1 Medial Axis Extraction and Thickness Measurement of Formed Sheet Metal Parts Mittelflächenbestimmung und Wanddickenmessung an gekrümmten umgeformten Blechbauteilen Der Technischen Fakultät der Universität Erlangen-Nürnberg zur Erlangung des Grades D O K T O R I N G E N I E U R vorgelegt von Nataša Petrović Erlangen 2010
2 Als Dissertation genehmigt von der Technischen Fakultät der Universität Erlangen-Nürnberg Tag der Einreichung: Tag der Promotion: Dekan: Prof. Dr.-Ing. Reinhard German Berichterstatter: Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c. mult. Albert Weckenmann Prof. Dr.-Ing. Rainer Tutsch
3 Abstract Optical scanning measuring systems enable probing of workpieces with a high point density and offer possibilities to create adequate evaluation solutions for different complex measurement tasks in the geometrical metrology. This work deals with the evaluation of scanned data in order to extract parameters which can contribute to improvement of sheet metal forming processes in the early production phase. These parameters are form of a medial surface and thickness values. An experimental measuring system for scanning formed sheet metal parts in certain areas of interests was developed. It consists of two fringe projection systems located opposite to each other. A solution for an accurate fusion of two independent measuring data in a unified coordinate system was found. Data obtained in a such manner reliably represent scanned sheet metal area. An innovative method for the extraction of medial surface points from the data and the calculation of large number of local thickness values in the direction orthogonal to the medial surface was designed and implemented. The extracted medial surface can be used for an assessment of the obtained sheet metal form by its comparison with a CAD model or with the form of a nominal sheet metal medial surface respectively. A visualization of the distribution of thickness values over the extracted medial surface enables localizing thinning and thickening places. Furthermore, the data obtained can be used for an evaluation of numerical simulations of thickness distributions offering possibility to compare simulated data with measuring data.
5 Zusammenfassung Optische Messgeräte ermöglichen eine flächenhafte Antastung von Werkstückoberflächen mit sehr hoher Punktdichte. Dies bietet die Möglichkeit zur Bestimmung neuer Strategien bei der Auswertung von Messergebnissen und zum Entwickeln passender Lösungen für komplexe Messaufgaben im Bereich der Fertigungsmesstechnik. Die vorliegende Arbeit befasst sich mit der Auswertung von optisch erfassten Punktewolken und der Bestimmung von Parametern, die zur Verbesserung von Blechumformprozessen in früheren Phasen der Prozessauslegung beitragen können. Diese Parameter sind die Form der Mittelfläche und die Wanddicke des Blechbauteils. Im Rahmen der Arbeit wurde ein experimentelles Messsystem zum Erfassen ausgewählter interessanter Bereiche gekrümmter umgeformter Blechbauteile entworfen und aufgebaut. Es besteht aus zwei gegenüberliegenden Streifenprojektionsgeräten. Eine Lösung für das Problem der Transformation von zwei unabhängigen Messdatensätzen in ein gemeinsames Koordinatensystem wurde bereitgestellt. Der erfasste Blechbauteilbereich wird dadurch zuverlässig abgebildet. Eine innovative Methode zur Bestimmung von Mittelflächenpunkten und zur Berechnung der lokalen Wanddicken, die senkrecht zur Mittelfläche bestimmt wurden, wurde entworfen und realisiert. Die Extrahierte Mittelfläche kann zur Bewertung der Blechbauteilform durch den Vergleich mit dem CAD-Modell, beziehungsweise mit der Soll-Mittelfläche des Blechbauteils, verwendet werden. Durch die Darstellung des Blechdickenverlaufs über die Mittelfläche ist eine genaue Lokalisierung von Ausdünnungen und Materialanhäufungen möglich. Die erhaltenen Daten sind außerdem zum Vergleich mit simulierten Daten und folglich zur Beurteilung der Zuverlässigkeit numerischer Simulationen verwendbar.
7 Danksagung Die vorliegende Arbeit entstand während meiner Tätigkeit als wissenschaftliche Mitarbeiterin am Lehrstuhl Qualitätsmanagement und Fertigungsmesstechnik der Friedrich-Alexander-Universität Erlangen-Nürnberg. Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c. mult. Albert Weckenmann, Inhaber des Lehrstuhls, gilt mein besonderer Dank für die Anregung zu dieser Arbeit, für die wohlwollende Unterstützung und Förderung meiner wissenschaftlichen Tätigkeit, sowie für die Übernahme des Hauptreferats. Prof. Dr.-Ing. Rainer Tutsch, Inhaber des Instituts für Produktionsmesstechnik der Technischen Universität Braunschweig, danke ich für die Übernahme des Korreferates. Allen jetzigen sowie ehemaligen Mitarbeiterinnen und Mitarbeiter des Lehrstuhls Qualitätsmanagement und Fertigungsmesstechnik danke ich für die sehr gute Zusammenarbeit. Bei Dipl.-Ing. (FH) Johannes Weickmann bedanke ich mich für die freundliche Übernahme des Korrekturlesens. Ich bedanke mich auch bei allen meiner Studienarbeiter und Hilfsassistenten. Insbesondere gilt mein Dank Arne Haas und Marko Plesinac, die einen direkten Beitrag zu dieser Arbeit geleistet haben. Meinem Freund Almir Uzunović danke ich liebevoll für sein Verständnis, viel Geduld, Diskussionsbereitschaft und das erste Korrekturlesen. Nicht zuletzt möchte ich erwähnen dass ich meinen Eltern und meinem Bruder für immer dankbar sein werde, denn sie haben mir diesen Weg ermöglicht. Erlangen, im Januar 2010 Nataša Petrović
9 1 Contents 1 Introduction State of the art and shortcomings D free form scanning Scanning the two sides of formed sheet metal parts Medial surface Medial axis transformation Chordal axis transformation Medial axis of sheet metal cross section obtained by MAT D skeletons, medial and chordal surfaces of smooth solids Thickness measurement Aims of the research carried out Measuring system and data merging Experimental set-up Data merging Calibration of the sensors against each other Implementation of obtained transformation parameters in merging two sheet metal point clouds Evaluation of the merging procedure Method for the detection of deviations Artefact for merging procedure evaluation, its calibration and measurement Results of the evaluation Methods for eliminating limitations regarding industrial usage of the system caused by the calibration procedure Extraction of a medial surface and thickness determination Sheet metal medial surface Method of extraction of a medial surface and thickness determination Medial line extraction Approximations in the 2D case Extraction of medial surface points and thickness determination Input data - triangulated surface models of scanning points Method for the determination of inscribed spheres Implementation of the procedure for extracting medial surface points and thickness determination... 93
10 Input data Implemented algorithm Increasing program efficiency Possible variations in the procedure implementation Results analysis Calculation results Self-evaluation of the procedure Comparison of the results obtained by the implemented computing algorithm with the results of other methods Comparison of the obtained results with the results of the method for calculating minimal sheet metal wall thickness Analysis of the calculated thickness values and a comparison with calibrated point-to-point distances Summary and outlook References List of abbreviations Appendices
11 Introduction 3 1 Introduction Formed sheet metal parts are produced from thin and flat metal pieces or coiled strips (sheet metal semi-finished products), which are cut and formed in to a variety of different shapes. Many everyday objects are made of formed sheet metal parts. They are widely used in the mechanical engineering, communications, electronics and light industries [LIU 2004], for example for car bodies (Figure 1-1), automotive fuel tanks, aeroplane fuselages, different housings and enclosures, pots, and many other objects. Figure 1-1: Formed sheet metal parts used for car body All sheet metal processes can be divided into two groups: cutting processes - shearing, blanking, punching, parting, shaving etc., and plastic deformation processes - bending, stretch forming, deep drawing, spinning, necking and various other forming processes. The second group of processes involves partial or complete plastic deformation of the material [BOLJANOVIC 2004]. According to the DIN 8582 standard, forming processes from this group are further assigned to tensile-compression forming, tensile forming and bending forming processes [FRITZ 1995]. In the design of deep drawn sheet metal parts it is assumed that the two sides of the sheet metal stay parallel after forming, and that the sheet metal thickness is constant over a workpiece. In order to form working material in to the required shape, a change in thickness is unavoidable. The change in material thickness is not an intention of the processes mentioned above, but rather a consequence. The two sides of manufactured sheet metal parts are also not parallel, or the inner and outer curvature arcs are not concentric, as the case may be. In the design of parts made by some other forming processes, for example by stretch forming, material thinning is expected. A change in shape is achieved at the expense of sheet metal thickness. Owing to various influencing
12 4 Introduction factors, material thinning is not necessarily uniformly distributed over the workpiece, and parallelism of the two sides of the sheet metal after forming is not assured. However, conventional methods for thickness measurement are based on the assumption of parallelism of the two sides. The determination of thickness (distances between the two sides) of sheet metal using conventional measurement methods such as ultrasonic measurement, laser thickness measurement, radiographic measurement, measurement by tactile coordinate measuring machines etc. [VELGAN 2007] [TUTSCH 2006A], leads to results that are associated with at least one predefined direction. These direction-related results arise from the functional principle of the equipment used (laser thickness measurement) or from dependence on a selected workpiece coordinate system (e.g. measurement on tactile CMM using single point mode) [WECKENMANN 2009]. The measurement direction is oriented orthogonally to one side of the plane sheet metal part, and thus with negligible deviation also to the other side. Such a solution is on the one hand not easily applied to curved formed sheet metal parts (Figure 1-2-a), and on the other hand, since there is no parallelism of the two formed sheet metal sides, a direction orthogonal to one side is not necessarily orthogonal to the other (Figure 1-2-b). Theoretically, measurement direction should be orthogonal to some ideal surface of symmetry, but for real sheet metal parts this surface is unknown. a) b) Figure 1-2: a) Difficulties in searching directions orthogonal to curved surfaces and b) different distances measured orthogonally to two different non-parallel surfaces There is also an additional disadvantage of conventional methods of thickness measurement when they are used for formed sheet metal parts. The measurement methods mentioned above are appropriate only for tasks for which low information density is sufficient. In free form metrology, for example in inspection of curved sheet metals, it is of fundamental importance to measure a large number of points distributed on surfaces [SAVIO 2007]. Sheet metal part thickness can vary significantly across
13 Introduction 5 different areas of the part. In some areas, local material thinning or material accumulations can occur [KLOCKE 2006], which can be overlooked by measurement only at a few sampling points. To overcome problems of conventional thickness measurement methods, scanning measuring systems that enable workpieces probing with high point density can be used. Systems of this type include, for example, tactile coordinate measuring machines used in a scanning mode, 3D laser scanners, or fringe projection systems [SAVIO 2007]. The result of scanning using these devices are point clouds (sets of points coordinates of scanned surfaces), which can be further processed. Hence, possibilities for creating adequate evaluation solutions for different complex measurement tasks appear. Software applications for data processing has been developed parallel to the development of scanning devices. However, many problems of data evaluation considering specific measurement tasks remain unsolved. This work introduces an innovative method for the extraction of a medial surface of formed sheet metal parts and for the determination of a large number of local thickness values, determined in a direction orthogonal to this surface. A visualisation of the distribution of thickness values over the extracted medial surface is an additional important result, which can significantly contribute to the optimisation of the forming processes. It enables rapid analysis of a formed part, offering the possibility to localize areas of thinning and thickening precisely. It is particularly important to identify local areas of thinning since they represent the weak areas of a sheet metal part. Material accumulations can occur in some areas as well. If thinning and areas of material accumulation are recognized in a development phase of the forming processes, the processes can be optimised. Hence, material flow during forming can be directed to areas of thinning, and in areas of material accumulation can be better exploited [KLOCKE 2006]. Extraction of a sheet metal medial surface enables not only a reference for the direction of distances for thickness determination but has other important advantages. It enables accurate assessment of the sheet metal form by comparing it with a CAD model or with the form of a nominal sheet metal medial surface respectively. In the design of sheet metal an infinitely thin surface body is sometimes used as a model of the sheet metal [LOMBARD 2008] and its thickness is only given as a numerical value.
14 6 Introduction Such a body can be equivalent to one of the two sheet metal surfaces (inner or outer) or to a theoretical (nominal) medial surface. In the last case, incorrect results would be obtained by comparing the model surface with one of the scanned sheet metal sides. The comparison can be made only if the actual medial surface is available. By comparing a medial surface extracted from measured data with a CAD medial surface model a colour error map can be generated. In this way deviations of measured or extracted surfaces to the nominal surface can be determined. It enables exact identification of how the current product differs in form from the original design intent. Furthermore, the medial surface can be used for verification of sheet metal process simulation. Numerical simulations are widely used in development phases of formed sheet metal production for testing the feasibility of the designed forming processes [KLOCKE 2006]. The reliability of the simulations must be verified. Often, the validation consists of comparing the simulation results from different methods with each other and not with real inspection data [DAHLSTRÖM 2007]. A better solution would be to compare simulated and experimental results, such as for example, simulated and measured thickness values. Hence, by enabling the extraction of the medial surface and also thickness determination, it would allow a reliable comparison, which would contribute indirectly to the improvement of forming processes in an early production phase. In FEM simulations, so called shell-elements [GOULD 1988] are often used for the analysis of sheet metal and other thin-walled parts [KLOCKE 2006]. Shell elements are surfacebased elements. In FEM programs they are represented as infinitely thin polygons in space. The thickness used for the calculation of stresses is defined as a numerical parameter. The surface represented by shell elements is often placed in the middle of the representing volume [BLECH 2007]. The values of thickness obtained through simulation are thus assigned to the theoretical medial surface. In order to enable a reliable comparison, thickness values of the actual formed sheet metal workpiece assigned to the actual medial surface are needed. Since future deformation processes should satisfy demands for greater precision, any contribution to their development could be of significant importance. An innovative method for calculation of the formed sheet metal medial surface and thickness values is represented in this work. Furthermore, a new procedure for scanning the two opposite sheet metal surfaces in certain areas of interest is introduced.
15 State of the art and shortcomings 7 2 State of the art and shortcomings This chapter gives a brief introduction to methods currently used for scanning sheet metal surfaces, and elaborates the selection of fringe projection systems as the most suitable method for capturing (probing) surface points of the two sides of a sheet metal workpiece in certain areas of interest. Furthermore, the chapter includes an overview of methods of computational geometry relating to medial surface extraction. It also deals with the existing procedure for sheet metal thickness estimation and its measurement using scanned point clouds D free form scanning In free form metrology it is fundamental to measure a large number of points distributed on the surface to be inspected [SAVIO 2007]. For capturing (probing) a large number of points different tactile and optical measuring devices can be used. Tactile scanning can be performed using tactile coordinate measuring machines (CMMs) equipped with scanning probing systems [WECKENMANN 1999]. The disadvantage of devices of this type when probing free form surfaces is the complex programming of the scanning paths. A considerable effort for programming complex scanning paths is acceptable when inspecting serially produced sheet metal parts. However, it is possibly not acceptable for inspecting single parts, where the scanning paths have to be programmed for each part separately. Compared with optical measuring methods, scanning by tactile CMMs takes significantly more time, since they can measure up to 200 points/s at speeds as high as 150 mm/s [SAVIO 2007]. Furthermore, for tactile probing of formed sheet metal parts clamping can be very complicated and time consuming. The clamping must be on the one hand sufficiently rigid to avoid even small displacements of parts during probing, and on the other hand, too tight clamping can deform a sheet metal part and yield wrong results about parts form deviation. For these reasons it is better to use optical scanning devices for probing formed sheet metal parts. The first condition for selecting an appropriate measuring system should be the size of the typical sheet metal parts, such as for example car body parts. Hence, optical measuring systems that allow scanning of areas from several cm 2 up to several hundred cm 2 should be used. Laser scanners and fringe projection systems both have this
16 8 State of the art and shortcomings measuring range. Both systems enable non-contact, highly accurate (specified 3D accuracy is several tens of µm), and very fast scanning (several seconds up to one minute) with large point density (several tens of points per mm) [BAUER 2007]. The functional principle of both methods is a principle of optical triangulation [DGZFP 1995], and both enable 2,5D scanning of workpiece surfaces [HOFFMANN 2009]. The functional principle of laser scanners can be described in the following way: a projection unit projects a line of light (laser light) onto the measured surface. A detection unit detects the image of the projected line. CCD-detectors (cameras) are usually used as the detection units. Measured coordinates of the workpiece are generated from the image of the line deformed by the surface topography [SCHWENKE 2002]. Further details of the measuring principles can be found for example in [BAUER 2007] and [FHG 2003]. In order to enable the capture of points on the whole surface, the sensor (projection and detection unit) must be moved relatively to the workpiece. During scanning, movement is performed along one axis, approximately orthogonally to the projected line. The sensors are usually mounted on manually operated articulated arms or on CMMs [SAVIO 2007], which enable a controlled displacement of sensors. All captured surface points are directly assigned to one global coordinate system [FHG 2003]. The disadvantage of laser scanning devices is that the accuracy of a whole system depends not only on the accuracy of a sensor, but also on the accuracy of the system for controlled displacement. In contrast to laser scanning, fringe projection systems do not require controlled displacement. The capture of the measuring points on the workpiece surface using fringe projection systems is obtained without any mechanical movement or the sensor or the workpiece during scanning. Fringe projection systems (FPS) or white light scanners belong certainly to the most popular sensors for high point density scanning of workpieces in industry [BÖHM 2005]. Fringe projection is an umbrella term for various generally related measuring methods, using projection of different fringe patterns on the workpiece viewed by one ore more cameras [KOCH 1998]. The method can be seen as an upgraded laser scanning approach. Instead of projecting a single line and moving it, several lines are projected simultaneously [BÖHM 2005]. An example of fringes projected on a formed sheet metal part is shown in Figure 2-1. More details about different approaches for determination of object coordinates from camera views of fringe patterns (for example, Gray-code, Phase-
17 State of the art and shortcomings 9 shift, Phase-code, Line-shift) can be seen for example in [BÖHM 2005], [GÜHRING 2002], [BAUER 2007], [FHG 2003]. Figure 2-1: Fringe pattern projected on a formed sheet metal part Considering the number of camera views used in the method, there are two approaches to be distinguished: 'The first one is based onto the projection of a pattern on the surface and at least two different camera views. In this case, the pattern only serves to generate homologous points for triangulation within the camera images. The second approach evaluates the deformation of the pattern itself. In this case, the projector of the pattern takes up the role of one camera [SAVIO 2007].' A typical lateral measurement volume of a fringe projection system is in the range between 0.1 m and 1 m, with a relative accuracy of up to 10-4 of the measurement volume [SAVIO 2007]. Systems deliver results in the form of (x, y, z) coordinates of the surface points with a density of up to several tens of points per mm. Advantages of fringe projection systems comparing to laser scanners are the higher speed of measurement per captured single point [BAUER 2007] and the constant lateral density of points in a point cloud. The lateral point density of the laser scanners mounted on a manually operated articulated arm depends on the speed of moving the scanning line over the workpiece surface. A constant lateral point density can be obtained if the sensors are mounted on the CMM, but the costs of such a system are significantly larger than those for fringe projection systems. Since scanning data using fringe projection systems can be achieved without any mechanical movement, it can be assumed that more accurate data can be obtained with these systems than with laser scanners.
18 10 State of the art and shortcomings Scanning the two sides of formed sheet metal parts Fringe projection systems can be used for the given task of scanning two sheet metal sides, with the main problem being to ensure that the scans are in the same global coordinate system. A possible solution for scanning the two sides of a sheet metal part at an area of interest and of obtaining a subsequent data fusion to recreate the shape of the sheet metal is given in [MURMU 2003], [ERNST 2003], [WECKENMANN 2004] and [VELGAN 2007]. It is proposed to use a fringe projection system with an integrated precision rotary table to enable the scanning of both sides. After one side has been scanned the sheet metal is rotated by 180 and the other side is scanned. In order to transform obtained data in to a unified coordinate system, the angle of rotation and the relative position and orientation of the rotary axis to the coordinate system of the fringe projection system must be known. It is difficult to determine these data with adequate accuracy. The disadvantage of this method is that the accuracy of data transformation depends on the accuracy of these data. Any deviation in the determination of angle of rotation, or position of the rotary axis, or any rotary axis orientation inaccuracies, result in a deviation in the relative position of two point clouds after transformation. Furthermore, rotating the sheet metal part is timeconsuming. To overcome these disadvantages the two sheet metal sides can be scanned using two fringe projection systems located opposite to each other. Within this research work an experimental measuring system for scanning interesting areas of sheet metal parts was designed, tested, and a solution to the problem of achieving accurate data fusion was found. 2.2 Medial surface The concept of a sheet metal medial surface can be simply explained using the example of sheet metal bending. A sheet metal medial surface is equivalent to the so called neutral axis. This axis undergoes neither compression nor extension of the material. Inward and outward tensions are balanced at this axis, which shifts, for typical bending cases towards the centre of the bending radius. Hence it is closer to the inside of the bend than a theoretical medial or neutral axis [TSCHÄTSCH 2008].
19 State of the art and shortcomings 11 In manufacturing metrology an extracted medial surface is currently defined only for plane surfaces. [DIN EN ISO :1997] defines the extracted medial surface in the following way: 'The locus of centre points between sets of opposite points of the opposite extracted surfaces, where: the connecting lines of sets of opposite points are perpendicular to the associated median plane; the associated median plane is the median plane of two associated parallel planes obtained from the extracted surfaces'. Figure 2-2 shows the geometrical elements needed for understanding the definition. Such an interpretation of the medial axis is applicable when the coordinate measuring machines are used, and it is suitable for plane sheet metal parts but not for curved parts. This method assumes an operation with associated features that are commonly used for standard geometrical elements but not for free form surfaces. However, using scanning measuring devices, point density is so large that there is no need to operate with the associated features. Surfaces are reliably represented by captured points and by polygonal surface models (polygonal meshes) [KARBACHER 1997] obtained from point clouds. Figure 2-2: Extracted medial surface [DIN EN ISO :1997] However, one important conclusion can be derived from the standard definition: equidistance of medial surface to the boundary must be defined orthogonally to the medial surface, whether to associated or, if possible, to the medial surface to be extracted. This is further explained in Chapter 5, where the definition of the formed sheet metal medial surface is introduced. There is no generally accepted definition of medial surface of real free form sheet metal parts, let alone the method for its calculation. Nevertheless, there are several related
20 12 State of the art and shortcomings theories and methods in computational geometry for the calculation of medial line approximations, solid body skeletons, and medial surfaces of solid models used for extraction of shell elements in FEM analyses (see sections 2.2.1, and 2.2.3). The main reason for the non-applicability of these approaches to the problem of the medial axis of formed sheet metal parts is the unsuitability of the proposed realisation methods, characterised by a lot of approximations. Furthermore, they are principally designed for planar 2D shapes. Variations of 2D methods, adjusted for 3D cases, were introduced in the past few years, but they are mainly implemented for smooth solids only. Within the following sections the main concepts of axial descriptions and skeletons will be briefly explained. Furthermore, the only known attempt to implement the abovementioned approaches to measured sheet metal surfaces, described in [MURMU 2003], [IVAKHIV 2005] and [VELGAN 2007], will be explained Medial axis transformation The medial axis of an object is the set of points having more than one closest point on the boundary of the object. It was introduced in [BLUM 1967] as a tool for biological shape recognition. As stated in [HULIN 2006] the more precise definition was proposed by Pfalz and Rosenfeld. According to this definition the medial axis of a two dimensional (planar) boundary shape is the locus of the centres of inscribed discs that are tangent to a boundary curve at two or more points. For a sheet metal cross section the boundary consists of two curves, and the condition must be supplemented with the request that discs are tangent to both of them, at one or more points. Only in this way can a medial axis conform to the shape of both boundary curves. The medial axis, together with the associated radius function of the maximal inscribed discs, is called the medial axis transform (MAT) or a skeletal representation of the boundary shape. The skeleton follows the shape of the boundary, while the radius function captures the local thickness [SURESH 2003], [RAMANATHAN 2003]. There are several MAT continuous approaches, but commonly medial axes (MA) are approximations of continuous MA extracted from the Voronoi diagram (Appendix A) of the sampled contour (set of sample boundary points) [HULIN 2006]. Such approaches are so called semi-continuous approaches [HULIN 2006]. An approximation of the medial axis is
21 State of the art and shortcomings 13 in such cases defined from a Voronoi diagram as a Voronoi subcomplex [TAM 2003], [YOSHIZAWA 2003], [DEY 2002A], [DEY 2002B], [HISADA 2001]. Principally, Delaunay circles are defined for a set of sample boundary points. Centres of circumcircles of selected Delaunay triangles (Voronoi nodes, Appendix A) connected in a continued line represent the medial axis. However, Delaunay circles are not inscribed circles, and the method deviates from the theoretical principles of MA extraction. Such medial axes are very sensitive to small perturbations of the objects boundary [TAM 2003], [YOSHIZAWA 2003]. The resulting skeleton is not ideal in most cases. It may have several small branches and spurs induced by minor undulations or noise present in the boundary [PRASAD 2002]. Trimming and post-processing the MA is often needed, to bring the MA in conformity with the free form object [RAMANATHAN 2003]. There are certain geometric operations to reduce skeletons to a continuous smooth medial line [SURESH 2003], but they can be computationally expensive. MAT is useful for object characterisation and recognition, and is needed for various applications such as pattern and image analysis, mould design, tool path planning, etc., However, MAT is only an approximative method and not an optimal solution for usage in metrology Chordal axis transformation In order to correct sensitivity to noise in sampled contours for a semi-continuous medial axis, a new approach for its representation, called chordal axis transformation (CAT) was introduced [HULIN 2006]. The approach was designed for sparsely and non-uniformly discretized boundaries of shapes at any prescribed resolution [PRASAD 2002]. The principles of CAT were defined by Prasad in 1997 [HULIN 2006]. A new term maximal chord of tangency (MCT) was defined in connection with CAT. An MCT of a planar shape is a chord of a maximal disc within the shape, which separates the boundary of the disc in to two arcs such that at least one is not tangent to the boundary of shape (see [PRASAD 2005]). The chordal axis is hence the set of midpoints of the MCTs plus the set of centres of maximal discs having at least three MCTs (Figure 2-3) [HULIN 2006]. Prasad defines the chordal axis transform (CAT) as a semi-continuous method to extract the CA. He starts with a discrete sample of the boundary of a given shape S, then calculates subsequently the Delaunay triangulation of these points inside S. The MCTs of
22 14 State of the art and shortcomings S are approximated by internal edges of the triangulation (non-boundary edges of S). A skeleton based on the semi-continuous CA is constructed by connecting the midpoints of two or three internal edges inside each triangle [HULIN 2006]. The method is extensively described in [PRASAD 2002]. Figure 2-3: Three maximal balls inside a shape and their MCTs, chordal axis, and Delaunay triangulation of sampled contour and CAT skeleton, [HULIN 2006] Principally the method differs from MAT for discrete boundaries in the selection of other points of Delaunay triangles for axis determination. While MAT uses Voronoi nodes (centres of circumcircles of Delaynay triangles), CAT uses mostly the middle points of internal edges of Delaunay triangles. Chordal axes are applied in similar problems as medial axes, for example shape analysis, characterisation and recognition [HULIN 2006], but also in FEM analysis for quadrilateral meshing of thin or narrow, two-dimensional domains [YAMAKAWA 2002]. Both methods are based on constrained Delaunay triangulations and result in an approximated medial line. Their application on the measurement data in 2D cases for obtaining workpiece cross section medial axes is theoretically possible. In order to illustrate disadvantages of methods based on constrained Delaunay triangulation, the only known method that is proposed for approximation of the medial axes of scanned sheet metal parts, described in [MURMU 2003], [IVAKHIV 2005] and [VELGAN 2007], will be analysed in the next section. The method serves primarily for the approximation of local thicknesses in sheet metal cross-sections, and it is based on MAT Medial axis of sheet metal cross section obtained by MAT The method proposed in [MURMU 2003], [IVAKHIV 2005] and [VELGAN 2007] assumes the usage of point clouds obtained by scanning the two sides of the sheet metal workpiece. It
23 State of the art and shortcomings 15 suggests 2D medial axis transformation for several data cross-sections and their separate analysis [IVAKHIV 2005]. The method does not deal with the computation of medial surface polygons. The method is based on the theoretical definition of medial axis by centre points of inscribed discs, and intends to use diameters of such circles as thickness information. However, implementation of the method is based on constrained Delaunay triangulation. Thereby, only the circumcircles of Delaunay triangles with corner points (vertices) belonging to both boundary lines are extracted, and their centre points used for medial axis creation. Since the used centre points of Delaunay circles are also nodes of corresponding Voronoi cells, the method was named 'Voronoi diagram method'. In [IVAKHIV 2005] it is stated that the accuracy of such method depends on the density of points used in boundaries. Accuracy increases with increasing density of boundary points. In order to illustrate the general uncertainty of the method, an example of the low density case that shows deviation of the medial line derived in such a way, is shown in Figure 2-4. The medial axis obtained by Voronoi nodes is coloured red. As it can be seen in the figure, the approximated medial axis is not equidistantly from the boundaries. If for the same case a chordal axis is constructed, better approximation is obtained (green line in Figure 2-4). Medial axis derived by Voronoi nodes (centre points of Delaunay circles) Selected Delaunay circles Delaunay triangles and Voronoi cells Boundary lines Chordal axis Figure 2-4: Generation of the medial axis using Delaunay circles (Voronoi nodes) and its deviation from the chordal axis
24 16 State of the art and shortcomings Delaunay circles cannot be used to obtain thickness values since they are obviously not inscribed circles: they protrude significantly from the outlying boundary profiles approximated by line segments. Therefore in [IVAKHIV 2005] and [VELGAN 2007] additional steps are proposed. Projections of discrete contour points onto the medial axis are found. The projection points are assumed to be centres of new circles with radii corresponding to the distances from the boundary points to the medial axis. The diameters of such circles, named inscribed circles, are used as sheet metal cross section thickness values. But, these circles also protrude from outlying boundary profiles, even for a larger point density, and they are also not actual inscribed circles. Thickness values obtained in such a way are only approximate. Which ratio of the sample point density to the nominal boundary line distance is to be respected in order to get a certain medial line and thickness approximation accuracy, is not precisely defined. Furthermore, the method described was implemented for 2D cases only, and it cannot be straightforwardly practicable in 3D cases. It must be noted that the usage of cross sections is connected with additional approximations as well. A cross section is always assigned to a certain direction. That means, the derived medial axis is not necessarily located on the certain global workpiece medial surface, but can only be attributed to that cross section. The same can be said for thickness values. MAT and CAT for 3D cases result in medial and chordal surfaces representations. These methods have been introduced only recently, in the last couple of years. Principles, applications and disadvantages of the methods are given in the following section D skeletons, medial and chordal surfaces of smooth solids Medial surfaces (or so called 3D skeletons) and chordal surfaces are results of various methods of implementation of described medial and chordal axis transformations on 3D polygons. They are of importance in many diverse areas such as computer vision, robot path planning, evaluation of moulds and dies, feature recognition, medical diagnostics and mesh generation [QUADROS 2002]. They have applications in surface reconstruction for physical models and in the dimensional reduction of complex models as well. The general advantage of using the chordal surface rather than the medial surface is that the generation of the chordal surface is computationally less expensive [QUADROS 2002].
25 State of the art and shortcomings 17 Chordal surface transformation is therefore used as a tool for generating optimal CAD solid model meshes in FEM analysis (discretization of thin section objects), for example for creating multiple-layered hexahedral elements [QUADROS 2002] and tetrahedral meshes of thin-walled solids [YAMAKAWA 2005]. A chordal axis transformation for various shell structures is generated by making a onelayered tetrahedral mesh of the thin wall solid model and cutting the obtained tetrahedral elements. Cutting the internal edges of tetrahedrons results in triangular chordal surface facets [QUADROS 2002]. The implementation of a 3D medial axis transformation is also based on tetrahedral meshes. Delaunay tetrahedra are used for the approximation of medial surfaces circumcentres. Nevertheless, medial and chordal surfaces are not optimal solutions for application in metrology. As it is explained for 2D cases and implied for 3D cases, medial and chordal axis transformations result in approximations of medial lines or surfaces. In [TAM 2003] even the following is stated for MAT: The main problem in such an approach is that, unlike in 2D, the Voronoi vertices (circumcentres of the tetrahedra) in 3D do not converge to the medial axis as the sampling density approaches infinity. Medial surfaces (midelements) obtained by MAT are ill-defined for irregular thin solids [SURESH 2003]. Furthermore, methods used for generating chordal surfaces, applied on CAD models of thin-walled solids, cannot be used for the determination of medial surfaces obtained from measured data. Scanned point clouds can be meshed and polygonal models of surfaces generated, but 3D CAT methods require volumetric models. Furthermore, the measured surfaces are not smooth. Even if two opposite surfaces could be used for creating a volumetric model, a calculation of the tetrahedral mesh of such a model would be impossible. Any smoothing of measured surfaces would be unacceptable from the metrological point of view since it would affect the accuracy of results. Since previously explained methods for extraction of medial and chordal axes and surfaces are characterized by many approximations, they were not followed up within the scope of this work. Instead, a method for extraction of accurate medial surface points from polygonal surface models has been devised. The only allowed approximation in this case was the usage of polygonal boundaries for the approximation of sheet metal surfaces. The method is principally closer to the theoretical approach of CAT than to
26 18 State of the art and shortcomings MAT. Chordal axis principles defined for planar shapes are adapted to 3D cases of sheet metal parts, and an implementation approach which is not based on Delaunay triangulation has been elaborated. 2.3 Thickness measurement Besides conventional methods for measuring sampling thickness values in certain predefined directions, briefly described in Chapter 1, there are only a few other methods for determining the thickness of formed sheet metal parts. A method for estimating thickness reduction as described in [GOM 2009] and [VIALUX 2008] is sometimes used in practice. The method is based on forming analysis of a sheet metal. Prior to the forming process, a pattern of circular dots or grid lines is applied to the original plane sheet metal (by electrochemical etching, laser marking, or printing) [GALANULIS 2007], [VIALUX 2008]. The applied grid deforms in accordance with the deformation of the part. Photogrammetric measurements using one or more high-quality cameras fixed together are performed on the same side of the formed sheet metal part. Comparing the original and deformed mesh geometry, local strain values are determined at thousands of local data points. The thickness reduction is calculated from the major and minor strain assuming a constant volume [GALANULIS 2007]. Information about thickness reduction is expressed as a percentage of original sheet metal thickness. It can be calculated and assigned to the measured sheet metal side, or to a roughly approximated medial surface [GALANULIS 2007]. The results can be used as assessment values when analysing thickness change over the different sheet metal areas, but they do not reliably represent actual sheet metal thickness. Considering available measurement and evaluation methods, only a single method is known which can be used for measuring and evaluating areas with critically reduced sheet metal thickness. The original approach is described in [ERNST 2003] and [VELGAN 2007]. Optimisation of the method was made within initial research activities of this work. It is described in detail in [WECKENMANN 2009]. The method proposes using two point clouds obtained by scanning the two opposite sides of the sheet metal workpiece. The algorithm applied for the calculation of minimal distances presents the iterative search for the minimal distance from each measuring
27 State of the art and shortcomings 19 point of one sheet metal side to triangles of the opposite polygonal surface. The result of the calculation is a sequence of the minimal distances corresponding to appropriate measuring points of the parent point cloud [WECKENMANN 2009]. The calculation of shortest distances must be made twice, changing thereby two sides source (points side) and target side (polygon surface side). A schematic representation of the measuring feature is given in Figure 2-5. The data obtained can be visualised in 3D by assigning calculated distance values to points on the 'source' sheet metal side. Polygonal model of the surface Triangle from the polygonal model Shortest distance from the selected point Point from the point cloud Point cloud Figure 2-5: The shortest distance between the measuring point on one sheet metal side and the polygonal surface on the opposite side [Weckenmann 2009] The main problem of the method is non-commutativity the difference in results for the two sides of the sheet metal [ERNST 2003], [WECKENMANN 2004], [WECKENMANN 2009]. Two sets of results, obtained by changing 'source' sides are similar but not identical. The problem is an unavoidable effect caused by non-parallelism of the two sides of the actual sheet metal [WECKENMANN 2009]. Since only the minimal distances between sheet metal sides are calculated, the method is not an optimal solution in a case when relevant thickness values in areas of high thickness need to be quantified. To overcome the problems of ambiguous results and the calculation of minimal distances alone, a method enabling the calculation of distances between the two sides of formed sheet metal in a direction orthogonal to an extracted medial surface has been devised and represented within this work.
28 20 Aims of the research carried out 3 Aims of the research carried out The aim of this work was to enable an accurate extraction of the medial surface points and a simultaneous calculation of thickness values of formed sheet metal parts in areas of interest. The first task was to design and realise an experimental set-up that allows scanning of both sides of a sheet metal part in areas of approximately 100 cm 2. It was decided to use two fringe projection systems located opposite to each other for that purpose. Since the data obtained by two completely independent measuring systems was assigned to two independent coordinate systems, it was important to find a method to merge them, in order to enable the reliable reconstruction of the scanned sheet metal area. The next task was to design a procedure for the verification of the realised point clouds merging method, and to perform the verification on data captured using the realised experimental set-up. Subsequently a method for further processing of the data in accordance with the given task was devised. Firstly, it was important to provide plausible definition of medial surface of curved thin wall parts that originally did not exist. Furthermore an applicable method for the extraction of medial surface points in accordance with the given definition had to be designed and implemented as a prototype in MATLAB. The aim of the method was to enable the calculation of thickness values assigned to extracted medial surface points as well. At the same time it was the intention as far as possible to avoid any approximations. Finally, the data obtained by calculations had to be evaluated. This was a particularly challenging problem since there was no possibility of making any comparison of the data obtained with reference data as, for data obtained by the implemented innovative procedure, no adequate reference data was available. Therefore, it was important on the one hand to enable a reliable qualitative assessment of the extracted medial surface, and on the other hand to devise and implement specific methods for the quantitative assessment of thickness values and indirectly of medial surface points as well.
29 Measuring system and data merging 21 4 Measuring system and data merging In order to enable scanning of the area of interest of the curved formed sheet metal parts from both sides, two fringe projection systems located opposite to each other can be used. This chapter deals with the experimental set-up, designed and realised for the purpose of scanning both sides of the sheet metal. The implemented method for the calibration of the whole system will be explained as well as the solution for the challenging problem of the accurate and reliable transformation (fusion/merging) of two point clouds, which have no redundant/overlapping scanned areas, in to a unified coordinate system. 4.1 Experimental set-up For the construction of the experimental set-up two fringe projection systems MikroCAD40 and TopoCAM50 were used, both from GFMesstechnik GmbH. MikroCAD40 is a measuring system with one fringe projector (DMD 1024 x 768 Digital) and one CCD camera (with normal camera objective, model BASLER A641f). Light to the projector is provided by the external halogen light source (model Schott KL 1500 LCD). The system is schematically shown in Figure Projector 2 - Camera 1 Light beam 2 2 ca. 600 mm 30 Figure 4-1: Schematic representation of MikroCAD40 The nominal measuring range of MikroCAD40 is 90 mm x 70 mm x 40 mm (x, y, z) and the working distance approximately 600 mm. Its vertical resolution (z-resolution) is 1 µm, and lateral (x-y resolution) 56 µm. The measuring system is controlled and manipulated using ODSCAD 6.0 control and evaluation software (also from GFMesstechnik GmbH).
30 22 Measuring system and data merging Measurements are performed by a combination of the 'Phase shift' and the 'Gray-code' method. The accuracy of the system is verified by the sphere-distance-error tests according to the guideline [VDI/VDE :2002], with maximal permissible error of ±100 µm (specified by manufacturer) not being exceeded. The measuring system TopoCAM50 also has one fringe projector (DMD 1024 x 768 Digital), but it differs from MikroCAD40 by having two CCD cameras (both of the model BASLER A631f). The same model of the external halogen light source as for MikroCAD40 is used. TopoCAM50 has a slightly larger measuring range 100 mm x 80 mm x 50 mm (x, y, z), and the working distance of approximately 300 mm (Figure 4-2). Its vertical resolution (z-resolution) is 2 µm, and lateral resolution (x-y resolution) 60 µm. The system is manipulated using the control and evaluation software TopoXenios2.1 (GFMesstechnik GmbH). The measurement is performed by the phase code method. The specified and verified accuracy of the system TopoCAM50, defined by the maximal permissible error limits for the sphere-distance-error tests [VDI/VDE :2002], is ±10 µm Projector 2 - Camera Light beam 15 2 ca. 300 mm 2 1 Figure 4-2: Schematic representation of TopoCAM50 Sensors are mounted on a granite plate over the guide pillars, which enable movement of the sensors in a vertical direction (Figure 4-3). The Z-axes of the sensors are oriented horizontally, not vertically as it is the common practice in sensor usage. Therefore, in order to avoid confusions in further explanations, terms like 'vertical resolution' of the sensor, or 'lateral measuring range', will be replaced by terms including axis names like 'z-resolution' or 'x-y measuring range'. A precondition for scanning both sides of the sheet metal workpiece, is that the
31 Measuring system and data merging 23 measurement ranges of the sensors overlap. Placed within the common sensor measurement range, the selected area of the sheet metal workpiece can be scanned from two sides without moving either the measuring equipment or the workpiece. Therefore the measurement systems are placed and oriented relative to each other in such a way to maximize the possible common range, which is, in a case of total overlapping, the range of MikroCAD40, 90 mm x 70 mm x 40 mm (Figure 4-3 and 4-4-b). Guide pillar MikroCAD40 Guide pillar TopoCAM50 Reference plane Granite plate Common measurement range Figure 4-3: Schematic representation of the experimental set-up, the reference plane and the common measurement range In order to enable total overlapping of the measuring ranges, a specially designed artefact is used. It serves as the reference object for alignment of the sensors relative to each other. It is a 3 mm thick sheet metal plate with the same grid line pattern on both sides (Figure 4-4-a). Generally, the plate helps in positioning the sensors, providing a basis for focusing the sensors at approximately the same place along the z-direction. Thus, approximately the same position of the sensors reference planes in this direction is achieved. Marked grid lines in the x-y directions help in adjusting the same orientation of the sensors regarding rotational movements around the z-axis. The enumerated points facilitate an equal positioning of the sensors in the x-y directions. The disposition of the sensors measuring ranges relative to each other, and positions of the sensors coordinate systems after alignment, are shown schematically in Figure 4-4-b.
32 24 Measuring system and data merging Figure 4-4: a) artefact used for alignment of the sensors relative to each other b) sensors measurement ranges and positions of the coordinate systems after alignment A more accurate sensors alignment than that obtained by the procedure described is not required and not practicable. For the purpose of merging the point clouds, the sensors must be previously metrologically adjusted (calibrated) against each other. The procedure is explained in the following section. MikroCAD40 Measured workpiece TopoCAM50 Figure 4-5: Experimental set-up and measured sheet metal part
33 Measuring system and data merging 25 Figure 4-5 shows the realised experimental set-up and the used sheet metal workpiece. This workpiece is scanned as an example for further explanations of the merging procedure and the implemented method of data evaluation. Scanning data obtained by the fringe projection systems can include 'gaps', i.e. areas without any information. These are so called 'shadow areas', which cannot be scanned in certain relative positions of the sensors to the workpiece. Furthermore, some areas cannot be optimally scanned since undesirable reflections appear. Data can also include randomly distributed single outliers. However, data in the area, for which the medial surface needs to be determined and thickness calculated, should be characterised by continuous point clouds, with no outliers that could affect results. This can be achieved by: careful selection of the workpiece position relative to the sensors, data filtering [SAVIO 2007], accomplished directly after scanning (filtering options are integrated in common measuring software), and removing outliers and trimming uninteresting areas manually using data visualisation and analysis software. Single outliers can be removed and data can be trimmed either before or after merging. On the scanned area of the demonstration workpiece this was carried out before the data merging. This workpiece can be seen in Figure 4-6. The scanned area is marked by a dotted white line, and the area that will be further processed, by a continuous grey line. Figure 4-6: Measured sheet metal area from the front (a) and the back side (b)
34 26 Measuring system and data merging 4.2 Data merging Using two independent fringe projection sensors scanning of the two sides of the workpiece (surface digitalisation) can be performed. Point clouds obtained in such a way are located in two independent coordinate systems. In other words, the coordinates of the measuring points are assigned to two different coordinate origin points. The point clouds of the sample sheet metal part, shown in Figure 4-6, are visualised in a unified coordinate system, retaining the original coordinates of the points in both cases. This is shown in Figure 4-7. The green point cloud is a result of scanning using the MikroCAD40 fringe projection system, and the blue one, of the TopoCAM50. This colouring convention will be kept in the whole work. It can be seen that the point clouds are oriented in a random way to each other. Their orientation depends on mutual orientation and position of the coordinate systems of the sensors. The position and orientation of the two coordinate systems relative to each other is only approximately known. Therefore, it cannot be used for correcting the mutual position and orientation of two point clouds. Figure 4-7: Point clouds before data merging (visualisation made in PolyWorks ) In order to represent a scanned sheet metal area reliably, captured point clouds must be positioned and oriented relatively to each other as if they were scanned originally in one coordinate system. Such a positioning of point clouds relative to each other can be obtained by the mathematical transformation of only one or both point clouds. Common alignment and merging procedures used for two or more point clouds, which are integrated in commercial measurement and visualisation software, cannot be used in this case.
35 Measuring system and data merging 27 Such procedures are employed when several scans are used to capture an object or a workpiece. In order to align and merge scans in a global coordinate system, all methods used (approximate or refining alignment methods) require scans to share some redundant information (to overlap) with adjacent scans [INNOVMETRIC 2007]. In the case of scanning two opposite sheet metal sides by two oppositely placed sensors, there is no possibility of having common (overlapping) areas. Hence, existing merging options integrated in common software cannot be directly used to align the scans. The option for scans alignment using a common reference object which would be placed on the sheet metal borders and would be a part of scan-image of both sensors is also inapplicable. The most interesting sheet metal areas, which need to be scanned, are usually placed somewhere in the middle of the object, not at its borders. Alignment can be made only according to some previously determined pattern. The main condition is then to keep the sensors' position and orientation relative to each other unchanged once the alignment pattern is determined. A constant relative position of the sensors can be easily ensured by avoiding any relative movement between the sensors. The problem of pattern determination can be precisely defined as the determination of transformation parameters, which can be repeatably applied to all point clouds. It is sufficient to make only one reliable and regular reference transformation in order to derive the transformation parameters. All subsequent transformations can be made in the same way using the same parameters. The implementation of this idea, a solution for the reference transformation problem, and the procedure for calculating reference transformation parameters are all represented in the following section Calibration of the sensors against to each other The procedure of determining transformation parameters can be designated as a calibration of the opposite sensors against to each other. The main problem thereby is to implement the reliable reference transformation. To perform a reference transformation means to bring scanned point clouds of a certain reference workpiece, from the original position to the 'right position'. Thereby, the 'right positions' of scans of a reference workpiece must either be previously determined or already known.
36 28 Measuring system and data merging General principles of calibration of sensors and merging of point clouds The measurement of one hypothetical reference workpiece is schematically shown in Figure 4-8. The form of the reference workpiece can be freely chosen. Any form of the reference workpiece is acceptable, but symmetrical workpieces should be avoided. FPS1 and FPS2 represent two fringe projection systems located opposite to each other. Measurement for the calibration Measuring range FPS 1 Reference FPS 2 workpiece Actual measurement FPS 1 Workpiece M z n Measuring range FPS 2 Figure 4-8: Schematic representation of measurement of a reference workpiece in the calibration procedure, and actual measurement of a workpiece whose scans of two opposite sides need to be merged The measurement of the workpiece whose scans need to be merged using parameters obtained from the previously performed calibration procedure is also shown in the figure. Within this example, instead of a sheet metal, as a measured workpiece the workpiece that is similar to the reference is selected. It simplifies explanations of the calibration procedure and the principle of application of transformation parameters. The precondition for locating scans of the reference workpiece in the 'right positions' relative to each other is to have something that can be used as a model for it. A model can be a point cloud, obtained by scanning the reference workpiece using some accurate measuring system. For example, measuring systems which enable the capture of all
37 Measuring system and data merging 29 probing points in one global coordinate system (e.g. a laser scanner, or a tactile coordinate measuring machine) can be used. A scanned model could be obtained even by a fringe projection system, making several scans sharing redundant information. In this case the workpiece model would be obtained through alignment and final merging (matching together) of the point clouds. Theoretically, a CAD-model of the reference workpiece can be used for performing the initial transformation in the calibration procedure. In practice, it should be avoided. Deviations of the real reference workpiece from the CAD model could cause unrealistic alignment of the workpiece scans, and consequently the wrong transformation parameters. Once the model of the reference workpiece is available it can be imported into any commercial measurement and visualisation software. The scans of the reference workpiece, made by two fringe projection systems, imported into the same coordinate systems, will have a random orientation, not only relative to each other, but also to the model. In order to explain the general principle an idealised case can be analysed. For example, it can be assumed the scans of the reference workpiece (shown in Figure 4-8), are shifted relative to the reference workpiece model only in the z direction - there are no x-y translations, and no rotational deviations. This case is shown in Figure 4-9. The assumption used is intended to simplify the explanation of the whole procedure. In practice, the scans can also have any other position and orientation to the model, the remaining part of the procedure would be the same. Using methods implemented in the measurement and visualisation software used, the scans can be easily aligned to the model regardless of their original position. The common algorithm implemented in measurement software is the Iterative Closest Point (ICP) algorithm [ZACHER 2004]. It minimizes the distances between two clouds of points. Given in the new positions, aligned to the workpiece model, the scans of the reference workpiece are also represented in Figure 4-9. They are marked by dotted lines. After alignment of the reference workpiece scans to the model the next task would be the extraction of the transformation parameters. This can be performed mathematically, using the original and final coordinates of the points from the point clouds. For the example given in Figure 4-9, transformation parameters are represented by
38 30 Measuring system and data merging translations parameters, namely Tz1 for the FPS1 point cloud, and Tz2 for the FPS2 point cloud. These parameters can be further implemented for transformation of point clouds obtained by scanning any workpiece by the same two fringe projection systems. The mutual position of the sensors must not be changed. Calibration procedure FPS 1 point cloud T z1 Alignment 1 Model of the reference workpiece T z2 Alignment 2 FPS 2 point cloud Determination of the transformation parameters (here T z1 and T z2 ) Data merging T z1 T z2 FPS 1 point cloud FPS 2 point cloud Applying the transformation parameters (here T z1 and T z2 ) z y x Merged data representing the workpiece M z n Figure 4-9: Principle of calibration using the reference workpiece and application of determined transformation parameters for the example of the workpiece 1, shown in Figure 4-8 In order to illustrate how such a transformation would look like in a specific case, a schematic representation of the merging procedure for the sample workpiece (workpiece M, Figure 4-8) is given in Figure 4-9. As it can be seen in Figure 4-8, an idealised workpiece position within the measuring range is selected. The workpiece is aligned on its left side with the left-hand position of the reference workpiece. It can be also observed that the measured workpiece is narrower for a certain value zn than the reference workpiece (Figure 4-8). In Figure 4-9 it can be perceived that the right scan of the workpiece M has original coordinates, which are shifted for the value zn from the position of the scan of the right-
39 Measuring system and data merging 31 hand side of the reference workpiece. That means that any difference in a shape of the measured workpiece to the reference workpiece will be reciprocally reflected on differences in their original coordinates. Therefore, transformation parameters derived from calibration can be applied for the transformation of scans in the subsequent merging regardless of the workpiece shape. Considering the specific example, applying the transformation parameters Tz1 and Tz2, both scans will be shifted to the new positions. Such merged scans represent the sides of the workpiece M, as if they were scanned in a single coordinate system (Figure 4-9). In cases when the transformation is characterised by additional translational and rotational parameters the same concept can be applied. Modification of the general method of calibration of sensors The calibration and merging procedure is described in previous section with a purpose of understanding the general concept of the method. Such concept is acceptable, but hat also following disadvantages: The reliable point cloud of the reference workpiece (model of the reference workpiece) is needed. The accuracy of the method is affected not only by the accuracy of the fringe projection systems used but also by the accuracy of the scanning systems used for obtaining a reference point cloud (model of the reference workpiece). The method requires the alignment and transformation of both point clouds, in the calibration procedure as well as in all subsequent merging tasks, and that is time consuming. In order to avoid these disadvantages and to improve and simplify the method, the general calibration concept has been modified. Instead of aligning both scans to the model of the reference workpiece, one of the two point clouds is aligned to the other one, which stays in the original position. Thereby, the derived associated geometrical elements, common to both scans, are used as references for the alignment. These elements are the centre points of the spheres of a ball plate, which have very small sphericity deviations, and which are scannable by both fringe projection systems. The advantages of the modified method, compared to the previously explained one, are:
40 32 Measuring system and data merging No model of the reference workpiece is required and the reference workpiece doesn't have to be previously scanned by some other measuring system for this purpose. This means that the accuracy of the method is not affected by any other measurement system, but only by the fringe projection systems used. The method requires the transformation of only one point cloud. The other one stays in the original position during the calibration procedure, and also in all subsequent steps. The implementation of the method is described in the following sections. Implementation of the method of calibration of sensors As a reference workpiece, or a calibration artefact, the ball plate, shown in Figure 4-10, is used. The only metrological requirement on the artefact is to be equipped with spheres having as small sphericity deviations as possible. Figure 4-10: Ball plate used for the determination of transformation parameters (calibration of sensors against each other) The artefact shown in the figure was specially designed and manufactured for the experimental set-up. It consists of four calibration spheres made of titanium alloy 6AL4V. Each sphere has a diameter of mm, a diffuse reflecting surface and very small sphericity deviation (less than µm, according to the spheres manufacturer [BAL-TEC 2009]). The spheres are fixed (glued) into the 2 mm thick steel plate. The plate is used for fixation of spheres, and has no relevance in the calibration. In order to avoid the negative influence of indirect reflection on the scanning of the half-spheres, the plate is provided
41 Measuring system and data merging 33 with diffuse reflecting surface finish. Each sphere is symmetrically mounted on the plate, with one half protruding from one plate side, and the other half from the other. Thus, the spheres can be simultaneously scanned with both fringe projection systems. The artefact must be dimensioned considering the common measuring range of the measuring systems. The arrangement of the spheres within the measuring ranges in the specific case, is schematically shown in Figure 4-11-b. Figure 4-11: a) Schematic representation of the ball plate, b) arrangement of the spheres within the measuring ranges Scanning one half of the sphere with a fringe projection system enables the determination of its centre point. Since the sphere has negligible sphericity deviation, two determined centre points of two hemispheres belonging to the same sphere can be conditionally considered as one and the same point. In this manner, derived spheres centre points can be used as references in the transformation of one point cloud within the calibration procedure. Generally, the calibration procedure consists of: Scanning the spheres from both sides with the fringe projection systems. Generation of associated Gauss-spheres. This can be performed using any commercial data processing and visualisation software. In this project PolyWorks was used. Determination of the Gauss-sphere centre points (also made using PolyWorks )
42 34 Measuring system and data merging Aligning the point clouds, using the centre points of the spheres as references. One of two point clouds must be kept in the original position, and the other one aligned in such a manner that the distances between four corresponding spheres centre points are optimally minimized. Calculation of rotation and translation parameters using original and new coordinates of the transformed sphere centre points. The calculation program for this purpose is implemented in MATLAB. The alignment of the point clouds within the calibration procedure is schematically shown in Figure B T (x BT,y BT,z BT ),z AT ) C M a) T (x' BT,y' BT,z' BT ) z b) Figure 4-12: a) Arrangement of the point clouds before the alignment, b) point clouds after the alignment
43 Measuring system and data merging 35 The hemispheres marked green, shown in Figure 4-12, represent the scan data obtained by the MikroCAD40 fringe projection system, and the blue hemispheres the data obtained by TopoCAM50. The derived spheres centre points are designated as A M, B M, C M, D M, and A T, B T, C T, D T. Those points are used as references in the alignment of the point clouds. The green point cloud is thereby selected to be fixed, and the blue one is transformed. In the ideal case the spheres centre points A T, B T, C T, and D T would overlap perfectly with the points A M, B M, C M, and D M after the alignment. In practice, this is impossible. Small differences in points positions will always be present. The software used for alignment enables optimal minimisation of these distances, but differences are unavoidable. The results of multiple measurements on several calibration settings showed that these distances between corresponding centre points after alignment are not larger than 15 µm. Generally, these deviations are caused by differently sized and positioned best-fit spheres (Gauss-spheres) at the two sides of the artefact, which in consequence have differently positioned sphere centre points. There are several reasons for this: The sphericity deviations of the spheres, although very small, would cause minor deviations of centre points positions, even when the measuring systems would be perfectly accurate. There are slight deviations of point clouds caused by inaccuracy of the measuring systems. Only small parts of the point clouds of two corresponding hemispheres are used for determination of Gauss-spheres (best-fit spheres). Therefore spheres could slightly deviate in position and size, and could be different for two sides of the same sphere. Boundary areas of the spheres cannot be scanned accurately enough, since those surfaces are significantly inclined to the measuring systems. For this reason significant parts of the point clouds must be trimmed and are not used in Gauss-sphere determinations (Figures 4-13-a and 4-13-b). However, the accuracy of the whole calibration method is not affected significantly by position deviations of the sphere centre points. It must be only obligatory considered in the procedure for calculating the transformation parameters. Although small deviations arise when one point cloud is shifted to the 'right position' relative to the other one the parameters which characterise its transformation can be
44 36 Measuring system and data merging used with sufficient reliability in subsequent transformations. These parameters need to be determined. The software used for the alignment of point clouds in calibration does not provide information on these parameters. They need to be calculated from the original and new coordinates of the transformed points. Hemispheres scanned by TopoCAM50 Hemispheres scanned by MicroCAD40 a) b) Figure 4-13: a) Point clouds and derived best-fit spheres before the alignment (visualised in PolyWorks ), b) point clouds and derived best-fit spheres after the alignment (visualised in PolyWorks ) In Figures 4-12-a and 4-12-b red markings denote the coordinates of four points from the transformed point cloud that are used in the parameter calculation. Generally any four points from the transformed point cloud which do not lie in one plane could be used for this purpose. The point cloud transformation can be seen as the transformation of a rigid body. Distances between the points belonging to the same point cloud stay unchanged. However, the original and new coordinates of the sphere centre points can be extracted in a simply manner. Therefore they will be used in parameters calculations. In order to prevent false conclusions it must be emphasised that it is not possible to replace in the calculation the new coordinates of the transformed points (for example coordinates of the points A' T, B' T, C' T, and D' T ), with the corresponding points from the other point cloud (A M, B M, C M, and D M ) and hence possibly evade the whole alignment process. As previously explained, there is always a slight deviation in position between those points. The distances between points A' T, B' T, C' T, and D' T are not the same as the distances between A M, B M, C M, and D M and, even after an optimal alignment, these points do not overlap. With such a replacement, the calculation of the transformation parameters would be impossible.
45 Measuring system and data merging 37 Determination of the transformation parameters Results from the alignment of the two sides of the artefact are original and new coordinates of four points, from one point cloud. According to the example represented schematically in Figure 4-12, the blue point cloud is assumed to be transformed within the reference alignment procedure. Extracted coordinates of the centre point of the sphere A T for example are designated as x AT, y AT, z AT (original coordinates, prior to transformation) and x' AT, y' AT, z' AT (the new coordinates). Analogous designations are used for the coordinates of the centre points of the spheres B T, C T and D T. These coordinates represent also components of point radius vectors. These vectors are represented in the form of a column matrix. The general transformation equation can thus be written as: [ x ' '] [ x z] y ' =R y T z (4.1) where the translation parameter T is a vector characterised by its components in three dimensions x T=[T T z] y T (4.2) and the rotation parameter is characterised by 3 x 3 matrix which represents components of rotations about three axes. In a general form it can be written as R=[R 11 R 12 R 13 R 21 R 22 R 23 R 31 R 32 R 33]. (4.3) The transformation equations of the radius vectors of the sphere centre points A T, B T, C T and D T are thus [x ' AT ' y AT ' z AT ]=R [x AT y z AT] T (4.4) ' [xbt ' y BT ' z BT ]=R [xbt y z BT] T (4.5)
46 38 Measuring system and data merging ' [xct ' y CT ' z CT ]=R [x CT y z CT] T (4.6) and ' [xdt ' y DT ' z DT ]=R [x DT y z DT] T. (4.7) By substituting equations 4.2 and 4.3 for T and R in equations 4.4, 4.5, 4.6 and 4.7 the following is obtained: [x ' AT ' y AT ' z AT ]=[R11 x AT R12 y AT R13 z AT T x R 21 x AT R 22 y AT R 23 z AT T z] y R 31 x AT R 32 y AT R 33 z AT T (4.8) ' [xbt ' y BT ' z BT ' [xct ' y CT ' z CT ' [xdt ' y DT ' z DT ]=[R 11 x BT R12 y BT R 13 zbt T x R 21 x BT R 22 y BT R 23 z BT T z] y R 31 x BT R 32 y BT R 33 z BT T ]=[R11 xct R12 y CT R13 zct T x R 21 x CT R 22 y CT R 23 z CT T z] y R 31 x CT R 32 y CT R 33 z CT T ]=[R11 xdt R12 y DT R13 zdt T x R 21 x DT R 22 y DT R 23 z DT T z] y R 31 x DT R 32 y DT R 33 z DT T (4.9) (4.10). (4.11) The matrix equations 4.8 to 4.11 can be represented as a system of twelve linear equations with twelve unknowns, or as three systems, each of four equations with four unknowns. Considering only the first rows in equations 4.8 to 4.11, the first system of equations can be written ' x AT =R 11 x AT R 12 y AT R 13 z AT T x ' x BT =R 11 x BT R 12 y BT R 13 z BT T x ' x CT =R 11 x CT R 12 y CT R 13 z CT T x ' x DT =R 11 x DT R 12 y DT R 13 z DT T x The system can be solved by Cramer's rule [WOLFRAM 2009].. (4.12)
47 Measuring system and data merging 39 The main determinant of the first system of equations (4.12), designated as 1, is AT y AT z AT 1 1 = x x BT y BT z BT 1 x CT y CT z CT 1 x DT y DT z DT 1. The determinants of the unknowns R11, R12, R13 and Tx are (4.13) AT ' x R 11 = x BT ' ' x CT ' x DT y AT z AT 1 y BT z BT 1 y CT z CT 1 y DT z DT 1 (4.17) AT R 12 = x x BT x CT x DT ' x AT ' x BT ' x CT ' x DT z AT 1 z BT 1 1 z CT 1 z DT (4.18) AT y AT x AT ' R 13 = x x BT y BT x BT ' x CT y CT x CT ' x DT y DT x DT and ' (4.19) ' AT y AT z AT x AT ' x T x = x BT y BT z BT x BT ' x CT y CT z CT x CT ' x DT y DT z DT x DT. (4.20) The solutions of the system are thus R 11 = R 11 1 (4.21) R 12 = R 12 1 (4.22) R 13 = R 13 1 (4.23) and
48 40 Measuring system and data merging T x = T x 1. (4.24) Analogously, considering the second or the third rows respectively in equations 4.8 to 4.11, two more equation systems can be defined and solved for the remaining unknowns R21, R22, R23, Ty, R31, R32, R33, and Tz. The rotation matrix and translation vector (equations 4.2 and 4.3) can for the purpose of clearer representation of the calculation principle be written as: R12 R ] R21 R22 R23 R=[ R R31 R32 R (4.25) T=[ T x 1 T y 2 T z 3 ]. (4.26) With the calculation of the R and T parameters the procedure for calibrating the positions of sensors relative to each other is completed. Once calibrated, the measuring systems can be used for measurements until the mutual position and orientation of the sensors is not changed, otherwise the calibration must be repeated. The calculation program for the determination of transformation parameters has been implemented in MATLAB. In order to attain further improvement of the procedure it is advisable to use the artefacts with more than four spheres. It can be supposed that more spheres, and hence more reference points, can increase the accuracy of the alignment for the calibration. In any case, for the calculation of the transformation parameters not more than four sphere centre point couples are required, regardless how many spheres are used for alignment. As already mentioned, which side will be fixed, and which aligned (moved), can be arbitrarily selected within the calibration procedure. However, the same disposition of sides must be retained in the merging of the sheet metal point clouds.
49 Measuring system and data merging Implementation of obtained transformation parameters in merging two sheet metal point clouds The calculated transformation parameters are used to determine new coordinates of points from a point cloud of one side of the sheet metal workpiece. This is also implemented in MATLAB. The point cloud which is placed on the same side as the point cloud moved in the alignment within the calibration procedure has to be transformed. The point clouds represented in Figure 4-7 are merged by transformation of the point cloud obtained by scanning using TopoCAM50 (Figure 4-14). Figure 4-14: Point clouds after data merging From the figure it can be seen that the point clouds in the new arrangement really do represent two opposite sheet metal sides. They are located at a certain, quite constant distance from each other. Making a subjective assessment, it can be concluded that they are properly positioned with respect to each other. The objective tests for confirmation of accuracy and reliability of the merging procedure are represented in the following section. The calculated new coordinates of points can be saved in the form of a numeric matrix in an ASCII data file and, together with the opposite sheet metal point cloud (also matrix ASCII data file) can be further processed, in order to determine thickness and sheet metal medial surface.
50 42 Measuring system and data merging 4.3 Evaluation of the merging procedure In order to evaluate the accuracy of the merging procedure and to quantify deviations in position of two merged point clouds, measurements of one specially designed and calibrated artefact are performed. But prior to describing performed experiments and achieved results, possible deviations in position and orientation of two point clouds, and solutions for their detection must be discussed Method for the detection of deviations Deviations in the position and orientation of two point clouds can be generally seen as combinations of translational and rotational deviations. Considering possibilities of their detection they can be divided into two groups. The first group covers cases of a translational deviation along the z-axis, and rotational deviations around the x- and y-axis. These deviations are represented schematically using the example of a thick plane sheet metal plate, as shown in Figure Figure 4-15: Deviations in positions and orientations of two opposite workpiece scans along the z-axis, and around the x- and y-axis Two sides of the plate, A and B, are thereby shown in the correct and deviating positions relative to each other. It is assumed that one of the sides is always in the right position (side A), and the other one deviates eventually in position and orientation from the nominally right position (side B). Deviations which are represented in Figure 4-15, or a combination of the given deviations, can be easily detected using one plate with two plane parallel sides. The distances between the planes should be calibrated. Deviations
51 Measuring system and data merging 43 of measured distances between two point clouds with respect to the calibrated values can be used in evaluation of the merging procedure. Generally, it can be assumed that the merging procedure is accurate enough if the detected deviations do not exceed the limit of expected deviations, depending on the accuracies of the systems used. However, there are deviation components that cannot be detected using distances between parallel planes as a reference. This is designated as the second group of deviations. This group covers translational deviations along the x- and y-axis, and rotational deviation around the z-axis (Figure 4-16). Figure 4-16: Deviations in positions and orientations of two opposite workpiece scans along the x- and y- axis, and around the z-axis Deviations in these directions are just as disadvantageous regarding merging the curved sheet metal sides as those from the first group. The detection of such deviations must also be included in the merging evaluation process. In a hypothetical case in which merging deviations consist of combination of a components from the second group only, the two parallel planes test would not be able to detect the deviations. To detect such deviations, additional geometrical elements on the plate are needed. These elements can be for example spherical calottes grooved in the plane. Each of the three deviation components, appearing independently, or combined with other two, can be detected using plates with grooved calottes. As an example, the case represented in Figure 4-17 can be analysed. As it is shown in the figure, it can be assumed that a deviation in position of two point clouds of the plate sides A and B is characterised by only one deviation component,
52 44 Measuring system and data merging namely the translation in the +y direction. The B side point cloud would thus be slightly shifted to the left after the merging of the point clouds. In Figure 4-17, the correct position of the B point cloud is represented by a continuous line, and the deviating position by a dotted line. The point-to-point distance d2, measured in a certain defined position with respect to the workpiece coordinate system (d), will deviate from the calibrated value d1. Since the distances are measured at the location of the spherical calotte, the deviation can easily be detected. Figure 4-17: Detection of deviations in position of two opposite artefact scans using the plate with the calotte grooved on one plate side It is important is to emphasise that the task of evaluation of the merging procedure is not to distinguish but only to detect deviations, regardless of components by which they can be characterised. The implemented method includes solutions for detecting any possible deviation. Possibilities for distinguishing deviation components are not considered and not included in the method. The implementation of the evaluation method is represented in the following section. Details about the artefact specially designed for the evaluation, the procedure for its calibration, and the strategy for performing actual point-to-point measurements are explained in that section as well Artefact for merging procedure evaluation, its calibration and measurement An evaluation of the merging procedure will be made by measuring point-to-point distances between the point clouds. Prior to such measurement, these distances will be measured on a tactile coordinate measuring machine (CMM). The results of two
53 Measuring system and data merging 45 measurements will be compared, whereby the results of measurement on the CMM will be regarded as calibrated values. The main precondition for comparability of the calibrated and measured results is the same way of alignment of the workpiece coordinate system in the calibration procedure and in actual measurements on the point clouds. In this way, the measurement of distances between opposite points can be performed by probing the point couples placed at the same positions on the artefact in both cases. Furthermore, measurements should be made by measurement strategies which are as similar as possible. In this case the calibrated and measured distances can be compared with each other. The principles for detection of merging deviations have been considered during the design of the evaluation artefact. The other important criterion was enabling the usage of the same geometrical elements for determination of the workpiece coordinate system during the calibration, and also when the point clouds were measured. Evaluation artefact The artefact designed for the evaluation of the merging procedure is shown in Figure This is a 6 mm thick plate with diffuse reflecting surfaces. The plate is made of nickel steel alloy Fe64Ni36 (Invar ). This material has a very low coefficient of thermal expansion. In this way the influence of the artefact thermal expansion on the measurement results is eliminated. Figure 4-18: Artefact for the evaluation of the merging procedure from the front (a) and the back side (b)
54 46 Measuring system and data merging Four spherical calottes are grooved on the plate surfaces. Three calottes are placed on the back side of the artefact, and one on the front side. Measurements at the calottes enable the detection of deviations that cannot be detected by measurements on the planar plate areas. The front side of the plate is used as a base for aligning the workpiece coordinate system, and is actually equivalent to the A side in Figures 4-15 to Therefore two spheres with very small sphericity deviations, and with diffuse reflecting surfaces are fixed on this side. These spheres enable the same alignment procedure at calibration and at measurement of point cloud distances. An exact description of where the coordinate system is placed, and how it is determined, is given in the next subsection. The plate is dimensioned considering the measuring range of the experimental set-up, so that all significant geometrical elements stay within the range during the scanning from both sides. Detailed artefact dimensions are given in Appendix B. Measurement of the artefact on a CMM In order to provide reliable evaluation of the merging of point clouds, forty-two point-topoint distances on the represented artefact are measured (calibrated) on a CMM. The calibration was made in Precision Measurement Centre of the Chair QFM (Quality Management and Manufacturing Metrology) of the University Erlangen-Nuremberg, on the tactile coordinate measuring machine Zeiss UPMC 1200 CARAT S-ACC. Measurement uncertainties for typical measurement tasks performed on this CMM are less than a few micrometers. Opposite points selected for distance measurements (calibration) are uniformly distributed on the plate surfaces. The disposition of the points, represented on the front side of the artefact, is shown in Figure Twenty points are placed on the plane areas of the artefact (points 1 to 20), and twenty-two on the calotte areas (points A to V). The nominal coordinates of points are determined with respect to the workpiece coordinate system, or the so called workpiece datum system [COLEMAN 1997]. The origin of the coordinate system is placed in the centre point of one of the artefact spheres (Figure 4-19). Alignment of the coordinate system was made in the following way: Thirty-five points were captured on the plane areas of the front side of the artefact. The best fit plane (Gauss plane) was fitted to the points. The primary axis [NEUMANN
55 Measuring system and data merging ], the z-axis in this case, was defined as a normal to the fitted plane, with the positive direction showing outside the front side of the artefact. The secondary axis [NEUMANN 2000], the x-axis in this case, was defined by a vector connecting two centre points of Gauss spheres. They are obtained by fitting derived spheres in the twenty-five randomly distributed points captured on both alignment spheres. It was defined that the positive direction of the x-axis protrudes from the lower towards the upper sphere (Figure 4-19). The origin is defined in the derived centre point of the lower sphere. G E H J I O U K T S 10 9 L V M N 8 7 P Q R x B A C E D y z 100 Figure 4-19: Disposition of the measuring points on the artefact for evaluating the merging procedure The identical alignment procedure was repeated during measurement of the distances between point clouds, with the only difference being in the number of points captured for the determination of the derived elements. In order to provide as similar derived centre points of spheres as possible, the alignment spheres, fixed on the artefact, have very small sphericity deviations (less than µm, according to the spheres manufacturer [BAL-TEC 2009]). Ensuring equal workpiece coordinate systems in the calibration and the actual measurement is of importance, since
56 48 Measuring system and data merging even small differences in the coordinates of the measuring points can lead to deviations in the results, and hence to false conclusions regarding the evaluated procedure. Regarding the calottes grooved in the back and front side of the artefact, it should be noted that deeper calottes or ones with a smaller radius, could cause the CMM styli to slip slightly from the nominal positions when the points are probed. Deviations of point coordinates could cause deviations in the results. For this reason, grooved calottes are quite shallow and have a relatively large radius. Two different radii for the calottes are selected, in order to recognise eventual deviations caused by stylus slippage during calibration should this happen. As already mentioned, three calottes are located on the back side of the artefact. They enable detection of deviations according to the principles described in Section The fourth sphere, located on the front side of the artefact is not of major importance. It enables the detection of the same deviations as the planar artefact areas. Nevertheless, a small overlapping area of this callote with the opposite calotte (looking in the z- direction) is assumed to be extremely sensitive in detecting any deviation component. This calotte area is equally significant as the other calottes when assessing the implemented computing algorithm for a sheet metal thickness determination and extraction of a medial surface, where the artefact is also used (Section 6.3.2). The actual distances are measured using the couples of opposite points, probed in such a manner to have identical nominal x and y coordinates. Finally, the 3D distances between each two captured points are calculated. Measurement of the point-to-point distances on the point clouds In order to provide comparability of results, measurements of point-to-point distances on point clouds must be made with the same measurement strategy as the calibration measurement. Two point clouds of the artefact were previously merged according to the described procedure. The measuring points should be equally distributed on the artefact as in the calibration. Since accurate capturing of these points depends mainly on the coordinate system, the most important consideration is to enable its alignment in the same way as it was done in the calibration. This means that the same geometrical elements must be used, and the axes and the origin points of the coordinate systems must be determined in the same
57 Measuring system and data merging 49 way. It would be also advisable to probe the same points on the artefact for determination of those geometrical elements. Nevertheless, it is not possible to select only some of the points for scanning, but the whole point cloud, or certain areas of the cloud must be used in determination of the elements. Therefore the elements used for alignment must have very small form deviations. The spheres have, as already mentioned, a small sphericity deviation and the front surface of the artefact, used for the reference plane determination, a small flatness deviation. Consequently it is assumed that regardless of the number and distribution of the captured points, the best fit elements (Gauss elements) derived from the points will be stable [WECKENMANN 1999], what means, not significantly and hence unacceptably different from those used in the calibration procedure. Regarding the point-to-point measurements, after alignment of the coordinate system it is only important to select two points with the same nominal x and y coordinates (points 1 to 20, and A to V, Figure 4-19). These points are opposite to each other in the z direction, and distances between them are required measuring features that need to be compared with calibrated values. Measurements for evaluating the merging procedure were made in PolyWorks. The rectangle in red shown in Figure 4-19 represents the approximate area that was scanned. The alignment was made in the previously described way. Point-to-point measurements were performed for the points, where derived vectors in the z direction, placed in the nominal x and y positions of the points 1 to 20, and A to V, intersect the surface represented by point clouds. The measurement results obtained are listed and described in the following section Results of the evaluation Generally speaking, the evaluation results depend not only on the accuracy of the merging process, but also on the accuracy of the two measuring systems used. The merging process can be approved, or considered as accurate, if deviations in the performed test are not larger than the arithmetical sum of two maximal permissible 3D measurement deviations of the measuring systems. As mentioned in Section 4.1, the accuracy of MikroCAD40, verified by sphere-distanceerror tests according to [VDI/VDE :2002], is ±100 µm and that of TopoCAM50, ±10 µm. The deviations in the merging tests performed for the experimental set-up used,
58 50 Measuring system and data merging should thus not be larger than ±110 µm if there are no additional significant deviations with sources other than the measuring systems. This value of maximal permissible limit can be increased for certain expected values of merging deviation. This is not done in the specific case for the following reason: The point-to-point measurements, performed within the evaluation test are not real 3D measurements, but they are performed in the z-direction only. Therefore, it cannot be expected that deviations caused by measuring systems during tests have maximal values. On the other hand, the measuring systems certainly have an influence on the geometrical elements used for the determination of the coordinate system. Consequently, because of the alignment procedure, the measurements results are indirectly affected by 3D deviations of the systems. However, it is not to be expected that the whole deviation range arising from the system inaccuracies can be 'exploited'. The deviations caused by inaccurate merging 'must fit' to this virtual gap of differences between really expected deviations arising from the measuring systems and the maximal possible deviations defined by given limits. The results of the tests are summarised in Table 4-1. As already mentioned, the task of evaluation is not to distinguish components of position and orientation deviations of two opposite sides after their merging, or to consider how the components are superimposed, but only to assess if deviations are below the limits defined by the maximal permissible values. For the experimental set-up used these limits are ±110 µm. It can be concluded from the results from Table 4-1 that these values have not been exceeded, and a regular, accurate merging of point clouds is confirmed. A tendency to systematic deviations in the minus z direction can be recognised. That means the point clouds are slightly closer to each other than one would expect from the calibration data. By repeating measurements in several artefact positions within the measuring range, systematic deviations could possibly, if they occur repeatedly, be reliably quantified. That would be a precondition for correcting such deviations. However, within this work such experiments were not performed. It is interesting to point out the assumption that by usage of two more accurate measuring systems than those used in the experimental set-up, the merging procedure accuracy could be improved and the deviations values could be less than 20 µm. Using more
59 Measuring system and data merging 51 accurate measuring systems, not only the test measurements would be more accurate, but also the foregoing merging procedure. Table 4-1: Point-to-point distances measured on the artefact within the evaluation of the merging procedure Measuring points Measurements made on planes Calibrated value in mm Measured value in mm Distance deviations in µm Measurements made on calotte-areas Measuring points Calibrated value in mm Measured value in mm Distance deviations in µm , A 2,900 2, ,016 5, B 2,936 2, ,016 5, C 2,104 2, ,017 5, D 2,877 2, ,013 5, E 2,907 2, ,016 5, F 2,884 2, ,016 5, G 2,927 2, ,015 5, H 2,067 1, ,015 5, I 2,863 2, ,015 5, J 2,903 2, ,014 5, K 4,280 4, ,017 5, L 3,571 3, ,016 5, M 4,358 4, ,014 5, N 4,298 4, ,016 5, O 4,278 4, ,012 5, P 4,285 4, ,015 5, Q 3,613 3, ,012 5, R 4,301 4, ,015 5, S 3,585 3, ,015 5, T 3,542 3, U 4,222 4, V 4,251 4, Methods for eliminating limitations regarding industrial usage of the system caused by the calibration procedure The represented procedure of calibration of sensors against each other is sufficiently reliable and accurate, but from a certain point of view can be regarded as too complicated and time-consuming. Therefore, for industrial usage of the system, it would be necessary to reduce the number of repetitions of the calibration to a minimum. This can be achieved
60 52 Measuring system and data merging by mounting the fringe projections sensors in permanently fixed positions. Hence, the calibration would be performed only once or repeated only in at long time intervals. Nevertheless, for the complete 3D measurement of some workpieces the scans must be made from different directions and point clouds stitched/matched together. The workpiece and the measuring system must be thereby moved relative to each other [TUTSCH 2006B]. Depending on the form of the measured sheet metal area, such a case can also appear when measurement is made by two opposite fringe projection systems. There can be areas on the sheet metal that are completely scannable only from different directions. In order to enable such scanning and thereby not affect the calibrated transformation parameters necessary for merging the two sides of the sheet metal workpiece, the following solution can be applied: Two fringe projection sensors can be mounted together to have a constant position and orientation relative to each other, but in such a way that their common movement is not obstructed. Such a concept can be realised in many different ways, for example industrial robots can be used which have the appropriate number of degrees of freedom of movement and on which two mutually fixed fringe projection systems are mounted (Figure 4-20). In this case rotation and translation movements do not need to be very precisely controlled. For example, the system manipulation could be realised using even manually operated equipment. z y x x Fringe projection sensor 1 y Measured workpiece Fringe projection sensor 2 z x y Figure 4-20: Schematic representation of the two fringe projection systems mounted on the industrial robot
61 Measuring system and data merging 53 Some advantages of the general solution to fix sensors relative to each other, and to enable them to move together, are the following: Calibration of the fringe projection sensors against each other have to be performed only once, after the sensors are mechanically aligned and fixed together, and repeated at long time intervals. Scanning of sheet metal areas that could not be accomplished in one scan operation would be possible through repeated scanning from different directions If needed, a larger sheet metal area than that limited by the maximal measuring range of sensors could be scanned. It can be assumed that, with this implementation, the measuring system would be more acceptable for industrial usage.
62 54 Extraction of a medial surface and thickness determination 5 Extraction of a medial surface and thickness determination As described in the previous chapter, two merged point clouds represent reliably the two sides of a scanned sheet metal part. They can be further processed in order to obtain the required sheet metal medial surface and local thickness values. If data filtering, outliers elimination and trimming of uninteresting areas or areas which were scanned with insufficient accuracy were not carried out before the merging operation, these operations should be done at the latest before the final processing (see section 4.1). The procedure designed for the determination of sheet metal medial surface and the thickness calculation is described in this chapter. The method of medial surface determination is emphasised in the given explanations. It was more important, but also a more complex problem to be solved, than the problem of thickness determination. However, the implemented method allows determination of both sets of data. Coordinates of medial surface points and sheet metal thickness values assigned to those points can be determined in a single calculation procedure. 5.1 Sheet metal medial surface The medial surface is the surface which lies equidistant from the two surfaces of a sheet metal workpiece. The problem of defining this appears since the real surfaces are neither parallel nor symmetrical. It is then very difficult even to imagine the equidistantly placed surface. In order to simplify the problem an analogous 2D case can be considered. In this case the medial surface is reduced to a medial line/curve. In Figures 5-1 a, b, and c the cases of two parallel or symmetrical boundary lines are shown. A medial lines in these special cases can be defined easily it is a simple set of points lying at equal distances from the lower and upper boundary lines. Thereby, these distances can have any orientation in relation to the boundary lines if these lines are parallel the same line will be derived as the medial. In the case of symmetrical boundary lines the distances must have an orientation orthogonal to the axis of symmetry which in this case is the medial line. In order to devise an acceptable definition of the medial line of two curved boundary lines which are randomly shaped and oriented relative to each other one common
63 Extraction of a medial surface and thickness determination 55 characteristic of the special cases must be emphasised: All normal distances from a medial line to the boundary lines must be equal if the line is medial. a) b) c) d) Figure 5-1: Medial lines It can be assumed that even if the boundary lines are differently shaped it is possible to find enough points which define a line (or an indefinite number of curved line segments) whose normals intersect the boundary lines in points equidistant from that medial line. In Figure 5-1-d two randomly curved lines with the medial line having this characteristic are shown. Analogously, for 3D cases, a medial surface is a surface of which the normal distances from any point to the boundary surfaces are equal. It will not be further explored if the medial line/surface, defined in this way, can be explicitly mathematically defined in all cases, or how this can be done. It is accepted that for sheet metal medial surface determination, it is enough to extract as many points as possible for which the condition is satisfied, and which certainly lie on a medial surface. This decision is based on an unavoidable principle of manufacturing metrology: A measurement system enables the capture of a certain number of measuring points. Any further result is only an attempt to avoid wrong conclusions assigned to the captured points. Hence the assumption is that wrong conclusions about totality (the remaining areas, for which information is not available) are also avoided. There is no way to
64 56 Extraction of a medial surface and thickness determination achieve an absolute accuracy, but instead only good practice (logically acceptable or conventionally agreed) in approaching task-related accuracy. More concretely this means that it can be assumed that in certain areas the medial surface defined as an orthogonally equidistant surface from the boundary surfaces cannot be explicitly, or even not at all, mathematically determined. But such areas are equivalent to so called unknown areas between captured points of a surface at probing. When operated with sheet metal surface approximations (polygonal surface models), an orthogonally equidistant surface can be explicitly defined in most areas. By extracting a certain number of points equidistant to the boundary surfaces, for representing the actual medial surface, the same level of accuracy can be attained as that by which the captured points represent the actual sheet metal surfaces. The point density in such an extracted medial point cloud must be at least approximately equal to the number of points that define one of the boundary surfaces. 5.2 Method of extraction of a medial surface and thickness determination The method of extraction of a medial surface is based on inscribing the spheres of the maximal size between two boundary surfaces. The line which connects two tangential points of a certain inscribed sphere, the chord, is the normal line of the medial surface at a certain point. Thus, the medial surface would be a set of medial points of chords of spheres inscribed between two border surfaces. Figure 5-2: Medial line of the curves approximated by line segments, extracted using a set of inscribed circles Boundary surfaces can be represented by triangulated surfaces of the scanned sheet metal area. In the 2D case, free formed boundary lines can be approximated by line
65 Extraction of a medial surface and thickness determination 57 segments. A schematic representation of the medial line extraction in such a case is shown in Figure 5-2. In this example only a few randomly selected inscribed circles are shown. The method is related to basics of the CAT concept defined for 2D polygons, as explained in Section Nevertheless, methods used for implementation of CAT are approximate and inapplicable for real 3D sheet metal data. An aim of the method realised in this work was to apply the concept for measured sheet metal surfaces and to avoid approximations as far as possible. Avoiding approximations applies particularly to the points of a medial surface which are to be extracted. The medial line or the surface points are not approximated by midpoints of internal edges of Delaunay triangles or by Voronoi nodes as is the case with CAT or MAT methods, neither by midpoints of internal edges or circumcentres of Delaunay tetrahedra, as is the case with points for determination of 3D skeletons. In order to extract a sheet metal medial surface, actual inscribed spheres which do not cross boundary surfaces are determined. Chords connecting appropriate tangency points of spheres are used for the extraction of medial surface points. Chord lengths of inscribed spheres are used for the determination of local sheet metal thicknesses. Deviations of a CAT and MAT axis from the actual medial axis obtained by a method of inscribed circles in 2D case are illustrated in Figure 5-3. Medial line derived using inscribed circles Selected Delaunay circles Chordal axis Delaunay triangles and Voronoi cells Figure 5-3: Deviations of the medial axis derived using Voronoi nodes and the chordal axis from the actual medial line Medial axis derived by Voronoi nodes (centre points of Delaunay circles)
66 58 Extraction of a medial surface and thickness determination Medial line extraction As already mentioned the main principles for extraction of medial surface can be derived from the analogous 2D case a medial line extraction. Surfaces obtained by triangulation of two captured point clouds are used as boundary surfaces in the 3D case. Analogously, in 2D cases the boundary lines can be represented by line segments obtained from two sets of points. In Figure 5-4 one line segment (l1) belonging to one border line, and two neighbour line segments (l2a and l2b) of the opposite border line, are shown. The cord of any circle inscribed between line segments is perpendicular to the required medial line. Since points of a medial line are orthogonally equidistant from two boundary lines, middle points of cords of inscribed circles can be designated as medial line points. In order to define unambiguously the medial line, only two inscribed circles for each couple of opposite line segments are needed. In the example in Figure 5-4 the medial line (designated yellow) is defined by the cord midpoints of the following circles: the circle k12a (marked green), tangency circle between the line segments l1 and l2a the circle k12b (marked blue) between the line segments l1 and l2b, and the circle k12ab (marked red) which is tangential to both couples of line segments. k12amax l2a l2b k12a RC1 k12ab RC R R12amax RC RC2 k12b l1 Figure 5-4: Line segments, inscribed circles and appropriate medial line Dotted lines shown in the figure are shortest and longest chords, belonging to two opposite line segments. If circles to which those lines correspond cross one of the adjacent line segments they are not acceptable as inscribed circles regarding the whole line area. Also some slightly bigger or smaller inscribed circles of one couple of line segments can cross neighbouring line segments. In that case, they cannot be accepted
67 Extraction of a medial surface and thickness determination 59 as inscribed circles. Moreover, middle points of chords of such circles do not belong to the common medial line. Such a circle is the circle k12max in Figure 5-4. It can be concluded, that only those inscribed circles of one couple of line segments which do not cross any other line segment can be used for construction of a medial line. Conditionally, diameters of such circles can be even used as reliable values for local distances between two border lines. The idea of using inscribed circles centre points for the construction of the medial line, is acceptable only if the opposite boundary line segments are concave to each other (as it is the case in Figure 5-4). In the case of convex boundary lines, as shown in Figure 5-5, centre points of inscribed circles do not lie necessarily on a medial line. Such a point is, for example, the red coloured centre point belonging to the red inscribed circle. However, chord middle points lie always on a medial line. Therefore, only those points are used for the determination of a medial line, or a medial surface in the 3D case. Figure 5-5: Line segments convex to each other, inscribed circles and appropriate medial line Approximations in the 2D case Using two acceptable inscribed circles, the smallest and largest, for each couple of opposite line segments the medial line can be unambiguously and precisely defined through connecting the chord centre points by straight lines (Figure 5-6). However, there are narrow segments where the medial line can only be approximated by a straight line connecting two neighbour points. Using the described method it is not possible to define
68 60 Extraction of a medial surface and thickness determination the 'real' medial line in these segments. For example, in the case shown in Figure 5-4, the real medial line part RC-R-RC would be replaced by the line segment RC-RC. Deviations imported by such approximations are very small, but unavoidable. Medial line segments approximated in this way will be narrower if the angle between the neighbour line segments is smaller. This is usually the case when a smaller number of line segments is used to represent a highly curved boundary line. Nevertheless, deviations introduced in this way are larger. Vice versa, larger angles between neighbouring line segments result in negligible deviations, but in longer approximation segments. Although approximations in the construction of medial lines are unavoidable, they are greatly reduced using the method described, compared for example to the method conceptualised in [MURMU 2003], [IVAKHIV 2005] and [VELGAN 2007]. It is important to emphasize that the extracted points are completely reliable medial line points. Segments of missing information Figure 5-6: Medial line points, medial line and segments of missing reliable information In the construction of a medial surface it must be strived towards obtaining at least as many uniformly distributed reliable medial surface points as there are vertices or measuring points in the boundary surfaces. 5.3 Extraction of medial surface points and thickness determination The extraction of the medial surface of the sheet metal should result in a point cloud which lies equidistant between two triangulated approximations of the scanned face surfaces. The higher the point density is in the extracted point cloud, the more accurate is the triangulated medial surface model. In the comparison of the extracted surface with a CAD medial surface, the point cloud
69 Extraction of a medial surface and thickness determination 61 should be used, not the surface model. High point density is therefore desirable, but not of primary importance much more significant is to provide accurately and reliably extracted medial surface points. The method presented in this work is based on the following principles: Two triangulated surface models derived from scanning points are used as boundary surfaces. A finite number of inscribed spheres will be defined in position and size. The centre points of chords (lines which connect appropriate tangency points of the inscribed spheres) will be extracted as medial surface points. Chord lengths assigned to appropriate centre points are used as information about local sheet metal thicknesses Input data - triangulated surface models of scanning points used as boundary surfaces The main aim of the method is to extract as accurately as possible a high number of relatively uniformly distributed medial surface points. But already in the beginning one unavoidable approximation is introduced in the implemented method. This approximation originates from the use of triangulated models (meshes) of boundary surfaces, generated from point clouds of scanned sheet metal surfaces. Such surface approximations are actually very adaptive to the local surface curvature, but deviate slightly from the real surfaces. However, for extraction of a medial surface by the method of inscribed circles, neither point clouds nor parametric surfaces, e.g. B-spline or NURBS (Non-Uniform Rational B-Spline) surfaces, offer any advantages, as explained further below. Point clouds and parametric surfaces as boundaries The idea of operating with spheres inscribed between two point clouds is not acceptable, since in this case inscribed spheres would actually be Delaunay spheres with determination points belonging to both sides of the sheet metal part. Neither the centre points nor the corresponding middle points of chords belonging to such spheres would represent medial surface points. This would be particularly emphasised in cases of lower point density. Parametric methods of surface description (parametric re-creations of faces) are avoided, since the methods that can enable exact modelling of highly structured surfaces are not
70 62 Extraction of a medial surface and thickness determination based on point clouds but on triangle meshes [KARBACHER 1997]. They need surfaces reconstructed by triangle meshes (polygonal models) as a preprocessing step and thus represent actually 'an approximation of an approximation'. From the metrological point of view, parametric surface re-creation (reconstruction) cannot be designated as more accurate than a triangulated mesh model of the surface. The deviations from the actual surface or the uncertainty of approximation remain thereby unknown, only a smoother surface is attained. For example, in the working module IMEdit in PolyWorks, NURBS surfaces are built on polygonal models by first fitting a curve network on its triangles, and then fitting NURBS surfaces on the polygonal model based on the curve network [INNOVMETRIC 2007]. Summarized, the parametric methods do not yield metrologically more accurate surface models than polygonal meshes. Additionally, their construction is more complicated, since one further modelling step must be performed. Polygonal models as boundary surfaces For the accurate creation of medial surface using polygonal approximation of sheet metal surfaces it is very important to generate meshes that recreate actual surfaces with deviations as small as possible. There are different surface triangulation methods. Most of them enable the creation of smooth polygonal meshes by fitting triangles to a set of points, and by subdividing triangles and moving their vertices in 3D space. Such methods are useful in reverse engineering, but for the purpose of measurement such a modification of scanned data can yield certain measurement inaccuracies. An approximation of the areas between the measuring points by triangle planes is unavoidable and can be acceptable, since those areas are unknown-areas anyway. On the contrary, the captured points are informationcarriers. They are the only guarantee for accuracy of measurement results, and the coordinates of measuring points should not be changed. The number of captured points can even be drastically reduced, and single outliers removed or filtered, but captured points should not be moved from their positions. Only in this way can it be ensured that measurement results are based on real data. Therefore, triangulation methods that do not allow movement of the measuring points are preferable (for example Delaunay triangulation). The range of introduced inaccuracies caused by applying common
71 Extraction of a medial surface and thickness determination 63 triangulation methods compared to those of simple triangulation, has not been investigated within the scope of this work. Figure 5-7: Triangulated models (meshes) of two scanned sheet metal sides Surface triangulation can be obtained using for example MATLAB or some commercial software for data visualisation and analysis. PolyWorks was used in this work. Such triangulated meshes have inner and outer sides and consist of triangle faces, vertices and edges. In Figure 5-7 triangulated models of two scanned sheet metal sides are shown. The mesh models are thereby oriented in such a manner that their outer sides correspond to the outer sides of a sheet metal part. The input variables that are used in calculations are radius vectors of triangles vertex points and normal vectors of triangles Method for the determination of inscribed spheres The algorithm for medial surface extraction implemented in this work represents basically the search for a certain number of inscribed spheres for each couple of triangles, belonging to the two surface models of a sheet metal workpiece, for which such spheres can be inscribed. The spheres can touch, but do not trim any other triangle. In principle, there are always two triangles to be considered. Thereby it must be analysed: if any sphere can be inscribed between the triangles which sphere diameters are generally suitable what the criteria are for defining the most relevant spheres
72 64 Extraction of a medial surface and thickness determination which positions of such spheres are possible, and which ones should be selected and finally, when a certain sphere has been defined in its position, if the sphere trims other triangles of both meshes and if it does not, whether it can be finally accepted as an inscribed sphere. As schematically shown in Figure 5-8-a, single triangles on one side (designated as the 'outgoing' or 'source' side) will be selected (e.g. ABC triangle) and iteratively processed with single triangles on the opposite side (triangles 1, 2,... n). So, there is always one couple of triangles (ABC and BCD, Figure 5-8-b) for which suitable inscribed spheres must be defined (in size and position). An indefinite number of spheres can be inscribed between two triangles if they overlap (if their projections on the common symmetry plane overlap). Therefore some discretizations must be accepted either the space discretization regarding positions of spheres, or discretization of spheres dimensions, or both. 3 F E 1 C 2 D C A B A B a) b) Figure 5-8: Selection of triangles for calculations in one iteration Since discretized values can intentionally be varied within the interval constraints (maximal and minimal values), and thus used as input values during calculations, it is reasonable to consider which size would, for certain discretization resolution (increment), result in a lower number of possible variations. A lower number of possible variations would result in turn in a lower number of program iterations. According to this criterion it can be decided which size should be discretized. That means that size ranges (intervals) should represent decisive factors. Consequently, it can be concluded that the spheres' dimensions are more suitable to be limited by an infinite number of possible solutions. The maximal range of sphere dimensions is only several millimetres, or less, and it can be defined considering the
73 Extraction of a medial surface and thickness determination 65 nominal sheet metal thickness. The space range (area range) is significantly larger (typically several tens of mm). Moreover, a space range is more difficult to discretize since its discretization would be three-dimensional. The discretization resolution (increment) must be selected in such a manner to avoid a significant influence on the accuracy of calculations. A resolution of several µm or less is preferable. The concept of resolution selection can be completely understood only after the whole calculation method is known. Examples of resolution selection are given in Chapter 6 and Appendix F. The values which must also be predefined, are the minimal and maximal sphere diameters. Values slightly larger and smaller than nominal sheet metal thickness including thickness deviations to be expected are recommended. As shown schematically in Figure 5-9, there are three ways to vary the size of sphere, when searching for acceptable dimensions: starting from the minimal sphere diameter and gradually increasing the dimensions of the spheres until the first sphere which satisfies all conditions is found vice versa, and varying spheres dimensions in both directions. Minimal sphere... Figure 5-9: Varying spheres dimensions from minimal to maximal and vice versa The best solution would be the third one. Using this method, two spheres, the minimal and maximal inscribed sphere, could be determined. In the 2D case, it would even be the optimal solution, enabling an almost unambiguous definition of the medial line. In the 3D case there is also an additional problem of sphere positioning, or the number of spheres of a certain size used in medial surface extraction and thickness determination. The quality of results that this solution can provide depends on further steps in the procedure as well.
74 66 Extraction of a medial surface and thickness determination In order to simplify the procedure, it will be assumed that enough good results can be achieved when sphere size is varied in one direction only, from maximal to minimal, until the first acceptable sphere for a couple of triangles is found. Regarding sphere positioning, only one position will be selected as representative. If such a sphere is defined in size and position and satisfies the condition of not trimming any other triangle of the two surfaces, the sphere will be accepted as the solution for a certain couple of triangles. If not, sphere size must be further varied and the whole procedure repeated until the first sphere which satisfies all conditions is found. Such a sphere will then be accepted as the maximal inscribed sphere for a certain couple of triangles. The number of inscribed spheres and medial surface points which can be subsequently extracted by such a procedure is approximately equal to the number of vertices representing one of the scanned sheet metal surfaces. The medial surface determined in this manner can be even accepted as a very good result, with a sheet metal thickness information density that is higher then necessary. Several examples that confirm this assertion are given in Chapter 6 and Appendix F. Distinguishing triangle couples between which the spheres can be inscribed from those for which it is not possible In order to distinguish triangle couples between which spheres can be theoretically inscribed, from those for which this is not possible, the relative position of the triangles must be analysed. There are several simple geometrical relations to be observed for this. It is possible to define the symmetry plane between two planes in which the triangles lie (Figure 5-10-a). If projections of the triangles on the symmetry plane overlap, inscribed spheres exist. The mathematical procedure for the elimination of triangle couples for which inscribed spheres are not possible, consists of two steps. The first will be performed within the procedure of defining suitable sphere sizes. If triangles projections are shifted relative to each other in a direction orthogonal to the intersection line of the planes (Figure 5-10-b), the triangle couple will be discarded from the further procedure. If it is confirmed that triangles are shifted in the direction along the intersection line of the planes it can definitely be concluded that they do not overlap. This analysis step will be performed within the procedure of defining sphere positions.
75 Extraction of a medial surface and thickness determination 67 Intersection line of the planes Symmetry plane b) Figure 5-10: Symmetry plane between two triangle planes (a), and projections of the triangles on the symmetry plane (b) The case of parallel triangle planes, when no intersection line of the planes exists, almost never occurs for the two surfaces of the sheet metal, either for curved nor for plane sheet metal. Therefore, such couples of triangles are not processed within the algorithm. As a precaution, when detected in the procedure they are immediately discarded. Similarly, triangles parallel to any of the coordinate system planes very seldom occur in real data sets, and they are discarded as well. The reason for this is that parameters used to determine such planes can lead to insufficiently defined mathematical expressions and to the impossibility of calculating some necessary variables. Since such cases appear very rarely, it can be concluded that the information loss is negligible. Spheres sizes For the mathematical determination of sphere sizes, the coordinates of vertices of the two triangles, and the normal vectors of triangle planes are used as input values. The calculation principles can be explained for the 2D case. For example, the acceptable
76 68 Extraction of a medial surface and thickness determination sizes of inscribed circles need to be defined for two opposite line segments. All parameters required for calculations are represented schematically in Figure For each circle temporarily considered in the procedure of gradual variation of circle sizes, the main condition for the circle being inscribed must be checked. As shown in Figure 5-11 there are four limiting chords which need to be determined. These are the longest and the shortest chords for both line segments. In the 2D case they are equivalent to four possible chords of line segments vertices. A circle satisfies the fundamental condition for being inscribed if its chord, connecting two tangential points, is larger than two minimal chords, and also smaller than two maximal chords. This condition can be written as: l hipmin1 l hipmin2 l hipmax1 l hipmax2. (5.1) The area marked grey in Figure 5-11 represents the area of acceptable chords for associated acceptable inscribed circles. Figure 5-11: Determination of acceptable sizes for circles inscribed between two line segments The chord of the circle, inscribed between two lines, is always perpendicular to the line of symmetry. The lines connecting tangential points and the circle centre point are perpendicular to the boundary lines. Consequently, for each circle size under consideration, the length of chord l, for a given sphere radius r, corresponding to two line segments, can be calculated as: l=2 sin 2 IzdT (5.2)
77 Extraction of a medial surface and thickness determination 69 where, IzdT= r tan 2. (5.3) The angle between two intersecting lines can be calculated as the angle between two normal vectors mn tn =arccos mn tn. (5.4) Since, in discussions, it is mostly the sine value of the angle α used, it is not important if the calculated angle is acute or obtuse. The sine values of both angles are the same. When used for the calculation of distances, the sine values are used as absolute values. Nevertheless, vector directions are necessarily carefully selected. In Equation 5.4, directions of the plane vectors are selected to calculate the angle between two internal plane sides in the 3D case. For the 2D case, this can be equivalently designated as 'internal line sides'. Hence, the adjacent angle of the chords σ can be easily calculated as = 2. (5.5) This angle is required for calculating the vertex chord lengths (hipa, hipb, hipc and hipd). For the reasons applicable also for the angle α, it is not considered if the calculated angle σ is acute or obtuse. Considering the specific case (Figure 5-11), the minimal and maximal chords can be determined in the following way: hipmin1 = min(hipa, hipb) = hipa (5.6) hipmin2 = min(hipd, hipe) = hipd (5.7) hipmax1= max(hipa, hipb) = hipb (5.8) hipmax2= max(hipd, hipe) = hipe. (5.9) In order to determine hipa, hipb, hipc and hipd not only the angle σ adjacent to chords, but also the shortest distances from the line segment vertices to the opposite line segment, denoted as ha, hb, hc and hd, must be known. Knowing the normal vectors of line segments and vertex radius vectors, the shortest distances from the vertices to the opposite line segment could be calculated. Such a
78 70 Extraction of a medial surface and thickness determination distance for a certain vertex is equal to the length of projection of the vector between this vertex and any of vertices which belong to the opposite line segment on the appropriate normal vector of that line segment. For example, the distance of the vertex B, in Figure 5-11, to the opposite line segment is hb= tn D B and of the vertex E, he= mn B E (5.10). (5.11) Distances from the other two vertices can be expressed analogously. Hence, all necessary variables for calculating the vertex chord lengths are available. Hence, for example for the vertex B hipb= hb sin and analogously hipa, hipd, and hipe. (5.12) Using Expression 5.1, it can be checked if each considered circle of a certain radius can be inscribed between two line segments. In the 3D case, the procedure is almost identical. The only difference is that in Equations 5.6 to 5.9 minimal and maximal values must be determined considering three appropriate vertex chord lengths for both triangles. For example: hipmin1 = min(hipa, hipb, hipc). (5.13) If for a certain triangle couple no sphere from the selected size range satisfies the condition determined by the expression 5.1, the triangles are shifted from each other in the direction normal to the intersection line of the planes. Their projections on the symmetry plane definitely have no overlapping area and this triangle couple is discarded from the further procedure. Sphere positions theoretical approach Considering the 2D case, the position of a circle with a certain radius which can be inscribed between two line segments can be explicitly defined using known parameters and parameters defined for the size acceptance tests. In the 3D case, there is one more degree of freedom for spheres positioning. As schematically represented in Figure 5-12, the sphere of a certain acceptable size can be moved in a direction parallel to the intersection line of the planes π1 and π2, in which two boundary triangles lie. The axis in
79 Extraction of a medial surface and thickness determination 71 this direction will be named 'rolling axis'. Hence, there is an infinite number of possible positions along this direction that can be selected for a sphere. Figure 5-12: Possible positions of the sphere with a certain size inscribed between two triangles It was explained in Sections 5.1 and that extracting only one medial surface point per triangle couple, could result in a point density for the sufficiently accurate determination of a sheet metal medial surface. Depending on the accepted procedure for sphere size acceptance tests, one or two acceptable sphere sizes will be defined for each triangle couple (Section 5.3.2) the first largest and/or the first smallest sphere, which satisfy all procedure conditions. That means that only one sphere position for a considered sphere size needs to be selected. The number of points per triangle couple would be conditionally one or two, and the overall point density of extracted points will be adequate. The position of the sphere in question needs to be selected considering the subsequent procedure steps. It can easily be concluded that the sphere has the best chance not to trim the neighbour triangles if it is placed 'medially' (centrally) along the rolling axis. Medially means to be in a certain way protected by its own triangles from coming in contact with other triangles, or to be located as far away as possible from edges of both triangles. Observing the projections of two triangles on the symmetry plane (Figure 5-13) and its intersections with the rolling axis, the medial position of the sphere can be defined. The position of intersection point of the sphere chord (which connects two tangential points)
80 72 Extraction of a medial surface and thickness determination with the symmetry plane is relevant. Generally, this point (Figure 5-13, yellow point) should be located equidistantly from two internal intersections of the edges of projected triangles with the rolling axis (red points) regardless of the way the triangle projections overlap. Subsequently, the position of the sphere centre point can be easily defined. Planes intersection line Rolling axis Symmetry plane Figure 5-13: Projections of the triangles on the symmetry plane, and selected position of the sphere chord line middle point along the rolling axis direction In the subsequent steps of the procedure it will be checked if such a sphere with the defined size and position satisfies further necessary conditions. If not, it could be theoretically possible to find a better position for the sphere, or to vary sphere positions until the optimal one is found. Since such a procedure could lead to an enormous number of iterations it should be avoided. The sphere size will be changed instead and the procedure will be repeated. Selected position of intersection point of sphere chord line with the symmetry plane Sphere positions mathematical approach The procedure for the mathematical determination of sphere positions is much more complicated than the procedure for defining acceptable spheres sizes. It consists of the following steps: Step 1: determination of six intersection points of edge lines of the triangles with the plane which is located orthogonally to the symmetry plane and along the rolling axis Step 2: selection of four points of the previously determined six, which are on the edges of the triangles Step 3: determination of two 'internal points' in the case of four previously determined
81 Extraction of a medial surface and thickness determination 73 points being projected in the plane of one triangle Step 4: determination of the midpoint of the relevant chord, its length, and the centre point of the inscribed sphere. Step 1 Intersection points of lines of triangle edges with the plane which is located along the rolling axis and orthogonally to the symmetry plane can be similarly defined as: Intersection points of rolling axis projections on the triangle planes (π1 and π2), with lines of triangle edges. It is assumed that the rolling axis is projected on π1 and π2 in the direction orthogonal to the symmetry plane. Figure 5-14: Upper and lower triangles, their planes, projections of the rolling axis on those planes, and position of the point P where the intersection line brakes through the xy coordinate system plane For example, for the line segment AB of the green triangle (Figure 5-14), the intersection point is the point T1. The other five intersection points will be designated as T2, T3, T4,
82 74 Extraction of a medial surface and thickness determination T5 andt6 respectively. Points T1, T2 and T3 belong to the green triangle (or henceforth called 'lower' side triangle), and the points T4, T5 and T6, to the blue triangle ('upper' side triangle). The first step in the process of the determination of these points is determining the unit vector s, which defines the direction of the plane intersection line. This vector is needed for defining the point P where the intersection line of the planes intersects the xy plane of the Cartesian coordinate system (red marked point in Figure 5-14). The unit vector s can easily be calculated as the cross product of the normal vectors of triangles planes π1 and π2 (which are unit vectors as well): s= mn tn. (5.14) As shown in Figure 5-14, the point P is actually the point of intersection of three planes: π1, π2 and xy. The vector of the point P can thus be derived from the equations of those three planes. Planes π1 and π2 could be defined by plane normal vectors mn and and by one known point of each plane, for example points A and D. The needed plane π1 is thus the set of all points X such that: mn AX =0. (5.15) Analogously, the three required equations of planes intersecting in the point P are: mn AP =0 (5.16) tn DP =0 (5.17) k OP =0. (5.18) In Equation 5.18 the point O is the origin of the coordinate system, and k is one of three standard basis vectors - the normal vector of the plane xy. Since the dot product of two vectors is equivalent to multiplying the row vector representation of the first vector by the column vector representation of the second vector, Equation 5.16 can be written as: tn [ x mn y mn P x A z mn ] [x y P y A. (5.19) z P z A]=0 Equation 5.19 can be further developed in x mn x P x A y mn y P y A z mn z P z A =0 (5.20) and finally in x mn x P y mn y P z mn z P =x mn x A y mn y A z mn z A. (5.21)
83 Extraction of a medial surface and thickness determination 75 Analogously, Equation 5.17 can be written: x tn x P y tn y P z tn z P=x tn x D y tn y D z tn z D (5.22) and Equation 5.18 z P =0. (5.23) Equations 5.21 to 5.23 written in matrix notation: [x mn y mn x tn y tn z z mn ] [xp tn y P z P]=[x mn x A y mn y A z mn z A ] x x tn D y y tn D z z tn D (5.24) 0 can be easily solved for all considered couples of triangle planes. The special cases, for which the equation would not be sufficiently defined, are eliminated in previous procedure steps. Such cases are mutually parallel triangle planes, or the case when one of those planes is parallel to some of the coordinate system planes. When the coordinates of the point P are known, the radius vectors of the points T1 to T6 can also be determined. Figure 5-15: Parameters for calculating the vector from the origin O (0,0) to the point T1 In order to derive all parameters required for the determination of the point T1, the lower triangle plane is shown again in Figure 5-15 with a detailed designation of parameters required for calculating T1. The determination of this point will be a representative example for the determination procedure used for the other five points. Vectors already known are shown in the figure as unbroken red arrow lines. These are the radius vectors of the points A, B and P (represented schematically by shortened lines), and the unit vector s. Other vectors, which are necessary for the determination of
84 76 Extraction of a medial surface and thickness determination T1, are represented by dotted red arrow lines, or simply by unit vectors. From Figure 5-15 can be derived AB= B A (5.25) AP= P A. (5.26) The unit vector of the vector AB is abj= AB. (5.27) AB The unit vector apj can be determined in the same way. The angle between the direction AP and the intersection line of the planes, β A is thus: A =arccos apj s apj s. (5.28) The angle α MAB, between the direction AB and the intersection line of the planes can be calculated as: MAB =arccos abj s. (5.29) abj s Normal distances of the point A to the intersection line of the planes line can be determined using these parameters: IzdA= sin A AP. (5.30) The normal distance of the point T1 is already known from the procedure of sphere size determination. This is actually the distance IzdT, the lot distance (perpendicular distance) from the rolling axis projection on the plane π1 or π2, to the intersection line of the planes (Figure 5-11). This distance is the same for all points T1, T2, T3, T4, T5 and T6: IzdT =IzdT1=IzdT2=IzdT3=IzdT4=IzdT5 =IzdT6. (5.31) The direction of the vector AT1 can be defined using the unit vector abj. Its magnitude can be easily calculated from the geometrical relations of known parameters. Hence, if the length IzdA is shorter then IzdB, the vector AT1 is: AT1= IzdT IzdA sin MAB abj. (5.32) In the other case, when IzdB is smaller than IzdA, the negative value of abj must be used, Figure Finally, the vector T1 is: T1= A AT1. (5.33)
85 Extraction of a medial surface and thickness determination 77 Following the same procedure, position vectors of the points T2, T3, T4, T5 and T6 can be determined. Figure 5-16: Example of the case when a negative value of subtraction (IzdT-IzdA) can lead to faulty determination of the T1 point, if the case is not considered in the right way It is necessary to emphasize that for calculations of distances the absolute values of sine functions of calculated angles are used (Equations 5-14 to 5-33). Figure 5-16 shows an example of the case where the negative sign of the calculated subtraction (IzdT-IzdA) used in Equation 5.32 leads to the wrong direction of the vector AT1, and thus to the faulty determination of the point T1. Therefore the correct sign for abj must be used. Step 2 Only four of the six intersection points determined (T1 to T6) are needed for further processing. These points are intersections of the triangle edges (line segments) with the projections of the rolling axis on the planes of the triangles. For each case separately it must be determined which four points of the six available points lie on the triangle edges. Regardless of where the rolling axis projection intersects the triangle only two possible intersections with the triangle edges exist. For the specific example shown in Figure 5-17, intersection points of the given rolling axis projection with the triangle edges, are points T1 and T3. After selection, these points are designated as points G and H for the triangle belonging to the lower data set (Figure 5-17), and as V and W points for the triangle from the upper data set.
86 78 Extraction of a medial surface and thickness determination Figure 5-17: Intersection points of the rolling axis projection with the triangle edges Prior to the explanation of the mathematical procedure for the determination of these four points, the following facts must be emphasised: For triangles from the triangle couple considered there are in each case two intersection points of the rolling axis projection with the triangle edges. There is no exception to this rule, since only the spheres with acceptable sizes are considered in this procedure step. The acceptable size is defined by the distances of the spheres from the triangle intersection line, which are not allowed to be smaller or larger than the orthogonal distances of the 'internal' vertices of the triangle couple (Section 5.3.2, Sub-section Sphere sizes) Only cases with minimal and maximal acceptable spheres for a certain triangle couple can be seen as an exception. The projection of the rolling axis in such two cases intersects the triangle at its vertices and theoretically at one point only (Figure 5-17). Nevertheless, since two edges intersect at this point as well, the rolling axis projection intersects two corresponding edges at two points with the same coordinates. Hence such cases are actually not exceptions, and they are not considered as special cases within the mathematical procedure. The selection of two required T-points for a triangle is based on distinguishing two points belonging to triangle edges from one point which does not. The number of operations thereby should be minimized and all unnecessary calculations avoided. For example, for the 'lower' triangle, it will be checked if the point T1 belongs to the line segment AB : T1 AB and if it does, then G=T1 (5.34). (5.35)
87 Extraction of a medial surface and thickness determination 79 Further on, it will be checked if T2 BC If it does, then H=T2 and if T2 BC then H=T3 (5.36) (5.37) (5.38). (5.39) If the point T1 does not lie on the line segment AB T1 AB then G=T2 and H=T3 (5.40) (5.41). (5.42) For example, points G and H, for the case represented in Figure 5-17 would be determined by Expressions 5.34, 5.35, 5.38 and The same procedure must be repeated for the upper triangle, and when doing so the points V and W will be defined. The calculation procedure for determining whether a certain point belongs to a certain line segment (edge), can be explained using the example of the point T1. Hence if AT1 BT1 AB the condition, given by Expression 5.34, will be satisfied. Actually the case that sum of the lengths AT1 and (5.43) BT1 is less than length of AB is theoretically impossible. But, since the calculation is performed using finite numbers, whose values are rounded in previous calculations, the variables have a magnitude slightly different from what is expected. Randomly combined, slightly smaller values can give rise to this case. The negative influences of rounding variables values is possible in the opposite way as well. For example, it can occur that the sum of lengths AT1 and BT1 results in a value slightly greater than the length of AB, although the point T1 lies actually on the line segment AB. This could lead to faulty 'conclusions' within the procedure and to complete incorrect results. Therefore Expression 5.43 is expanded by a 'safety appliance factor' ε,
88 80 Extraction of a medial surface and thickness determination with a value of expected maximal error caused by rounding variables. Consequently, Inequality 5.43 will be used in form AT1 BT1 AB (5.44) where ε is freely selectable for different specific cases, and is of the order of several nm to several µm. The missing value BT1 can be calculated as BT1 = T1 B. (5.45) If the condition given by Inequality 5.44 is not satisfied, it will then be assumed that the condition given by the Expression 5.40 is satisfied. F C E B a) T2 A T3 T6 T1 D T4 G = T1 H = T3 V = T4 W= T6 T5 F B E A b) T1 E T2 T6T4 D C T3 G = T2 H = T3 V = T4 W= T6 T5 C D B c) T2 T5 A T3 T1 T6 G = T1 H = T3 V = T5 W= T6 T4 D F A C F T3 d) T4 E B T1 T2 T5 G = T1 H = T2 V = T4 W= T5 T6 Figure 5-18: Different ways of triangles overlapping, and determined points G, H, V and W
89 Extraction of a medial surface and thickness determination 81 Several different ways of triangles overlapping, observed from the direction orthogonal to the triangle symmetry plane, are shown in Figure The points G and H determined for the lower triangle, and V and W for the upper triangle for the given cases are also shown. Step 3 In order to define the intersection point of the given sphere chord with the symmetry plane, the determined points G, H, V and W should be projected onto the symmetry plane. But, it is simpler to project only two intersection points, belonging to one triangle (for example to the upper side triangle), onto the plane of the other one (lower side triangle) and then to define two 'internal points'. These points would be equivalent to 'internal points' of the projections onto the symmetry plane (Figure 5-13), only shifted for a half of the chord length in the direction of the lower side triangle. Such newly defined 'internal points' need to be determined in this procedure step, and will be designated as points M and N. F E B T5 T4 T6 T3 T2 R2 T1 Rolling axis C D a) D A b) Figure 5-19: Two triangles observed from the direction orthogonal to the triangle symmetry plane (a) and from the direction parallel to the symmetry plane and orthogonal to the intersection line of the planes (b)
90 82 Extraction of a medial surface and thickness determination In Figure 5-19-a two triangles are shown, observed from the direction orthogonal to the triangle symmetry plane. The intersection points of triangle edges with the rolling axis projections are also shown. The same triangles are shown in Figure 5-19-b, but observed from the direction parallel to the symmetry plane and orthogonal to the intersection line of the planes. As shown in the figure, the points V and W are projected onto the plane of the triangle ABC. The projections are designated as V1 and W1, and they lie on the rolling axis projection onto the same plane. Radius vectors of these points can be determined from the geometrical relations between triangles considered and the sphere with the given radius r. Such a sphere, located in the searched target position is shown in Figure 5-19-b. The same situation, observed from the direction marked with the thick arrow in the figure 5-19-b, is shown again in Figure Thereby, only the disposition of elements required for calculating the position of the point V1 is shown. Figure 5-20: Disposition of elements, required for calculating the position of the point V1 The chord middpoint R2 of the sphere shown in Figure 5-20 lies on the rolling axis, and the sphere centre point R lies on the axis designated as 'R-axis', which is parallel to the rolling axis. The directions of distance vectors form the point V to the R-axis, and from the R-axis to the point V1 are thus determined by the normal vectors of triangle planes tn and mn. Their magnitude is determined by the radius of the analysed sphere r. The vector of the point V1 is thus: V1= V r tn r mn. (5.46)
91 Extraction of a medial surface and thickness determination 83 Similarly, the position vector of the point W1 is:. (5.47) In order to define 'internal points' possible dispositions of the points G, H, W1 and V1 must be analysed, and the following two main cases are to be distinguished: dispositions of points in the case when projections of two triangles on the symmetry plane do not overlap, Figure They are shifted from each other in the direction parallel to the intersection line of the planes, the rolling axis and the R-axis. dispositions when projections of triangles overlap (for example, the cases shown in Figure 5-18). If the triangles are shifted mutually, as it shown in Figure 5-21, the line segments GH and W1= W r tn r mn V1W1 do not overlap. This implies that neither the line segment point H nor the point G of the line segment GH lie on the line segment V1W1, and vice versa. The opposite case will be used as one of the relevant conditions in confirmation that the triangles overlapp. It will be designated as the 'additional overlapping condition'. D F B W1 V1 H G E C A Figure 5-21: Two triangles, observed from the direction orthogonal to the triangle symmetry plane, whose projections on that plane are shifted from each other in the direction parallel to the intersection line of the planes The first step in the procedure of the determination of the 'internal points' (points M and N) and confirmation of overlapping triangle projections is the determination of the six distances between four relevant points: GV1 = V1 G HV1 = V1 H GH = H G (5.48) (5.49) (5.50)
92 84 Extraction of a medial surface and thickness determination GW1 = W1 G HW1 = W1 H V1W1 = W1 V1 (5.51) (5.52) (5.53) Generally, the method is based on the assumption that the distance between two external points is the largest one of all six distances calculated using equations 5.48 to In this manner it can be easily determined which two points are internal. For example, it can be concluded that points G and H are the internal points if the line segment V1W1 is the largest one (Figure 5-22). Similarly, if the largest segment is GH it is obvious that the internal points are V1 and W1. For such dispositions of points no additional conditions are required. For the remaining four disposition groups, shown in Figure 5-22, the additional 'overlapping condition' must be satisfied as well. That means, it must be also checked if the point H, or the point G, lie on the line segment V1W1. Inequalities that represent the 'overlapping conditions' for each group of point dispositions are highlighted green (Figure 5-22). The 'safety appliance factor' ε has the value of expected maximal error caused by rounding. The specific point dispositions for which these additional conditions are not satisfied are marked red (Figure 5-22). It is not required to know which condition, if any, is not satisfied. If in the whole checking procedure the M and N points cannot be defined, it can be concluded that triangles do not overlap in the certain area. In this case the triangles may not be discarded from further consideration but the new sphere radius must be selected, and the procedure repeated. For each new sphere the new rolling axis will be defined. It will be parallel to the previous one, but shifted closer to the intersection line of the planes. Hence, there is a possibility that the area where triangles overlap is intersected by some of these lines, and that points M and N for the triangle couple can be defined. Only if no internal points can be defined for all possible sphere radii, can it be concluded that the triangles are shifted from each other in the direction along the intersection line of the planes. The new triangle from the so called 'opposite side' will be selected. The new triangle couple will be processed in the same way, and the same procedure steps repeated. For the triangle couples, for which the 'internal points' M and N can be defined, the procedure is continued.
93 Extraction of a medial surface and thickness determination 85 Dispositions of the points V1 G H W1 V1 H G W1 W1 G H V1 V1W1 HV1 V1W1 GV1 V1W1 HW1 V1W1 GW1 V1W1 GH M = G N = H W1 H G V1 G V1 W1 H H G V1 W1 G W1 V1 H GH HV1 GH GV1 GH HW1 GH GW1 GH V1W1 M = V1 N = W1 H W1 V1 G G H W1 V1 Additional condition not satisfied! V1 W1 H G V1 H W1 G GV1 HV1 GV1 GH GV1 HW1 GV1 GW1 GV1 V1W1 HW1 V1W1 HV1 G W1 H V1 M = H N = W1 H G W1 V1 Additional condition not satisfied! V1W1 G H V1 G W1 H H W1 G V1 HV1 GV1 HV1 GH HV1 HW1 HV1 GW1 HV1 V1W1 GW1 V1W1 GV1 M = G N = W1 G H V1 W1 Additional condition not satisfied! W1 V1 H G W1 H V1 G G V1 H W1 GW1 GV1 GW1 GH GW1 HW1 GW1 HV1 GW1 V1W1 HV1 V1W1 HW1 M = H N = V1 H G V1 W1 Additional condition not satisfied! W1 V1 G H W1 G V1 H H V1 G W1 HW1 GV1 HW1 GH HW1 GW1 HW1 HV1 HW1 V1W1 GV1 V1W1 GW1 M = G N = V1 Figure 5-22: Conditions for the determination of the 'internal points' M and N
94 86 Extraction of a medial surface and thickness determination Step 4 According to the concept for sphere positioning, the sphere centre point should be located at equal distances from the points M and N. The simplest way to determine radius vector of the sphere centre point R and the middle point of the sphere tangent chord R2 is to determine previously the point R1 (Figure 5-23). This point is a tangential point of the sphere on the lower (green) triangle. It should be located on the direction MN and should be equidistant from these two points. The radius vector of the point R1 can be easily calculated from the geometrical relations shown in Figure 5-23-a: R1= M MN 2 mnj where mnj is the unit vector of the vector MN given by (5.54) mnj= N M N M. (5.55) a) Figure 5-23: Disposition of elements required for calculating the positions of the points R2 (the middle point of the sphere tangent chord) and R (the sphere centre point) observed from the direction orthogonal to the rolling axis (a) and the direction of the rolling axis (b) The distance from the point R1 to the point R is already known. This distance is represented by the sphere radius r. As shown in Figure 5-23-b the direction of the vector R1R is inverse direction of the normal vector of the lower triangle mn. Consequently: R= R1 r mn. (5.56) The radius vector of the point R2 can be determined in a similar way. The vector R1R2 is orthogonal to the symmetry plane (Figure 5-23-b). Its unit vector has the same direction
95 Extraction of a medial surface and thickness determination 87 as the unit vector V1V (Figure 5-20). The vector of these two points V1V= V V1 v1vj and its magnitude is equal to half of the magnitude of the vector V1V can be calculated using the known radius vectors. (5.57) Hence R2= R1 V1V 2 v1vj (5.58) where v1vj= V1V V1V. (5.59) The point R2 is one of the required sheet metal medial surface points if the sphere does not trim any other triangle from the upper and lower triangle mesh surfaces. The length of the sphere chord, which connects two tangential points, is equal to the magnitude of the vector chrd= V1V =2 R1R2 V1V, or double the magnitude of the vector R1R2 :. (5.60) In the case that the sphere does not trim any other triangle, and the point R2 is confirmed as the sheet metal medial surface point, the length chrd will represent the local sheet metal thickness assigned to the point R2. Procedure for final acceptance of a sphere as inscribed In order to accept a sphere as an inscribed sphere between two sheet metal surface approximations, it must be confirmed that such a sphere does not intersect any triangle from two surfaces. Geometrical relations between a sphere and single triangles from both sheet metal sides are considered in this procedure step. The probability that the sphere in question trims some of triangles is much higher than the probability that the sphere can be accepted as inscribed. Therefore the procedure of the assessment is based on rejection rather than acceptance of the sphere. It is sufficient to detect that the sphere trims only one of triangles, to automatically discard the sphere from further steps of the procedure. Hence, unnecessary calculations within the implemented program are avoided. Parameters of an accepted sphere will be taken to be a part of the
96 88 Extraction of a medial surface and thickness determination final output variable. A sphere most likely trims the triangles which are adjacent to the two the triangles used for the sphere determination (further designated as 'base triangles'), but not necessarily. The sphere can trim any triangle from both data sets. Only the base triangles are excluded from processing. In the following examples only triangles from the 'lower' data set (green side) will be considered. Conditions defined for this side are implemented for triangles from the opposite side as well. At the same time, only if the sphere does not trim any of the triangles from the 'lower side', will the sphere be checked with the triangles from the 'upper side'. It can be suspected that the sphere trims the triangle under consideration if the distance of the sphere centre point to the triangle plane h is less than the radius of the sphere r. An example of such a situation is given in Figure 5-24, where the base triangle ABC is shown along with the adjacent triangle AABBCC to be considered and also the sphere. Figure 5-24: Base triangle ABC and its adjacent triangle trimmed by the sphere The distance of the sphere centre point to the triangle plane (plane ρ1) can be calculated using the following equation [WOLFRAM 2009]: h= mn 1 R1 AA Hence, if. (5.61)
97 Extraction of a medial surface and thickness determination 89 r h (5.62) the sphere undoubtedly does not trim the triangle. The next iteration step can be performed - the next triangle from the data set can be considered. In case the sphere radius r is larger than the distance h, further steps must be performed in order to investigate if the sphere trims the triangle or not. For an illustration of different possible cases that can occur, three triangles 1, 2 and 3, lying on the same plane ρ1 and the sphere whose radius r is larger than the distance h, are shown in Figure It is assumed that triangles 1, 2 and 3 lie in the same plane, since this enables a better understanding of the problem and simplifies the explanation of the procedure. Figure 5-25: Base triangle ABC and three triangles, lying in one plane, from which two are trimmed by the sphere and one is not Triangles 1 and 2 are trimmed by the sphere, and the triangle 3 is not. In principle, if a triangle intersects a sphere-plane intersection circle, it can be concluded that it is trimmed by the sphere. Thus the problem can be seen as two-dimensional. Since the radius of the plane-sphere intersection circle and the centre point of the circle can be determined, it is also possible to define conditions required for checking if there is a circle-triangle intersection. The radius of the plane-sphere intersection circle rr is: rr= r 2 h 2. (5.63) The centre point of the circle PP is the point of intersection of the shortest distance line h and the triangle plane. Hence, the position vector of the point PP is:
98 90 Extraction of a medial surface and thickness determination PP= R h mn 1. (5.64) Considering the triangles 1 and 2 from Figure 5-25 in can be seen that, in the first case, the point PP lies within the triangle. For the triangle 2 this is not the case. However, both triangles intersect with the intersection circle. A condition derived from the case of the triangle 1, which can be applied to all equivalent cases, can be stated in the following way: If the point PP lies inside the triangle it can be concluded that the sphere trims the triangle. No further tests are required in this case. The inequality used for checking if the point PP lies inside the triangle can be explained by analysing Figure The procedure is equivalent in principle to the procedure used to determine if a certain point belongs to a certain line segment (explained in Step 2 within the previous section). BB Figure 5-26:Triangle areas used for the determination of the position of the point PP in the case when the point PP lies inside the considered triangle (a) and outside of it (b) Firstly, four triangle areas must be calculated: Hence, if a) b) pab= AA PP x BB PP 2 pbc= BB PP x CC PP 2 pca= CC PP x AA PP 2 pabc= BB AA x CC BB 2 pab pbc pca pabc (5.65) (5.66) (5.67). (5.68) (5.69) it can be concluded that point PP lies inside the triangle AABBCC (Figure 5-26-a), and
99 Extraction of a medial surface and thickness determination 91 the sphere trims the triangle. In the opposite case, if the condition is not satisfied, the point PP lies outside of the triangle (Figure 5-26-b) and the investigation must be continued. In that case, it must be checked if the intersection circle intersects the triangle in the same way as it intersects the triangle 2 in Figure 5-25 (over one or more triangle edges). There are actually two different cases to be considered: the intersection circle can intersect triangle edges in such a way that some of the triangle's vertices or even all three vertices lie within the circle (Figure 5-27-a) the circle can intersect only the edges, with no vertices lying inside the circle (Figure 5-27-b). a) Figure 5-27: Sphere-triangle intersection in the vertex area (a), and in the edge area (b) The condition for confirmation of the first case can be stated as: The intersection circle intersects the triangle in the vertex area if the distance from the point PP to the vertex, is shorter than the circle radius rr Thereby, b) PPAA rr PPBB rr PPCC rr PPAA= AA PP. (5.70). (5.71)
100 92 Extraction of a medial surface and thickness determination PPBB and PPCC can be calculated in the same way. If the circle intersects only the edges of the triangle, with no vertices lying inside the circle area, an additional condition must be defined. Therefore several new variables must be calculated. It is required to calculate distances from the point PP to the triangle edges. For example, the distance from PP to the edge AABB [WOLFRAM 2009], designated as uu is: uu= AABB x PPAA AABB. (5.72) The other two equivalently calculated distances to the edges vv= BBCC x PPBB BBCC and ww= CCAA x PPCC CCAA BBCC and CCAA are: (5.73). (5.74) CC PPAA 2 uu 2 PPBB 2 uu 2 AA rr uu UU PP BB Figure 5-28: Intersection of the circle with one triangle edge and the geometrical elements required for its detection The precondition for intersection of the circle with the triangle is that any of the distances uu, vv or ww is shorter than the circle radius rr. Furthermore, the point at which the line in the direction of the certain distance intersects the appropriate edge line, must actually lie on that edge. These intersection points are designated as UU (Figure 5-27-b), VV and WW.
101 Extraction of a medial surface and thickness determination 93 Hence, the condition for a circle to intersect the triangle over the edge written as: uu rr PPAA 2 uu 2 PPBB 2 uu 2 AABB An illustration for better understanding is given in Figure AABB can be. (5.75) The intersection circle crosses the triangle over some edge if the condition 5.75, or any of the following two conditions for the other two edges, is satisfied: vv rr PPBB 2 vv 2 PPCC 2 vv 2 BBCC (5.76) ww rr PPCC 2 ww 2 PPAA 2 ww 2 CCAA. (5.77) The sphere definitely does not trim the triangle if all the described conditions from 5.69 and further are not satisfied. If the sphere does not trim any of triangles from the 'lower sheet metal side', the same tests will be performed with triangles from the 'upper side'. If it is confirmed that the sphere does not trim any of triangles from both sides, the determined chord midpoint (Equation 5.58) and the chord length (Equation 5.60) will be written to output matrices (output variables). Depending on the selected number of spheres which need to be determined for one triangle couple, the procedure will be continued by the consideration of new spheres for the same triangle couple, or alternatively the new couple will be observed. In the procedure implemented within this work, spheres are varied only in one direction, from maximal to minimal, until the first acceptable sphere for a couple of triangles is found. Hence, a further step in the procedure will be consideration of the new triangle couple and performing previously described steps again. The process will be repeated iteratively for all possible combinations of triangles from two opposite sides. 5.4 Implementation of the procedure for extracting medial surface points and thickness determination The algorithm for extracting medial surface points and thickness determination is implemented in MATLAB. This language features many of preprogrammed functions, which makes it much easier to implement an algorithm and test it in a visual way. Once the algorithm works correctly under MATLAB it can be ported easily to some other language (for example C or C++) if required.
102 94 Extraction of a medial surface and thickness determination Input data In the calculation processes, triangulated meshes (Figure 5.7) are introduced in the form of ASCII STL data. STL format contains information about coordinates of vertices of each mesh triangle and its normal vector. Triangles of the polygonal surface are imported into the calculation program as two sets of variables - the matrix of coordinates of triangles vertices and the matrix of triangles normal vectors. Both data sets are given in matrix form with three columns representing the x, y and z components of vertex-points or components of vectors respectively. In the case of a matrix of triangle normal vectors, the number of rows is equal to the number of triangles. For the vertex-matrix, the number of rows is equal to the threefold number of triangles. The medial axis calculation function operates with four variables two 3n x 3 vertex-matrices, and two n x 3 matrices of normal vectors of triangles, where n is the number of triangles. The value of n differs for the two surface data sets Implemented algorithm The main procedure steps, their order of execution, and mutual connections can be seen in the rough flow chart, given in Appendix C. The procedure details, explained in Section are too complex to be shown in a figure covering just one page. Therefore, the flow chart is supplemented with the references to sections, in which these steps are explained in details. Procedure steps that have not been previously mentioned, must be explained. In the flow chart in Appendix C they are marked green. These are the steps: Elimination from the calculation procedure of triangles which are parallel to any of the coordinate system planes. Elimination of triangle couples consisting of mutually parallel triangles. Elimination of unnecessary calculations by avoiding attempts to determine inscribed spheres between the triangles being too far away from each other. The reasons for avoiding mutually parallel triangles, and triangles parallel to the coordinate system planes, have already been explained in Section In order to test if a certain triangle is parallel to any of the coordinate system planes, one could check if any of the triangle normal components is equal to zero. Thus, the condition for one triangle from the 'lower' data set is:
103 Extraction of a medial surface and thickness determination 95 x mn =0 y mn =0 z mn =0. (5.78) Analogously, for the triangle from the 'upper' data set the condition can be written as: x tn =0 y tn =0 z tn =0. (5.79) If a triangle is parallel to any of the coordinate system planes it will be discarded from the further procedure. It has already been emphasised that information loss in such cases can be neglected, since they appear extremely seldom or never (Section 5.3.2). Similarly, it can be tested if one triangle from the 'lower' data set is parallel to the triangle from the 'upper' data set, which is temporarily selected within the superordinate iterative procedure. Triangles are parallel if x mn =x y =y z =z x = x y = y z = z tn mn tn mn tn mn tn mn tn mn tn. (5.80) Subsequent steps in the case of two parallel triangles, can be seen in the flow chart in Appendix C. Nevertheless, this case appears almost never for real sheet metal data sets. The condition for avoiding unnecessary calculations, regarding the opposite triangles which are too far away from each other, is not as simple as the two previously explained conditions. An illustration of the concept used is given in Figure y Figure 5-29: Constraint x-y area around a triangle in question from the 'lower' data set and the triangles from the 'upper' data set, of which at least one vertex is located within this lateral area The concept is based on the requirement that not all triangles from the 'upper' data set should be processed together with the 'lower' side triangle in question, but only those for which there is a realistic possibility of having common inscribed sphere. Only those
104 96 Extraction of a medial surface and thickness determination triangles from the 'upper' data set whose x and y coordinates for at least one vertex belong to the previously defined lateral area (grey area in Figure 5.29) of the 'lower' side triangle considered are accepted for further calculation steps. In this way, many unnecessary iterations are avoided and the time efficiency of the algorithm is increased. The most important thing to do is to select optimal dimensions of the lateral constraint area around the triangle from the 'lower' data set. Dimensions of the constraint area are characterised by distances of its boundaries to the nearest triangle vertices, and these values must be individually selected for each data set. The distance from a certain boundary line to the nearest triangle vertex should approximately be equal to the nominal sheet metal thickness value. This value should be defined considering the sheet metal curvature, the lateral data set point resolution (maximal adjacent point distances) and the inclination of the probed sheet metal surface to the fringe projection sensor as well. Distance parameters, selected for one exemplary sheet metal area, are shown in Appendix F. In order to define the conditions for determination if the triangles are located in sufficiently close each other, several variables must first be determined. These values of minimal and maximal x and y coordinates of three vertices from the 'lower' data set triangle ABC must be determined: gminx=min x A, x B, x C gmaxx=max x A, x B,x C gminy=min y A, y B, y C gmaxy=max y A, y B,y C (5.81) (5.82) (5.83). (5.84) The following two conditions must be satisfied for the triangle in question DEF from the 'upper' data set, if the triangle lies near to the considered triangle ABC from the 'lower' data set: x D gminx gx x D gmaxx gx x E gminx gx x E gmaxx gx x F gminx gx x F gmaxx gx and y D gminy gy y D gmaxy gy y E gminy gy y E gmaxy gy y F gminy gy y F gmaxy gy (5.85). (5.86) gx and gy represent previously selected values of the distances of the area boundaries to the nearest triangle vertices, in the x and y directions.
105 Extraction of a medial surface and thickness determination 97 Theoretically, a similar condition could be applied in the procedure step for spheretriangle 'trimming tests'. Nevertheless, there is a realistic possibility that the sphere trims triangles located far away, even if it does not intersect triangles located in its vicinity. Such cases can be imagined for two mutually concave boundary surfaces, or boundary surfaces whose form is locally significantly variable. In order to improve algorithm efficiency further, a reduction of the size of calculation area is made. Several new areas are defined, which are large enough to avoid reliable 'trimming tests' being influenced, but also significantly smaller than the previous whole area, so that the number of iterations is decreased. This method is described in the following section Increasing program efficiency The implemented procedure for sheet metal medial surface extraction and thickness determination is quite complex. Principally, calculations take more time in cases of the larger input variables, or when a larger number of triangles in boundary surfaces and also smaller sphere size resolution are selected. For this reason additional procedure steps for increasing of program efficiency are introduced. To reduce the time of calculation the input data are divided into calculation blocks in the x and y directions [WECKENMANN 2009]. The program procedure is repeated for each block independently. Only triangles that belong to the same calculation block are used as boundary surfaces in one program iteration (Figure 5-30). Hence many unnecessary calculations for the triangles that clearly lie far away from each other, are avoided. The block constraints are defined considering minimal and maximal values of the coordinates of the triangle's vertices in the x and y directions and selected numbers of area divisions in both directions. The number of divisions in the x and y directions must be selected by the user for each data set anew. Different factors must be considered for this such as, for example, the form of the sheet metal area, the size of the triangles of the surface models (meshes), the nominal sheet metal thickness, etc. If the number of calculation blocks is increased the block sizes decrease and the algorithm becomes more efficient. Nevertheless, blocks that are too small can affect accuracy of results (Section 5.4.2), and should be avoided.
106 98 Extraction of a medial surface and thickness determination Figure 5-30: Calculation blocks The criteria used for distinguishing if triangles belong to the certain block are the x and y coordinates of their vertices. If any of the vertices of the triangle in question belong to the constrained block area, the triangle is assigned to that calculation block. Some triangles can belong to two different neighbouring blocks. That means that blocks unintentionally overlap slightly over these triangles. If one triangle couple happened to be considered twice within the whole calculation procedure, the smaller calculated chord length will be taken as a relevant result for the triangle couple. Smaller chord is result from more rigorous conditions within the block regarding sphere 'trimming tests', what is the reason for this practice. Nevertheless, within the procedure for the determination of block constraints, the overlapping areas of blocks are sometimes introduced intentionally. The user can select the dimension of these overlaps for each data set individually. It can be shown in an illustrative example why these overlapping areas are required. A typical case that can occur in the block border zone is shown in Figure It is assumed that the triangles considered in the couple are triangles 1 and 2, which belong to block n. Conditionally it is possible, that the certain inscribed sphere determined for the couple, which does not trim any of the triangles from the block n, does actually trim one or more triangles from the neighbouring block. For example, this could be triangle 3 from block n+1. In order to avoid accepting this sphere as inscribed, the initial constraint of
107 Extraction of a medial surface and thickness determination 99 block n+1 is intentionally shifted back for some value in the area of block n. Within block n+1, the triangle couple 1-2 will be reconsidered, and the smaller sphere, which does not trim triangle 3 will be determined. As previously mentioned, only the shortest chord for one triangle couple will be selected for the output variable. In this way, the initially determined wrong result for the triangle couple 1-2 is rejected. Figure 5-31: Overlapping of the calculation blocks The size of the block overlaps can be freely selected, however overlaps of less than 20% of a block size are usually sufficient. Although not implemented within this work, the division of scanned area into calculation blocks could also enable the parallelization of this algorithm. The nature of the problem is such that processing the data from one block can be done independently, and would not affect calculations in other blocks. Since nowadays typical PCs contain multiple CPUs, remarkable speed-ups can be achieved through parallelization Possible variations in the procedure implementation Results of the calculations are two output variables. One is the matrix of medial surface points with three columns for x, y and z components, and the other, the single column matrix of thickness values. The number of rows in both matrices is the same. As previously mentioned, the lengths of the chords of the inscribed spheres are taken as the thickness values, and midpoints of those lines represent the sheet metal medial surface points.
108 100 Extraction of a medial surface and thickness determination Optionally, the diameters of inscribed spheres can be used as thickness information, but in this case they must be assigned to centres of spheres. The centres of spheres tend to lie on the medial surface, but not in all cases. A medial surface represented by these points is only an approximation, but the thickness information connected with it is slightly more useful if areas of thickening need to be detected. Considering possible procedure optimisations, three things can be named as priorities. First, the enhancement of numbers of acceptable sphere positions can be made. This means that not only 'internally' located spheres but also spheres in additional positions, near the border of the overlapping areas of two triangles (see Section 5.3.2), could be accepted if they do not trim other triangles. Furthermore, it may be required to enable additional steps for selecting spheres sizes. This can be achieved by introducing additional iterative selection of sphere sizes, enlarging the sizes from the smallest possible value, until the first acceptable sphere is found. As explained in Section 5.3.2, the sphere size variation is implemented in the direction from the largest to the smaller radii only, but the optimal solution would be to vary sizes in both directions. In the design of the algorithm the goal was to try to avoid all unnecessary calculations. Possibly this has not been achieved in the best possible way and some additional optimisations considering procedure flow could be required. Nevertheless, it must be emphasised that the method described enables accurate and reliable results, with a satisfactory information density, what will be shown in several different examples in the following chapter.
109 Results analysis Results analysis The implemented method for determining sheet metal medial surface points and calculating local thickness values is a completely new and original method. An evaluation of the accuracy of the achieved results is difficult. A typical calculation produces several thousands of point coordinates and thickness values. It is difficult even to imagine how this amount of data could be evaluated. The most acceptable solution would be to select only a certain number of values to be evaluated. Even then, an accuracy assessment by a comparison of the results with some reference or calibrated values is only conditionally applicable. For example, thickness values calibrated using common methods (Section 2.3) are not comparable with thickness results obtained by the implemented method, since the measurement and evaluation strategies are not the same. There are no equivalent distances that could be mutually compared. Values could be compared only integrally. An extracted medial surface is particularly difficult for assessment. There is no alternative method to be used for comparison. Theoretically, it would be possible to extract the medial points of the point-to-point distance lines, measured on the artefact with two slightly inclined surfaces with a very small flatness deviation, and to compare it with the results of the implemented method. Nevertheless, confirmation of the accuracy of the implemented method when used on curved surfaces would not be achieved in this way. Within this work results of several different measurement tasks were analysed, and a qualitative assessment of the procedure made. A rough quantitative assessment of reliability and accuracy of the method was made by comparison of the obtained thickness results with values obtained by two other methods. Additionally, a specific method for procedure self-evaluation was developed and carried out. All evaluation results are shown in this chapter. 6.1 Calculation results In the first example, the results of calculations made on data of the demonstration sheet metal part are represented. The extracted medial surface points for the triangulated meshes of boundary surfaces (Figure 5-7) are shown in Figure 6-1 (visualisation made using PolyWorks ). It can be seen that the extracted point cloud has a very similar shape
110 102 Results analysis to two boundary surfaces. It can also be seen that point density and distribution are satisfying, although only one inscribed sphere per triangle couple was defined. Figure 6-1: Extracted medial surface points of the demonstration workpiece The triangle mesh (surface model) obtained by triangulation of the extracted point cloud is shown in Figure 6-2. As in the figure shown, the surface model is smooth, without extreme peaks or valleys. Figure 6-2: Triangulated model (mesh) of the extracted medial surface The colour-coded thickness distribution assigned to the extracted medial surface points is shown in Figure 6-3. The diagram of values sorted in ascending order is represented in Figure 6-4. The number of calculated values is shown on the abscissa. From the diagram it can be seen how many calculated values lie in a certain interval of values.
111 Results analysis 103 Both visualisations are made using MATLAB. Figure 6-3: Colour-coded thickness distribution on the extracted medial surface Figure 6-4: Diagram of calculated thickness values sorted in ascending order
112 104 Results analysis It can be seen that the thickness values vary from approximately 0.6 mm to 2.2 mm for the whole measured area. As expected, the distribution of values is locally quite uniform (Figure 6-3). Through a simple data analysis it is not possible to known if the calculated thickness values are accurate. However, a qualitative evaluation of the reliability of the medial surface shape and its position relative to the boundary surfaces can be made. For this purpose the boundary surfaces and the extracted medial surface are visualised in PolyWorks. A cross-section of the surfaces with a plane parallel to the xz coordinate system plane was defined, Figure 6-5. Figure 6-5: Position of the cross-section plane A 2D view of the cross-section is shown in Figure 6-6. In this figure the green line represents the intersection line of the plane with the lower surface, blue the upper surface, and dark yellow the medial surface. From the figure it can be seen that: The mutual distances of the boundary surfaces vary equivalently to the thickness distribution shown in Figure 6-3. The extracted sheet metal medial surface follows the common shape of the boundary surfaces, and it is located equidistantly from the boundary surfaces. The quantitative assessment of this equidistance is given in the example represented in the following section.
113 Results analysis 105 Figure 6-6: 2D view of the cross-section 6.2 Self-evaluation of the procedure The self-evaluation of the procedure is based on the idea that distances between the medial surface and one boundary surface must be equal to the distances between the medial surface and the other boundary surface. For this purpose new medial surfaces, called support surfaces, located between the upper boundary surface and the medial surface and between the lower boundary surface and the medial one, will be constructed using the same algorithm. Thickness information will be extracted and assigned to the support surfaces. If the medial surface is accurately calculated, thickness values obtained in these two calculations should be equal. Distribution of the thickness values on the two new medial surfaces must also be approximately the same. In order to perform such an assessment, a small area (approximately 5 mm x 5 mm) of the boundary surfaces from the previous example is selected. The area is shown in Figure 6-7 (dark coloured). It is assumed that results of the calculation performed on such a small data set can be more descriptive than those performed on a large area. The smaller number of points is identified, which enables better comparability of the results in a self-evaluation test. Furthermore, possible irregularities in thickness distributions over the support surfaces can easily be investigated on smaller areas.
114 106 Results analysis Figure 6-7: Selected area to be used for calculations in self-evaluation tests Nevertheless, the point density in the new selected boundary areas was higher than that used in previous calculations. Triangulated meshes were made using all selected points. The calculated medial surface points as well as the boundary surfaces of the selected area are shown in Figure 6-8. As before, it can be seen from this figure that the density of calculated medial points is satisfactory, and that the form and the position of the surface relative to the boundary surfaces are correct. Figure 6-8: Calculated medial surface points and the boundary surfaces of the selected area
115 Results analysis 107 Before the final results of the self-evaluation tests are shown, a short analysis of the obtained small-area medial surface will be given. The calculated medial surface is consistent with the medial surface of the whole area. This can be seen in Figure 6-9. It can be concluded that the point density of boundary surfaces does not influence results significantly. Even a coarse point density can be used without having a significant negative influence on the reliability of results. Figure 6-9: Consistency of calculated medial surfaces for the small and the larger calculation area a) b) Figure 6-10: Position of the cross-section plane regarding the whole medial surface and the small area considered (a), cross-section profiles obtained (b)
116 108 Results analysis The position of the calculated small-area medial surface relative to the boundary surfaces can be seen on the example of one cross section. The position of the cross-section plane is shown in Figure 6-10-a, and the cross-section profiles obtained, in Figure 6-10-b. Based on a visually evaluation of the cross-section profiles, it can be concluded that the medial line (yellow line) is located centrally relative to the boundary lines. Whether the position of the extracted medial surface is really medial or not can be tested by calculating two new support medial surfaces (see introduction of this section). Two new point clouds, consisting of calculated medial surfaces points (dark red points), can be seen in Figure Figure 6-11: Boundary surfaces, and point clouds of the main surface and two support medial surfaces Surfaces obtained by triangulation of these points are shown in Figure 6-12-a. Profiles of the cross-section of the surfaces are shown in Figure 6-12-b. The cross-section was made using the same plane as shown in Figure The profile lines of both support medial surfaces are located medially relative to the boundaries. Nevertheless, it is much more interesting to analyse the obtained thickness values and their distributions on the support medial surfaces. This is shown in Figures 6-13 to 6-15.
117 Results analysis 109 a) b) Figure 6-12: Boundary and medial surfaces (a), and cross-section profiles (b) Figure 6-13: Diagram of calculated distances between the lower/upper sheet metal surface and the main medial surface, sorted in ascending order. The steps are caused by selected sphere size resolution (increment), which was 0,005 mm
118 110 Results analysis Figure 6-14: Colour-coded thickness distribution on the lower support medial surface Figure 6-15: Colour-coded thickness distribution on the upper support medial surface
119 Results analysis 111 Figure 6-13 is a diagram of calculated distances. One curve on the diagram represents distances between the lower sheet metal surface and the main medial surface, and the other between the upper sheet metal surface and the main medial surface. Percentages of calculated values are shown on the abscissa instead of the number of calculated values, since the numbers of calculated distances in these two cases differ (4893 and 5614 points/values). Changes of distances represented by the two curves hence can be compared with each other. The precondition for the reliability of such a comparison is that the calculated points (distances) in both cases have a relatively uniform and similar distribution. From the figure it can be seen that the results of both calculations are almost equal. This means that the distances between the main medial surface and the lower boundary surface are the same as the distances between the main medial surface and the upper boundary surface. The diagrams are almost identical, they differ only in the number of calculated values. The minimal and maximal values as well as the form of the curves are the same. A direct comparison of the obtained thickness values is not possible since they are assigned to differently distributed points. Distances are measured at different places and have different directions since different triangle meshes are used. The same applies for the thickness distributions (Figures 6-14 and 6-15). It can be also noted that the calculated values are equal to half-values of the original thicknesses calculated for this area (see figures in the Appendix D), as expected. Finally, it can be concluded that the main medial surface is indeed located medially and that sheet metal thickness values are accurately determined. Hence, the implemented calculation algorithm can be evaluated as reliable. Additional tests performed by comparison of thickness values with the distances obtained by other available methods are shown in the following section. 6.3 Comparison of the results obtained by the implemented computing algorithm with results of other methods Generally it can be assumed that if the accuracy of the determined thickness values is confirmed, the medial surface points have also been reliably determined. Considering the implemented procedure and the method of obtaining both sets of information, this assumption is completely acceptable. Both sets of information are derived from the same
120 112 Results analysis chords of certain inscribed spheres, and a logical conclusion is that the accuracy of one set of information implicates the accuracy of the other. At the same time, the last steps of the procedure, which are different, are very simple, and incorrect calculations in this phase are excluded (see Section 5.3.2). In order to check the reliability of the calculated thickness values and, implicitly, the reliability of the extracted medial points as well, two further tests were performed. The calculated values were compared with the results of thickness measurements obtained using two different methods. Since there is no possibility of a comparison of certain selected values, the tests are not optimal. However there is no alternative to these tests Comparison of the obtained results with the results of the method for calculating minimal sheet metal wall thickness The sheet metal area shown in Figure 6.16 is evaluated in the examples within this section. The area is quite curved and relatively small. Its x-y-dimensions are approximately 27 mm x 20 mm. Evaluated area a) b) Figure 6-16: Evaluated sheet metal area viewed from the front (a) and from the back side (b) The point cloud consisting of extracted medial points for the evaluated sheet metal area is shown in Figure A relatively uniform point distribution can be observed. The triangulated medial surface, represented together with the boundary surfaces and viewed sidewise, is shown in Figure 6-18.
121 Results analysis 113 Figure 6-17: Calculated medial surface points of the sheet metal area considered Figure 6-18: Boundary surfaces and calculated medial surface of the sheet metal area analysed It is assumed that a confirmed accuracy of the calculated thickness values implicates a confirmation of the accuracy of the extracted sheet metal medial surface points, and vice versa. The extracted medial surface shown in Figures 6-17 and 6-18 looks plausible. Hence, based on a qualitative subjective assessment, it could be concluded that the thickness values obtained are accurate as well. Nevertheless, more extensive tests were performed to confirm this. Thickness values obtained by the medial surface method can be compared with the results of the method of minimal sheet metal wall thickness calculation. This method was already explained in Chapter 2. In short, the algorithm for calculating minimal sheet metal wall thickness represents the iterative search for the minimal distance from each measuring point (or vertices) of one
122 114 Results analysis sheet metal side, to triangles of the opposite polygonal surface. The result of the calculation is a sequence of minimal distances corresponding to appropriate measuring points (vertices) of a parent point cloud [WECKENMANN 2009]. It is obvious that thickness values calculated using the procedure implemented within this work and values obtained by the method for calculating minimal sheet metal wall thickness are calculated in completely different ways and that they cannot be directly compared. Nevertheless, the distribution of obtained values over the evaluated area must be approximately the same for both methods. That means that at least areas of thinning and thickening must be equally distributed over the total area, regardless which method is used. A smaller sheet metal area having dimensions 27 mm x 20 mm is analysed, since eventual differences in thickness distributions are more obvious for smaller parts. Furthermore, diagrams of thickness values sorted in ascending order can also be compared better if a smaller number of values is considered. The of colour-coded thickness values distributed on the medial surface are given in Figure Figure 6-19: Colour-coded distribution of thickness values calculated by the medial surface method represented on the extracted medial surface
123 Results analysis 115 Figure 6-20: Colour-coded distribution of thickness values calculated by the minimal sheet metal wall thickness method represented on the upper sheet metal surface Calculations of shortest distances using the method of minimal sheet metal wall thickness must be made twice with the two sides source (points or vertices side) and target side (polygon surface side) being changed. Due to the non-parallelism of the sides of the sheet metal, the two results are similar but not the same [WECKENMANN 2009], Figure There are hence two 3D thickness distributions, one assigned to the lower and one to the upper sheet metal surface. One of them is selected for comparison with the results of the medial surface method. A 3D thickness distribution, calculated by the minimal sheet metal wall thickness method with the upper surface as the base ('source') side, is shown in Figure Observing two thickness distribution representations obtained by the medial surface method and the minimal sheet metal wall thickness method (Figures 6-19 and 6-20), it can be seen that they are very similar. The same can be said for diagrams of sorted thickness values, shown in Figures 6-21 and The results obtained by the medial surface method (Figure 6.21) are compared with two results of minimal sheet metal wall thickness calculations (Figure 6.22).
124 116 Results analysis Figure 6-21: Diagram of thickness values calculated by the medial surface method and sorted in ascending order. The curve is step-formed due to the selected sphere size resolution (increment), which was 0,005 mm Figure 6-22: Diagram of thickness values calculated by the minimal sheet metal wall thickness method (values are sorted in ascending order)
125 Results analysis 117 There is no significant difference between the results shown in the diagrams. The curves of calculated thickness values have a very similar form and almost equal extreme values. Hence, the accuracy of results obtained by the implemented method is confirmed to a certain degree by comparison with the minimal sheet metal wall thickness method. Indirectly, the accuracy of the extracted sheet metal medial surface is also approved. Nevertheless, due to differences in the methods, primarily due to differences in the directions of determined distances between two sheet metal sides, it cannot be known with certainty what the real range of mutual deviations of results is. Within this work this was not further investigated. Deviations between the two sets of result are assumed to be in the range of several µm. It is left open to discussion which method can be considered the better of the two, or which one gives more eligible results for specific measuring tasks. The obvious advantage of the medial axis method is that the thickness results obtained are unambiguous, since they are calculated in directions orthogonal to the medial surface common to both sheet metal sides. In order to evaluate accuracy of thickness results even further, values obtained by the medial axis method are compared with several calibrated point-to-point distances. This is shown in the following section Analysis of the calculated thickness values and a comparison with calibrated point-to-point distances The most regular method for assessment of calculated thickness values is their comparison with some calibrated distances between the two sides of a sheet metal workpiece. However, a conventionally calibrated distance presupposes that the distance is measured in a known predefined direction. For example, in the case of the forty-two calibrated point-to-point distances on the artefact used for evaluation of the merging procedure (Section 4.3.2), all have a direction orthogonal to the associated feature plane used for the alignment of the workpiece coordinate system. For comparison of the calculated thickness values with the calibrated values the directions of calculated distances must be identical to those of calibrated distances. Their absolute positions on the sheet metal must be the same. However, thickness values or distances between the two sides of a sheet metal workpiece that are calculated by the
126 118 Results analysis medial axis method have different orientations and positions that cannot be controlled. Therefore, they cannot be directly compared with conventionally calibrated point-to-point distances. Several thousand thickness values originating from a thickness calculation using the medial axis method performed on the artefact mentioned above all have different directions and positions from the calibrated point-to-point distances. Hence, there is no acceptable criterion which would enable selection of thickness values for direct comparisons. Nevertheless, the results can be roughly compared. As described in Section 4.3.2, the artefact consists of plane areas and grooved calottes. Hence, an evaluation of algorithm reliability for plane and curved surfaces is possible. Furthermore, data obtained by the medial axis method can simply be analysed separately. Due to the specific and precisely known form of the artefact (Figure 4-18 and Appendix B) a theoretical form of the medial surface and corresponding thickness values can be assumed and compared with the results obtained. Triangulated model meshes of the artefact surfaces used in the calculation are shown in Figure Areas of two spheres and narrow strips of point cloud borders were trimmed since they are not relevant for further analysis. The selected point density was approximately 1 point pro mm. In order to avoid frequent parallelism between triangles of the lower and the upper data sets in plane artefact areas, the two surfaces were triangulated using two different triangulation methods. On the lower data set simple triangulation without removal and displacement of points was made (see Section 5.3.1). As an exception, triangulation with minimal points removal and displacements was selected for the upper data set. Nevertheless, triangles of two data sets are approximately of the same size (Figure 6-23). As shown in Figure 6-24, calculated medial surface points are concentrated in the calotte areas. The density of points calculated for plane artefact areas is significantly smaller. Counted using the PolyWorks software, there are only 6950 of a total of calculated medial surface points which can be assigned to plane areas. Counted points are marked dark grey in Figure e-1 in Appendix E. It is supposed that the reason for the smaller point density in plane areas is the mutual parallelism of a certain number of considered triangles from two opposite surfaces that
127 Results analysis 119 could not be avoided. These triangle couples were discarded from calculations within the procedure, and there were no output values in such cases. Generally, it can be concluded that the medial surface and thickness calculation algorithm is more suitable for curved sheet metal surfaces than for plane and parallel surfaces. An optimisation of the algorithm for parallel triangles of boundary meshes is needed in some future work. Figure 6-23: Triangulated model meshes of the artefact surfaces Figure 6-24: Calculated medial surface points of the artefact
128 120 Results analysis Figure 6-25: Boundary surfaces and calculated medial surface of the artefact viewed from the +x direction Figure 6-26: Boundary surfaces and calculated medial surface of the artefact viewed from the -y direction Figure 6-27: Diagram of calculated thickness values for the artefact, sorted in ascending order
129 Results analysis 121 Figure 6-28: Colour-coded thickness distribution represented on the extracted medial surface of the artefact A triangulated model mesh of the medial surface observed from above is shown separately in Figure e-2 in Appendix E. Together with the boundary surfaces, viewed sidewise, the medial surface is represented in Figures 6-25 and The form of the extracted medial surface is equivalent to the theoretically expectable form. In order to assess the accuracy of thickness calculations, obtained thickness values and their distribution over the medial surface must be analysed. The diagram of sorted thickness values is shown in Figure 6-27, and a 3D representation of the colour-coded distribution of the values over the medial surface in Figure Plane surfaces are areas of maximal artefact thickness. Out of a calculated total of points, 6950 lie in a plane area. If only the last 6950 points from the output variable are considered it can be seen that thickness values related to these points are distributed between mm and mm. On the thickness values diagram (Figure 6-27) these 6950 values can be identified as the straight part of the value curve. The dispersion ranges of values around the nominal thickness of 6 mm are 95 µm in the negative direction, and only 5 µm in the positive direction. However, only several outlier values exceed the limit of 6 mm. If the calibrated point-to point distances on the artefact plane area (Table 4-1) are seen, it can be concluded that the calculated values deviate in
130 122 Results analysis general from the calibrated distances. Calibrated point-to-point values lie between mm and mm (Table 4-1). There is no calculated values in this range. If point-to-point distances, measured on the scanned point clouds (Section 4.3.2), are considered it can be seen that these values lie between mm and mm (Table 4-1). Even 5034 calculated thickness values (approximately 72%) of the total values for plane areas lie within this range too. Hence, it can be concluded that the calculated results deviate much more from the calibrated point-to-point distances than from those measured using point clouds scanned on the experimental set-up. The verified accuracy of the merging procedure is ±100 µm. That means the deviations of calculated thicknesses are probably caused by an inaccurate merging procedure, or by inaccurate scanning, rather than by the calculation procedure. To confirm such a hypothesis calotte areas should be considered as well. Nevertheless, a similar analysis of calotte areas is difficult to perform, since the values from the output list cannot be easily assigned to the particular calotte areas. Only the minimal calculated values could possibly be compared with the calibrated and measured point-to-point distances. It can be assumed that these values are assigned to the deepest points of the type 1 calottes (Appendix B). The two minimal calculated thickness values are both mm. Since the x and y coordinates of the medial surface points to which those two thickness values are assigned differ significantly it can be concluded that the values belong to two different calottes. Those two calculated minimal values are compared with the calibrated and measured point-to-point distances of points C and H (Figure 4-19) from Table 4-1. Again it can be concluded that the calculated values deviate more from the calibrated values than from those measured on the artefact point clouds. The assumption that the deviations of calculated values are caused by the merging procedure, or by inaccurate scanning, and not by the implemented calculation algorithm, is confirmed once again. For further comparisons of calculated thickness values of the calotte areas with the pointto-point distances, the colour-coded thickness distribution (Figure 6-28) and the remaining values from Table 4-1 can be used. For example, for two points, T and S, which belong to the overlapping area of the calottes of the type 2 (Appendix B), point-topoint values have a magnitude of approximately 3.5 mm. The same colour-coded value
131 Results analysis 123 for areas of those points can be extracted from the 3D thickness distributions, shown in Figure Such comparisons can only very roughly confirm that no significant inaccuracies are introduced in the thickness calculation procedure. Analysing 3D thickness distribution separately, it can be concluded that it is equivalent to the distribution to be expected, what also contributes to the positive assessment of the procedure. All analyses represented in this and previous sections confirmed the reliability of the results obtained by the implemented calculation procedure. Although the calculation deviations cannot be precisely determined or the calculation accuracy quantified, all tests showed that there are no reasons to doubt the accuracy of the obtained calculation results or the reliability of the implemented procedure.
132 124 Summary and outlook 7 Summary and outlook Within this work a method for extracting medial surface points and the simultaneous calculation of thickness values of formed sheet metal parts in areas of interest was implemented. An experimental set-up that allows scanning of both sides of a sheet metal part in areas of approximately 100 cm 2 was designed and realised. The scanning system consists of two fringe projection systems located opposite to each other. A solution for merging data which were initially assigned to two independent coordinate systems was found. For this purpose a calibration of sensors against each other was performed using a specially designed ball plate with four spheres with negligible sphericity deviations. The calibration enables an extraction of transformation parameters that can be applied on scans of the two sides of a sheet metal workpiece in order to bring them in the proper position relative to each other. Hence a reliable reconstruction of the scanned sheet metal area was achieved. The accuracy of merging was verified using a specially designed and produced calibrated artefact that enables the detection of position and orientation deviations of merged point clouds. The assessment procedure showed that the accuracy of the mutual position of merged point clouds depends mainly on the accuracy of the fringe projection systems used. Although the merging procedure can be evaluated as reliable, it is assumed that it could be even more accurate using ball plates with more than four spheres. The benefits of such optimisation were not further investigated within this work, and it should be a task of further research. Since the position of the sensors relative to each other may not be changed after calibration, an innovative solution for the industrial usage of the system was suggested. After enabling a positioning of the point clouds as if they were scanned originally into a unified coordinate system, a method for further processing of the data in accordance with the given task was designed. Firstly, a definition of the medial surface of curved thin walled parts was derived. It was defined as a surface whose normals intersect boundary surfaces at points equidistant from this medial surface. A method for extracting medial surface points in accordance with the given definition was devised. It is based on the determination of inscribed spheres which do not intersect boundary surfaces of a sheet metal part, with triangulated models (meshes) obtained from point clouds being used as
133 Summary and outlook 125 boundary surfaces. Chords connecting appropriate tangency points of inscribed spheres are used for extracting medial surface points. Chord lengths are used for the determination of local sheet metal thicknesses assigned to extracted medial surface points. The solution was implemented as a prototype using MATLAB. In the design and implementation of the algorithm it was attempted to avoid all unnecessary calculations, but some further optimisations considering procedure flow are eventually needed. The data obtained by the calculations were evaluated. Cross sections made through the boundary surfaces and the extracted medial surface of the area of a scanned demonstration sheet metal workpiece showed an accurate central position of the medial surface relative to the boundary surfaces. The distribution of obtained thickness values over the medial surface was compatible with expectations based on cross section analyses. In order to verify such conclusions based on a subjective assessment, which presupposes a regular position of the medial surface and the reliability of thickness values obtained, a method for self-evaluation of procedure was implemented. The selfevaluation of the procedure is based on the idea that distances between the medial surface and one boundary surface must be equal to the distances between the medial surface and the other boundary surface. The results of such tests performed on a small area of the demonstration sheet metal workpiece showed good results as well. Two new medial surfaces calculated between the main medial surface and the boundary surfaces were located in regular positions. The thickness value distributions over two new medial surfaces were almost identical. This confirmed the accuracy of the position of the main medial surface and hence indirectly the accuracy of the calculated thickness values as well. Furthermore, the thickness values obtained were compared with values obtained by the method for calculating minimal sheet metal wall thickness, and the results were compatible to each other. A comparison of thickness values obtained on the artefact for the verification of the merging procedure with calibrated and measured point-to-point distances was performed as well. It showed that the minimal deviations of calculated thicknesses from calibrated values were caused by inaccurate scanning, rather than by the calculation procedure. Due to the lack of any appropriate reference, calculation deviations could not be precisely
134 126 Summary and outlook determined, but all performed tests showed that there are no reasons to doubt the accuracy of the obtained calculation results or the reliability of the implemented procedure.
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141 References 133 [VELGAN 2007] VELGAN, R. B.: Вимірювальна система для контролювання формованої бляхи на основі трианґулювання зображень (Image Triangulation Measuring system for Formed Sheet Metal Inspection). Львів: Національний університет "Львівська політехніка", Дисертація на здобуття наукового ступеня кандидата технічних наук, УДК : 531.7, [VIALUX 2008] VIALUX MESSTECHNIK + BILDVERARBEITUNG GMBH (EDITOR): Formänderungsanalyse-System AutoGrid [WECKENMANN 1999] WECKENMANN, A.; GAWANDE, B.: Koordinatenmeßtechnik. Flexible Meßstrategien für Maß, Form und Lage. München, Wien, Carl Hanser Verlag, ISBN: [WECKENMANN 2004] WECKENMANN, A.; ERNST, R.; VELGAN, R., NALBANTIC, K., GALL, P.: Detection of defects in sheet metal by using fringe projection system. In: Proceedings of 8 th International Symposium on Measurement and Quality Control in Production. (Erlangen, ). VDI Berichte , p [WECKENMANN 2009] WECKENMANN, A.; PETROVIC, N.: Detection of areas with critically reduced thickness of formed sheet metal parts using two oppositely positioned fringe projection systems. In: 11 th International Conference on Computer Aided Tolerancing 'Geometric Variations within Product Life-Cycle Management' (Annecy, ). - Proceedings, b3.3 [WOLFRAM 2009] WOLFRAM RESEARCH INC. (EDITOR): Wolfram MathWorld. The web's most extensive mathematics resource. Internet page, called on
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143 References 135 Internet sources [BAL-TEC 2009] BAL-TEC MICRO SURFACE ENGR. INC.: Bal-tec Products Catalog. page last called on [BLECH 2007] BLECH: Blechstrukturen effizient berechnen. nachrichten/archiv, [BLUM 1967] Blum, H.: A transformation for extracting new descriptions of shape. In: Dunn, W. (editor): Models for the Perception of Speech and Visual Form. MIT Press, 1967, p page last called on List of software packages used MATLAB, The MathWorks Inc., ODSCAD 6.0, GFMesstechnik GmbH, PolyWorks, InnovMetric Software Inc., TopoXenios2.1, GFMesstechnik GmbH,
144 136 List of abbreviations 9 List of abbreviations Table 9-1: List of abbreviations Abbreviation Meaning 2,5D Three-dimensional data with z(x,y) 2D Two-dimensional 3D Three-dimensional CA Chordal axis CAD Computer-aided design CAT Chordal axis transformation CCD Charge-coupled device CMM Coordinate measuring machine DMD Digital mirror device FEM Finite element method FPS Fringe projection system MA Medial axis MAT Medial axis transformation MCT Maximal chord of tangency
145 Appendices Appendices A Delaunay triangulation and Voronoi diagram B Dimensions of the artefact for evaluation of the merging procedure C Flow chart of the main procedure steps D Diagram of calculated thickness values and colour-coded thickness distribution on the extracted medial surface of a sheet metal area which is used as an example in Section E Medial surface points and triangulated medial surface model of the artefact used for verification of the merging procedure F Example of selection of parameters: Scanned and evaluated area, resulting medial surface points, colour coded thickness distribution, diagram of thickness values and selected parameters used in calculation
146 138 Appendices Appendix A Delaunay triangulation and Voronoi diagram (Source: Wikipedia, The Free Encyclopedia) Definitions In mathematics, and computational geometry, a Delaunay triangulation for a set P of points in the plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). (see Figure b-1) Connecting the centres of the circumcircles of Delaunay triangles produces the Voronoi diagram. Centres of the circumcircles of Delaunay triangles are actually Voronoi nodes. (see Figures b-2 and b-3) In mathematics, a Voronoi diagram is a special kind of decomposition of a metric space determined by distances to a specified discrete set of points. (see Figure b-3) Figure a-1: Delaunay triangulation in the plane with circumcircles shown (Source: Wikipedia, The Free Encyclopedia) Figure a-2: The Delaunay triangulation with all the circumcircles and their centers - Voronoi nodes (Source: Wikipedia, The Free Encyclopedia)
147 Appendices 139 Figure a-3: Delaunay triangulation and Voronoi diagram (Source: Wikipedia, The Free Encyclopedia)
148 140 Appendices Appendix B Dimensions of the artefact for evaluation of the merging procedure Calotte type 1 Plane A Ø 19,05 Sphere 2 Calotte type 2 Sphere 1 35,00 25,00 25,00 25,00 120,00 Calotte type 1 14,50 35,00 4,00 Calotte type 2 2,50 30,00 20,00 15,00 30,00 100,00 8,025 6,00 Plane B Figure b-1: Dimensions of the artefact for evaluation of the merging procedure
149 Appendices 141 Appendix C Flow chart of the main procedure steps
150 142 Appendices Appendix D Diagram of calculated thickness values and colour-coded thickness distribution on the extracted medial surface of a sheet metal area which is used as an example in Section 6.2 Figure d-1: Diagram of calculated thickness values for the evaluated sheet metal area Figure d-2: Colour-coded thickness distribution on the extracted main medial surface
151 Appendices 143 Appendix E Medial surface points and triangulated medial surface model of the artefact used for verification of the merging procedure Figure e-1: Medial surface points of the artefact used for verification of merging procedure Figure e-2: triangulated medial surface model (mesh) of the artefact used for verification of merging procedure
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