#97 Curvature Evaluation of Faceted Models for Optimal Milling Direction

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1 #97 Curvature Evaluation of Faceted Models for Optimal Milling Direction Gandjar Kiswanto, Priadhana Edi Kreshna Department of Mechanical Engineering University of Indonesia Kampus Baru UI Depok 644 Phone : , Fax : gandjar_kiswanto@eng.ui.ac.id, gandjar.kiswanto@ui.edu Abstract. One of the most important things which can not be separated in the current milling processes are determination and development of tool paths. Tool paths are made in order to guide the cutting tool to cut the material into a shape that is similar to the model that has been designed. These tool paths are usually planned and then generated in a CAM (Computer Aided Manufacturing) system. However, current commercial CAM (Computer Aided Manufacturing) systems have limited functionality in milling strategy. They simply provide a user to select only single milling strategy (parallel, contour, radial, or spiral) to machine a whole area of a part to be machined. The Laboratory of Manufacturing Technology of Department of Mechanical Engineering University of Indonesia, has been intensively developing an advance CAMsystem based on faceted models which is not only enhance the tool path generation reliability but also milling strategy functionality. The previous result of the part investigations showed that the milling process has its optimum direction based on the global curvature of the surface area. The estimations of global curvature are conducted and compared using several methods to determine the shape of the surfaces. In this paper, we propose a combination of GaussBonnet Scheme, or sometimes called angle deficit, and Spherical Image methods to calculate the estimation of the curvature, then to determine the shape of a surface area. The surface shapes are divided into concave, convex, saddle, and flat. The milling direction and the tool path is then automatically determined and generated based on the shape and the curvature that has been estimated. The implementation in the CAMsystem shows the surfaces to be machined are segmented for its best milling direction based on its curvature characteristic. Keywords: global curvature, faceted models, milling processes. INTRODUCTION In an intelligent advanced CAMsystem, self determining of best milling direction on creating the tool paths based on the shapes of the surface model is crucial. In this case, the need for user to decide which milling direction or pattern is suitable for a segment and other direction for other segment/area of a surface to be machined can be completely eliminated. Therefore, the method to accurately estimate the shape of the surface is needed. In this work, milling direction is decided based on the estimation and evaluation of the local curvature value of the surface model. Shape of a surface model can be clearly seen using bare eyes. All the thing one has to do is visualizing the file which contains the information of the model. Unfortunately, for a computer, the shape could not easily be detected, though it does not mean it is impossible to do. Thus, some methods to estimate the curvature has been developed in order to calculate and do the evaluation, so that the shape at any location can be recognized. For a model which is represented by an equation, such as parametric model, or Non Rational BSpline Surface (NURBS), analytical curvature computation to find the shape at every point on the surface can be done easily. Only by applying the differential formula of the equation that represents the model, then the curvature is found. But the problem is, the model which is used as input for the CAM System being developed is 3D faceted models, in the form of triangular meshes. This model is composed by limited numbers of small facets in the shape of triangles which are related each other and have some information about their normal vectors and some discrete points as their vertices. Some methods to estimate the global curvature were known, Paraboloid Fitting, Circular Fitting, The Gauss Bonnet Scheme, The Watanabe and Belyaev Scheme, and

2 The Taubin Approach. The study that Surazhky et. al., 003, had done was comparing those 5 methods on some triangular meshes. On other study that had been done by Meek et. al, there are some analysis about Spherical Image method and an Angle Deficit algorithm, which was called GaussBonnet Scheme. In this research, the curvature estimation methods, Spherical Image and GaussBonnet Scheme, are combined and implemented to determine the shape of the model as the characteristic of curvature and then segment it based on the shape of surface model, where an area would be put in the same segment if it has the same curvature characteristics and adjacent location. This paper consists of several parts; Chapter contains the description about the methods which are used in the research and development as well as the division of input (3D faceted model) into several computation areas. The result and the visualization example, that makes one easier to understand, are shown in chapter 3. In the chapter 4, there are explanations about how the methods which have been implemented are combined in order to determine the shape of the surface accurately.. METHODS Approximations to the surface normal and Gaussian curvature of a smooth surface are often required when the surface is defined by a set of discrete points rather by a formula (Meek et al., 000). In this study, the used objects are 3D faceted models where they are composed by discrete points forming mesh of triangles. This approximation is only valid for smooth surface; a surface that is continue, is not coarse, and does not make angle in any area (Figure ). Figure : 3D faceted models that were used for curvature estimation tests. Surface Curvature Roughly, curvature is a deviation of an object or of space from a flat form and therefore from the rules of geometry codified by Euclid. At any point on the surface, one may consider surface curves passing through that point; locally such a curve lies in some plane intersecting the surface at that point (Kresk P et al., 00). Theorem. Meusnier s theorem : k C = k n cosθ () where : k c k n θ = curve curvature; curvature of the curve = normal curvature; the curvature of any curve lying in an intersection plane which contains the surface normal direction at the given point = the angle between the intersection plane and the surface normal. Principal curvatures are primary and secondary directions and magnitudes of curvature. A given direction in the tangent plane is chosen and θ is defined as the angle between this preferred direction and the intersection plane. For θ (0, π) the function kn = f(θ) has two extrema. These extrema are called the principal curvatures (Kresk P, et al).

3 Theorem. Euler s theorem : k n = k cos α + k sin α () where : k & k = principal curvatures k n = normal curvature α = the angle between the direction of k and desired direction Gaussian curvature (global curvature) is the product of principal curvatures k and k as follows: K = k k (3) And mean curvature is the average of principal curvatures: ( + ) k k H = (4). GaussBonnet Scheme GaussBonnet Scheme is usually called Angle Deficit Method, because this method uses the total of angle which is surrounding a vertex in an area, compared to the full circumference angle 360 o. If an area, or a curvature calculation zone, is convex or concave, then the surrounding angle is less then 360 o. Furthermore, if the area is flat, then the angle is +/ 360 o. On the other hand, if the area is rather wavy or saddle, or formed a hyperbolic shape, then the angle is more than 360 o. The total angle of the triangles on the vertex V (in figure 5) is angle deficit of the polyhedron which are formed from the related triangles, angle deficit = π Θi, i+ (6) Then total is the angle deficit in equation (6), compared with the area related to the vertex V in equation (5), π Θ i, i+ K = (7) S i, i+ 3.3 Spherical Image The estimation of the curvature value using Spherical Image method is conducted by creating a closed curve on the surface, surrounding a vertex which is being a chosen point to determine the value of the curvature. On the path that has been created, some points should be put (Figure 3), whereas the surreounding points are adjacent with the center point. Figure 3: Path of spherical image Vertices which compose the path and the center vertex will form a polyhedron that consists of triangles of the faceted model, just like the triangles that are used in GaussBonnet Scheme (Figure 4). Figure : Surface of angle deficit method Area of every triangle (which is represented by symbol S x,y in Figure ) can be partitioned into three equal parts, one corresponding to each of of its vertices, so that the total area of a path on the surface around V on the polyhedron is total area = S i, i+ (5) 3 where S i, i+ is the area of triangles, formed by vertices V, V i, dan V i+ (Meek, et al., 000). After the vertices for curvature estimation have been acquired, then the next step is creating normal vectors on the surface for every vertex, and that is center vertex, as well as the surrounding vertices. If the tails of the unit normals to the surface along the path are placed at the origin, then the heads of those unit normals are the vertices located on the top of the normals. If the vertices on the top are connected, then a path called spherical image of the vertices located on the surface is formed.

4 on the model but, it doesn t mean that the vertices are immediate neighbor each other (don t not lie in the same triangle). For the cleared one can be seen in figure 5 and 6. Figure 4: Surface of spherical image method The theorem states that the curvature at the given point on a surface (in figure 8, the vertex is characterized with O ) is the limit of the area of the spherical image of the path divided by the area of the path as the path shrinks around the point (Meek, et al). In Figure 4 above, a path that forms spherical image is a closed curve which surrounds a nonplanar polygon P P PnP. And then spherical image of this path is a polygon n n..n n n, where n i is unit normal of P i. The ratio of the area of all triangles n i n 0 n i+ over the area of all the triangles is an approximation to the total curvature at point O. Lninoni + K = (8) L Pi Po Pi +.3 Choosing Curvature Calculation Points A faceted model consists of several triangles, that makes it called triangular meshes, and the size of the triangles are different, up to the tolerance and the resolution that has been decided by the user. The more complex the model is, then the tolerance should be smaller to reach better resolution of the model, and that means the triangle would be smaller and smaller. Figure 5: Chosen vertices where the center vertex surrounded by 4 vertices Figure 6: Chosen vertices where the center vertex surrounded by 8 vertices The areas of the curvature estimation are divided based on radius of the tool as well. The divided area will become pixels which are the smallest segment to calculate curvatures. The center point of the calculation is the chosen vertex of the pixel, and it is surrounded by vertexes to make some triangles that have been made before (the modification triangle). The explanation picture can be seen in Figure 7. Very small triangles sometimes cause the accuracy of the curvature estimation reduced, since the numbers of the computer can handle are not unlimited. A computer stores numbers in the floating point form, and it has limitation on storing numbers. If the numbers which are being operated are in very small value, then the computer will round it, and makes an error which is usually called rounding error. Due to the limitation that the computer has, then another approach is developed, which use a reduced resolution triangles so called modification triangles. A modification triangle is a non real triangle which is made based on the tool radius, and the vertices of the triangle are real vertices

5 Table : The example result of spherical image method for the faceted model in Figure (A) bari s\ko lom Figure 7: Dividing the curvature estimation area 3. RESULTS The experiment is conducted to the several faceted models. Every model represents the shape which has been divided, concave, convex, saddle and flat. The experiment models can be seen in figure. The value of curvature estimation method is determined in every point of calculation. The value of curvature shows the shape of surface on the specific location In Figure 8, the result of the points which show value less than means the shape is concave, are colored by blue, whereas the result which value is more than (convex) is colored by red. If the value is +/ (the shape is flat or saddle) then the color would be white. The presentation by using colors are meant to make easier to understand what the value is trying to tell, since visual approach is often clearer than the series of numbers for a human. Using the Spherical Image method, the value of +/ shows that the shape of the surface is flat or saddle, while the value which is less than means that the surface is concave. If the value from the calculation is more than, then the surface which is located at that point is convex. The example of the result using the spherical image method for the model that has been shown in Figure (A) can be seen in Table. Every cell represents the location of every point of calculation of the model. The visualization by using some colors in every point of the model which represents the result of the calculation can be seen in Figure 8. Figure 8: Segmentation of models using Spherical Image

6 For GaussBonnet Method, the value of +/ 0 indicates that the shape of the surface is flat, and the negative value indicates that the shape is saddle. If the value is positif, then it means that the shape is convex or concave. The example of the result using GaussBonnet Scheme for the model that has been shown in Figure (A) can be seen in Table. Same as above, every cell represents the location of every point of calculation of the model. The visualization of the result by using colors can be seen in Figure 9. Table : The example result of GaussBonnet Scheme for the faceted model in Figure (A) baris\kolom In Figure 9, the red color represent that the result of the curvature calculation is positive, which means the shape of the surface is concave or convex. On the other hand, if the color is blue, then then the value of the curvature is negative and the shape is saddle. The shape of flat will be indicated by white color and the value is +/ Figure 9: Segmentation of models using GaussBonnet Segmentation will be done based on the result of the curvature estimation. For each point which has the same / similar curvature value and is adjacent to each other will be included in the same segment. An example is shown in the visualization of the result in Figure 8 and 9. In the figure, some points in an area which has the same color can be included in a segment. On the other hand, the neighboring area which is covered by the same color, but different color with its neighbor will be included in the other segment. From this result, there will be one or more segment(s) in a model which will become a hint to make a tool path and determine each segment s the best milling direction based on its curvature characteristic which has been known. 4. DISCUSSION A shape of the surface can be determined exactly by the combination of GaussBonnet and Spherical Image method whereas the value produced by those methods give a different meaning. GaussBonnet Scheme gives the value 0, if the shape is flat, and it gives value negative when the shape is estimated saddle. But when it shows the positive value, which means the shape can be convex or concave. It can not determine whether the shape is convex or concave. On the other hand, Spherical Image can determine whether the shape is concave or convex by giving the result > for convex, and < for convex. But Spherical Image can t determine if the shape is flat or saddle since it gives the same result. By combining the two of them, the shape can be determined by following several conditions,

7 . if the result of GaussBonnet Scheme is 0 and the result of Spherical image is, then the shape is flat,. if the result of GaussBonnet Scheme is negative and the result of Spherical image is, then the shape is saddle, 3. if the result of GaussBonnet Scheme is positive and the result of Spherical image is more than, then the shape is convex, 4. if the result of GaussBonnet Scheme is positive and the result of Spherical image is less than, then the shape is concave. The detailed and cleared one of the condition, can be seen in Table. Table 3: Combination of Spherical Image and Gauss Bonnet to determine the surface shape Surface Shape Estimation method GaussBonnet Spherical Image Flat +/ 0 +/ Sadel < 0 +/ Convex > 0 > Concave > 0 < 5. CONCLUSION Estimating curvature value is an important thing to do to create segmentation on the model and determining its best milling direction, which is one of the advantages of the CAMsystem that are being developed. In this study, several curvature estimation methods have been implemented, they are GaussBonnet Scheme and Spherical Image. By doing the curvature evaluation and determining surface of the model at every specific area, segmentation can be conducted based on an assumption that every area which has similar shape and is adjacent (has near location from the area that has similar shape) can be included in one segment. The shape of the surface can be accurately estimated by combining the GaussBonnet Scheme and Spherical Image. For the CAMsystem, the milling direction will automatically be determined based on the segmentation that has been done and the shape of every segment which has been estimated. ACKNOWLEDGEMENT The authors would like to thank Indonesia Toray Science Foundation (ITSF) for partial research fund support of this research work. Choi, B., Jerard, R.B. (998). Sculptured Surface Machining. Dordrecht : Kluwer Academic Publishers. Gatzke, T., Grimm, C. (003). Tech Report WUCSE004 9 : Improved Curvature Estimation on Triangular Meshes. Eurographics Symposium on Geometry Processing. Kilgard, M. (996). The OpenGL Utility Toolkit (GLUT) Progarmming Interface. Silicon Graphics. Kiswanto, G. (005). Pengembangan dan Pembuatan Sistem CAM (Computer Aided Manufacturing) yang Handal Berbasis Model Faset 3D untuk Pemesinan Multiaxis dengan Optimasi Orientasi Pahat dan Segmentasi Area dan Arah Pemesinan. Laporan Kemajuan RUT XII Tahap II. Kresk, P., Lukacs, G., & Martin, R.R. (00). Algorithm for Computing Curvatures from Range Data. Technical Report Prague Computer and Automation Research Institute, Budapest Cardiff University. Meek, D., & Walton, D. (000). On Surface Normal and Gaussian Curvature Approximations Given Data Sampled from A Smooth Surface. Computer Aided Geometric Design, (7), Neider, Jackie, Davis, & Tom. (997). OpenGL Programming Guide. AddisonWesley Publishing Company. Surazhky, T., Magid, E., Soldea, E., Elber, G., & Rivlin, E. (003). Comparison of Gaussian and Mean Curvatures Estimation Methods on Triangular Meshes. IEEE Transaction on Robotics and Automation, (), AUTHOR BIOGRAPHIES Gandjar Kiswanto is presently a researcher at the Department of Mechanical Engineering University of Indonesia. He has been a head of Laboratory of Manufacturing Technology since 004 and become a research head for many manufacturing research areas. His research areas include the robotics gesture, CAD/CAM and their applications to the entire domain of Mechanical Engineering. He has published a lot of papers in National and International Journal as well as in proceedings of several conferences he has participated in. Currently he is developing an intelligent advanced multi axis CAMsystem based on faceted models, which is being funded by BPPT and partially by Indonesia Toray Science Foundation (ITSF). REFERENCES Anton, H. (000). Introduction to Linear Algebra, volume. Jakarta : Interaksara.

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