Geometry Optimisation for 2D Cutting: A Quadratic Programming Approach
Florian Sellmann*, Titus Haas**,†, Hop Nguyen*, Sascha Weikert**, and Konrad Wegener*
*Institute for Machine Tools and Manufacturing
Tannenstrasse 3, 8092 Zürich, Switzerland
A novel approach to geometry optimisation in the field of 2D cutting is presented in this paper. Set point generation inside of state of the art CNCs is divided in the preparation of the geometry and the feed rate generation. The feed rate generation is influenced by parametric derivatives of the given geometry. Due to this fact, the shaping of a B-Spline is carried out by optimisation of the weighted sum of parametric derivatives while the given manufacturing tolerances are maintained. For the sake of robustness, the arising optimisation problem is formulated as a quadratic program with linear constraints, one which can be solved with great efficiency by using an interior point method. In contrast to state of the art methods, the discrete formulation of the problem allows for a pointwise specification of the manufacturing tolerance. Depending on the manufacturing process, the given manufacturing tolerance is shared by different axes, which is shown for a 2D cutting geometry. An application example shows that the geometry optimisation leads to an increase in machining productivity over state of the art methods.
-  Y. Aminov, “Differential Geometry and Topology of Curves,” CRC Press, 2000.
-  R. Bartels, J. Beatty, and B. A. Barsky, “An introduction to Splines for use in Computer Graphics & Geometric Modeling,” Morgan Kaufmann Publishers, Inc, Los Altos, California 94022, 1987.
-  X. Beudaert, P.-Y. Pechard, and C. Tournier, “5-Axis tool path smoothing based on drive constraints,” International Journal of Machine Tools and Manufacture, Vol.51, No.12, pp. 958–965, 2011.
-  M. Bouard, V. Pateloup, and P. Armand, “Pocketing toolpath computation using an optimization method,” Computer-Aided Design, Vol.43, No.9, pp. 1099–1109, 2011.
-  Y. Boz, O. Demir, and I. Lazoglu, “Model Based Feedrate Scheduling for Free-Form Surface Machining,” International Journal of Automation Technology, Vol.4, pp. 273–283, 2010.
-  J. Bretschneider and T. Menzel, “Virtual Optimization of Machine Tools and Production Processes,” International Journal of Automation Technology, Vol.1, pp. 136–140, 2007.
-  T. Haas, “Diskrete Geometrieoptimierung und Interpolationsmethoden füur die Laserschneidbearbeitung,” Master’s thesis, ETH Züurich, Switzerland, 2012.
-  M. Hadorn, “Optimal Set Point Generation for Machine Tools under Nonlinear Constraints,” PhD thesis, ETH Züurich, Switzerland, 2007. newblock DISS. ETH Nr. 1132.
-  H. Kano, H. Fujioka, and C. F. Martin, “Optimal smoothing and interpolating splines with constraints,” Applied Mathematics and Computation, Vol.218, No.5, pp. 1831–1844, 2011.
-  K. Morishige and M. Kaneko, “Tool Path Generation for Five-Axis Controlled Machining with Consideration of Motion of Two Rotational Axes,” International Journal of Automation Technology, Vol.5, pp. 412–419, 2011.
-  M. H. Nguyen, S. Weikert, and K. Wegener, “Toolbox for control system analysis of machine tools,” In Proceedings of the 13th Mechatronic Forum International Conference, Linz, Austria, 2012.
-  H. Park and J.-H. Lee, “B-spline curve fitting based on adaptive curve refinement using dominant points,” Computer-Aided Design, Vol.39, No.6, pp. 439–451, 2007.
-  L. Piegl and W. Tiller, “The NURBS Book,” Springer, 1996.
-  L. Piegl and W. Tiller, “Least-Squares B-Spline Curve Approximation with Arbitary End Derivatives,” Engineering with Computers, Vol.16, pp. 109–116, 2000.
-  M. Steinlin, “Model based Feed-rate Optimization for Machine Tool Trajectories,” PhD thesis, ETH Züurich, Switzerland, 2012.
-  M. Weck and C. Brecher, “Werkzeugmaschinen,” Springer Berlin, Heidelberg, New York, 2005.
-  G. Yu, “Optimale Steuerung der Bewegung und der Geschwindigkeit füur das drei- und fünfachsige Fr”asen,” PhD thesis, ETH Züurich, Switzerland, 1996.
-  H. Zhao, L. Zhu, and H. Ding, “A real-time look-ahead interpolation methodology with curvature-continuous B-spline transition scheme for CNC machining of short line segments,” International Journal of Machine Tools and Manufacture, Vol.65, No.0, pp. 88–98, 2013.
This article is published under a Creative Commons Attribution-NoDerivatives 4.0 International License.