A realtime curvature-smooth interpolation scheme and motion planning for CNC machining of short line segments

Abstract G01 codes generated by CAM (Computer Aided Manufacturing) system are the most common form of tool path in CNC (Computer Numerical Control) machining. For the piecewise linear path, tangential and curvature discontinuities bring about large fluctuation of feedrate and acceleration, which produces vibration of machine tool. In recent studies, the methods for G 2 (curvature-continuous) tool-path smoothing and jerk-limiting feedrate scheduling were developed. However there still exist the deficiencies when these methods are employed in CNC machining. It is difficult to simultaneously ensure that the tool path is chord-error-constrained and G01-point-interpolated in real time. In addition, heavy computational load hinders realtime processing in CNC system. Recently the scholars experimentally found the potential of G 3 (curvature-smooth) trajectory and jerk-continuous motion in reducing the vibration of machinery. This work proposes a realtime tool-path smoothing algorithm, generating G 3 interpolative tool path composed by mixed linear and quartic Bezier segments. The purpose of the smoothing scheme is the simultaneous considerations of G 3 continuity, confined chord error, G01 points interpolated, and realtime performance. And the tool path generated is optimized in curvature variation energy (CVE) and analytical curvature extrema is available. To reduce the vibration, a high-efficient algorithm of jerk-continuous (JC) feedrate scheduling for G 3 tool path is provided. Finally, a realtime tool-path processing scheme is developed, including G 3 interpolation and motion planning functions. As shown in the simulation, the contour error, curvature of tool path, feedrate fluctuation and machining time are reduced compared with G 2 transition scheme. The experiment on a machine tool is conducted to demonstrate the advantages of the proposed algorithm in vibration reduction and surface quality, compared with G 2 transition scheme.

[1]  Xiao-Shan Gao,et al.  Time-optimal interpolation for five-axis CNC machining along parametric tool path based on linear programming , 2013 .

[2]  Vito Vitrih,et al.  On interpolation by Planar cubic G2 pythagorean-hodograph spline curves , 2010, Math. Comput..

[3]  A. Piazzi,et al.  G3-splines for the path planning of wheeled mobile robots , 2003, 2003 European Control Conference (ECC).

[4]  Xavier Beudaert,et al.  Feedrate interpolation with axis jerk constraints on 5-axis NURBS and G1 tool path , 2012 .

[5]  Hong-Tzong Yau,et al.  Real-time NURBS interpolator: application to short linear segments , 2009 .

[6]  H. Brussel,et al.  Real-Time NURBS Interpolator for Distributed Motion Control , 2002 .

[7]  Mei Zhang,et al.  Curve fitting and optimal interpolation on CNC machines based on quadratic B-splines , 2011, Science China Information Sciences.

[8]  Malcolm A. Sabin,et al.  High accuracy geometric Hermite interpolation , 1987, Comput. Aided Geom. Des..

[9]  Jamaludin Md. Ali,et al.  G3 Transition Curve Between Two Straight Lines , 2008, 2008 Fifth International Conference on Computer Graphics, Imaging and Visualisation.

[10]  Rida T. Farouki,et al.  Algorithms for time-optimal control of CNC machines along curved tool paths , 2005 .

[11]  Huashan Liu,et al.  Time-optimal and jerk-continuous trajectory planning for robot manipulators with kinematic constraints , 2013 .

[12]  Larry I. Schumaker.  ON SHAPE PRESERVING QUADRATIC SPLINE INTERPOLATION , 1983 .

[13]  Wendelin L. F. Degen.  High accurate rational approximation of parametric curves , 1993, Comput. Aided Geom. Des..

[14]  Yuwen Sun,et al.  A novel adaptive-feedrate interpolation method for NURBS tool path with drive constraints , 2014 .

[15]  Jingyan Dong,et al.  Feed-rate optimization with jerk constraints for generating minimum-time trajectories , 2007 .

[16]  Steven Pruess,et al.  Shape-preserving interpolation by parametric piecewise cubic polynomials , 1991, Comput. Aided Des..

[17]  An-Chen Lee,et al.  The feedrate scheduling of NURBS interpolator for CNC machine tools , 2011, Comput. Aided Des..

[18]  Jiing-Yih Lai,et al.  On the development of a parametric interpolator with confined chord error, feedrate, acceleration and jerk , 2008 .

[19]  Alessandro Bardine,et al.  A real-time configurable NURBS interpolator with bounded acceleration, jerk and chord error , 2012, Comput. Aided Des..

[20]  Rida T. Farouki,et al.  The elastic bending energy of Pythagorean-hodograph curves , 1996, Comput. Aided Geom. Des..

[21]  Muhammad Abbas,et al.  Monotonicity-preserving C2 rational cubic spline for monotone data , 2012, Appl. Math. Comput..

[22]  D. Renton,et al.  High speed servo control of multi-axis machine tools , 2000 .

[23]  Hong-Tzong Yau,et al.  Fast Bezier interpolator with real-time lookahead function for high-accuracy machining , 2007 .

[24]  Wei Yan,et al.  Interpolation of parametric CNC machining path under confined jounce , 2012 .

[25]  Li Bing Zhang,et al.  The transition algorithm based on parametric spline curve for high-speed machining of continuous short line segments , 2011 .

[26]  Meng-Shiun Tsai,et al.  Real-time NURBS interpolation using FPGA for high speed motion control , 2006, Comput. Aided Des..

[27]  Chongyang Deng,et al.  C-shaped G2 Hermite interpolation with circular precision based on cubic PH curve interpolation , 2012, Comput. Aided Des..

[28]  Robert J. Renka.  Shape-preserving interpolation by fair discrete G3 space curves , 2005, Comput. Aided Geom. Des..

[29]  Jingyan Dong,et al.  A Generalized Time-Optimal Bidirectional Scan Algorithm for Constrained Feed-Rate Optimization , 2006 .

[30]  Dereck S. Meek,et al.  G 2 blends of linear segments with cubics and Pythagorean-hodograph quintics , 2009, Int. J. Comput. Math..

[31]  Rene de Jesus Romero-Troncoso,et al.  Methodology for obtaining C3 continuity on tool trajectory featuring acceleration and jerk constraint on computer numerical control machine , 2011 .

[32]  H. Ding,et al.  A real-time look-ahead interpolation methodology with curvature-continuous B-spline transition scheme for CNC machining of short line segments , 2013 .

[33]  Chandrajit L. Bajaj,et al.  A-splines: local interpolation and approximation using Gk-continuous piecewise real algebraic curves , 1999, Comput. Aided Geom. Des..

[34]  Richard Bearee,et al.  Dissociated jerk-limited trajectory applied to time-varying vibration reduction , 2013 .

[35]  Lizheng Lu,et al.  An explicit method for G3 merging of two Bézier curves , 2014, J. Comput. Appl. Math..

[36]  Richard Bearee,et al.  New Damped-Jerk trajectory for vibration reduction , 2014 .

[37]  Lyle Noakes,et al.  Bézier curves and C2 interpolation in Riemannian manifolds , 2007, J. Approx. Theory.

[38]  Erich Hartmann,et al.  Parametric Gn blending of curves and surfaces , 2001, The Visual Computer.

[39]  Zhiwei Lin,et al.  A fine-interpolation-based parametric interpolation method with a novel real-time look-ahead algorithm , 2014, Comput. Aided Des..

[40]  Guozhao Wang,et al.  Geometric construction of energy-minimizing Béezier curves , 2011, Science China Information Sciences.

[41]  Ming Chen,et al.  Design of a real-time adaptive NURBS interpolator with axis acceleration limit , 2010 .

[42]  Jun-Hai Yong,et al.  Geometric Hermite curves with minimum strain energy , 2004, Comput. Aided Geom. Des..

[43]  Lian Fang.  G3 approximation of conic sections by quintic polynomial curves , 1999, Comput. Aided Geom. Des..

[44]  Ke Zhang,et al.  Time-optimal interpolation for CNC machining along curved tool pathes with confined chord error , 2013, J. Syst. Sci. Complex..

[45]  Manabu Sakai,et al.  Osculatory interpolation , 2001, Comput. Aided Geom. Des..

[46]  Zvi Shiller,et al.  Robust computation of path constrained time optimal motions , 1990, Proceedings., IEEE International Conference on Robotics and Automation.

[47]  Gudrun Albrecht,et al.  Convexity preserving interpolatory subdivision with conic precision , 2012, Appl. Math. Comput..

[48]  D. J. Walton,et al.  G2 Hermite interpolation with circular precision , 2010, Comput. Aided Des..

[49]  Kazuo Yamazaki,et al.  An accurate adaptive NURBS curve interpolator with real-time flexible acceleration/deceleration control , 2010 .

[50]  Gerald E. Farin,et al.  Geometric Hermite interpolation with circular precision , 2008, Comput. Aided Des..