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Analytical curvature-continuous dual-Bézier corner transition for f i ve-axis linear tool pathQingZhen Bi, Jing Shi, YuHan Wang, LiMin Zhu, Han DingnState Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240,People's Republic of Chinaa r t i c l e i n f oArticle history:Received 13 June 2014Received in revised form4 February 2015Accepted 5 February 2015Available online 7 February 2015Keywords:5-axis InterpolatorCurvature-continuousPath-smoothingBézier transitiona b s t r a c tA novel analytical f i ve-axis path-smoothing algorithm is developed for the high speed machining of alinear f i ve-axis tool path. Segment junctions of the linear tool path in the machine tool coordinatesystem, which are tangent-discontinuous points, are all blended by two transition cubic Bézier curves.One cubic Bézier curve is used to smooth the segment junction of the translational path, and the otherBézier curve is used to smooth the segment junction of the rotational path. The tangency and curvaturecontinuities are both guaranteed in the new path. The dual-Bézier transition algorithm has three ad-vantages: (1) Compared with the path-smoothing method in the workpiece coordinate system, the newdual-Bézier transition method directly and simultaneously smooths the machine tool axis trajectories ofboth translational path and rotational path. The feed speed and stability will both be improved becausethe tool path discontinuities are the most important source of feed f l uctuation. (2) The constraints ofapproximation error and the synchronization of parametrization of two smoothed curves, which are themost challenging problems in the smoothing of 5-axis tool path, are both considered. (3) The transitioncubic Bézier curve pair has an analytical solution and can be easily integrated in the real-time inter-polator. Computational examples and the cutting experiment of an impeller blade show that the novelpath-smoothing method has obvious advantages in both feed smoothness and cutting eff i ciency over theoriginal linear interpolator.& 2015 Elsevier Ltd. All rights reserved.1. Introduction5-axis NC machining is a widespread way to eff i ciently ma-chine sculptured shapes, such as dies, molds and aero-engineparts. Compared with the conventional 3-axis machining, 5-axismachining has many advantages in cutting eff i ciency. High mate-rial removal rate machining method, such as f l ank milling [1,2]and curvature matched machining [3,4], can be used by controllingthe tool orientation of the 5-axis machine tool. Another importantway to improve part quality and cutting eff i ciency is 5-axis highspeed machining (HSM). 5-axis HSM requires smooth feed speedsof each axis of the machine tool.The widespread linear tool path cannot satisfy the require-ments of smooth feed speed in high-speed and high-accuracymachining. Linear segments provide an easy way to approximatethe complex curve under a predef i ned tolerance. Many eff i cientalgorithms have been developed to generate linear 5-axis tool path[5]. The 5-axis linear tool path is usually described by a set ofcutter locations, as shown in Fig. 1. Each cutter location consists ofa position vector and a unit orientation vector. The position vectorrepresents the trajectory of the tool tip, and the orientation vectorrepresents the trajectory of the tool axis. The cutter locations inthe workpiece coordinate system (WCS) are usually transformedinto NC code by a post-process system. Most of the f i ve-axis NCcode is still def i ned by G01 blocks (Linear interpolation), thoughsome spline representations of the NC code have been supportedby some industrial CNCs. The NC unit f i rst converts discrete cutterlocations into a continuous trajectory, and the feedrate inter-polator plans the motion law under the kinematical constraints ofthe machine tool. The real-time servo commands (such as positionand velocity) of every axis are all generated based on the motionlaw. The tool is supposed to be moved in a straight line betweentwo successive positions of linear tool path. Both the tangency andcurvature discontinuities then appear at every segment junction. Ifthe cutter exactly passes the linear tool path, the tangential velo-city will be equal to zero at the linear segment junction because oflimited acceleration of the machine tool. The discontinuities inplenty of short linear segments will cause frequent start and stop.In a practical machining, the actual feed speed is generally lowerthan the programmed feed speed due to the physical restrictionsContents lists available at ScienceDirectjournal homepage: www.elsevier.com/locate/ijmactoolInternational Journal of Machine Tools & Manufacturehttp://dx.doi.org/10.1016/j.ijmachtools.2015.02.0020890-6955/& 2015 Elsevier Ltd. All rights reserved.n Corresponding author.E-mail address: mehanding@gmail.com (H. Ding).International Journal of Machine Tools & Manufacture 91 (2015) 96–108五轴直线刀具轨迹的解析曲率-连续双贝塞尔转角过渡