module uses the ICP algorithm proposed by Besl and McKay 关14兴 to find the rigid transformation matrix minimizing the Euclidean distance between two point clouds. Before beginning the ICP algorithm, nodes in S that could potentially be in a zone where the manufactured part P has a profile deviation are removed from S. Section 4.6 details the method to identify potenFEBRUARY 2011, Vol. 133 / 011009-3


tially problematic points. Additionally, a random sampling of Nsampling nodes of S is used in the ICP to lower its computing time, as proposed by Masuda and Yokoya 关15兴. 4.4 Nonrigid Registration. Deforming the meshed CAD model into a shape that more closely resembles the scanned point cloud of the manufactured part so as to enable their comparison is a complex process. It must be accomplished without masking the profile variation of the manufactured part. This is achieved through judicious usage of Allen et al.’s 关25兴 nonrigid registration algorithm. Allen et al.’s 关25兴 nonrigid registration algorithm finds an affine transformation matrix xi, of size 3 ⫻ 4, for each point si to deform the mesh composed of S to a form S⬘ closer to the scanned part P. This process takes place while simultaneously safeguarding the smoothness of the new mesh S⬘. The quality of the mesh composed of S⬘ is likewise maintained by a distance error function Ed and a smoothness error function Es within the method. The unknown parameters xi are organized in a 4NS ⫻ 3 matrix, X = 关x1, . . . ,xNS兴T

共3兲

As previously discussed, the first registration criterion is the distance error function, NS

Ed共X兲 =

兺 w 储x · s 储 i

i

i

共4兲

i=1

where each node of S is described as si = 关x , y , z , 1兴T. The influence on the displacement field of each point si is controlled by the corresponding weight wi. Points identified as potentially in a zone with a profile deviation on the manufactured part are given a weight wi of 0, while points outside this zone are assigned a weight wi of 1. The weight factor of 0 of a point si prevents this point from being forced on its corresponding point ci, if ci is within an area with a profile deviation. The smoothness function, when this is the case, acts to constrain the displacement vector applied to si. This entails then that si will be displaced only if the neighboring points are also displaced, so that the mesh maintains a smooth shape. The smoothness error Es function regulates the displacement field by minimizing the difference between two affine transformations xi of connected nodes in the mesh. This function limits the displacement of each node so that the process is dependent on the displacement of their connected nodes. This function is what simultaneously guarantees the smoothness of the mesh and maintains its structure, Es共X兲 =

兺

兵i,j其苸l

储xi − x j储F2

共5兲

where 储储2F is the Frobenius norm. The nonrigid registration objective function to minimize is then formulated as E共X兲 = Ed共X兲 + ␣Es共X兲

ci⬘ = si + ␦Dsi,

1 ⱕ i ⱕ NS

共7兲

where Dsi is the projection of the point-point distance on the normal of the mesh at point si. This will be described further in detail in Sec. 4.5. ␦ is the percentage of the displacement to apply. Figure 3 illustrates this concept. Amberg et al. 关26兴 started with a high ␣ value to recover the global deformation between the sets. The value was progressively reduced to allow the mesh to match its target locally. However, convergence on a local minimum of the objective function E共X兲共Eq. 共6兲兲 is possible if the differences between the parts are substantial. To reduce the chance of such convergence, the addition of the ␦ factor in Eq. 共7兲 was necessary for a more progressive approach to the deformation. The nonrigid registration module presented in this section allows for the deformation of the meshed CAD model until it matches the scanned part, without taking into consideration the profile deviation of the latter. Schematically, Fig. 4 illustrates the deformation imposed on the CAD mesh nodes S to create S⬘ closer to the point cloud P. In this figure, the nonrigid registration displaced points s7 and s8 to points s⬘7 and s⬘8 in a smooth manner, meaning without forcing s⬘7 and s8⬘ on the points 共c7 and c8兲 in the profile deviation zone of the scanned part. Identifying these points in the profile deviation zone from the part’s deformation itself is a distinct challenge, and one that is not trivial. Section 4.6 will introduce an innovative method to precisely address this challenge. 4.5 Correction of the Point-Point Distance. Throughout the registration modules, the point combinations si-ci and their respective distances can be computed using the dsearchn function in Matlab® based on the quickhull algorithm 关35兴. It is necessary, however, to introduce a correction of the point-point distance in order to reduce the influence of the density of points within each point cloud; the corrected distance is given by the Dsi metric. This metric Dsi represents the projection of the point-point distance dsi onto the normal vector nsi of the mesh at point si 共see Fig. 3兲. In order to evaluate Dsi using Eq. 共8兲, the normal vector nsi is estimated as shown in the Appendix,

共6兲

The parameter ␣ influences the smoothness quality of the deformed mesh. An ␣ with a high numerical value will impose a strong correlation between the affine transformation matrix xi of connected nodes. In contrast, a low ␣ value will make the displacement of each node only constrained by the term Ed, and its assigned xi will be independent of those of their connected nodes. Consequently, a low ␣ value would result with an irregularly deformed mesh. In the proposed algorithm, the resolution of Allen et al.’s 关25兴 objective function E共X兲共Eq. 共6兲兲 is solved, as proposed by Amberg et al. 关26兴. In order to reduce the influence of the distance error caused by the point-point metric, each point of C is replaced with ci⬘ formulated as 011009-4 / Vol. 133, FEBRUARY 2011

Fig. 3 Substitution of each target point ci with ci⬘

Fig. 4 Construction of a deformed meshed CAD model „S⬘… closer to the scanned part „P… even though points c7 and c8 are in a zone with a profile deviation

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Fig. 5 Schematic diagram of „a… a part without a zone with profile deviations and „b… a part with a zone with profile deviations „points c8 and c9…

D si =

n si · d si 储nsi储

2

共8兲

n si

4.6 Identification Method. As stated in Sec. 4.4, a weight wi is attributed to each node of the meshed CAD depending on whether or not its corresponding point in the scanned part is in a zone with a profile deviation. Thus, the identification method developed in this subsection introduces a mechanism for identifying points si whose corresponding points ci are in a zone of profile deviation on the manufactured part. The method is inspired by the work of Merkley 关30兴, who emphasized the difference between material covariance and geometric covariance. Material covariance refers to the interdependence of the displacement between points of a mesh when subjected to external forces, whereas geometric covariance describes the correlation of surface errors at neighboring points of a surface. Surface areas are strongly correlated when examined at a microscopic scale due to their derivation from a continuous surface. The identification method thus uses this correlation as the basis for distinguishing the behavior of the global deformation from that of the local deformation, which is typical of profile deviation. This distinction is crucial to bringing the two models closer together and is achieved by comparing the distance Dsi of si with the distances Ds j of its neighboring nodes. An example will help clarify the above discussion. Figure 5共a兲 presents a case where the surface P has no profile deviation. In this case, the displacement that has to be applied to each node si to get closer to P is strongly correlated as a result of the material covariance. This characteristic translates itself as a minor difference between the distance Dsi of si and those of its neighboring nodes. In contrast, Fig. 5共b兲 illustrates a case in which the surface P has a profile deviation in the zone of points c8 and c9. In this case, the displacement that has to be applied to nodes s8 and s9 to get them closer to c8 and c9 does not follow the global displacement imposed by the material covariance. Therefore, points s8 and s9 are distinguished because their distances Ds8 and Ds9 are substantially different from the distance Dsi of their neighboring nodes. As a result of these characteristics, it is possible to identify nodes si with corresponding points ci that fall in a zone with a profile deviation. Consequently, when comparing the meshed CAD model with the scanned part, this method separates the deformation owing to the part’s flexibility from the profile deviation. The next paragraphs will detail the identification method itself. Considering that the level 1 neighborhood of a node si is composed of all nodes constituting the elements connected to node si, level 2 is composed of all nodes constituting the elements connected to the nodes of level 1 共see Fig. 6兲. Additionally, let Vs1 = 关兵Vs1 其 ¯ 兵Vs1 其兴 and Vs2 = 关兵Vs2 其 ¯ 兵Vs2 其兴 be, respectively, the 1

NS

1

its neighboring nodes. The first step of the two-step calculation of Ii uses an averaging filter, like those commonly used in image processing 关36兴, to reduce the measurement noise. Each distance Dsi is substituted by Fsi, which is equivalent to the average of the distances Ds j of the nodes in level 1 of si, 1 NV

si

兺D F si =

j=1

1 兲 s共Vsi j

共9兲

NV1 si

where NV1 is the number of nodes in the level 1 neighborhood si of si. The list of neighboring nodes is denoted as 1 兵Vsi1 其 = 关Vsi1 Vsi1 ¯ Vsi1 兴储 Vsi1 苸 R3⫻NVsi. 1

2

NS

Subsequently, the confidence indicator Ii 苸 I of the node si is determined by squaring the average of the difference between Fsi and each Fsi of the nodes from the level 1 neighborhood of si,

Ii =

冤

1 NV

si

兺F j=1

si

− F s共V1 兲

NV1 si

si j

冥

2

,

1 ⱕ i ⱕ NS

共10兲

Finally, nodes with a confidence indicator of ␶ times larger than the 95th percentile of I, which are also not on the outline of the part, are considered to potentially have their corresponding point ci in a zone with profile deviation. Here, the 95th percentile threshold is used to find the extreme limit of the half-normal distribution 关37,38兴 of confidence indicators, thus identifying nodes that behave distinctly from the rest of the population. Moreover, a multiplier factor ␶ is added to minimize the influence of measurement noise and the effect of the distance metric accuracy due to the shape curvature of the part 共i.e., normal vector evaluations兲 during the identification process. Following the identification process, the weights wi of the level 2 neighborhood of the points identified are then placed at 0 in the nonrigid objective function 共Eq. 共6兲兲. Points along the outline of the part are omitted as they lack sufficient points in their neighborhood to properly evaluate their associated normal vector. To summarize, this section presents a mechanism to differentiate the behavior of the global deformation of the part from the behavior of the local deviation of the surface, which is typical of profile deviation. As explained, a confidence indicator Ii is used to identify points si with corresponding points ci as within a zone

NS

lists of points of levels 1 and 2 of a node in S. Ds = 关Ds1 ¯ DsN 兴 S represents the corrected Euclidean distance between each pair of points si-ci. Using the aforementioned definitions, we introduce here a confidence indicator Ii 苸 I, with 1 ⬍ i ⬍ NS, which allows a node si to be distinguished if its displacement does not follow the material covariance. That is to say, the confidence indicator represents the difference between the distance Dsi of si with the distances Ds j of Journal of Manufacturing Science and Engineering

Fig. 6 Neighborhood of node 742: „a… level 1 and „b… level 2

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Fig. 7 Flow chart of the identification method Fig. 8 Flow chart of the IDI algorithm

with profile deviation. This identification is essential to decreasing the influence of certain points in the computation of the displacement field. Figure 7 illustrates this identification method.

4. Find point ci and the point-point distance dsi corresponding to each node si. 5. Compute the normal vectors 共nsi兲. 6. Compute the corrected distance Dsi. 7. Compute each confidence indicator Ii and identify the nodes with an Ii indicator equal to ␶ multiplied by the 95th percentile I. 8. If K ⬎ Kmax, terminate the algorithm and evaluate the profile deviations. 9. If K = 1, subtract the identified nodes from the point set S and restart at step 3. 10. If K ⬎ 1, using the nonrigid algorithm of Sec. 4.4, compute the matrix X and apply the transformation to S. 11. Set K = K + 1. 12. If 兩XK − XK−i兩 ⬍ ⌬X and ␣ ⬍ ␣F, set ␣ = ␣ − step␣. 13. If K ⬎ K␦, set ␦ = 1, where the symbol K is the iteration in course, Kmax is the maximum number of iteration of the IDI algorithm, K␦ is the number of iteration for which the factor ␦ is applied, ␣D and ␣F are, respectively, the initial and minimum values of the ␣ parameter, ⌬X is the convergence criteria, and step␣ is the step value subtracted from ␣.

4.7 Algorithm. Having explained rigid and nonrigid registration techniques, as well as an identification method, it is now possible to present the IDI algorithm, which is the core purpose of this paper. IDI works by comparing the meshed CAD model to the scanned part through an iterative displacement of nodes si of the mesh until the shape of the meshed CAD model matches the scanned part, but without concealing the profile deviations. To reflect the metric modification of Sec. 4.5, the point-point metric 共dsi兲 used in the objective function 共Eq. 共1兲兲 must be replaced by the corrected distance Dsi, NS

f=

兺储D 储 i=1

si

2

共11兲

The following steps describe the IDI algorithm, schematized in Fig. 8. Starting with the set of nodes S of the meshed CAD model, the point cloud P of the manufactured part, the set of points of corresponding landmark points, and the lists of levels 1 and 2 neighborhood points for each node, the algorithm works as follows. 1. Make a pre-alignment between S and P using the landmark points, as discussed in Sec. 4.2. 2. This is followed by the initialization phase with K = 1 and ␣ = ␣D. Set the ␦ value, as well as the number of sampling nodes 共Nsampling兲. 3. While ␣ ⬎ ␣F, align S and P using the rigid registration shown in Sec. 4.3 using the Nsampling sampling nodes. 011009-6 / Vol. 133, FEBRUARY 2011

5

Case Studies

This section presents a series of case studies using the IDI algorithm. Three shapes are examined, representing a sample of common parts in the transport industry: a quasi-constant surface, a U-shape, and a freeform surface. The quasi-constant surface serves as a starting point for the study and represents an aerodynamic skin. The U-shape, representing an extruded part, introTransactions of the ASME


Fig. 9 Descriptions of the case studies: „a… quasi-constant surface, „b… U-shape surface, and „c… freeform surface. Points PDCSi, with i = 1 , 2 , 3, indicate the 3-2-1 fixing layout used to build the simulated manufactured part. They are also used as landmark points in the pre-alignment step.

duces a difficulty with its sharp shape changes. The third shape is a freeform surface, representing a body panel. These case studies have three objectives, which are as follows: to validate, evaluate the performance, and find the parameters influencing the robustness of the algorithm. These goals explain the use of simulated manufactured parts for such parts allow for the control of all aspects of the studies in question. Therefore, starting with the CAD model of the part, the scanned manufactured parts are simulated using the following steps. 1. A local modification in the nominal surface, representing a zone with a profile deviation with a dent shape, is created using the Freeform module in PROENGINEER® WILDFIRE 2. 2. A finite element analysis of the model created in the previous step is generated using ANSYS®. This analysis takes into account both the mounting fixture configuration3 during a presumed digitalization and the force of gravity. Additionally, an external force is applied to the model to simulate unknown deformations to the part from the manufacturing process, such as residual stress induced distortion. This additional deformation guarantees that the simulated part behaves differently than the theoretical model of the part submitted to the gravity load. A point cloud representing the mesh of the part with a profile deviation zone and also with a deformed shape is created following this last step. 3. A measurement noise that follows a normal distribution N共0,0.1 mm兲 is introduced on the newly created point cloud. This new point cloud then represents the scanned part. With the aforementioned protocol, 32 scanned parts were built consisting of 12 quasi-constant shapes, 8 U-shapes, and 12 freeform shapes. All 32 parts are in aluminum gauge 14 共0.7213 mm兲 with a Young modulus of 7 ⫻ 1010 N / m2 and a density of 2700 kg/ m3. The 12 quasi-constant shapes were constructed using one quasi-constant surface with six different types of profile deviations, each one with two possible deformations induced. The eight U-shapes were constructed using one U-shape surface with 3 Three well-positioned clamps were used during the simulation in order to ensure the deformation of the part.


four different types of profile deviations, each one also with two possible deformations induced. Finally, the 12 freeform shapes were constructed using one freeform surface with six different types of profile deviations, each one again with two possible deformations induced. Figure 9 illustrates the three types of surfaces used, as well as the position and the combinations of dent-shape zones used to create the different types of profile deviations. Table 1 describes the external constraints 共in addition to the force of gravity兲 imposed in the protocol with its resulting maximum induced deformation as well as the number of zones with imposed profile deviation, the area of each zone, the maximum amplitude of the profile deviation in each zone, and the number of nodes in the nominal mesh with corresponding points in profile deviations zones. These manufactured scanned parts are then compared with their corresponding meshed CAD model using the IDI algorithm, with the parameter values in Table 2. These values were determined in a series of trials in which combinations of ␣ and ␶ values were tested to ensure both the smoothness of the mesh after each iteration and the proper identification threshold. Each value is dependent on the rigidity and the shape of the part. Automatic meshing software was used to derive the meshed CAD model, enabling a mostly uniform grid. Each meshed CAD model comprises 1364, 3550, and 2952 nodes for the quasi-constant surface, the U-shape, and the freeform surface, respectively. In order to avoid the misclassification of nodes on rounded corners of the U-shape surface, the weight wi of these nodes is set to 0 in the nonrigid objective function 共Eq. 共6兲兲 as it is difficult to properly evaluate their associated normal vector 共i.e., their associated confidence indicator Ii兲. 5.1 Results. This subsection discusses the results using the IDI algorithm to inspect the case studies. An analysis of the convergence behavior, the accuracy, and the speed of the proposed algorithm is presented. For the validation of the algorithm’s convergence, Fig. 10 shows the average corrected distance Dsi per iteration for each case study. Figure 10 demonstrates the method’s convergence. Figure 10 also highlights a proportional relationship among the number of iterations required for convergence, the complexity of the part, and the induced deformation of the scanned part. FEBRUARY 2011, Vol. 133 / 011009-7


Table 1

Case studies Quasi-constant: overall dimensions, 350⫻ 800 mm2, 0.311 m2; point cloud, 14,841 points; nominal mesh, 1364 nodes

Abbreviation

U-V0F1 U-V1F1 U-V2F1 U-V3F1 U-V0F2 U-V1F2 U-V2F2 U-V3F2

Freeform: overall dimensions, 350⫻ 1000 ⫻ 250 mm3, 0.171 m2; point cloud, 16,500 points; nominal mesh, 2952 nodes

C-V0F1 C-V1F1 C-V2F1 C-V3F1 C-V4F1 C-V5F1 C-V0F2 C-V1F2 C-V2F2 C-V3F2 C-V4F2 C-V5F2

Number of zone Maximum with induced profile deformation deviations 共mm兲

Simulation configuration

A-V0F1 A-V1F1 A-V2F1 A-V3F1 A-V4F1 A-V5F1 A-V0F2 A-V1F2 A-V2F2 A-V3F2 A-V4F2 A-V5F2

U-shape: overall dimensions, 300⫻ 500 ⫻ 150 mm3, 0.209 m2; point cloud, 22,000 points; nominal mesh, 3550 nodes

Table of case studies

2 N forceb on point A

22

3 N forceb on points A and B

34

10 N forceb on point C

12

5 N forceb on points C and D

36

2 N forcec on point E

33

10 mm displacementb of curve F-E

13

None 1 1 1 2 3 None 1 1 1 2 3 None 1 1 4 None 1 1 4

Area 共AD兲 of each zone 共m2兲

Peak 共HD兲 profile deviation in each zone 共mm兲

Nb deviationsa

n/a 0.014 0.015 0.015 0.015/0.015 0.015/0.019/0.015 n/a 0.014 0.015 0.015 0.015/0.015 0.015/0.019/0.015

n/a 3.1 2.8 2.4 2.3/2.8 2.3/2.3/2.8 n/a 3.1 2.8 2.4 2.3/2.8 2.3/2.3/2.3

n/a 27 56 67 96 138 n/a 27 56 67 96 138

n/a n/a 0.014 3.0 0.014 3.0 0.009/0.008/0.007/0.006 1.5/1.7/2.5/2.5 n/a n/a 0.014 3.0 0.014 3.0 0.009/0.008/0.007/0.006 1.5/1.7/2.5/2.5

None 1 1 1 2 3 None 1 1 1 2 3

n/a 0.004 0.007 0.006 0.004/0.007 0.006/0.004/0.007 n/a 0.004 0.007 0.006 0.004/0.007 0.006/0.004/0.007

n/a 3.5 3.7 2.5 3.5/3.6 2.5/3.5/3.5 n/a 3.5 3.7 2.5 3.5/3.6 2.5/3.5/3.5

n/a 76 73 112 n/a 76 73 112 n/a 52 62 103 110 211 n/a 52 62 103 110 211

a

Number of nodes in the nominal mesh corresponding to points with imposed profile deviations in the simulated manufactured part. Applied on the inverse direction of axis z. c Applied on the y axis direction. b

Given that the profile deviations of the manufactured part are known, the method’s accuracy is simply determined by the difference between the imposed deviations and those found through the IDI method. Points on the part’s outline are excluded from the accuracy analysis as they have an insufficient number of neighboring nodes to permit a useful evaluation of the normal vectors. Figure 11 summarizes the method accuracy as a function of the simulation characteristics of the manufactured part. This figure shows that the method’s accuracy is approximately 10% of the imposed peak profile deviations on the manufactured part and is therefore well beneath the value of the deformation. Current inspection methods would have simply identified the deformation of

the manufactured part as a profile deviation since they are based on rigid registration methods and thus do not compensate for the part’s deformation. Furthermore, no systematic lower accuracy values were found in zones with known profile deviations; small concentrations of lower accuracy were found, however, on nodes in rounded corners of the U-shape surface due to the high curvature on the zone 共i.e., accuracy of the normal vector evaluation兲. Figure 12 gives an overview of the IDI method accuracy by dividing the accuracy found for nodes in zones with and without known profile deviations. This figure demonstrates that the distributions of the accuracy values in nodes in zones with and without known profile

Table 2 Table of parameters for the IDI algorithm used in the case studies. In all cases, step␣ equals 50, ␦ equals 0.5, and 1200 sampling nodes „Nsampling… are used. Parameter

␣D

␣F

␶

⌬X

K␦

Kmax

6 500 10 000 17 000

4 500 8 000 15 000

2.5 1.5 6.0

0.75 2.00 1.25

10 20 10

150 500 300

Case studies Quasi-constant U-shape Freeform

011009-8 / Vol. 133, FEBRUARY 2011

Transactions of the ASME


Fig. 10 Convergence of the average corrected distance

deviations behave similarly. The slight differences in the percentile values between Figs. 12共a兲–12共c兲 and Figs. 12共d兲–12共f兲 are caused by the low density of nodes in zones with known profile deviations, which gives each node’s value a heavy weight in the percentile value evaluation. Moreover, given that the injected measurement noise follows a normal distribution centered at zero with a standard deviation of 0.1 mm, its absolute value follows a half-normal distribution 关37,38兴. Figure 13 superposes the half-normal distribution of the measurement noise on the distribution of the accuracy results of the IDI algorithm. This figure demonstrates that the distribution errors follow closely the distribution of the measurement noise in a 95% confidence interval 共0.2 mm兲. Consequently, it can be inferred that the method accuracy results are primarily the consequence of the identification of the measurement noise by the algorithm. That is to say that the IDI method is able to create a meshed CAD model with the same induced deformation as the scanned part, but without creating the profile deviation or measurement noise of this same part. All case studies were performed on an Intel Core 2 Duo, 2.66 GHz, 4.00 Gbyte laptop using a 64 bit operating system. The 12 quasi-constant surface studies took an average of 6 s/iteration for a total of 16 min/case. The eight U-shape surface studies took an average of 21 s/iteration for a total of 174 min/case. Finally, the 12 freeform surface studies took an average of 13 s/iteration for a total of 64 min/case. The main computational demanding steps are the rigid registration, correspondence search, computation of the normal vector, solving of the nonrigid registration equation, and application of the nonrigid registration. Thus, the speed of the IDI method is proportional to the number of nodes in the meshed CAD model, as well as the number of iterations performed to achieve the desired accuracy. This is shown in the algorithm’s convergence figure 共Fig. 10兲.

6

Fig. 11 IDI method’s accuracy compared with the peak profile and the maximum deformation imposed on the simulated scanned part

Conclusions

This paper presents a new algorithm, IDI, for the inspection of surface profiles of flexible parts without the use of specialized jigs. The algorithm works through a comparison of two sets of points, one from the meshed CAD model and one from the scanned manufactured part, despite the significant difference in their respective geometries. It combines rigid with nonrigid

Fig. 12 Overview of the IDI method accuracy by dividing the accuracy found for nodes in zones with and without known profile deviations: „„a…–„c…… Nodes in zones without known profile deviations. „„d…–„f…… Nodes in zones with known profile deviations for the quasi-constant surface, the U-shape surface, and the freeform surface, respectively.


FEBRUARY 2011, Vol. 133 / 011009-9


Fig. 13 Distribution of the IDI method accuracy compared with the distribution of the noise added during the simulation of the scanned part: „a… the quasi-constant surface, „b… the U-shape surface, and „c… the freeform surface

Fig. 14 Illustration of the normal vector at a point

registration methods, as well as a newly developed identification method to distinguish deviations from deformations. The method outlined proceeds by smoothly deforming the meshed CAD model successively until it matches the scanned part. This is done without forcing the meshed CAD model to match the profile deviation or measurement noise of the scanned part. This gradual matching process is made possible with the introduction of the identification method developed here, which enables the effects of profile deviations to be distinguished from those of deformations due to the positioning of the part and its flexibility. Further work will focus on improving the IDI method so that it will be able to accommodate a variety of geometries with changing thicknesses, in addition to other types of form inspection. For instance, a sharp change in thickness dramatically changes a part’s behavior; this change still poses a challenge to the present method. As such, particular attention will first need to be given to develop more nuanced ␣ and ␶ parameters to handle such challenges. Second, additional research will be required to remove the trial and error aspect of estimating these values and to assess the measurement uncertainty of the method. Finally, pilot studies will need to move beyond parts synthesized on a computer and into parts that exist in the material world.

Acknowledgment The authors would like to thank the National Sciences and Engineering Research Council 共NSERC兲 and the École de Technologie Supérieure 共ÉTS兲 for their support and financial contribution.

Nomenclature AD ⫽ area of a dent-shape profile deviation ␣ ⫽ weight of the smoothness error Es 共Eq. 共5兲兲 in the nonrigid registration objective function 共Eq. 共6兲兲 ␣D ⫽ initial value of the ␣ parameter ␣F ⫽ minimum value of the ␣ parameter A − ViF j ⫽ quasi-constant surface case study with the ith type of profile deviation and the jth type of deformation 011009-10 / Vol. 133, FEBRUARY 2011

C ⫽ set of points in P closest to the set of nodes S C − ViF j ⫽ freeform surface case study with the ith type of profile deviation and the jth type of deformation DCS ⫽ design coordinate system ␦ ⫽ percentage of the necessary displacement applied during the Kth iteration dsi ⫽ point-point distance metric Dsi ⫽ corrected point-point distance; projection of the point-point distance on the normal of the mesh at point si ⌬X ⫽ convergence criteria Fsi ⫽ average of the distances Ds of the nodes in j level 1 of si HD ⫽ peak deviation of a dent-shape profile deviation zone ICP ⫽ iterative closest point algorithm IDI ⫽ iterative displacement inspection algorithm, as proposed Ii ⫽ confidence indicator associated with an si node Kmax ⫽ maximum number of iteration of the IDI algorithm K␦ ⫽ number of iteration for which the factor ␦ is applied MCS ⫽ measurement coordinate system N P ⫽ number of points representing the scanned part NS ⫽ number of nodes representing the meshed CAD model NV1 ⫽ number of nodes in the level 1 neighborhood si of si Nsampling ⫽ number of sampling nodes of S used during the ICP algorithm P ⫽ set of N P points representing the scanned part S ⫽ set of NS 兩 NS Ⰶ N P nodes of the meshed CAD model step␣ ⫽ step value subtracted from ␣ ␶ ⫽ factor used to set the upper limit of the confidence indicator Ii U − ViF j ⫽ U-shape surface case study with the ith type of profile deviation and the jth type of deformation Vs1 ⫽ list of points of the level 1 neighborhood of nodes in S Vs2 ⫽ list of points of the level 2 neighborhood of nodes in S wi ⫽ weight attributed to point si in the distance error function Ed 共Eq. 共4兲兲 X ⫽ matrix 共4NS ⫻ 3兲 of the unknown affine transformation matrix xi Transactions of the ASME


Appendix: A Normal Vector Estimation Triangular areas of the mesh around point si are used to compute the normal vector nsi, as proposed by Jirka and Skala 关39兴, NE

i

兺␾n

j j

j=1

n si =

NE

兩

共A1兲

i

兺␾n兩 j j

j=1

where NEi is the number of elements around si, ␾ j is the area, and n j is the normal vector of the triangle inside the element E j 共see Fig. 14兲.

References 关1兴 Savio, E., Chiffre, L. D., and Schmitt, R., 2007, “Metrology of Freeform Shaped Parts,” CIRP Ann., 56共2兲, pp. 810–835. 关2兴 Gu, P., and Huang, X., 1998, “CAD-Model Based Inspection of Sculptured Surfaces With Datums,” Int. J. Prod. Res., 36共5兲, pp. 1351–1367. 关3兴 Li, Y., and Gu, P., 2004, “Free-Form Surface Inspection Techniques State of the Art Review,” Comput.-Aided Des., 36共13兲, pp. 1395–1417. 关4兴 Li, Y., and Gu, P., 2005, “Inspection of Free-Form Shaped Parts,” Rob. Comput.-Integr. Manufact., 21共4–5兲, pp. 421–430. 关5兴 Yao, A. W. L., 2005, “Applications of 3D Scanning and Reverse Engineering Techniques for Quality Control of Quick Response Products,” Int. J. Adv. Manuf. Technol., 26共11–12兲, pp. 1284–1288. 关6兴 Gao, J., Gindy, N., and Chen, X., 2006, “An Automated GD&T Inspection System Based on Non-Contact 3D Digitization,” Int. J. Prod. Res., 44共1兲, pp. 117–134. 关7兴 Shi, Q., and Xi, N., 2008, “Automated Data Processing for a Rapid 3D Surface Inspection System,” IEEE International Conference on Robotics and Automation 2008 共ICRA 2008兲, Pasadena, CA, pp. 3939–3944. 关8兴 Ravishankar, S., Dutt, H., and Gurumoorthy, B., 2010, “Automated Inspection of Aircraft Parts Using a Modified ICP Algorithm,” Int. J. Adv. Manuf. Technol., 46共1–4兲, pp. 227–236. 关9兴 Karbacher, S., Babst, J., Häusler, G., and Laboureux, X., 1999, “Visualization and Detection of Small Defects on Car-Bodies,” Modeling and Visualization ‘99, Sankt Augustin, pp. 1–8. 关10兴 Leopold, J., Günther, H., and Leopold, R., 2003, “New Developments in Fast 3D-Surface Quality Control,” Measurement, 33共2兲, pp. 179–187. 关11兴 Eichhorn, A., Girimonte, D., Klose, A., and Kruse, R., 2005, “Soft Computing for Automated Surface Quality Analysis of Exterior Car Body Panels,” Appl. Soft Comput., 5共3兲, pp. 301–313. 关12兴 Dorïng, C., Eichhorn, A., Xiaomeng, W., and Kruse, R., 2006, “Improved Classification of Surface Defects for Quality Control of Car Body Panels,” 2006 IEEE International Conference on Fuzzy Systems, Vancouver, BC, Canada, pp. 1476–1481. 关13兴 Megahed, F., and Camelio, J., 2010, “Real-Time Fault Detection in Manufacturing Environments Using Face Recognition Techniques,” J. Intell. Manuf., online first. 关14兴 Besl, P. J., and McKay, H. D., 1992, “A Method for Registration of 3-D Shapes,” IEEE Trans. Pattern Anal. Mach. Intell., 14共2兲, pp. 239–256. 关15兴 Masuda, T., and Yokoya, N., 1994, “A Robust Method for Registration and Segmentation of Multiple Range Images,” Proceedings of the 1994 Second CAD-Based Vision Workshop, Champion, PA, pp. 106–113. 关16兴 Park, S. Y., and Murali, S., 2003, “A Fast Point-to-Tangent Plane Technique for Multi-View Registration,” Proceedings of the Fourth International Conference on 3-D Digital Imaging and Modeling 共3DIM 2003兲, Banff, AB, Canada, pp. 276–283.


关17兴 Pottman, H., and Hofer, M., 2002, “Geometry of the Squared Distance Function to Curves and Surfaces,” Institute of Geometry, Vienna University of Technology, Technical Report No. 90. 关18兴 Rusinkiewicz, S., and Levoy, M., 2001, “Efficient Variants of the ICP Algorithm,” Proceedings of the Third International Conference on 3-D Digital Imaging and Modeling 共3DIM 2001兲, Quebec City, QC, Canada, pp. 145–152. 关19兴 Gelfand, N., Ikemoto, L., Rusinkiewicz, S., and Levoy, M., 2003, “Geometrically Stable Sampling for the ICP Algorithm,” Proceedings of the Fourth International Conference on 3-D Digital Imaging and Modeling 共3DIM 2003兲, Banff, AB, Canada, pp. 260–267. 关20兴 Dawant, B. M., 2002, “Non-Rigid Registration of Medical Images: Purpose and Methods, A Short Survey,” 2002 IEEE International Symposium on Biomedical Imaging, Washington, DC, p. 4. 关21兴 Holden, M., 2008, “A Review of Geometric Transformations for Nonrigid Body Registration,” IEEE Trans. Med. Imaging, 27共1兲, pp. 111–128. 关22兴 Ferrant, M., Warfield, S. K., Guttmann, C. R. G., Vulkern, R. V., Jolesz, F. A., and Kikinis, R., 1999, “3D Image Matching Using a Finite Element Based Elastic Deformation Model,” Proceedings of the Second International Conference on Medical Image Computing and Computer-Assisted Intervention— MICCAI ‘99, Cambridge, UK, Vol. 1679, pp. 202–209. 关23兴 Feldmar, J., and Ayache, N., 1994, “Rigid and Affine Registration of Smooth Surfaces Using Differential Properties,” Proceedings of the Third European Conference on Computer Vision, Stockholm, Sweden, Vol. 2, p. 397. 关24兴 Feldmar, J., and Ayache, N., 1996, “Rigid, Affine and Locally Affine Registration of Free-Form Surfaces,” Int. J. Comput. Vis., 18共2兲, pp. 99–119. 关25兴 Allen, B., Curless, B., and Popovic, Z., 2003, “The Space of Human Body Shapes: Reconstruction and Parameterization From Range Scans,” ACM SIGGRAPH 2003, San Diego, CA, Vol. 22, pp. 587–594. 关26兴 Amberg, B., Romdhani, S., and Vetter, T., 2007, “Optimal Step Nonrigid ICP Algorithms for Surface Registration,” IEEE Conference on Computer Vision and Pattern Recognition 共CVPR ‘07兲, Minneapolis, MN, pp. 1–8. 关27兴 Liu, S. C., Hu, S. J., and Woo, T. C., 1996, “Tolerance Analysis for Sheet Metal Assemblies,” ASME J. Mech. Des., 118共1兲, pp. 62–67. 关28兴 Camelio, J. A., Hu, S. J., and Ceglarek, D., 2004, “Impact of Fixture Design on Sheet Metal Assembly Variation,” J. Manuf. Syst., 23共3兲, pp. 182–193. 关29兴 Liu, S. C., and Hu, S. J., 1997, “Variation Simulation for Deformable Sheet Metal Assemblies Using Finite Element Methods,” ASME J. Manuf. Sci. Eng., 119共3兲, pp. 368–373. 关30兴 Merkley, K. G., 1998, “Tolerance Analysis of Compliant Assemblies,” Ph.D. thesis, Department of Mechanical Engineering, Brigham Young University, Provo, UT. 关31兴 Bihlmaier, B. F., 1999, “Tolerance Analysis of Flexible Assemblies Using Finite Element and Spectral Analysis,” Ph.D. thesis, Department of Mechanical Engineering, Brigham Young University, Provo, UT. 关32兴 Markvoort, L., Gliniorz, S., and Deneux, D., 2005, “Prise en compte de la nature flexible des matériaux pour l’analyse de tolérance,” 9ième Colloque National AIP PRIMECA. 关33兴 Markvoort, L., Gliniorz, S., and Deneux, D., 2005, “Etat de l’art des méthodes d’analyse de tolérance,” 9ième Colloque National AIP PRIMECA. 关34兴 Markvoort, L., 2007, “Méthodologie d’analyse statistique de tolérances dans les assemblages impliquant des composantes déformables,” Ph.D. thesis, Université de Valenciennes et du Hainaut-Cambrésis, Valenciennes, France. 关35兴 Barber, C. B., Dobkin, D. P., and Huhdanpaa, H., 1996, “The Quickhull Algorithm for Convex Hulls,” ACM Trans. Math. Softw., 22共4兲, pp. 469–483. 关36兴 Gonzalez, R. C., and Woods, R. E., 2008, Digital Image Processing, 3rd ed., Prentice-Hall, Upper Saddle River, NJ. 关37兴 Pewsey, A., 2004, “Improved Likelihood Based Inference for the General Half-Normal Distribution,” Commun. Stat: Theory Meth., 33共2兲, pp. 197–204. 关38兴 Bland, J., 2005, “The Half-Normal Distribution Method for Measurement Error: Two Case Studies,” Department of Health Sciences, University of York, Technical Report. 关39兴 Jirka, T., and Skala, V., 2002, “Gradient Vector Estimation and Vertex Normal Computation,” Department of Computer Science and Engineering, University of West Bohemia, Pilsen, Czech Republic, Technical Report No. DCSE/TR2002-08.

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