from discrete points to continuous spline paths - Mechanical ...

(a)(b)Figure 4: Two sets of synthetic data used in the experiments: (a) a sinusoidal surface;(b) a wavy surface.(a)(b)Figure 5: Two sets of real data used in the experiments: (a) a compound surface; (b) aface like surface.12

4020CL pointsCL pathProjected CC pointsz0-200 25 50 75 100x(a)(c)(e)0.150.100.050.00-0.00-0.05-0.10-0.150.150.100.050.00-0.00-0.05-0.10-0.15(b)(d)(f)Figure 7: CL paths and CL points generated from a wavy surface: (a) isometric view ofinput points and CL paths; (b) the CL path, interpolated CL points, and correspondingCC points; (c) normal curvatures along the forward direction; (d) top view of CL pointsand CL paths; (e) normal curvatures along the side direction; (f) top view of CL paths.16

6040CL pointsCL pathProjected CC pointsz2000 25 50 75 100 125x(a)(b)(c)(e)0.200.150.100.050.00-0.00-0.08-0.16-0.24-0.320.280.210.140.070.00-0.00-0.08-0.16-0.24-0.32(d)(f)Figure 8: CL paths and CL points generated from a compound surface: (a) isometricview of input points and CL paths; (b) the CL path, interpolated CL points, andcorresponding CC points; (c) normal curvatures along the forward direction; (d) topview of CL points and CL paths; (e) normal curvatures along the side direction; (f) topview of CL paths.17

240CL pointsCL pathProjected CC pointsz220200(a)(c)(e)0.150.100.050.00-0.00-0.05-0.10-0.150.150.100.050.00-0.00-0.05-0.10-0.151800 50 100 150x(b)(d)(f)Figure 9: CL paths and CL points generated from a face like surface: (a) isometricview of input points and CL paths; (b) the CL path, interpolated CL points, andcorresponding CC points; (c) normal curvatures along the forward direction; (d) topview of CL points and CL paths; (e) normal curvatures along the side direction; (f) topview of CL paths.18

Maximum deviations of CL paths5.2 x 10-354.84.4Sequence number of CL pathsMaximum deviations of CL paths5.2 x 10-354.84.4Sequence number of CL paths40 25 50 75 10040 25 50 75 100(a)(b)Figure 10: Deviations of CL paths generated from the sinusoidal surface: (a) CL pathsobtained by enforcing the tolerance only at midspan points; (b) CL paths obtained byour method.7 x 10-3 Sequence number of CL paths6 x 10-3 Sequence number of CL pathsMaximum deviations of CL paths654Maximum deviations of CL paths5430 25 50 75 100 12530 25 50 75 100 125(a)(b)Figure 11: Deviations of CL paths generated from the wavy surface: (a) CL pathsobtained by enforcing the tolerance only at midspan points; (b) CL paths obtained byour method.19

Maximum deviations of CL paths0.0150.010.005Maximum deviations of CL paths6 x 10-354Sequence number of CL paths00 50 100 150 200Sequence number of CL paths(a)30 50 100 150 200(b)Figure 12: Deviations of CL paths generated from the compound surface: (a) CL pathsobtained by enforcing the tolerance only at midspan points; (b) CL paths obtained byour method.Maximum deviations of CL paths7 x 10-3 Sequence number of CL paths654Maximum deviations of CL paths6 x 10-3 Sequence number of CL paths5430 50 100 150 200 250 30030 50 100 150 200 250 300(a)(b)Figure 13: Deviations of CL paths generated from the face like surface: (a) CL pathsobtained by enforcing the tolerance only at midspan points; (b) CL paths obtained byour method.20

Number of interpolation points300250200150100B-spline pathsLinear pathsNumber of interpolation points20015010050B-spline pathsLinear pathsNumber of interpolation points500 25 50 75 100Sequence number of CL paths300250200150100(a)B-spline pathsLinear paths500 50 100 150 200Sequence number of CL paths(c)Number of interpolation points250200150100500 25 50 75 100 125Sequence number of CL paths(b)B-spline pathsLinear paths0 50 100 150 200 250 300Sequence number of CL paths(d)Figure 14: Bounded by the same interpolation tolerance, the number of interpolationpoints of B-spline CL paths and the number of interpolation points of linear CL paths:Deviations of CL paths generated from a face like surface: (a) the sinusoidal surface;(b) the wavy surface; (c) the compound surface; (d) the face like surface.As Fig. 14 shows, the number of interpolation points of B-spline CL paths is smallerthan the number of interpolation points of linear CL paths, where all the paths arebounded by the same tolerance. As Table 5 shows, the total number of interpolationpoints of our B-spline paths is less than 25% of that of linear paths for the sinusoidalsurface, less than 31% of that of linear paths for the wavy surface, less than 48% of thatof linear paths for the compound surface, and less than 42% of that of linear paths forthe face like surface.6 CONCLUSIONThis paper presents an approach for generating B-spline CL paths directly from discretepoints. The method for calculating the single tool path is suitable for any PSSsrepresented by implicit forms. An ODE representation of NC paths is introduced toproduce CL points on the driving plane directly from the implicit MLS surface. By theprocesses of inserting and removing interpolation points, a compact set of interpolation21

Table 5: Number of interpolation points of CL pathsB-spline pathsExamples Linear paths n In n InReSinusoidal surface 26,637 6,581 6,563Wavy surface 19,993 6,270 6,185Compound surface 39,951 20,276 18,785Face like surface 42,424 18,564 17,467points is obtained for each CL path such that the number of interpolation points isreduced and the B-spline path interpolated satisfies a given tolerance. Our approach ismore accurate than conventional methods since it checks the point with the maximumdeviation in each curve segment of CL paths instead of the point at the midspan of eachcurve segment. The method for calculating the side step between consecutive paths issuitable for part surfaces in different forms provided that the closest distance from apoint to the part surface can be calculated. The step size is determined such that themaximum scallop height along a scallop curve coincides with a given tolerance.Computational experiments demonstrate that the approach produces high-quality B-spline paths from a variety of noisy point cloud data. The resultant paths are curvatureadaptive and satisfy specified tolerances.Our future work would be to experimentally verify this method on high-speed CNCmachines and to run the formulations of the method on parallel processes to improvethe efficiency.ACKNOWLEDGMENTSWe gratefully acknowledge the financial support from NSF grants (#0900597 and #1030347).References[1] Tikhon, M., Ko, T., Lee, S., and Sool Kim, H., 2004. “NURBS interpolator forconstant material removal rate in open NC machine tools”. International Journalof Machine Tools and Manufacture, 44(2-3), pp. 237–245.[2] Tsai, M., Nien, H., and Yau, H., 2008. “Development of an integrated look-aheaddynamics-based NURBS interpolator for high precision machinery”. Computer-Aided Design, 40(5), pp. 554–566.[3] Lin, A., and Liu, H., 1998. “Automatic generation of NC cutter path from massivedata points”. Computer-Aided Design, 30(1), pp. 77–90.22

[4] Park, S., and Chung, Y., 2003. “Tool-path generation from measured data”.Computer-Aided Design, 35(5), pp. 467–475.[5] Park, S., 2004. “Sculptured surface machining using triangular mesh slicing”.Computer-Aided Design, 36(3), pp. 279–288.[6] Feng, H., and Teng, Z., 2005. “Iso-planar piecewise linear NC tool path generationfrom discrete measured data points”. Computer-Aided Design, 37(1), pp. 55–64.[7] Chui, K., Chiu, W., and Yu, K., 2008. “Direct 5-axis tool-path generation frompoint cloud input using 3D biarc fitting”. Robotics and Computer-Integrated Manufacturing,24(2), pp. 270–286.[8] Zhang, D., Yang, P., and Qian, X., 2009. “Adaptive NC path generation frommassive point data with bounded error”. Journal of Manufacturing Science andEngineering, 131, p. 011001.[9] Tam, H., Xu, H., and Zhou, Z., 2002. “Iso-planar interpolation for the machiningof implicit surfaces”. Computer-Aided Design, 34(2), pp. 125–136.[10] Lartigue, C., Thiebaut, F., and Maekawa, T., 2001. “CNC tool path in terms ofB-spline curves”. Computer-Aided Design, 33(4), pp. 307–319.[11] Jin, Y., Wang, Y., Feng, J., and Yang, J. “Research on consecutive micro-lineinterpolation algorithm with local cubic B-spline fitting for high speed machining”.In Mechatronics and Automation (ICMA), 2010 International Conference on, IEEE,pp. 1675–1680.[12] Erkorkmaz, K., and Altintas, Y., 2001. “High speed CNC system design. Part I:jerk limited trajectory generation and quintic spline interpolation”. InternationalJournal of machine tools and manufacture, 41(9), pp. 1323–1345.[13] Vijayaraghavan, A., Sodemann, A., Hoover, A., Rhett Mayor, J., and Dornfeld,D., 2010. “Trajectory generation in high-speed, high-precision micromilling usingsubdivision curves”. International Journal of Machine Tools and Manufacture,50(4), pp. 394–403.[14] Amenta, N., and Kil, Y., 2004. “Defining point-set surfaces”. ACM Transactionson Graphics, 23(3), pp. 264–270.[15] Choi, B., and Jerard, R., 1998. Sculptured surface machining: theory and applications.Kluwer Academic.[16] Lasemi, A., D., X., and Gu, P., 2010. “Recent development in CNC machining offreeform surfaces: A state-of-art review”. Computer-Aided Design, 42(7), pp. 641–654.23

[17] Levin, D., 2003. “Mesh-independent surface interpolation”. Geometric Modelingfor Scientific Visualization, 3, pp. 37–49.[18] Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., and Silva, C., 2003.“Computing and rendering point set surfaces”. IEEE Transactions on Visualizationand Computer Graphics, 9(1), pp. 3–15.[19] Yang, P., and Qian, X., 2008. “Adaptive slicing of moving least squares surfaces:toward direct manufacturing of point set surfaces”. Journal of Computing andInformation Science in Engineering, 8, p. 031003.[20] Sarma, R., and Dutta, D., 1997. “The geometry and generation of NC tool paths”.Journal of Mechanical Design, 119, pp. 253–258.[21] Yang, P., and Qian, X., 2007. “Direct computing of surface curvatures for point-setsurfaces”. In Eurographics symposium on point-based graphics, pp. 29–36.VITAEYu Liu is a postdoctoral of Huazhong University of Science and Technology and aresearch associate of Illinois Institute of Technology. He received an undergraduatedegree in mechatronic engineering from Tianjin University, China in 2000 and a doctor’sdegree in the mechatronic engineering of Huazhong University of Science and Technology,China in 2008. His research interests include reverse engineering, CAD/CAM, anddigital manufacturing.Songtao Xia is a PhD student at the department of Mechanical, Materials and AerospaceEngineering at IIT. He received his BS degree in mechanical engineering from HuazhongUniversity of Science and Technology, China in 2008 and MS degree from The GeorgeWashington University in 2010. His current research interests focus on geometric processingof massive point-cloud data.Xiaoping Qian is an associate professor in the Department of Mechanical, Materialsand Aerospace Engineering at IIT. He received his Ph.D. in mechanical engineering fromthe University of Michigan, Ann Arbor in 2001. His research interests lie in the generalarea of computational design.24