Guide to the 3 D pattern fitting in coordinate metrology

Physikalisch-Technische Bundesanstalt National Metrology Institute Guide to 3 D pattern fitting in coordinate metrology Version 1 | 2017-05-23 DOI 10.7795/530.20170606EN Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 1 Guide To 3 D pattern fitting in coordinate metrology Version 1 2017-05-23 Table of Contents 1. Introduction ..................................................................................................................................... 2 2. General Requirements .................................................................................................................... 4 2.1. Hole pattern fit in standardization .......................................................................................... 4 2.2. Determination of the CAD parameters for gauging ................................................................ 6 2.3. Degrees of freedom for fitting and datum features................................................................ 8 2.4. Gauging with coordinate measurement systems .................................................................... 8 2.5. Influence of measurement uncertainties on the fitting result .............................................. 14 3. Application example 1: flanges ..................................................................................................... 15 3.1. Inspection according to the standard ........................................................................................ 15 3.2. Mathematical modelling for datum features and fitting ........................................................... 17 3.4 Generalization of the test task for the simulation of assembly .................................................. 25 4. Application example 2: conical discs ............................................................................................. 26 4.1. Inspection according to the standards .................................................................................. 27 4.2. Mathematical model of datum creation and fit .................................................................... 28 5. Application example 3: cubes ........................................................................................................ 35 5.1. Inspection according to the standard ................................................................................... 37 5.2. Mathematical model of the 3D hole pattern fit .................................................................... 37 5.3. Generalization of the inspection task for the simulation of assembly .................................. 41 References ............................................................................................................................................. 42 Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 2 1. Introduction Physical gauges are very important for the inspection of products, as they were already used prior to the development of initial coordinate measuring machines (CMM). The inspection principle is called the Taylor principle. The gauge used – also called functional gauge [1] – consists of two parts. A go gauge determines the maximum permissible limit deviations for the shape of product features. For the inspection of a hole, this is, for example, a cylindrical gauge pin which is to be entirely inserted in the manufactured hole. If the pin gets jammed during insertion, the hole is a reject. The second feature of gauging is the no-go gauge. With this feature, local measures are inspected for permissibility, in contrast to the go gauge. For holes, this includes pins for the inspection of maximum permissible two-point measures of the hole's inner surface. In a similar way, snap gauges are used for the inspection of shafts. The inspection by means of functional gauges is not limited to individual geometric elements, but can also include patterns of several functional features of one product. In modern productions, the components of several products are usually manufactured at different sites or by suppliers. The permissible shape deviations for workpieces are agreed upon by means of technical drawings with tolerance frames in accordance with the ISO standards 1101 [2], ASME Y14.5M [3] or other specific in-house standards. The ease of assembly of the components can be ensured later on, only if the tolerances in the technical drawings are interpreted in a consistent manner by the client and by the supplier. Therefore, a production which is adapted to the function indispensably requires the inspection of the fitting capability prior to the assembly. This problem is illustrated by the flange in Figure 1. Figure 1 Outline of the assembly requirements for a flange connection with a bolt. Figure 1 shows a flange with 16 equidistant holes. This flange is to be connected with a second flange via individual bolts in the form of screws. It is evident that the individual bolts of different diameters fit into the appurtenant holes. However, flange connections require all bolts to fit into the appurtenant holes at the same time. The right side of the figure makes the problem even more obvious, as holes and the appurtenant bolts can also be skew to each other. Generally, all shape deviations of the products have an influence on the fitting capability. This includes size, form and position deviations. If the requirements on accuracy are very high, influences by ripple and roughness of the workpiece surface at the mating surfaces must also be taken into account. The implementation of gauging via CMMs is generally called "virtual" or arithmetical gauging. Here, the physical functional gauge is replaced by a CAD model. The CMM is also used to measure the workpiece surface by probing a finite number of points with the CMM. These data are also referred to as extracted workpiece geometry. Finally, the virtual gauging is implemented by the procedure of the 3D fitting of hole patterns between the extracted geometry and a CAD model of the gauge. Thereby, the term "hole pattern" refers to the 3D elements of size "cylinder" (hole, shaft) and pairs of parallel planes (slot, toung). When fitting and associating single geometric elements, there is a wide range of standards for a uniform specification of procedures, robust reference algorithms and tests for industry ([4], [5]). In Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 3 contrast to this, the geometric fitting with several elements in the form of hole patterns and the 3D fitting of hole patterns have hardly been uniformly documented and regulated by standards with regard to metrological and computing processes. Although drawing specifications are defined for the tolerancing of the 3D hole pattern fit in ISO standardization for geometric product specification (GPS), the necessary standard parts for the implementation in a uniform test procedure are lacking entirely. Guidelines on the correct use of different sensors for the metrological detection of measurement points are required, so that the highest points of the elements considered are very likely to be measured. They are significant for the calculation of the quantities and thus for the quality and reliability of product inspection. With this background, optical and CT measurement procedures are of particular importance for the 3D fitting of hole patterns via virtual gauging, as they are suitable to detect the entire surface of a product very rapidly. However, a large part of today's existing standards is only designed for tactile sensors which generally detect a product surface much more slowly. Most of the inspection tasks which are dealt with by physical gauging or arithmetical gauging with several geometric elements refer to the tolerancing according to ISO 2691 [6]. This standard specifies the maximum material condition (MMC) and least material condition (LMC) for components. The guideline for the application of the standard is that a manufactured workpiece must be able to mate with a counterpart defined by drawing specifications. The implementation of the inspection by MMC and/or LMC in a simulation of assembly is not documented. Consequently, literature provides different approaches (e.g.: [7], [8], [9]) for arithmetical hole pattern fit. The diversity of the algorithms is also reflected in the measuring machine software. Solutions by various manufacturers are incompatible if different algorithms are used or if the procedures to be used have not been determined in the forefront. The general correct procedure is the fitting with all measurement points to the virtual counterpart. Furthermore, very large data sets occur in state-of-the-art coordinate measuring systems using multi-sensor technology, optical-tactile measuring sensors or also CT measuring systems. If, for reasons of efficiency, the software reaches its limits of economic benefit, filters are often used to reduce the measurement data prior to the evaluation. In addition to the fitting algorithm, they also have an influence on the measurement results during hole pattern fit. The task of this Guide to 3D hole pattern fitting is the user-oriented presentation of inspection processes for the arithmetical simulation of assembly with virtual gauging. Thereby, the fundamental requirements of classical physical gauging according to ISO 1101 are transferred to coordinate metrology and recommendations for a software-based implementation are presented. Various coordinate metrology systems with different sensors are considered. Product measurements with dimensional computer tomography (CT) are of special interest, as they allow the entire product surface to be efficiently extracted. Apart from the description of a general procedure in Section 2, the recommendations also comprise details for three special application examples. The first example is a flange which, e.g., has to be manufactured during the construction of wind engines. This flange is presented in Section 3. Based on a technical drawing in which several parallel holes have a common position tolerance with maximum material condition, a rotatory hole pattern fit via cylinders is described. In the sense of the ISO GPS standardization, a geometric ideal datum system is attributed to the flange which determines a rotation axis of the gauge for fitting. In addition, Section 4 deals with tasks where miscellaneous geometric elements are fitted simultaneously. The example which is presented here is a conical disc with a slot consisting of parallel planes and several parallel holes. A datum is also available. The third application example in Section 5 specifies the fitting of a gauge made of cylinder elements in several holes in a cube whose axes are perpendicular to each other. Here, the case of a free fit will be considered, i.e. datum features which limit the fitting of the gauge geometry are not specified. Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 4 2. General Requirements The 3D hole pattern fit is used for the inspection of specific form and position tolerances of functional product features. This mainly includes the inspection of complete geometries with arithmetical gauging. In addition, there is a whole range of further applications where integral geometric elements, e.g. axes or median planes, can also be inspected according to the formal principles of 3D hole pattern fit. These principles are referred to in a later suitable section of this Guide. However, this will not be dealt with in greater detail as the necessary procedures and requirements for the inspection are sufficiently covered by the standardization or can simply be taken over by arithmetic gauging. In Subsections 2.1, 2.2 and 2.3, the most important requirements placed on the hole pattern fit are presented which must be fulfilled when applying the ISO 1101 and ISO 2962 standards. The principle interpretation and supplement of hole pattern fitting for coordinate metrology are shown in Section 2.4. It also includes requirements placed on the evaluation software and relevant numerical procedures. Finally, Section 2.5 deals with the measurement deviations occurring during 3D hole pattern fit. These measurement deviations are significant for the consistency or inconsistency between arithmetical gauging and an inspection during real physical gauging. 2.1. Hole pattern fit in standardization The subsequent considerations start with the technical drawings of products where the tolerancing according to the ISO 8015 [10] principle has been inserted. Hereby, the application of ISO 1101 and all attributed individual standards will be implied for the inspection of tolerances. The examples given in Figures 2-1 and 2-2 show a plate with two holes that are tolerated in four different ways. For each example, a 3D hole pattern fit will be used for inspection. Figure 2-1 (left) (Example a) shows a position tolerancing of the median lines of the holes. DIN EN ISO 14660 Part 2 [11] regulates the procedure for the extraction of these lines for cylindrical geometric elements. The tolerance frame is used to define two cylindrical tolerance zones which are represented below the technical drawing. These cylindrical regions have an ideal shape. Their axes are parallel with a nominal distance of 20 mm. In addition, both cylinders are orthogonal to the datum plane A. When inspecting the plate, first the lateral surface A and the holes are extracted in order to attribute the datum surface and the median lines of the holes. During fitting, the tolerance zones are allowed to be shifted along the planes A and to be rotated orthogonally to A. The workpiece is permissible if both median lines are within the tolerance zones at the same time. As the median lines of the holes are constructed, i.e. are not available as directly measurable geometry at the manufactured product, the inspection can only be carried out by means of arithmetical hole pattern fit. Example b) (Figure 2-1, on the right) inspects the position of the holes according to the maximum material condition. This has been specified with the aid of the drawing specification Ⓜ which is directly behind the value for the position’s tolerance. Normatively, the requirements for the inspection via MMC are regulated in ISO 2962. Next, the plate must be fittable by means of a geometrically ideal counterpart which is shown below the engineering drawing. The counterpart is composed of a level contact plane. Two bolts are orthogonal to the contact plane. The axes of the bolts have the nominal distance of the holes and are parallel. The bolt diameter of 4.9 mm is calculated on the basis of the lower size limit of a 5.0 mm hole and the position tolerance. The inspection aims to verify whether the holes of the plate lie within the admissible tolerance. Physical gauging as well as arithmetical gauging can be applied. For the arithmetical hole pattern fit, the level contact surface of the gauging is aligned to the datum surface A of the plate and/or an ideal plane is assigned. At the same time, the two bolts of the gauge must fit in the holes of the plate. Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 5 Figure 2-1 Two examples for tolerancing a plate with two holes. The virtual inspection of the tolerances presupposes the 3D hole pattern fit in both cases (Tolerance principle ISO 8015). Several products have specific functional requirements. In these cases, datum elements with the specification Ⓜ for the MMC are also occasionally entered in drawings. Figure 2-2 shows two examples for this. Figure 2-2 Tolerances with datum features including drawing specifications Ⓜ. In example c), the contact surface A is the primary datum. The left hole is defined as secondary datum B. The right hole has a position tolerance with the specification Ⓜ for the tolerance zone and the datum B. The form of hole B or its orientation to A is not tolerated. For this reason, the quantities are neglected during the toleration. The compliance with the tolerance can be inspected again by Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 6 classical physical gauging or arithmetical hole pattern fit. The nominal geometry of the go gauge assigned to it consists of a plane which is aligned to the datum surface A. On this datum surface, a gauge cylinder for the datum B is located having a diameter of 5.0 mm which is the lower size limit of the datum hole. The second gauge cylinder has a diameter of 4.4 mm which results from the lower limit of the hole minus the position tolerance. It stands for the position of the hole. Both cylinders are orthogonal to Plane A and their axes have the nominal distance of 20 mm. The classical inspection with physical gauging and the hole pattern fit in coordinate metrology can be applied in the example. Example d) in Figure 2-2 shows that datum elements with the specification Ⓜ can be identified for the MMC, but that the tolerated element will not be inspected according to the MMC. These drawing specifications only occur in rare applications. The inspection of the tolerance requires an arithmetical 3D hole pattern fit as the extracted median line at the tolerated element does not exist at the real product as directly measurable geometry. A gauge cylinder in datum hole B and the cylindrical position tolerance zone to the median line will be fitted together for this purpose. The cylindrical tolerance zone has a diameter of 0.1 mm. The gauge pin has a diameter of 5.0 mm (the lower size limit of the datum hole). The bolts and the tolerance zone for the extracted axis are orthogonal on the datum plane A. The axes of gauge bolt and tolerance zone of the median line have a nominal distance of 20 mm. In each of the four examples shown, the 3D hole pattern fit for tolerance inspection has three degrees of freedom. They include two orthogonal translations and one rotation within the datum surface A. Further explanations refer to applications with the drawing specification Ⓜ for position tolerances and data. The following basic requirements apply for the correct application of drawing specifications according to ISO 2962.  Only form and position tolerances can be supplemented by the specification Ⓜ.  Tolerances and datum elements with the specification Ⓜ must refer to integral geometric elements from elements of size, such as the axis of a cylinder or of the median plane of a slot.  Further elements of size include spheres and – in the two-dimensional case – circles and pairs of parallel lines. In this context, cones and wedges are not elements of size.  Datum features can be supplemented by the specification Ⓜ. In the case of datum systems, individual datum elements can occur with and without the specification Ⓜ, as long as a Ⓜ element is not followed by an element which is not identified by Ⓜ at the evaluation sequence according to the drawing specification within the tolerance (justification according to ISO 5459 [12] for datum evaluation – uniqueness requirement). 2.2. Determination of the CAD parameters for gauging The determination of go gauge geometric parameters is relevant for the hole pattern fitting. This means that the CAD model of a geometrically ideal counterpart must be constructed and manufactured as correctly as possible. When inspecting the fitting capability, an attempt is made to fit this part into the product without jamming. In Figure 2, these gauges are shown as sketches in the lower part of the picture, using the elementary example of the hole plate. Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 7 When defining geometrically ideal counterparts for gauging according to ISO 2692 in the case of drawing specifications numerous rules must be considered. The created ideal geometric elements which form a CAD model of a gauge are denoted as maximum material virtual condition (MMVC). The size of a geometric element of the gauge is referred to as maximum material virtual size (MMVS). Size, form and position tolerances can be equally integrated in the calculation of the MMVS. The following hand rules can be applied for the determination of the parameters required for the construction of the gauge:  Gauge geometry for form and position tolerances with Ⓜ o External elements of size (shaft, toung): The MMVS of the gauge geometry will be formed as sum from the upper size limit of the tolerated geometric element Go and the tolerance for the form or likewise position deviation t. MMVS = Go + t o Internal elements of size (hole, slot): The MMVS of the gauge geometry will be formed as a difference from the lower limit of the tolerated geometric element Gu and the tolerance for form or likewise position deviation t. MMVS = Gu - t  Gauge geometry for datum elements with Ⓜ o External elements of size (shaft, toung):  The MMVS of the gauge geometry is the upper size limit Go of the datum element for negligible form deviations. MMVS = Go  If the datum element has an additional form or position tolerance (tolerance value referred to as t) it must be taken into account for the MMVC of the datum element. MMVS = Go + t o Internal element of size (hole, slot):  The MMVS of the gauge geometry is the lower size limit Gu of the datum element for negligible form deviations. MMVS = Gu  If the datum element has an additional form or position tolerance (tolerance value referred to as t) it must be taken into account for the MMVC of the datum element. MMVS = Gu - t The value for MMVS only determines the measures of the geometric elements which are plugged into or have to encircle the product during the gauging. As a gauge generally accounts for several of such coupling elements, the positions to each other and to a gauge coordinate system must be determined in the following. The distances between centre points, axes and median planes of the gauge’s geometric elements are clearly determined in the technical drawing by means of exact local measures. However, the assignment of a gauge coordinate system is not always obvious and the decision is left to the user. If datum elements are available according to ISO 5459 they can be used to determine the x, y and z axes of Cartesian coordinates, as local measures refer to the datum features in case of a correct drawing. If datum systems are incomplete or if no data are available, individual degrees of freedom remain available for the positioning of the gauge. For example, a single datum plane only defines the direction of one axis and one zero point on this axis. The user can then arbitrarily determine two further coordinate axes in the datum plane and their position. However, it must be ensured that – in Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 8 the coordinate system provisions – the nominal distances between the gauge elements comply with the requirements in the technical drawing. Only then is it ensured that the fitting results between the different coordinate systems are still compatible with each other. Further information for the use of the MMC including the drawing specification Ⓜ as well as the correct calculation of gauge complying with the standards are presented in Jorden [1], for example. 2.3. Degrees of freedom for fitting and datum features So far, the question of how the term "fitting" ("coupling" or "plugging together") of a product and gauge can be formally interpreted has not been answered. For this purpose, the classical physical gauging is initially considered. A tester will try to plug the product together with the gauge. Thereby, he/she puts the product on the edges of the gauge and by simply "shaking" and "tilting" he causes both parts to slide into each other. If a product and a gauge can be plugged together by this procedure to such an extent that the requirements of the inspection are complied with, the product is within the tolerance which is referred to as complete fitting capability. The CAD model of the virtual counterpart for arithmetical 3D gauging is available in the gauge coordinate system (𝑥𝐺,𝑦𝐺,𝑧𝐺). The measurement points of the extracted product are given in the workpiece coordinate system(𝑥𝑊,𝑦𝑊,𝑧𝑊). Both systems are Cartesian coordinates. The fitting of the gauge to the extracted geometry is described by a linear transformation. 𝑇:(𝑥𝐺,𝑦𝐺,𝑧𝐺)⟶(𝑥𝑊,𝑦𝑊,𝑧𝑊) This transformation maps the gauge geometry into the workpiece coordinate system. Overlapping or empty space can be quantified between the extracted product geometry and the transformed gauge. The virtual gauge and the extracted geometry are completely fitted if there is a transformation which represents a plugged state where there is still empty space between gauge geometry and the measurement points. In the three-dimensional case, the transformation is defined by rotations – e.g. by means of the Euler angle – and translations towards the three coordinate axes 𝑥𝐺,𝑦𝐺 and 𝑧𝐺. Generally, six parameters are available for the transformation of gauge geometry. The unrestricted selection of a parameter value is referred to as degree of freedom for the fitting. If there are no restrictions for the parameter selection, they are referred to as full degrees of freedom to determine the transformation T. In many cases, the transformation permitted for the fitting is limited by constraints. In the context of position tolerancing including hole pattern fit there are datum elements which block the degrees of freedom during transformation. If, for example, a datum axis of the workpiece is given, the workpiece coordinate system is initially oriented towards this. The rotation around and a translation along the datum axis then remain as degrees of freedom for the fitting. An exception for the restriction of the degrees of freedom is the datum elements labelled with the symbol Ⓜ. 2.4. Gauging with coordinate measurement systems In Figure 3, eight individual steps are represented for the implementation of gauging or – in general – 3D hole pattern fit by means of coordinate measurement systems. They are split up in two groups. The first group is the workpiece measurement. It comprises the steps of extraction via the recording of measurement points at the functional surfaces of the product, partitioning and/or a reduction of the measurement points. The second group comprises all steps for the arithmetical evaluation of the hole pattern fit by means of given measurement points. Hereby, data and datum systems are initially set up which are required for the definition of product and gauge coordinate systems. By means of transformation of the measurement points into the nominal position of the gauge the procedure Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 15 3. Application example 1: flanges The example of "assembly of a flange", as it occurs during the construction of wind engines shows that a seemingly simple geometrical product cannot always be treated in a trivial way. Characteristic for the flange at modern facilities are the large dimensions of more than 5 m of flange diameter as well as the high number of holes on the flange. The arithmetical simulation of assembly requires the extraction of almost the entire external surface of the flange. This is possible, for example with the aid of surface measuring optical sensors. Due to the geometry, the arising data amount is particularly large and too complex for a description of the hole pattern fit. To show the core requirements for the arithmetical assembly simulation of the flange including 3D hole pattern fit we therefore resort to the simplified model in Figure 6. The principal set-up is equivalent to a flange for wind engines. The measures and the number of the holes are clearly smaller. Furthermore, different measuring systems can be used for the extraction of this type of products. Figure 6 Application example of flange The application example for the flange is shown in Figure 6. The top surface of the flange is described by two parallel planes at a distance of 8.0 mm. The external surface represents a cylinder with a diameter of 70.0 mm. The internal surface of the flange is described as cylinder with a diameter of 50.0 mm. Both cylinders are coaxial and perpendicular to both top surfaces. On the bolt circle with a diameter of 58.0 mm there are 5 individual holes. Each of them is staggered by 72°. Each hole has a diameter of 4.0 mm. 3.1. Inspection according to the standard In order to be able to plug the flange, the contact surface (assembly surface) must be level. Furthermore, the holes are required to be orthogonal to the assembly surface. In addition, the bolt circle is concentrical to the axis of the cylinder of the external lateral surface. Thereby, there will be no eccentricity during operation later on which would lead to an increased wear of mobile parts for wind engines. Top surface (Assembly surface) Bolt circle Internal lateral surface External lateral surface Hole for assembly with bolts Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 16 Figure 7 ISO 1101 compliant inspection of the flange. According to these requirements, an unambiguous unique test task including tolerances of the manufactured flange can be derived. Figure 7 shows a technical drawing according to ISO 1101. The arithmetical inspection results in the following task: An ideal geometrical plane – identified as datum A – must be aligned to the real assembly surface at a minimum distance. Datum B specifies an ideal cylinder that is orthogonal to plane A and whose diameter is the smallest possible diameter where the external lateral surface of the product is completely enclosed, i.e. all measurement points lie within the assigned cylinder. The datum assignment is represented in Figure 8. Finally, the size and position of the individual holes to the datum surface (datum point) and to the axes of the datum cylinder must be inspected. This is shown in Figure 9. The tolerance from Figure 7 with the symbol for the maximum material condition hereby defines 5 gauge cylinders with the diameter of 3.8 mm (MMVS). The axes of the cylinder are parallel to the datum axis B. They are regularly arranged on a bolt circle with a diameter of 58,0 mm at a distance of 72° segments. The bolt circle is parallel to the datum plane A. Its centre point lies on the datum axis B. A transformation is allowed for the fitting which rotates the gauge around the axis of the datum cylinder. One must check if there is a rotation angle where all gauge bolts are within the holes without overlapping with the material of the flange ring. In this case, the product is within the range of tolerance. On the other hand, the product cannot be assembled due to the inspection result. Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 17 Figure 8 Datum assignment at the manufactured flange. Figure 9 3D hole pattern fit for the flange (on the left: perspective view, on the right: top view); the gauge cylinder highlighted in red overlaps with the hole. By rotating all gauge cylinders around the datum axis, no position is found where all cylinders lie within the holes. 3.2. Mathematical modelling for datum features and fitting In order to be able to carry out the 3D hole pattern fit arithmetically, a correct and reliable mathematical modelling is fundamental in order to meet the requirements and targets of the inspection. Figure 10 gives an overview at the process of the 3D hole pattern fit for the flange. Here, the calculation of measurands on the basis of the extracted data is shown. The 3D hole pattern fit is carried out in six consecutive steps. Every step is individual and requires an independent calculation method for the finalization. The starting point is the calculation of the datum elements 𝐴 and 𝐵. They include single geometric elements that are assigned to the extracted flange according to the Chebyshev and likewise Minimum circumscribed condition. A workpiece coordinate system is derived from the datum elements which – apart from a rotation at the workpiece – is uniquely determined. The lacking degree of freedom is determined by a special transformation of the measurement points at the initial position into the workpiece coordinate system. The modelling at the hole pattern fit includes the calculation of the gauge geometry (ideal counterpart for the hole pattern fit), the calculation of an initial solution of the fitting and, finally, the implementation of the hole pattern fit. The goal is to determine the position of the virtual gauge in such a way that maximum empty space and/or minimum overlapping is achieved between the gauge geometry and measurement points. By means of the Gaussian method for the calculation of the initial solution, this target is attained only roughly. The desired position at the gauge can only actually be determined by applying the Chebyshev criterion. Manufactured product with deviations Datum surface A Assigned cylinder for datum B Datum point (position) Datum axis (direction) Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 18 Figure 10 Flowchart of the 3D hole pattern fit for the flange. The following subsections present details on the mathematical modelling and on the tasks to be solved for each of the 6 steps. 3.2.1. Assignment of the datum plane The datum plane is assigned to the extracted top surface of the flange as an adjacent Chebyshev plane (minimum zone criterion). The input data for the calculation of the plane are the extracted points of the top surface. 𝑃≔{𝑃1,…,𝑃𝑚}, 𝑃𝑖∈ℝ3 Every point has the form 𝑃𝑖=(𝑥𝑖,𝑦𝑖,𝑧𝑖)𝑇. The coordinates 𝑥,𝑦 and 𝑧 are specified in the measurement point coordinate system. The assigned plane has an ideal geometrical form. It has been parameterized via the normal vector 𝑣=(𝑣𝑥,𝑣𝑦,𝑣𝑧)𝑇∈ℝ3 and a point on the plane. 𝐶=(𝐶𝑥,𝐶𝑦,𝐶𝑧)𝑇∈ℝ3 With this definition, the representation of an ideal plane is not yet unambiguous. For example, for 𝐶, every arbitrary point of the plane is possible. Thus, further constraints are made on the parameters. On the one hand, the normal vector is supposed to have the length 1. 〈𝑣,𝑣〉=1. (3) Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 19 On the other hand, the point of the plane must be located so that the distance to the centroid 𝐺 of the data 𝑃 is at a minimum. For this purpose, 𝑣1,𝑣2∈ℝ3 shall be two vectors with a length of 1 which are orthogonal to the normal 𝑣. For those, 〈𝑣1,𝑣2〉=0 applies. Then, point 𝐶 is the projected centroid, if 〈𝐺−𝐶,𝑣1〉=0 〈𝐺−𝐶,𝑣2〉=0 (4) is met. The centroid is calculated as an arithmetic mean. 𝐺=1 𝑚∑𝑃𝑖 𝑚 𝑖=1 . For the correct assignment of the datum plane the orthogonal distances between the ideal plane and the measurement points are considered. These are as follows: 𝑓𝑖(𝐶,𝑣)=〈𝑃𝑖−𝐶,𝑣〉 The assignment finally takes place according to the following mathematical model. Assignment task: Chebyshev plane as adjacent datum plane If 𝑃 is the extracted geometry of a plane surface. The assigned Chebyshev plane has the parameters 𝐶 and 𝑣 which solve 𝑚𝑖𝑛 𝐶,𝑣 𝑚𝑎𝑥 𝑖|𝑓𝑖(𝐶,𝑣)| (5) and meet the constraints (3) and (4). For 𝑠=𝑚𝑎𝑥 𝑖|𝑓𝑖(𝐶,𝑣)|, the adjacent Chebyshev datum plane with the parameters 𝐶󰆹,𝑣 is calculated by means of the translation 𝐶󰆹=𝐶+𝑠∙𝑣. (6) The sign of 𝑣 must be determined in such a way that this vector points away from the material side of the extracted geometry for the calculation of (6). 3.2.2. Assignment of the datum cylinder The input data for the cylinder calculation are the points measured at the outer flange surface. In the case of the planes, they are simply called 𝑃. The assigned cylinder has a geometrically ideal form. It is parameterized by a parameter for the direction of the cylinder axis 𝑣=(𝑣𝑥,𝑣𝑦,𝑣𝑧)𝑇∈ℝ3, a parameter for the position of the cylinder axis 𝐶=(𝐶𝑥,𝐶𝑦,𝐶𝑧)𝑇∈ℝ3 and the radius of the cylinder lateral surface 𝑟>0 As the cylinder is assigned as secondary datum element, the following constraints apply for the parameters. The direction of the cylinder axis corresponds to the normal vector of the datum plane Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 20 previously determined. For the unambiguous calculation of the position, a further criterion for the position of the point on the cylinder axis must be given. This point can be the intersection between the axis and the datum plane. The assignment of the cylinder parameters is carried out according to the minimum circumscribed element criterion. For its mathematical formulation the orthogonal distances 𝑓𝑖(𝐶,𝑣,𝑟)=‖(𝑃𝑖−𝐶)×𝑣‖−𝑟 between the measurement points from 𝑃 and the lateral surface of the ideal cylinder are considered. The following task must be solved for the correct calculation of the cylinder parameters. Assignment task: envelope cylinder as secondary datum vertically to a plane: Determine the point 𝐶 and the radius 𝑟 in such a way that 𝑚𝑖𝑛 𝐶,𝑟 𝑟 𝑠.𝑡. 𝑓𝑖(𝐶,𝑣,𝑟)≤0 𝑓𝑜𝑟 1≤𝑖≤𝑚 (7) is obtained. 3.2.3. Workpiece coordinate system and virtual gauge In the case of the flange considered here, the virtual gauge for the hole pattern fit must only be rotated around the axis of the datum cylinder. As the axis can generally be tilted in space, this can only be solved with great technical effort. By suitably assigning a workpiece coordinate system and the suitable geometry of the virtual gauge, the fitting can be technically realized more easily. Assignment of the workpiece coordinate system The axis of the envelope cylinder (7) is the basis for the definition of the workpiece coordinate system. This example specifies that the direction vector 𝑣 defines the z axis 𝑧𝑊 of the workpiece. Likewise, point 𝐶 from (7) is defined as the centre point of the workpiece coordinate system. The remaining axes 𝑥𝑊 and 𝑦𝑊 are not clearly determined by the datum system. It is only during the transformation of the extracted geometry into the workpiece coordinate system – which is still incomplete at that point in time – that they are specified. The transformation shown here is a possible variant for this. There are also other approaches which lead to the same fitting result later on, however, they will not be discussed here. The conversion of point coordinates 𝑃𝑖∈ℝ3 into the workpiece coordinate system is implemented by the linear transformation 𝑃𝑖=𝑅(𝑃𝑖−𝐶). The 3x3 matrix 𝑅 defines a rotation which map the direction vector 𝑣 of the datum cylinder on the basis vector 𝑒𝑧=(0,0,1)𝑇. As 𝑣 corresponds to the z axis of the coordinate system, the following must be valid 𝑅𝑣=𝑒𝑧. By means of the Tait-Bryan angles for rotations around the x and/or y axis 𝑅𝑥=(1000 cos(𝛼) sin(𝛼)−0 sin(𝛼) cos(𝛼)) and 𝑅𝑦=(cos(𝛽) 0 −sin(𝛽)010sin(𝛽) 0 cos(𝛽)) Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 21 𝑅≔𝑅𝑦𝑇𝑅𝑥𝑇 is created. The point coordinates 𝑃𝑖 then lie in the workpiece coordinate system with the origin point 𝑊0=(0,0,0)𝑇, x axis 𝑥𝑤=(1,0,0)𝑇, y axis 𝑦𝑤=(0,1,0)𝑇 and z axis 𝑧𝑤=(0,0,1)𝑇. The following algorithm provides a numerically more stable procedure for the calculation of the two Euler rotation angles 𝛼 and 𝛽. Algorithms for the calculation of the rotation angles for the coordinate transformation Step 0: Set a positive accuracy 𝜀≪1. Step 1: Calculate 𝑡=√𝑣𝑦2+𝑣𝑧2 If 𝑡<𝜀, set 𝑐𝑜𝑠(𝛼)=1 and sin(𝛼)=0. If not, set 𝑐𝑜𝑠(𝛼)=𝑣𝑧 𝑡, sin(𝛼)=𝑣𝑦 𝑡 and 𝑣𝑧′=t. Step 2: Calculate 𝑡=√𝑣𝑥2+𝑣′𝑧2 If 𝑡<𝜀, set 𝑐𝑜𝑠(𝛽)=1 and sin(𝛽)=0. If not, set 𝑐𝑜𝑠(𝛽)=𝑣′𝑧 𝑡 and sin(𝛽)=𝑣𝑥 𝑡. The limit for the precision 𝜀 determines when a component of a vector 𝑣 already lies close enough to the direction achieved of the z axis. In this case, no further rotation of the data is done. The transformation must be implemented for all measurement points which have been assigned to the cylindrical holes on the flange. Each hole is available with its own set of measurement points. 𝑃(1)={𝑃1(1),…,𝑃𝑚1 (1)}, 𝑃(2)={𝑃1(2),…,𝑃𝑚2 (2)}, 𝑃(3)={𝑃1(3),…,𝑃𝑚3 (3)}, 𝑃(4)={𝑃1(4),…,𝑃𝑚4 (4)}, 𝑃(5)={𝑃1(5),…,𝑃𝑚5 (5)} For simplification reasons, the index 𝑘 with 1≤𝑘≤5 will be introduced for the assignment of the measurement point sets to the individual holes in the following. It is written as 𝑃(𝑘) and/or 𝑃𝑖(𝑘) with the coordinate values 𝑃𝑖(𝑘)=(𝑥𝑘𝑖,𝑦𝑘𝑖,𝑧𝑘𝑖)𝑇. Furthermore, the values 𝑚𝑘 stand for the number of points of the extracted geometric element with the index k. The transformation of the measurement data is 𝑃𝑖(𝑘)=𝑅(𝑃𝑖(𝑘)−𝐶). The extracted geometry is outlined with the workpiece coordinate system in Figure 11. Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 22 Figure 11 Workpiece coordinate system and extracted geometry at the flange. To simplify things, the further notation 𝑃𝑖(𝑘) is used instead of 𝑃𝑖(𝑘) for the point coordinates in the workpiece coordinate system referring to the respective context. Specification of the virtual gauge: The virtual gauge consists of 5 cylinders with an ideal geometrical form. All cylinder axes are parallel with the common direction vector 𝑣=(0,0,1)𝑇. Likewise, every cylinder has the same radius 𝑟=1.9 mm. However, the positions of the individual cylinder axes 𝐶1,…,𝐶5 are different. They are also referred to as 𝐶𝑘 (𝑘=1,…,5). The nominal position is carried out by means of the specified bolt circle with the radius 𝑟𝐿=29 mm and the angular distance 𝜏=72° to 𝐶𝑘=(𝐶𝑘𝑥 𝐶𝑘𝑦 𝐶𝑘𝑧)=(𝑟𝐿cos(𝑘𝜏) 𝑟𝐿sin(𝑘𝜏) 0) The following matrix formulation is suitable for the storage of the geometric parameters of the gauge. 𝑀= ( 𝐶1𝑥 𝐶1𝑦 𝐶1𝑧 𝑣𝑥𝑣𝑦𝑣𝑧𝑟 𝐶2𝑥 𝐶2𝑦 𝐶2𝑧 𝑣𝑥𝑣𝑦𝑣𝑧𝑟 𝐶3𝑥 𝐶3𝑦 𝐶3𝑧 𝑣𝑥𝑣𝑦𝑣𝑧𝑟 𝐶4𝑥 𝐶4𝑦 𝐶4𝑧 𝑣𝑥𝑣𝑦𝑣𝑧𝑟 𝐶5𝑥 𝐶5𝑦 𝐶5𝑧 𝑣𝑥𝑣𝑦𝑣𝑧𝑟 ) (8) Note: The formulation of (8) requires that the normal vector of the datum plane A points towards the material side of the flange. However, if the datum direction is turned, i.e. if the normal vector points away from the material, then the sequence of the entries in the parameter matrix changes. Then, 𝐶5 to 𝐶1 must be entered instead of 𝐶1 to 𝐶5. 3.2.4. Initial value and 3D hole pattern fit for the flange The starting point for the modelling of the 3D hole pattern fit for the flange are the measurement points of the holes 𝑃(1),…,𝑃(5) in the workpiece coordinate system and the matrix with the parameters of the gauge geometry 𝑀. Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 23 The free parameter of the fitting is the rotational angle 𝜑 which turns the gauge or, likewise, the gauge parameter around the z axis in the coordinate origin of the coordinate system. The angle determines a special rotary matrix 𝐻. 𝑀(𝜑)=𝑀∙𝐻(𝜑) There is 𝐻(𝜑)= ( 𝑐𝑜 𝑠𝑖 0000 0 −𝑠𝑖 𝑐𝑜 0000 0 0 0 1 000 0 0 0 0 100 0 0 0 0 010 0 0 0 0 001 0 0 0 0 000 1 ) (9) with the values 𝑠𝑖=sin(𝜑) and 𝑐𝑜=cos(𝜑). In addition, 𝐶𝑘(𝜑)=(𝐶𝑘𝑥,𝐶𝑘𝑦,𝐶𝑘𝑧)𝑇∙(𝑐𝑜 −𝑠𝑖 0𝑠𝑖 𝑐𝑜 0001) is used for the rotated position points of the gauge cylinders in order to simplify the notation for the mathematical modelling of the fitting task. In the following subsections, the mathematical models are described for a consistent 3D hole pattern fit. The procedure starts with a Gaussian-type coarse fitting which is used as the starting solution for the exact fit according to the Chebyshev criterion (1). Starting solution with a Gaussian fitting In a first step, the centroid 𝑄𝑘=(𝑞𝑘𝑥,𝑞𝑘𝑦,0)𝑇 is calculated for every point set 𝑃(𝑘). Thereby the following applies: 𝑞𝑘𝑥=1 𝑚𝑘∑𝑥𝑘𝑖 𝑚𝑘 𝑖=1 and 𝑞𝑘𝑦=1 𝑚𝑘∑𝑦𝑘𝑖 𝑚𝑘 𝑖=1 . The calculation is also efficient for large data sets. In order to determine the starting position of the gauge, the angle 𝜑0 is calculated by means of a best fitting which solves the minimization program min φ0 12∑ ‖𝐶𝑘(𝜑0)−𝑄𝑘‖2 5𝑘=1 (10) The sum of the distance squares between the centre point of the gauge cylinder axes and the point cloud centroids is hereby minimized. This coarse fitting is shown in Figure 12. Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 24 Figure 12 Coarse fitting at the flange On the left, the initial situation where the workpiece coordinate system and the gauge are superimposed is shown. Measurement points and gauge cylinders are shown in the top view on the x-y-plane. The gauge cylinders coloured in blue are clearly staggered to the black dotted extracted holes. On the right side of the figure, the distances between the gauge centre points and the point cloud centroids were minimized. Thereby, the gauge coordinate system was rotated around the angle 𝜑0. Calculation of the 3D hole pattern fit The measurement points at the 5 individual holes describe cylinders which – due to measurement and manufacturing deviations – do not have an ideal geometric shape. In general, there are deviations of measure, shape and position. They influence the result of the previous Gaussian fit. The angle 𝜑0 calculated from the initial value (10) is not yet the angle with the smallest possible overlap or largest possible empty space between the gauge cylinders and the measurement points available. For the calculation of the 3D hole pattern fit, the orthogonal distances between the gauge cylinders and the measurement points of the holes are defined as follows: 𝑓𝑘𝑖(𝑀(𝜑))=𝑟−‖(𝑃𝑖(𝑘)−𝐶𝑘(𝜑))×𝑣‖ Index 𝑘 is the number of the hole and index 𝑖 specifies the number of the measurement points to the hole 𝑘. The norm is the Euclidean standard norm in the ℝ3. The application of the general fitting program (2) from Chapter 2.4 thereby provides the 3D hole pattern fit for the flange (11). min φ∈ℝ,𝑠∈ℝ𝑠 𝑠.𝑡. 𝑓𝑘𝑖(𝜑)≤𝑠 ∀ 𝑘=1,..,5 𝑎𝑛𝑑 ∀ 𝑖=1,…,𝑚𝑘 (11) The rotation angle 𝜑 calculated from this program and the maximum distance 𝑠 is the determination of the quantity searched – minimum overlapping or maximum empty space – and easy to realize. The following statements are valid  If 𝑠>0, there is an overlap between the gauge and the measurement points. It has the value 𝑠.  If 𝑠<0, there is empty space between the gauge and the measurement points. It has the value 𝑠. Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 31 parameter is identical to the procedure in section 3 (algorithm for the calculation of the rotation angle for the coordinate transformation). In the case of the conical disc, the geometry parameter 𝐶 and 𝑣 are derived from the datum cone axis. The assigned coordinate system is sketched in Figure 20. Figure 20 Workpiece coordinate system for the conical disc For the conical disc, special measurement point amounts are available 𝑃(1)={𝑃1(1),…,𝑃𝑚1 (1)}, 𝑃(2)={𝑃1(2),…,𝑃𝑚2 (2)}, 𝑃(3)={𝑃1(3),…,𝑃𝑚3 (3)}, 𝑃(4)={𝑃1(4),…,𝑃𝑚4 (4)}, 𝑃(5)={𝑃1(5),…,𝑃𝑚5 (5)} for the five holes. Furthermore, 𝑃(6)={𝑃1(6),…,𝑃𝑚6 (6)} and 𝑃(7)={𝑃1(7),…,𝑃𝑚7 (7)} stand for the lateral surfaces situated opposite of the slot. When defining 𝑃(𝑘), 𝑚𝑘 is the respective number of points per data set. The transformation of the measurement points into the workpiece coordinate system is formally calculated by 𝑃𝑖(𝑘)=𝑅(𝑃𝑖(𝑘)−𝐶). In the following, the designation 𝑃𝑖(𝑘) is used for the points in the workpiece coordinate system to simplify the notation. Specification of the virtual gauge The virtual gauge consists of five cylinders and a pair of parallel planes with an ideal geometrical form. All cylinder axes are parallel to the common direction vector 𝑣=(0,0,1)𝑇. Likewise, every Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 32 cylinder has the radius 𝑟=3.9 mm (diameter 7.8 mm). However, the positions of the individual cylinder axes are different: 𝐶1,…,𝐶5 They are also referred to as 𝐶𝑘 (𝑘=1,…,5). The calculation of the nominal position is carried out by means of the specified bolt circle with the radius 𝑟𝐿=22 mm,𝜏0=−30°0 and the angular distance 𝜏=60° 𝐶𝑘=(𝐶𝑘𝑥 𝐶𝑘𝑦 𝐶𝑘𝑧)=(𝑟𝐿cos(𝜏0+𝑘𝜏) 𝑟𝐿sin(𝜏0+𝑘𝜏) 0). The pair of parallel planes for the slot is initially defined by the normal vector 𝑛=(1,0,0)𝑇 in its orientation. Thus, the planes are parallel to the conical axis and symmetrical to the five gauge cylinders. For each plane, the sign of the normal is selected in such a way that it points away from the theoretical material side in the technical drawing and/or at the real product. Furthermore, the position is defined by a point which is situated centrally between the two planes. For 𝑘=6 and 𝑘= 7 this point is 𝐶𝑘=(𝐶𝑘𝑥 𝐶𝑘𝑦 𝐶𝑘𝑧)=( 0 −22 0). This is exactly the intersection point of the theoretically exact bolt circle and the y axis of the gauge coordinate system which is situated centrally in the slot. The orthogonal distance of the lateral planes to the median plane via point 𝐶𝑘 with the normal vector 𝑛 is 𝑑=7.925 mm. The matrix formulation for the storing of the gauge geometry parameters is 𝑀= ( 𝐶1𝑥 𝐶1𝑦 𝐶1𝑧 𝑣𝑥𝑣𝑦𝑣𝑧 0 0 0 𝑟 𝐶2𝑥 𝐶2𝑦 𝐶2𝑧 𝑣𝑥𝑣𝑦𝑣𝑧 0 0 0 𝑟 𝐶3𝑥 𝐶3𝑦 𝐶3𝑧 𝑣𝑥𝑣𝑦𝑣𝑧 0 0 0 𝑟 𝐶4𝑥 𝐶4𝑦 𝐶4𝑧 𝑣𝑥𝑣𝑦𝑣𝑧 0 0 0 𝑟 𝐶5𝑥 𝐶5𝑦 𝐶5𝑧 𝑣𝑥𝑣𝑦𝑣𝑧 0 0 0 𝑟 𝐶6𝑥 𝐶6𝑦 𝐶6𝑧 0 0 0 −𝑛𝑥−𝑛𝑦−𝑛𝑧𝑑 𝐶7𝑥 𝐶7𝑦 𝐶7𝑧 0 0 0 𝑛𝑥 𝑛𝑦 𝑛𝑧𝑑 ) . (17) The first five lines are the parameters of the gauge bolts for the holes. The two remaining lines provide the parameters of the pair of planes for the fitting of the slot. 4.2.3. Calculation of the initial value and the 3D hole pattern fit The starting point for the modelling of the 3D hole pattern fit of the conical disc are the measurement points transformed into the workpiece coordinate system of the holes 𝑃(1),…,𝑃(7) and the matrix with the parameters of the gauge geometry 𝑀. The free parameter of the fitting is the angle 𝜑 which rotates the gauge in the coordinate origin around the z axis of the workpiece coordinate system. The angle determines a special rotary matrix 𝐻 which helps to calculate the parameters of the rotated gauge via 𝑀(𝜑)=𝑀∙𝐻(𝜑). Hereby, Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 33 𝐻(𝜑)= ( 𝑐𝑜 −𝑠𝑖 00000000 𝑠𝑖 𝑐𝑜 00000000 0010000000 0001000000 0000100000 0000010000 000000 𝑐𝑜 −𝑠𝑖 00 000000 𝑠𝑖 𝑐𝑜 00 0000000010 0000000001 ) (18) with the values 𝑠𝑖=sin(𝜑) and 𝑐𝑜=cos(𝜑). In 𝑀(𝜑), only the position entries (the x and y coordinates of the gauge bolts, plane centre) and the normal vectors of the pair of planes change. For a simplified notation, these entries are thus also referred to as 𝐶𝑘(𝜑)=(𝐶𝑘𝑥,𝐶𝑘𝑦,𝐶𝑘𝑧)(𝑐𝑜 𝑠𝑖 0 −𝑠𝑖 𝑐𝑜 0 0 0 1) and 𝑛𝑘(𝜑)=(𝑛𝑘𝑥,𝑛𝑘𝑦,𝑛𝑘𝑧)(𝑐𝑜 𝑠𝑖 0 −𝑠𝑖 𝑐𝑜 0 0 0 1). 4.2.3 Initial value and 3D hole pattern fit for the conical disc In this subsection, the mathematical models are described for the 3D hole pattern fit. In turn, a coarse fitting starts according to a suitable Gaussian criterion. Subsequently, the 3 D hole pattern fit is implemented according to the Chebyshev criterion. Starting solution with a Gaussian fitting As in the case of the flange, the calculation of a coarse fitting considered here is oriented towards the centres of the ideal gauge geometries and the measurement points. In a first step, the centroid is calculated for each extracted hole 𝑃(𝑘) with 𝑘=1,…,5. The z component of the centroid is set to 0, as it must not influence the fitting result. The other components are 𝑞𝑘𝑥=1 𝑚𝑘∑𝑥𝑘𝑖 𝑚𝑘 𝑖=1 and 𝑞𝑘𝑦=1 𝑚𝑘∑𝑦𝑘𝑖 𝑚𝑘 𝑖=1 . The calculation is efficient for large data volumes. Likewise, a centroid is calculated for the extracted pair of parallel planes of the slot. This centroid is 𝑄6=(𝑞6𝑥,𝑞6𝑦,0)𝑇 with 𝑞6𝑥=1 𝑚6+𝑚7(∑𝑥6𝑖 𝑚6 𝑖=1 +∑𝑥7𝑖 𝑚7 𝑖=1 ) and Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 34 𝑞6𝑦=1 𝑚6+𝑚7(∑𝑦6𝑖 𝑚6 𝑖=1 +∑𝑦7𝑖 𝑚7 𝑖=1 ). In order to determine the starting position of the gauge, the angle 𝜑0 is calculated in a second step which solves the minimization task min φ0 12∑ ‖𝐶𝑘(𝜑0)−𝑄𝑘‖2. 6𝑘=1 (19) Problem (19) is a Gaussian fitting. It minimizes the sum of the distance squares between the centroids of the point clouds and the arithmetically ideal centres of the gauge elements. The distances are only calculated in the x-y-plane of the workpiece coordinate system. The situation is shown in Figure 21 for a better understanding. Figure 21 Initial value of the conical disc fitting. In the top view, the gauge geometry is shown in the direction of the median axis (conical axis). The gauge geometry comprises the cylindrical bolts marked in blue and the slot area marked in green. The extracted geometry of the holes and slot lateral surfaces is sketched using black points. The gauge geometry and the extracted geometry are twisted towards each other prior to the fitting. This is shown in the left side of the figure. The ideal centres of the gauge 𝐶𝑘 and the centroids of the extracted geometry 𝑄𝑘 defined for the Gaussian fitting are clearly visible. After the rotation of the gauge by the calculated angle 𝜑0, the initial fit is available which is shown on the right side of the figure. Calculation of the 3D hole pattern fit The solution angle 𝜑0 from the initial value (19) is not yet the angle with the smallest possible overlapping and/or largest possible empty space between the gauge geometry and the measurement points. In order to apply the general Chebyshev fitting (2) from chapter 2.4, suitable distance functions for the gauge cylinders and the slot must be defined. As in case of the flange, the following definition is valid for the gauge cylinder with 𝑘=1,…,5 𝑓𝑘𝑖(𝑀(𝜑))=𝑟−‖(𝑃𝑖(𝑘)−𝐶𝑘(𝜑))×𝑣‖. 𝜑0 Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 35 In case of the slot, a distinction is made between the left and the right lateral surface. This is realized by a respective assignment of the point sets to the planes of the gauge geometry and the opposite determination of the normal direction. In this application example, 𝑃(6) is the extracted geometry of the left lateral surface and 𝑃(7) is the extracted geometry of the right lateral surface. For this, the following is valid: 𝑓𝑘𝑖(𝑀(𝜑))=𝑑−〈𝑃𝑖(𝑘)−𝐶𝑘(𝜑),𝑛𝑘(𝜑)〉. The following fitting procedure is set up. min φ∈ℝ,𝑠∈ℝ𝑠 𝑠.𝑡. 𝑓𝑘𝑖(𝜑)≤𝑠 ∀ 𝑘=1,..,7 𝑎𝑛𝑑 ∀ 𝑖=1,…,𝑚𝑘 (20) By means of the rotation angle 𝜑 calculated from this fitting program and the maximum distance 𝑠 the determination of the desired quantity for a minimum overlap or a maximum empty space is easy to realize. The following statements are valid:  If 𝑠>0, there is an overlap between the gauge and the measurement points. It has the value 𝑠.  If 𝑠<0, there is empty space between the gauge and the measurement points. It has the value 𝑠.  If 𝑠=0, the gauge is adjacent to the measurement points. There is neither empty space nor overlap. To be able to evaluate for which of the gauge geometry elements there is an overlap where 𝑠> 0, the distances to the locally assigned measurement points can be inspected for each individual element. These are as follows: 𝑠𝑘≔ max i=1,.,,,mk𝑓𝑘𝑖(𝑀(𝜑)) for all 𝑘=1,…,7. If 𝑠𝑘>0, there is an overlapping with the product. 5. Application example 3: cubes In the case of the application examples of flanges and conical discs considered above, the gauge elements are parallel. The inspection using a physical gauge is possible by means of a one-sided plugging of the gauge with the test specimen. In contrast to this, the application example for cubes considers the assembly of products on top of holes or bolts that are twisted toward each other. I.e., the geometry elements of the gauge are no longer parallel. As a result, the inspection by means of physical gauges that are made of only one part are no longer possible. Specifically in this case, virtual gauging by means of CMM and 3D hole pattern fit is a suitable means of testing with regard to feasibility. For the application, the cube sketched in Figure 22 is considered. Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 36 Figure 22 Application example for a cube Figure 23 ISO 1101 compliant inspection task for the cube. The cube has an edge length of 40.0 mm. In two of the six lateral surfaces, three holes respectively, have been integrated. The depth of all holes is 20.0 mm each. The diameters are 6.0 mm. All hole axes are vertical to the respective lateral surfaces. In order to better identify the position of the holes, the external surface of the cube was drawn as transparent (Fig. 22). Lateral surfaces Upper hole Side hole Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 37 5.1. Inspection according to the standard No specific function is attributed to the inspection of the cube. Therefore, a common inspection task is described here. This task is shown in Figure 23. When selecting the inspection task, a datum or datum system was totally dispensed with. Only the position of the six holes is tolerated. This leads to a gauging or 3D hole pattern fit with a maximum number of six degrees of freedom. At the same time, it places the highest demands on numerical stability and efficiency of mathematical procedures for an arithmetical fit. 5.2. Mathematical model of the 3D hole pattern fit The arithmetical 3D hole pattern fit is carried out by the four steps shown in Figure 24. Figure 24 Flowchart on the 3D hole pattern fit for the cube. At the beginning, the workpiece coordinate system and the gauge parameters were assigned. Subsequently, the discussion of a suitable initial value took place. The procedure is much more timeconsuming than for the previous applications. Furthermore, a general 3D transformation of the gauge geometry must be defined. Finally, the formal specification of the Chebyshev 3D hole pattern fit is carried out. Workpiece coordinate system and gauge parameters As we have the case of a general fit for the cube (no constraints due to datum elements), basically every Cartesian coordinate system can be used as a workpiece coordinate system. In the example, the Cartesian measurement point coordinate system is selected for the workpiece, therefore, we do not need to convert the point coordinates in this case. Figure 25 Coordinate system and model of the virtual gauge for the cube. Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 38 The gauge consists of six ideal cylinders. These cylinders are assigned to each hole of the cube. They have a radius of 𝑟=2.94 mm (corresponding to a diameter of MMVS = 5.88 mm). The position and direction of the gauge cylinders is presented in Figure 25. On the left side of the figure, the gauge coordinate system of the cube is shown. Furthermore, the holes are clearly numbered by the indices of 𝑘=1,..,6. In the figure on the right, the gauge geometry (the counterpart to the fitting) is shown. The centres 𝐶1,…,𝐶6 of the geometrically ideal cylinders have been plotted. Each of these cylinders is located on the axis of the gauge cylinder. The position is the centre between the ends of each hole in the technical drawing of the product. In the following, the parameter values are provided. 𝐶1=(𝐶1𝑥,𝐶1𝑦,𝐶1𝑧)𝑇=(−10,−30,−10)𝑇, 𝐶4=(𝐶4𝑥,𝐶4𝑦,𝐶4𝑧)𝑇=(−30,−10,−10)𝑇 𝐶2=(𝐶2𝑥,𝐶2𝑦,𝐶2𝑧)𝑇=(−30,−25,−10)𝑇, 𝐶5=(𝐶5𝑥,𝐶5𝑦,𝐶5𝑧)𝑇=(−10,−10,−25)𝑇 𝐶3=(𝐶3𝑥,𝐶3𝑦,𝐶3𝑧)𝑇=(−10,−10,−10)𝑇, 𝐶6=(𝐶6𝑥,𝐶6𝑦,𝐶6𝑧)𝑇=(−30,−10,−30)𝑇 The direction vector of the cylinders 1 to 3 is 𝑣1=(𝑣1𝑥,𝑣1𝑦,𝑣1𝑧)𝑇=(0,0,1)𝑇. The cylinders 4 to 6 have the direction vector 𝑣2=(𝑣2𝑥,𝑣2𝑦,𝑣2𝑧)𝑇=(0,1,0)𝑇. Here, the parameter values are also summarized in a joint matrix 𝑀. 𝑀= ( 𝐶1𝑥 𝐶1𝑦 𝐶1𝑧 1𝑣1𝑥 𝑣1𝑦 𝑣1𝑧 𝑟 𝐶2𝑥 𝐶2𝑦 𝐶2𝑧 1𝑣1𝑥 𝑣1𝑦 𝑣1𝑧 𝑟 𝐶3𝑥 𝐶3𝑦 𝐶3𝑧 1𝑣1𝑥 𝑣1𝑦 𝑣1𝑧 𝑟 𝐶4𝑥 𝐶4𝑦 𝐶4𝑧 1𝑣2𝑥 𝑣2𝑦 𝑣2𝑧 𝑟 𝐶5𝑥 𝐶5𝑦 𝐶5𝑧 1𝑣2𝑥 𝑣2𝑦 𝑣2𝑧 𝑟 𝐶6𝑥 𝐶6𝑦 𝐶6𝑧 1𝑣2𝑥 𝑣2𝑦 𝑣2𝑧 𝑟 ) (21) The column with the numerical values of "1" is of special importance for the transformation of the parameter matrix. This will be explained in the following. Specification of the transformation operator for the fitting The transformation comprises six different parameters. Initially, three translations of the gauge along the workpiece coordinate system are possible. These translations are denoted by the parameter. 𝑇=(𝑡𝑥,𝑡𝑦,𝑡𝑧)𝑇 The components indicate the respective fraction of the translation towards the coordinate axis with an identical index. Furthermore, three rotations around the workpiece coordinate axes are possible. This is realized here with the aid of the Euler rotation angle. 𝜑=(𝜑𝑥,𝜑𝑦,𝜑𝑧)𝑇 Each component defines a rotation matrix. These are Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 39 𝑅𝑥=( 1 0 0 0 cos(𝜑𝑥)−sin(𝜑𝑥) 0 sin(𝜑𝑥) cos(𝜑𝑥))𝑇, 𝑅𝑦=( cos(𝜑𝑦) 0 sin(𝜑𝑦) 0 1 0 −sin(𝜑𝑦)0 cos(𝜑𝑦))𝑇 and 𝑅𝑧=(cos(𝜑𝑧)−sin(𝜑𝑧) 0 sin(𝜑𝑧) cos(𝜑𝑧)0 0 0 1)𝑇. From the individual rotation matrices, the following matrix is formed by means of multiplication: 𝑅=𝑅𝑥𝑅𝑦𝑅𝑧=(𝑟11 𝑟12 𝑟13 𝑟21 𝑟22 𝑟23 𝑟31 𝑟32 𝑟33) Thereby, the matrix 𝐻(𝑇,𝜑) for the transformation of the gauge geometry 𝑀 in (21) is 𝐻(𝑇,𝜑)= ( 𝑟11 𝑟21 𝑟31 0 0 0 0 0 𝑟12 𝑟22 𝑟32 0 0 0 0 0 𝑟13 𝑟23 𝑟33 0 0 0 0 0 𝑡𝑥 𝑡𝑦 𝑡𝑧 1 0 0 0 0 0 0 0 0 𝑟11 𝑟21 𝑟31 0 0 0 0 0 𝑟12 𝑟22 𝑟32 0 0 0 0 0 𝑟13 𝑟23 𝑟33 0 0 0 0 0 0 0 0 1 ) . (22) The transformed parameter matrix is 𝑀(𝑇,𝜑)≔𝑀∙𝐻(𝑇,𝜑). During the calculation, the points indicating the position of the gauge cylinders are rotated and subsequently shifted. The calculation formula is 𝐶𝑘(𝑇,𝜑)=𝐶𝑘𝑅+𝑇. The direction vectors of the gauge cylinders are exclusively twisted. The rotated vectors are designated with 𝑣𝑙(𝜑)=𝑣𝑙𝑅. These transformation provisions for the gauge parameters are used for the specification of the fitting task later on. Formulation of an initial value according to the Gaussian criterion Again, a favourable initial orientation of the gauge is being sought for via a suitable Gaussian fitting. Initially, 𝑃(𝑘)={𝑃1(𝑘),…,𝑃𝑚𝑘 (𝑘)} with 𝑘=1,…,6 shall be the point clouds of the measured holes of the cube. The indication corresponds to the specification from Figure 25. The arithmetical centroid is assigned to each point cloud: 𝑄𝑘=1 𝑚𝑘∑ 𝑃𝑖(𝑘) 𝑚𝑘 𝑖=1 The initial value is then the solution of the Gaussian program Physikalisch-Technische Bundesanstalt Braunschweig u. Berlin Guide to 3D pattern fitting in coordinate metrology 40 min T0,φ0 12 ∑ ‖𝐶𝑘(𝑇0,𝜑0)−𝑄𝑘‖ 6𝑘=1 2. (23) In Figure 26, the initial value is illustrated once again. The upper part shows the gauge and the extracted holes in the starting position. The measured holes are shown in simplified manner by means of black dotted contours. The transformation operators are illustrated at the axes of the coordinate system. By means of the Gaussian fitting, the median gauge points 𝐶𝑘 are shifted as near as possible to the centroids of the point clouds. This is shown by the lower half of the figure. Figure 26 Initial value for the cube fitting Definition of the 3D hole pattern fit for the cube Also in this application example, cylindrical gauge elements are available. Therefore, the local orthogonal distances between gauge and the measured points near the elements 𝑘=1,…,3 are defined by 𝑓𝑘𝑖(𝑀(𝑇,𝜑))=𝑟−‖(𝑃𝑖(𝑘)−𝐶𝑘(𝑇,𝜑))×𝑣1(𝜑)‖ and for 𝑘=4,…,6 by Gaussian