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1 REVERSE ENGINEERING: AUTONOMOUS DIGITIZATION OF FREE-FORMED SURFACES ON A CNC COORDINATE MEASURING MACHINE C.-K. SONG# and S.-W. KIMt* (Received 15 December 1995) Abstract--We present an automatic algorithm of digitization on a CNC coordinate measuring machine for reverse engineering of an unknown free-formed surface. This algorithm supervises the movement of a contact type ball-tip probe in an autonomous manner in that the surface is discretely sampled to be fitted by a polyhedron surface model of triangular plane patches. Vertex points of the polyhedron are measured so that the surface is approximated with a minimum number of point data satisfying a specified geometric tolerance. An effective compensation algorithm of ball-tip radius is also proposed to obtain an actual surface profile from the digitized data Published by Elsevier Science Ltd 1. INTRODUCTION For the automatic machining of a free-formed surface, an effective tool of surface modeling is essential to generate the numeric data of the desired surface profile. Computer-aided modeling techniques are currently widely available, but this type of software modeling often turns out to be unproductive when the surface is of a complex shape designed from aesthetic viewpoints. In such a case, reverse engineering techniques can be usefully incorporated in that a l:l or reduced model of the surface is prefabricated with clay or wood and then the model is directly digitized to produce the actual surface profile. A CNC coordinate measuring machine equipped with a conventional contact type balltip probe may be used effectively for digitization. A simple method for performing digitization is to sample height data in a routine manner at fixed xy-coordinates predetermined either merely by equispaced rectangular grid points [1-3] or by some statistical distributions [4-7]. However, this passive sampling tends to result in a collection of data that is too dense with abundant points of no significance or too sparse to satisfy a specified geometric tolerance of approximation. This problem can be improved if sample points are selected adaptively during digitization by estimating the actual surface profile directly from the data being measured. We present an autonomous algorithm of digitization that can supervise the movement of a contact probe in an automatic manner against an unknown surface. The surface is discretely sampled and fitted using a polyhedron surface model of triangular plane patches. Vertex points of the polyhedron are adaptively annexed as digitization proceeds so that the polyhedron approximates the surface with a minimum number of data satisfying a specified geometric tolerance. An effective compensation algorithm of the ball-tip radius is also proposed to finally obtain an actual surface profile from the digitized data. 2. TRIANGULAR POLYHEDRAL APPROXIMATION Any surface with CLcontinuity may be approximated by a polyhedron of triangular plane patches. The actual surface profile is not known in the first stage of digitization. Hence the polyhedron is initially erected only with coarse NxM vertex points whose xycoordinates are selected to be equispaced such as is measured by moving the ball-tip probe to approach the surface in "down-touch-up" mode to avoid collision, as illustrated in Fig. 1. Each measured vertex point is then given an index number and stored in sequence:

2 Once digitization of the initial NxM vertex points is completed, triangular patches are to be generated to constitute the polyhedron. This procedure is done simply by connecting the three nearest neighboring points successively, as vertex points are evenly distributed in the xy-plane in this initial stage of digitization. The resulting triangular patches are then stored in the data structure shown in Fig. 2, which is especially suitable for triangle polyhedrons. This data structure is referred to as Lawson's triangle list [8, 9], which accommodates the index numbers of three vertices for each patch in counterclockwise order. Also included in the list are two groups of additional topological information on neighboring patches: three adjacent triangles possessing two common vertices, and another three proximity triangles having only one common vertex. With these additional data, all neighboring vertices of a certain vertex point can be easily identified and also the normal vector at a vertex point can be readily computed [10, I 1]. The next stage of digitization is to refine the initial polyhedron by annexing more vertex points for better approximation. Additional vertex points are selected by assessing whether each patch fits its corresponding surface within allowed tolerances. For this purpose, three vertex points of a triangular patch may be designated as v~, v2 and v3, respectively, omitting the index number i for convenience. Since the patch is of a planar triangular shape, its centroid is given by This centroid inevitably deviates from the actual surface that is in fact not planar. The deviation can be measured as shown in Fig. 3, the probe is positioned in the first place above the surface at the location of

4 where n is the normal vector of the patch that can be computed by and d is the allowance distance needed to avoid collision. The probe is then moved towards the surface along the negative direction of the normal vector n until contact is made between the probe and the surface. Let the contact point be represented by c', then the deviation is determined as If this deviation e exceeds the specified tolerance, i.e. e>t, then the centroid c is taken as a new vertex point. This assessment procedure for adding new vertices is performed on every patch so that the set of vertex points of Eqn (1) becomes augmented. With the augmented set of vertex points it is also necessary to generate new triangular patches to extend the Lawson's triangular list accordingly. New patches may be made simply by sectioning an existing patch into three smaller triangles if its centroid has been added as a new vertex point. However, as illustrated in Fig. 4(a), this easy method of refinement tends to result in abnormally slim triangles whose centroids may not effectively represent the corresponding parts of the surface. To avoid this problem, whole triangle connections including existing ones have to be reconfigured so that the resulting new polyhedron comprises as many near-equilateral triangles as possible. This is accomplished by adopting the criterion of minimum diagonal distance, which decides a locally better configuration by grouping four neighboring vertices, as described in Fig. 4(b). The above refinement process improves the polyhedron to represent the actual surface better with a smaller error of approximation. Therefore the refinement process of adding new vertices and regenerating subsequent patches has to be repeated until every patch satisfies the specified tolerance. The whole procedures of digitization is summarized in Fig PROBE RADIUS COMPENSATION

5 When digitization is completed, it is finally necessary to convert the digitized polyhedron into the actual surface profile since the polyhedron only represents the center trajectory of the bali-tip probe. This means that the measured data are offset by the amount of ball-tip radius from the surface as shown in Fig. 6 [12, 13]. It is therefore necessary to shift the measured point data in the normal direction by the amount of ball-tip radius to reconstruct the actual surface profile. For this purpose it may be assumed that at a sample point vi its neighboring surface may be approximated by a quadratic equation of

In order to decide the coefficients least-squares fitting may be adopted in which neighboring vertices of the adjacent and proximity triangles

6 In order to decide the coefficients least-squares fitting may be adopted in which neighboring vertices of the adjacent and proximity triangles contained in the Lawson's triangle list are used. Noting that the maximum number of vertices can be 12, i.e. m=12, least-squares fitting may be arranged as

It is often observed that the above shifting algorithm causes significant distortion between the compensated data and the original measured data, especially when the surface has a steep slope in the

10 It is often observed that the above shifting algorithm causes significant distortion between the compensated data and the original measured data, especially when the surface has a steep slope in the z-direction. This is because, as seen in Eqn (7), the error considered in least-squares fitting is measured in the z-direction and not in the normal direction of the surface. To avoid this undesirable problem, prior to performing least-squares fitting, it is necessary to rotate the measured data so that the normal direction of the surface coincides with the z-direction. This rotation can be implemented by

12 where n~, n>. and n: are the x, y and z components of the normal vector n of a triangular patch defined in Eqn (5). 4. EXPERIMENT AND DISCUSSION The digitization algorithm proposed in this study has been tested to verify its usefulness. The first test was implemented against a stamped metal surface, whose overall shape is a near half-sphere of 50 mm radius as shown in Fig. 7. The whole digitization procedure is illustrated in sequence in Fig. 8: Fig. 8(a) shows the polyhedron initially erected by using the equispaced xy-rectangular grid coordinates; Fig. 8(b) displays the probe path made to assess deviations of centroids of plane patches from the actual surface profile; Fig. 8(c) shows the refined polyhedron with newly added vertices where abnormal slim triangles are seen simply by sectioning existing patches; Fig. 8(d) shows new patches that are reconfigured by adopting the principle of minimum diagonal distance as explained in Fig. 4; Fig. 8(e) is the polyhedron further improved through iteration of refinement; and finally Fig. 8(f) is the completed result of digitization which satisfies the given tolerance

13 of approximation. As iteration of refinement proceeds the maximum error of approximation reduces. This tendency is shown in Fig. 9 where the total number of iterations performed was four in this case when the tolerance was given as 0.2 mm. Fig. 10 shows the distribution of the finally sampled point data in the xy-plane. It can be seen that samplings were concentrated on the spots of steep slopes and also on dents. Thus savings can be made in the total measuring time using a minimum number of significant data by adopting the proposed digitization algorithm as compared to conventional equispaced digitization. Another test result is illustrated in Figs 11 and 12, in which a more complex surface with convex and concave shapes is used. 5. CONCLUSIONS An automatic algorithm of digitization has been proposed which can be operated on a CNC coordinate measuring machine for reverse engineering of an unknown free-formed surface. This algorithm supervises the movement of a contact type ball-tip probe in an autonomous manner, in that the surface is discretely sampled to be fitted by a polyhedron surface model of triangular plane patches. Vertex points of the polyhedron are measured so that the surface is approximated with a minimum number of point data satisfying a specified geometric tolerance. An effective compensation algorithm of ball-tip radius has also been proposed to obtain an actual surface profile from digitized data. Finally, two test results have been discussed to verify that the proposed algorithm can save on the total digitization time and accuracy of approximation with a minimum number of significant data. REFERENCES [I] N. Duffle, J. Bollinger, R. Piper and M. Kroneberg, CAD directed inspection and error analysis using surface patch databa~s, Ann. CIRP 33(1), 347 (1984). 12] C.K. Song, S.W. Kim and J. Park, Automated measurement of three-dimensional profile of free-formed surfaces by using touch-trigger probes, J. Korean Soc. Mech. Engrs 17(2), 407 (1993). [3] S. Kawabe, F. Kimura and T. Sata, Generation of NC commands for sculptured surface machining from three-coordinate measuring data, Ann. CIRP 29(I), 369 (1980). [41 H.T. Yau and C.H. Menq, An automated dimensional inspection environment for manufactured parts using coordinate measuring machines, lnt. J. Prod. Res. 30(7), 1517 (1992). [5] C.H. Menq, H.T. Yau and G.Y. Lai, Automated precision measurement of surface profile in CAD-directed inspection, IEEE Trans. Robot Automn 8(2), 268 (1992). [6] T.C. Woo and R. Liang, Dimensional measurement of surfaces and their sampling. Comput.-Aid. De~. 25(4), 233 (1993). [7l H.J. Pahk, Y.H. Kim, Y.S. Hong and S.G. Kim, Development of computer-aided inspection system with CMM for integrated mold manufacturing, Ann. CIRP 42(I), 557 (1993). [8] C.L. Lawson, C' surface interpolation for ~attered data on a sphere, Rocky Mountain J. Math 14(I). 177 (1984). [91 B.K. Choi, H.Y. Shin, Y.I. Yoon and J.W. Lee, Triangulation of scattered data in 3D space, Comput.-Aid. Des. 20(5), 239 (1988). [10] J.P. Duncan and S.G. Mair, Sculptured Surfaces in Engineering and Medicine, Chap. I 1. Cambridge University Press, Cambridge, MA (1983). [I I I A.K. Cline and R.L. Renka, A storage-efficient method for construction of a Thiessen triangulation, Rocky Mountain J. Math. 14(I), 119 (1984). [12] L. Jie-Chi, N.A. Duffle and J.G. Bollinger, Two-dimensional tracing and measurement using touch trigger probes, Ann. CIRP 31(I), 415 (1982). [13] A.C. Lee, D.P. Chen and C.L. Lin, A CAD/CAM System from 3D coordinate measuring data. Int. J. Prod. Res. 28(12), 2353 (1990).

Incomplete two-manifold mesh-based tool path generation

Int J Adv Manuf Technol (2006) 27: 797 803 DOI 10.1007/s00170-004-2239-8 ORIGINAL ARTICLE Dong-Yoon Lee Su-Jin Kim Hyun-Chul Kim Sung-Gun Lee Min-Yang Yang Incomplete two-manifold mesh-based tool path