ANALYTICAL EVALUATION OF THE UNCERTAINTY OF COORDINATE MEASUREMENTS OF GEOMETRICAL DEVIATIONS. MODELS BASED ON THE DISTANCE BETWEEN POINT AND PLANE

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1 ADVANCES IN MANUFACTURING SCIENCE AND TECHNOLOGY Vol. 37, No. 3, 2013 DOI: /ast ANALYTICAL EVALUATION OF THE UNCERTAINTY OF COORDINATE MEASUREMENTS OF GEOMETRICAL DEVIATIONS. MODELS BASED ON THE DISTANCE BETWEEN POINT AND PLANE Władysław Jakubiec, Wojciech Płowucha S u a r y In this paper a series discussing the possibility of analytical evaluation of the uncertainty of coordinate easureents is presented. It presents odels of evaluation of uncertainty of soe geoetric deviations (such as flatness, perpendicularity of axes, position of point, axis and plane) based on the forula for the point-plane distance. An iportant eleent of the presented ethodology for deterining the uncertainty of easureent is the use of atheatical iniu nuber of characteristic points of the easured workpiece and expressing the deviation as a function of coordinates differences of the points. Keywords: coordinate easuring technique, easureent uncertainty, geoetrical deviations Analityczne wyznaczanie niepewności współrzędnościowych poiarów odchyłek geoetrycznych. Modele bazujące na odległości punktu od płaszczyzny S t r e s z c z e n i e W artykule oówiono ożliwość analitycznego określania niepewności poiarów współrzędnościowych. Przedstawiono odele wyznaczania niepewności niektórych odchyłek geoetrycznych (np. płaskości, prostopadłości osi, pozycji) bazujące na równaniach na odległość punktu od płaszczyzny. Istotny eleente tej etodyki wyznaczania niepewności poiaru jest przyjęcie inialnej liczby charakterystycznych punktów ierzonego przediotu oraz wyrażenie odchyłki jako funkcji różnic wartości współrzędnych tych punktów. Słowa kluczowe: współrzędnościowa technika poiarowa, niepewność poiaru, odchyłki geoetryczne 1. Introduction The previous paper [1] presented the theoretical background of the analytical evaluation of uncertainty of coordinate easureents and the odels based on the forula for the distance between a point and a line. In particular, it described the odels to assess the uncertainty of easureent of geoetric Address: Władysław JAKUBIEC, DSc Eng., Wojciech PŁOWUCHA, PhD Eng., University of Bielsko-Biała, Laboratory of Metrology, Bielsko-Biała, Willowa 2, Poland, phone: , fax: , e-ail: wjakubiec@ath.eu

2 6 W. Jakubiec, W. Płowucha deviations such as straightness, concentricity, parallelis of the axis (cylindrical tolerance zone) and the perpendicularity of axis in regard to the plane. This article presents odels based on the forula for the distance between a point and a plane. In particular, the odels to assess the uncertainty of easureent of geoetric tolerances such as: flatness, parallelis of axes in the plane noral to coon plane, position of a point, axis and plane in regard to datu plane, perpendicularity of axes in space, perpendicularity of a plane to the datu axis, total axial run-out, parallelis of axes in the coon plane, parallelis of axis in regard to datu plane, parallelis of planes, parallelis of a plane to datu axis, parallelis of planes and perpendicularity of planes. 2. General odel of easuring the distance between point and plane In the coordinate easuring technique the distance l of point S fro the plane p given by any point P belonging to this plane and the unit noral vector u is calculated as follows: l ( S p) = ( P S) u, (1) Particular tasks described below differ in the iniu nuber of points required for their deterination and a way to define the vector u. It should be noted that in all odels the deviation is a function of the difference of coordinates of points used to deterine the deviation. In all the forulae a siplified notation of the difference of coordinates of two points is used for exaple: x = x x. BA B A 3. Measureent odels of the flatness, the parallelis of axes in the plane noral to the coon plane and the position of the point in regard to the priary datu plane The flatness can be deterined as the distance l between the point S and the plane defined by three points A, B and C of this plane (Fig. 1). This odel refers to the siplified classical ethod of flatness easureent. The iniu nuber of points required to deterine the deviation is 4. Point S is noinally in the plane containing the points A, B and C. The parallelis of axes in the plane noral to the coon plane can be deterined by the distance l between the point S of the toleranced axis and the plane defined by the two points (A and B) of the datu axis and one point (C) of the toleranced axis. The iniu nuber of points required to deterine the

3 Analytical evaluation of the uncertainty... 7 deviation is 4 (Fig. 2). In this case, the point S noinally lies in the plane containing the points A, B and C. Fig. 1. Measureent of flatness: a design drawing with characteristic points, easureent odel Fig. 2. Measureent of parallelis of axes in the plane noral to the coon plane: a design drawing with characteristic points, easureent odel In the above tasks, plane p (fro the forula (2)) is given by the three points A, B and C. As the point P to the forula (1) you can adopt one of these points (A, B or C) and the noral unit vector u can be calculated according to the forula: AB AC u = (2) AB AC Depending on whichever point we assue as the point P, we get the following three odels:

4 8 W. Jakubiec, W. Płowucha axsa + bysa + czsa l1 = (3) axsb + bysb + czsb l2 = (4) axsc + bysc + czsc l3 = (5) where: a = ybazca zba yca b = xcazba x z BA CA c = xba yca xca y BA = a + b + c (6) As the uncertainty of easureent the sallest of the three cobined uncertainties calculated on the basis of the above forulae is assued: { u, u u } u = in (7) l 1 l 2, The position of the point, axis or plane in regard to the datu plane defined by three points (Fig. 3-5) can be deterined as the doubled difference between the distance l of this point fro the plane and the theoretically exact distance l TED. l 3 Fig. 3. Measureent of position of the point in regard to the datu plane: a design drawing with characteristic points, easureent odel

5 Analytical evaluation of the uncertainty... 9 = 2 l l TED (8) It can be proved that the uncertainty of the deterination of the position deviation is two ties bigger than the uncertainty of easureent of the distance of the characteristic point S fro the datu plane ABC. Fig. 4. Measureent of position of the axis in regard to the datu plane: a design drawing with characteristic points, easureent odel Fig. 5. Measureent of position of the plane in regard to the datu plane: a design drawing with characteristic points, easureent odel 4. Measureent odel of perpendicularity of axes, perpendicularity of plane in regard to axis and total axial run-out The perpendicularity of axes in space can be calculated as the distance l of the point S fro the plane perpendicular to the straight line AB and containing the point K (Fig. 6). The iniu nuber of points required to deterine the deviation is 4. The datu axis is represented by 2 points A and B. The axis the orientation of which is toleranced is represented by the points K and S. The perpendicularity of a plane in regard to a datu axis as well as total axial run-out (Fig. 7) can be calculated in the sae way as the perpendicularity of axes in space. The iniu nuber of points required to deterine the deviation is 4. The datu axis is represented by 2 points A and B. The plane the orientation (or the run-out) of which is toleranced is represented by the points K and S.

6 10 W. Jakubiec, W. Płowucha c) Fig. 6. Measureent of perpendicularity of axes in space:, design drawings with characteristic points, c) easureent odel c) Fig. 7. Measureent of perpendicularity of plane in regard to axis and the easureent of total axial run-out:, corresponding design drawings with characteristic points, c) easureent odel In the above described easuring tasks, the plane p fro the forula (1) is perpendicular to the straight line AB and contains the point K. As the point P in the forula (1) the point K is to be assued and the noral unit vector u can be calculated as: AB u = (9) AB Finally, the forula takes the for:

7 Analytical evaluation of the uncertainty xks xba + yks yba + zks zba l = (10) x + y + z 2 BA 2 BA 2 BA 5. Measureent odel for parallelis of axes in the coon plane The parallelis of axes in the coon plane is calculated as the distance l of the point S fro the plane parallel to the straight line AB, perpendicular to the plane ABK and containing the point K (Fig. 8). The iniu nuber of points required to deterine the deviation is 4. The datu axis is represented by two points A and B. The axis the orientation of which is toleranced is represented by the points K and S. Fig. 8. Measureent of parallelis of axes in the coon plane: a design drawing with characteristic points, easureent odel In this easuring task, the plane p fro the forula (1) is perpendicular to the plane ABK, parallel to the straight line AB and contains point K. As the point P in the forula (1) the point K is to be assued and the noral unit vector u can be calculated as the vector product of the noral vector of the plane ABK and the vector AB divided by its length: ( AB AK ) AB ( AB AK ) AB u = (11) Finally, the forula for calculating the parallelis of axes in the coon plane is:

8 12 W. Jakubiec, W. Płowucha where: 2 2 ( ax ) + ( by ) + ( cz ) SK SK SK l = (12) 2 ( xba yka ybaxka ) zba( zbaxka xbazka ) ( ybazka zba yka ) xba ( xba yka ybaxka ) ( z x x z ) y ( y z z y ) a = yba b = z BA c = x BA BA KA BA KA BA BA KA BA KA = a + b + c (13) 6. Measureent odel for parallelis of axis in regard to plane and for parallelis of planes The parallelis of axis in regard to plane can be deterined as the distance l between the point S and the plane parallel to the plane ABC and intersecting the point K (Fig. 9). The iniu nuber of points required to deterine the deviation is 5. The datu plane is represented by the three points A, B and C. The axis the orientation of which is toleranced is represented by the points K and S. Fig. 9. Measureent of parallelis of axis in regard to plane: a design drawing with characteristic points, easureent odel The parallelis of planes can be deterined as the distance l between the point S and the plane parallel to the plane ABC and intersecting the point K (Fig. 10).The iniu nuber of points required to deterine the deviation is 5. The datu plane is represented by the three points A, B and C. The plane the orientation of which is toleranced is represented by the points K and S.

9 Analytical evaluation of the uncertainty Fig. 10. Measureent of parallelis of planes: a design drawing with characteristic points, easureent odel In both easuring tasks, the plane p fro the forula (1) is given by the point K and the noral unit vector u calculated as: where: AB AC u = (14) AB AC The distance between the point S and the plane is calculated as: axks + byks + czks l = (15) a = yab z AC z AB y AC b = z AB xac xac z AC c = xab yac y AB x AC = a + b + c (16) 7. Measureent odel for parallelis of plane in regard to axis The parallelis of plane in regard to axis can be deterined as the distance l between the point S and the plane parallel to the axis AB intersecting points K and L (Fig. 11).The iniu nuber of points required to deterine the deviation is 5. The datu axis is represented by the two points A and B. The plane orientation of which is toleranced is represented by the points K, L and S.

10 14 W. Jakubiec, W. Płowucha Fig. 11. Measureent of parallelis of the plane in regard to axis: a design drawing with characteristic points, easureent odel In the task, the plane p fro the forula (1) is parallel to both the straight line AB and the straight line KL. The noral unit vector u of that plane: KL AB u = (17) KL AB Finally, we can assue the following forulae for deterining the perpendicularity of axis in regard to the plane: where: axks + byks + czks l1 = (18) axls + byls + czls l2 = (19) a = ybazlk zba ylk b = zbaxlk x z BA LK c = xba ylk ybax LK = a + b + c (20) As the uncertainty of easureent of parallelis the saller value of the two values calculated on the basis of the above forulae is assued.

11 Analytical evaluation of the uncertainty Measureent odel for perpendicularity of planes The perpendicularity of two planes can be deterined as the distance l between the point S and the plane perpendicular to the plane ABC and intersecting points K and L (Fig. 12). The iniu nuber of points required to deterine the deviation is 6. The datu plane is represented by the points A, B and C. The plane the orientation of which is toleranced is represented by the points K, L and S. Fig. 12. Measureent of perpendicularity of the planes: a design drawing with characteristic points, easureent odel In this task, the plane p fro the forula (1) is perpendicular to the plane ABC and intersects points K and L. The noral unit vector u of the plane can be calculated as: ( AB AC) KL ( AB AC) KL u = (21) As the point P in the forula (1) the point K or the point L can be assued. 9. Suary The accuracy evaluation is the key eleent of any easureent [2]. Using in the atheatical odel of the easuring task the atheatical iniu nuber of characteristic points of the easured workpiece and expressing the deviation as the function of difference of the coordinates of the characteristic points enables analytical evaluation of uncertainty of coordinate easureent fully consistent with the GUM requireents [3]. The presented algoriths were tested and used to develop the special software. The software is fully consistent with the rules of geoetrical product specifications described in ISO 1101 [4].

12 16 W. Jakubiec, W. Płowucha References [1] W. JAKUBIEC: Analytical estiation of uncertainty of coordinate easureents of geoetric deviations. Models based on distance between point and straight line. Advances in Manufacturing Science and Technology, 33(2009)2, [2] A. GESSNER, R. STANIEK: Evaluation of accuracy and reproducibility of the optical easuring syste in cast achine tool body assessent. Advances in Manufacturing Science and Technology, 36(2012)1, [3] JCGM 100:2008 Evaluation of easureent data. Guide to the expression of uncertainty in easureent. [4] ISO 1101:2012 Geoetrical product specifications (GPS). Geoetrical tolerancing. Tolerances of for, orientation, location and run-out. Received in Noveber 2013