Abstract
In this paper, using the specifications of nodal points on a nonuniform rational B-spline (NURBS) curve of three degrees with respect to NURBS curve parameter and defining the coefficients for the velocity and acceleration vectors on these points, a new method is presented to design a tool path via C2 PH spline curves. Values of the velocity/acceleration vector coefficients corresponding to the nodal points on the original NURBS curve are computed by pattern search algorithm. To this end, the normal distance between the constructed C2 PH spline curve and its corresponding original NURBS curve is considered as the objective function. Using combination of the time-dependent feed rate interpolation in the acceleration/deceleration phase of the motion and the constant feed rate interpolation in the middle region of the motion, the position commands of the designed NURBS-based C2 PH spline curve are generated. Several improved NURBS-based C2 PH spline curve following tasks were implemented with pseudo-derivative feedback feed forward (PDFF) controller. The experimental and simulation results confirm that the devised interpolator with designed PDFF controller is not only feasible for machining the complicated tool path represented in the improved NURBS-based C2 PH spline form but also yields satisfactory contouring performance under variable feed rate.
Similar content being viewed by others
References
Cheng M-Y, Tsai M-C, Kuo JC (2002) Real-time NURBS command generators for CNC servo controllers. Int J Mach Tool Manuf 42(7):801–813
Cheng C-W, Tsai M-C (2004) Real-time variable feedrate NURBS curve interpolator for CNC machining. Int J Adv Manuf Technol 23(11–12):865–873
Su K-H, Cheng M-Y (2008) Contouring accuracy improvement using cross-coupled control and position error compensator. Int J Mach Tools Manuf 48:1444–1453
Cheng M-Y, Su K-H, Wang S-F (2009) Contour error reduction for free-form contour following tasks of biaxial motion control systems. Robot Comput Integr Manuf 25(2):323–333
Tsai M-C, Cheng CW (2003) A real-time predictor–corrector interpolator for CNC machining. Trans ASME J Manuf Sci Eng 125(3):449–460
Farouki RT, Tsai YF (2001) Exact Taylor series coefficients for variable-feedrate CNC curve interpolators. Comput Aided Geom Des 33(2):155–165
Farouki RT, Sakkalis T (1990) Pythagorean Hodographs. IBM J Res Dev 34(5):736–752
Farouki RT, Sagar S (1996) Real-time CNC interpolators for Pythagorean-hodograph curves. Comput Aided Geom Des 13(7):583–600
Sir Z, Juttler B (2005) Constructing acceleration continuous tool paths using Pythagorean hodograph curves. Mech Mach Theory 40:1258–1272
Farouki RT, Manjunathaiah J, Yuan GF (1997) G codes for the specification of Pythagorean-hodograph tool paths and associated feedrate functions on open-architecture CNC machines. Int J Mach Tool Manuf 39(1):123–142
Jahanpour J, Imani B-M (2008) Real-time PH curve CNC interpolators for high speed cornering. Int J Adv Manuf Technol 39(3–4):302–316
Imani BM, Ghandehariun A (2011) Real-time PH-based interpolation algorithm for high speed CNC machining. Int J Adv Manuf Technol 456:619–629
Tsai YF, Farouki RT, Feldman B (2001) Performance analysis of CNC interpolators for time-dependent feed rates along PH curves. Comput Aided Geom Des 18(3):245–265
Pelosi F, Sampoli ML, Farouki RT, Manni C (2007) A control polygon scheme for design of planar C2 PH quintic spline curves. Comput Aided Geom Des 24(1):28–52
Albrecht G, Farouki RT (1996) Construction of C2 Pythagorean hodograph interpolating splines by the homotopy method. Adv Comput Math 5:417–442
Farouki RT, Kuspa BK, Manni C, Sestini A (2001) Efficient solution of the complex quadratic tridiagonal system for C2 PH quintic splines. Numer Algor 27:35–60
Farouki RT, Neff CA (1995) Hermite interpolation by Pythagorean-hodograph quintics. Math Comp 64:1589–1609
Juttler B (2001) Hermite interpolation by Pythagorean hodograph curves of degree seven. Math Comput 70:1089–1111
Farouki RT, Manjunathaiah J, Jee S (1998) Design of rational cam profiles with Pythagorean-hodograph curves. Mech Mach Theory 33:669–682
Farouki R.T, Manni C, Pelosi F., Sampoli M.L (2010) Design of C2 spatial PH quintic spline curves by control polygons. Proceedings of the 7th international conference on curves and surfaces, p 253–269
Jahanpour J, Tsai M-C, Cheng M-Y (2010) High-speed contouring control with NURBS-based C2 PH spline curves. Int J Adv Manuf Technol 49(5):663–674
Piegl L, Tiller W (1997) The NURBS Books, 2nd edn. Heidelberg, Springer, Berlin
de Boor C (1972) On calculating with B-splines. J Approx Theory 6:50–62
Lewis RM, Torczon V (1999) Pattern search algorithms for bound constrained minimization. SIAM J Optim 9(4):1082–1099
http://www.mathworks.com Direct Search Toolbox, (2006) MathWorks, Inc.
Ellis G. (1999) PDFF: An evaluation of a velocity loop control method, PCIM-Europe
Ellis G (2004) Control system design guide. Academic, San Diego
Hsieh M-F, Yao W-S, Chiang C-R (2007) Modeling and synchronous control of a single-axis stage driven by dual mechanically-coupled parallel ball screws. Int J Adv Manuf Technol 34(9–11):933–943
Epureanu A, Teodor V (2006) On-line geometrical identification of reconfigurable machine tool using virtual machining. CISE, Barcelona, pp 14–18
Erkorkmaz K, Altintas Y (2001) High speed CNC system design. Part I: Jerk limited trajectory generation and quintic spline interpolation. Int J Mach Tool Manuf 41:1323–1345
Erkorkmaz K, Altintas Y (2005) Quintic spline interpolation with minimal feed fluctuation. ASME J of Manuf Sci Eng 127(2):339–349
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jahanpour, J., Ghadirifar, A. The improved NURBS-based C2 PH spline curve contour following task with PDFF controller. Int J Adv Manuf Technol 70, 995–1007 (2014). https://doi.org/10.1007/s00170-013-5332-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00170-013-5332-z