Abstract
Zigzag pocket machining (or 2D-milling) plays an important role in the manufacturing industry. The objective is to minimize the number of tool retractions in the zigzag machining path for a given pocket (i.e., a planar domain). We give an optimal linear time dynamic programming algorithm for simply connected pockets, and a linear plus O(1)O(h) time optimal algorithm for pockets with h holes. If the dual graph of the zigzag line segment partition of the given pocket is a partial k-tree of bounded degree or a k-outerplanar graph, for a fixed k, we solve the problem optimally in time O(n logn). Finally, we propose a polynomial time algorithm for finding a machining path for a general pocket with h holes using at most OPT+εh retractions, where OPT is the smallest possible number of retractions and ε>0 is any constant.
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References
Arkin, E.M., Bender, M.A., Demaine, E.D., Fekete, S.P., Mitchell, J.S.B., Sethia, S.: Optimal Covering Tours with Turn Costs. SIAM Journal on Computing 35(3), 531–566 (2005)
Arkin, E.M., Fekete, S.P., Mitchell, J.S.B.: Approximation Algorithms for Lawn Mowing and Milling. Computational Geometry: Theory and Applications 17, 25–50 (2000)
Arkin, E.M., Held, M., Smith, C.L.: Optimization Problems Related to Zigzag Pocket Machining. Algorithmica 26(2), 197–236 (2000)
Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of Finding Embeddings in a k-tree. SIAM Journal on Algebraic Discrete Methods 8, 277–284 (1987)
Arnborg, S., Proskurowski, A.: Linear Time Algorithms for NP-hard Problems Restricted for Partial k-trees. Discrete Applied Mathematics 23, 11–24 (1989)
Arya, S., Cheng, S.-W., Mount, D.M.: Approximation Algorithm for Multiple-tool Milling. International Journal of Computational Geometry and Applications 11(3), 339–372 (2001)
Baker, B.: Approximation Algorithm for NP-complete Problems on Planar Graphs. Journal of the ACM 41(1), 153–180 (1994)
Bodlaender, H.L.: Some Classes of Graphs with Bounded Tree-width. Bulletin of the EATCS 36, 116–126 (1988)
Bruckner, L.K.: Geometric Algorithms for 2.5D Roughing Process of Sculptured Surfaces. In: Proc. Joint Anglo-Hungarian Seminar on Computer-Aided Geometric Design, October 1982, Budapest, Hungary (1982)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)
Guyder, M.K.: Automating the Optimization of 2.5 Axis Milling. In: Proc. Computer Applications in Production and Engineering. North-Holland, Amsterdam (1989)
Held, M.: A Geometry-Based Investigation of the Tool Path Generation for Zigzag Pocket Machining. Visual Computers 7(5-6), 296–308 (1991)
Held, M.: On the Computational Geometry of Pocket Machining. LNCS, vol. 500. Springer, Heidelberg (1991)
Held, M., Arkin, E.M.: Letter to Editor: An Algorithm for Reducing Tool Retractions in Zigzag Pocket Machining. Computer-Aided Design 32(10), 917–919 (2000)
Kim, B.K., Park, J.Y., Lee, H.C., Kim, D.S.: Determination of Cutting Direction for Minimization of Tool Retraction Length in Zigzag Pocket Machining. In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds.) ICCSA 2003. LNCS, vol. 2669, pp. 680–689. Springer, Heidelberg (2003)
Kramer, T.R.: Pocket Milling with Tool Engagement Detection. J. Manufacturing Systems 11(2), 114–123 (1992)
Lichtenstein, D.: Planar Satisfiability and Its Uses. SIAM Journal on Computing 11, 329–343 (1982)
Reed, B.: Finding Approximate Separators and Computing Tree-width Quickly. In: Proc. 24th Annual ACM Symp. on Theory of Computing, pp. 221–228 (1992)
Schwerdt, J., Smid, M., Janardan, R.: Computing an optimal hatching direction in layered manufacturing. International Journal of Computer Mathematics 79, 1067–1081 (2002)
Suh, Y.S., Lee, K.: Neural Network Modeling for Tool Path Planing of the Rough Cut in Complex Pocket Milling. J. Manufacturing Systems 15(5), 273–284 (1990)
Tang, K.: Geometric Optimization Algorithms in Manufacturing. Computer-Aided Design and Applications 2(6), 747–758 (2005)
Tang, K., Chou, S.-Y., Chen, L.-L.: An Algorithm for Reducing Tool Retractions in Zigzag Pocket Machining. Computer-Aided Design 30(2), 123–129 (1998)
Tang, K., Joneja, A.: Traversing the Machining Graph of a Pocket. Computer-Aided Design 35(11), 1023–1040 (2003)
Yao, Z.-Y., Gupta, S.K.: Cutter Path Generation for 2.5D Milling by Combining Multiple Different Cutter Path Patterns. Int. J. Prod. Res. 42(11), 2141–2161 (2001)
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Chen, D.Z., Fleischer, R., Li, J., Wang, H., Zhu, H. (2006). Traversing the Machining Graph. In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_22
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DOI: https://doi.org/10.1007/11841036_22
Publisher Name: Springer, Berlin, Heidelberg
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