Constant scallop-height tool path generation for three-axis sculptured surface machining

COMPUTER-AIDED DESIGN Computer-Aided Design 34 2002) 64±654 www.elsevier.com/locate/cad Constant scallop-height tool path generation for three-axis sculptured surface machining Hsi-Yung Feng*, Huiwen Li Department of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 Received 18 September 2000; revised 15 May 2001; accepted 25 May 2001 Abstract This paper presents a new approach for the determination of ef cient tool paths in the machining of sculptured surfaces using 3-axis ballend milling. The objective is to keep the scallop height constant across the machined surface such that redundant tool paths are minimized. Unlike most previous studies on constant scallop-height machining, the present work determines the tool paths without resorting to the approximated 2D representations of the 3D cutting geometry. Two offset surfaces of the design surface, the scallop surface and the tool center surface, are employed to successively establish scallop curves on the scallop surface and cutter location tool paths for the design surface. The effectiveness of the present approach is demonstrated through the machining of a typical sculptured surface. The results indicate that constant scallop-height machining achieves the speci ed machining accuracy with fewer and shorter tool paths than the existing tool path generation approaches. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Tool path planning; Sculptured surface; Scallop height; Machining error; Ball-end milling 1. Introduction * Corresponding author. Tel.: 11-519-6612111; fax: 11-519-6613020. E-mail address: sfeng@eng.uwo.ca H.-Y. Feng). Sculptured surfaces are widely used in the design of complex products with aerodynamic features. These freeform surfaces are often produced by 3-axis computer numerical control CNC) machine tools using ball-end milling cutters. The utilization of CNC machines to manufacture complex surfaces has driven extensive research work, especially in the area of tool path generation [1]. Two criteria are generally used to evaluate the generated tool paths. One deals with the validity of the tool paths, and the other with their optimality [2]. Research work on optimal tool path generation has been aiming at achieving two con icting objectives: quality and ef ciency. This has led to the determination of optimal intervals between successive tool paths to optimize the two con icting objectives. A large tool path interval results in a rough surface while a small interval increases machining time, making the process inef cient. In general, the methods of generating tool paths for the machining of complex surfaces can be classi ed into three categories. The method of iso-parametric machining takes advantage of the parametric representations of the sculptured surfaces and is most widely used. By keeping one of the two parameters constant, the iso-parametric curves are formed and employed as the tool paths [3]. The generation of tool paths is straightforward with this method. Each uniform tool path interval in the parametric space between adjacent tool paths is constrained by the scallop-height requirement. The generated iso-parametric tool paths are, thus, often much denser in one surface region than others due to the non-uniform transformation between the parametric and Euclidean spaces [2]. This results in varying scallop-height distribution on the machined surface and non-optimal machining time. The method of iso-planar machining uses parallel plane± surface intersection curves as the tool paths [4,5]. This method is characterized with a uniform interval between adjacent tool paths in the Euclidean space. Each interval is determined according to the scallop-height requirement. It should be evident that iso-planar tool paths are not optimal in general and the choice of the optimal plane orientation for a given surface remains an open issue [6]. The tool path generation method to achieve constant scallop height was rst reported by Suresh and Yang [6]. Redundant machining in the iso-parametric and iso-planar methods was minimized since the scallop height was kept constant. The overall tool path length can be reduced signi cantly compared with the other two methods. 0010-4485/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S0010-4485 01)00136-1

648 H.-Y. Feng, H. Li / Computer-Aided Design 34 2002) 64±654 Fig. 1. Geometric elements of a machined surface. Improvements on this preliminary work were later proposed by Lin and Koren [] and Sarma and Dutta [8]. Sarma and Dutta [8] used swept sections along the tool paths to calculate the tool path intervals. It was pointed out that the previous work had simply assumed that the corresponding swept sections on adjacent tool paths were coplanar. This assumption caused inaccuracy in the calculation of tool path intervals and compromised the generation of optimal tool paths. Nevertheless, the derivation by Sarma and Dutta [8] assumed that the undetermined swept sections of the following tool path were in planes perpendicular to the tangent vectors of the common scallop curve. This was in effect equivalent to the assumption that the two corresponding swept sections on adjacent tool paths were in the same plane. A new approach to generate tool paths for constant scallop-height machining is proposed in the present work. A scallop surface and a tool center surface are de ned based on the design surface from the scallop-height requirement and the cutter radius, respectively. Tool paths are established by analyzing the surface geometry in 3D such that the questionable assumptions involved in transforming the 3D cutting geometry to 2D are not required. 2. Relevant geometric elements The geometric elements relevant to the derivation in the present work are de ned in this section Fig. 1). As in most computer-aided manufacturing literature, the cutter location CL) path represents the trajectory of the cutter center for a particular tool path. The cutter contact CC) path represents the tangential trajectory between the ball-end mill and the design surface. In the machining of a 3D surface, the CL paths are actually on an offset surface that is generated by offsetting the design surface in the surface normal direction by an amount equal to the cutter radius [9,10]. This offset surface is called the tool center surface. As the cutting tool moves along the tool path, a tool envelope surface is created [11]. This envelope surface can be de ned by sweeping a circle of the cutter radius along the CL path. The horizontal distance between two adjacent tool paths is referred to as the tool path interval or side step, which results in the scallop on the machined surface. The scallop curve is de ned as the 3D curve tracing the machined scallop. The scallop height represents the distance between the scallop curve and the design surface. For constant scallopheight machining, the scallop curves are on an offset surface scallop surface) of the design surface with the scallop height as the offset distance. In fact, the scallop curve is the common intersection curve of the two tool envelope surfaces of adjacent tool paths and the scallop surface. This exact representation of the scallop geometry is employed in the present work while previous studies use the approximated scallop height formulated in the plane perpendicular to one of the two adjacent tool paths. 3. Tool path determination Tool paths in the present work for constant scallop-height machining are determined by following two sequential steps similar to those proposed by Sarma and Dutta [8]. First, a given tool path and the scallop-height requirement are used to identify the corresponding scallop curve in the side step direction. Second, the known scallop curve is used to establish the next tool path. Fig. 2 shows two adjacent CL paths and the common scallop curve for a machined surface. For a point C 1 on CL path 1, the plane perpendicular to its tangent will intersect the scallop curve at S 1 and S 1 will lie on a circle of the cutter radius centered at C 1 in the perpendicular plane. The tangent of the scallop curve at S 1 is in general not parallel to the tangent of CL path 1 at C 1. Similarly, the plane perpendicular to the scallop curve tangent at S 1 will intersect the next CL path CL path 2) at C 2 and C 2 as well as C 1 ) will lie on a circle of the cutter radius centered at S 1 in the perpendicular plane. The tangent of CL path 2 at C 2 is not parallel to the tangent of the scallop curve at S 1, either.

H.-Y. Feng, H. Li / Computer-Aided Design 34 2002) 64±654 649 Fig. 2. Typical adjacent CL paths and the common scallop curve. The assumption in previous studies that these tangent vectors are parallel is evidently not justi ed and introduces errors in tool path generation for constant scallop-height machining. 3.1. Identifying the scallop curve from a given tool path As described previously, a tool envelope surface is created as the tool moves along a given CL path. The corresponding scallop curve is the intersection curve of the tool envelope surface and the scallop surface in the side step direction. For a common parametric surface P u; v, the scallop surface P SC u; v can be de ned as P SC u; v ˆP u; v 1 nh where h is the scallop height and n the unit normal vector from the surface P. The unit normal vector can be calculated from n ˆ Pu P v 2 up u P v u where P u represents the partial derivative of P with respect to the parameter u and P v with respect to v. In 3-axis surface machining with ball-end mills, the given CL path is an offset curve of the corresponding CC path on the surface P in the direction of the surface normal by a distance equal to the cutter radius R. This relationship can be expressed using the parameter t as 2 3 X CL t CL t ˆCC t 1 nr ˆ P u t ; v t Š 1 P u P v up u P v u R ˆ 1 6 Y CL t 4 5 Z CL t 3 The parametric equation of the swept tool envelope surface is derived by formulating the generating curve in a movable work coordinate system WCS) along the given CL path [12]. Fig. 3 shows the work coordinate system in relation to the CL path and the tool envelope surface. This local Cartesian coordinate system x w y w z w is created by setting: 1) a point on the CL path such as C 1 ) as the origin; 2) the surface normal n corresponding to C 1 as the z w axis; and 3) the tangent of the CL path at C 1 as the y w axis. The generating curve of the tool envelope surface is a circle of the cutter radius R. This circle can be represented using the parameter u as 2 3 R w u ˆ 6 4 Rcosu 0 Rsinu 5 4 Transforming Eq. 4) to the xed model coordinate system MCS) XYZ, the parametric equation of the tool envelope surface is obtained: R M t; u ˆwM RŠ R w u 1 C 1 where 2 3 2 3 t 11 t 12 t 13 X CL t w M RŠ ˆ6 4 t 21 t 22 t 23 5 C 1 ˆ 6 Y CL t 4 5 t 31 t 32 t 33 Z CL t 2 3 2 3 t 13 t 12 6 4 t 23 5 ˆ n ˆ k 6 w 4 t 22 5 ˆ CL0 t ucl 0 ˆ j w t u 2 6 4 t 33 t 11 t 21 t 31 Fig. 3. Work coordinate system along the CL path. 3 5 ˆ j w k w ˆ i w t 32 In the above formulation, i w, j w,andk w are unit vectors in the positive x w, y w,andz w directions, respectively. With Eqs. 1) and 5), the scallop curve can be identi ed by 5

650 H.-Y. Feng, H. Li / Computer-Aided Design 34 2002) 64±654 Fig. 4. Scallop point identi cation geometry. solving the following equation: P SC 2 R M ˆ 0 Eq. 6) is solved by a numerical procedure that identi es the corresponding scallop point for a given CL point such as C 1. The numerical routine evaluates points on the circular generating curve R w u. The scallop point is found when its distance to the design surface is equal to the scallop height Fig. 4). The calculation of the distance between a point on R w u and the design surface requires nding the corresponding closest) point on the surface. Let the point on R w u be denoted as R w;i. The square of the distance between R w;i and any point on the design surface P u; v is f u; v ˆuP u; v 2 R w;i u 2 By using the Newton±Raphson iterative method, the closest point on the design surface, which minimizes the function value of Eq. ), is identi ed with the following recursive formula: v k11 ˆ v k 2 F 21 v k f v k where " v ˆ u # " # " # ; f v ˆ fu g u; v ˆ ; v f v h u; v " # F 21 1 h v u; v 2g v u; v v ˆ g u h v 2 g v h u 2h u u; v g u u; v The corresponding CC point is chosen as the initial guess for the Newton±Raphson iteration since its position is close to the nal solution. In Eq. 8), f v and F v are the gradient and the Hessian matrix of f at v, respectively. Their expanded mathematical expressions suitable for numerical implementations are shown in detail in Ref. [13]. The bisection search algorithm [14] is employed to identify the scallop point. Only one quarter of the generating circle starting at the CC point and toward the side step direction is taken as the initial bracketing interval to reduce the amount of the calculation. The scallop point corresponding to the given CL point can thus be identi ed and used to form the scallop curve. 6 8 3.2. Establishing the next tool path from a known scallop curve The scallop curve identi ed in the previous subsection represents the intersection curve of the tool envelope surfaces of the current tool path and the next tool path. Determination of the next tool path from the scallop curve is based on the condition that the vector from the tool center to the corresponding scallop point is normal to the tangent vector of the scallop curve at the scallop point. This condition is derived from the geometric relation that the vector from the tool center to any point on the corresponding circular swept section of the tool envelope surface is the normal vector of the tool envelope surface at the point [12]. The surface normal vector is normal to the tangent plane at the point and, thus, normal to the tangent vector of any curve on the tool envelope surface passing through the point. This important geometric condition is employed in the following procedure to determine the tool center point on the next tool path corresponding to a given scallop point. 1. Create a plane perpendicular to the tangent of the scallop curve at the given scallop pointðthe tool center has to be on this plane. 2. Draw a circle in this plane with the scallop point as the center and R cutter radius) as the radiusðthe tool center has to be on this circle. 3. Determine the intersection point between the circle and the tool center surface. Since the tool center point has to be on both the circle and the tool center surface, the intersection point is the tool center point to be determined. To implement the above procedure, a movable work coordinate system similar to that in the previous subsection is created along the scallop curve. The scallop point is set as the origin, the surface normal through the scallop point as the z w axis, and the tangent of the scallop curve at the scallop point as the y w axis Fig. 5). The numerical procedure to determine the intersection point between the scallop circle and the tool center surface is similar to that of determining the scallop point in the previous subsection. The only difference is that the tool

H.-Y. Feng, H. Li / Computer-Aided Design 34 2002) 64±654 651 Fig. 5. Next CL path determination geometry. center point is located on the tool center surface instead of the scallop surface) and the distance from the tool center point to the design surface is the cutter radius R instead of the scallop height h). The next CL path can, thus, be established point by point from given points on the common scallop curve. 4. Piecewise linear NC tool path generation Current CNC machine tools approximate a 3D tool path curve with a series of linear tool movements linear interpolation). The piecewise linear numerical control NC) tool path is generated by connecting discrete points on the tool path curve with linear line segments. This approximation results in a machining error which is to be constrained by the speci ed tolerance. The discrete CL points generated by the numerical procedures described in the previous section are most often not the optimal discrete points on the CL paths to meet the tolerance requirement. A further numerical procedure is, thus, required to generate the piecewise linear NC tool paths so that the developed tool path generation approach is readily applicable in practice. The machining error e of a linear CL tool path segment can be calculated from e ˆ R 2 d where d is the extremum minimum or maximum) distance between the linear CL tool path segment and the design surface. The machining error e can, thus, be positive or negative, depending on the shape of the design surface convex or concave). For a point on the linear line segment, the distance to the design surface is determined by using the Newton±Raphson iterative method as discussed in Section 9 3.1 to nd the closest point on the surface. Huang and Oliver [5] proposed a similar formulation to calculate the machining error. The linear line segment was sampled at uniform intervals to estimate d. The Golden Section search algorithm [14] is employed in the present work for improved numerical accuracy and ef ciency. Fig. 6 shows a typical line segment of a piecewise linear CL tool path between two end points, A 0 and B 0, in which the minimum distance between the linear line segment and the design surface is to be determined. The normal projection curve represents the trajectory of the corresponding closest point on the design surface for each point on A 0 B 0. The Golden Section search algorithm rst calculates the distance to the design surface at two intermediate points, A 1 and B 1. The intermediate points are chosen in such a way that the reduction ratios A 0 B 1 =A 0 B 0 and A 1 B 0 =A 0 B 0 ) equal the Golden Section Constant r, which has a value of 0.61803. If the distance from A 1 to the design surface, d A 1 ), is smaller than that from B 1, d B 1 ), then the `minimizer' the point on the linear line segment with the minimum distance to the design surface) must lie in the range of A 0 ; B 1 Š. If not, the minimizer is located in the range of A 1 ; B 0 Š. Repeating the above process will progressively narrow the uncertainty range for the minimizer until it is boxed in with suf cient accuracy. It is evident that after N steps of range reduction using the Golden Section Constant, the uncertainty range is reduced by the factor of r N. The optimal discrete points for a particular CL path to meet the speci ed tolerance requirement are calculated from the series of CL points obtained in Section 3.2. Fig. illustrates the algorithm for the determination of the optimal discrete CL points. The algorithm starts by evaluating the machining error between the rst two consecutive CL points, C 0 and C 1. The result is then compared with the

652 H.-Y. Feng, H. Li / Computer-Aided Design 34 2002) 64±654 Fig. 6. Calculation geometry of machining error due to linear interpolation. speci ed tolerance. If it is less than the tolerance, the evaluating range is increased from C 0 ; C 1 Š to C 0 ; C 2 Š and further until the machining error in C 0 ; C n Š exceeds the tolerance. An approximated CL point, C 0, within C n21 ; C n Š, is then obtained by interpolation according to the ratios e n21 =tol and e n =tol, where tol represents the speci ed tolerance and e n21 and e n the machining errors in C 0 ; C n21 Š and C 0 ; C n Š, respectively. The algorithm continues with the same process to determine the subsequent optimal discrete points for the CL path. 5. Implementation and results The tool path generation approach for constant scallopheight machining described in the previous sections was applied to a typical sculptured surface. The numerical algorithms were implemented using `C' routines. The sample surface to be machined was the bulb holder cover of a halogen desk lamp, which is characterized with a convex shape Fig. 8). The machining parameters used for the generation of the tool paths for the sculptured surface were: diameter of the ball-end mill 2R): 25.4 mm; scallop height h): 0.5 mm; and tolerance tol): 0.05 mm. The present approach requires the rst tool path be speci- ed to initiate the tool path generation procedure. The boundary curve on the front was chosen as the initial CC tool path for the sample surface, although any curve on the surface could be chosen as the rst tool path. A set of uniformly sampled points on the rst CC tool path was used to determine the successive sets of scallop points and CL points for the following tool paths. The number of the sampled points was determined on the basis of the evaluation of the resulting chordal deviations for the rst CC tool path and could be adjusted according to the geometric complexity of the entire design surface. The CC tool paths generated by the present work are shown in Fig. 9. Figs. 10 and 11 show the respective CC paths generated by the iso-parametric and the iso-planar methods to meet the same scallop-height and tolerance requirements. It is evident from these gures that isoparametric tool paths tend to be denser in one surface region and iso-planar tool paths are simply parallel paths Fig.. Algorithm to determine the optimal discrete CL points. Fig. 8. The bulb holder cover of a halogen desk lamp.

H.-Y. Feng, H. Li / Computer-Aided Design 34 2002) 64±654 653 Table 1 Comparison of overall tool path lengths Iso-parametric Iso-planar Present method Tool path length mm) 1828 160 150 without considering the surface geometry. Table 1 compares the total lengths of the tool paths generated by the three methods to machine the bulb holder cover surface. The iso-parametric tool paths were 21% longer than the constant scallop-height tool paths generated by the present method and the iso-planar tool paths were % longer. 6. Conclusions A new tool path generation approach to achieve constant scallop height for 3-axis sculptured surface machining is presented in this paper. The advantage of constant scallop-height machining compared with iso-parametric and iso-planar machining methods has been demonstrated for a practical convex surface. The constant scallopheight tool paths are ±21% shorter and more ef cient than those generated by the other two methods. The improvement percentage is dependent upon the design surface geometry. It is expected that considerable improvement in tool path length will be observed if a surface with large varying curvature is to be machined. It should be noted, however, that tool paths for constant scallopheight machining may lead to the development of cusps and self-interactions, especially when the variation of surface curvature is large [8]. A tool path conditioning routine is required to smooth the cusps and self-interactions and to facilitate the practical implementation of the present approach. Machining errors due to linear interpolation are determined by evaluating the position of the piecewise linear tool envelope surfaces with respect to the design surface. Interpolated CL points are calculated from the exact CL points on a particular CL path to meet the tolerance requirement. An iterative numerical procedure based on identifying the corresponding CL points on the rst tool path can be implemented if exact rather than interpolated CL points are to be calculated and used in the NC code. Acknowledgements Fig. 10. Iso-parametric tool paths. This work was in part supported by Natural Sciences and Engineering Research Council of Canada and The University of Western Ontario VP Research) Research Grant. Fig. 9. Constant scallop-height tool paths. Fig. 11. Iso-planar tool paths.

654 H.-Y. Feng, H. Li / Computer-Aided Design 34 2002) 64±654 References [1] Dragomatz D, Mann S. A classi ed bibliography of literature on NC milling path generation. Computer-Aided Design 199;29 3):239±4. [2] Elber G, Cohen E. Toolpath generation for freeform surface models. Computer-Aided Design 1994;26 6):490±6. [3] Loney GC, Ozsoy TM. NC machining of free form surfaces. Computer-Aided Design 198;19 2):85±90. [4] Bobrow JE. NC machine tool path generation from CSG part representations. Computer-Aided Design 1985;1 2):69±6. [5] Huang Y, Oliver JH. Non-constant parameter NC tool path generation on sculptured surfaces. International Journal of Advanced Manufacturing Technology 1994;9:281±90. [6] Suresh K, Yang DCH. Constant scallop-height machining of freeform surfaces. ASME Journal of Engineering for Industry 1994; 116:253±9. [] Lin RS, Koren Y. Ef cient tool-path planning for machining freeform surfaces. ASME Journal of Engineering for Industry 1996; 118:20±8. [8] Sarma R, Dutta D. The geometry and generation of NC tool paths. ASME Journal of Mechanical Design 199;119:253±8. [9] Chen YJ, Ravani B. Offset surface generation and contouring in computer-aided design. ASME Journal of Mechanisms, Transmissions, and Automation in Design 198;109:133±42. [10] Kim KI, Kim K. A new machine strategy for sculptured surfaces using offset surface. International Journal of Production Research 1995; 33 6):1683±9. [11] Blackmore D, Leu MC. Analysis of swept volume via lie groups and differential equations. International Journal of Robotics Research 1992;11 6):516±3. [12] Hoschek J, Lasser D. Fundamentals of computer aided geometric design. Wellesley, MA: A K Peters, 1993. [13] Li H. Constant scallop-height tool path generation for sculptured surface machining. Master's Thesis, The University of Western Ontario, London, Ontario, Canada, 2000. [14] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recipes in C: the art of scienti c computing. 2nd ed. Cambridge, UK. Cambridge University Press, 1992. Hsi-Yung Steve) Feng is an assistant professor in the Department of Mechanical and Materials Engineering at The University of Western Ontario, London, Ontario, Canada. He received a BSin Mechanical Engineering from the National Taiwan University in 1986, and an MSand a PhD in Mechanical Engineering from The Ohio State University in 1990 and 1993, respectively. His current research interests include sculptured surface machining, geometric tolerance evaluation and precision 3D laser scanning. Huiwen Li received his MESc in Mechanical and Materials Engineering from The University of Western Ontario in 2000. His BSin Mechanical Engineering is from the University of Electronic Science and Technology of China. He is now an MSc student in Computer Science at The University of Western Ontario and will join IBM Canada Ltd in September 2001. His research interests include computer-aided manufacturing and objectoriented programming.