Tool Orientation Optimization Based on Spatial Tractrix Method for Five-Axis CNC Machining with Ball End Cutters

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)705–737
Journal / PublicationCommunications in Mathematics and Statistics
Issue number4
Online published15 Sept 2022
Publication statusPublished - Dec 2022


When milling part surfaces with a ball-end tool in 5-axis CNC machining, maintaining a constant cutting speed by keeping a fixed inclination angle between the tool axis and surface normal is crucial to ensure safe operation and achieve high quality of the machined surface. Under this constraint, the variation of tool orientation is expected to be “smoothest possible” to reduce the angular speed of the rotary axes for the efficient and robust machining. To address this issue, the spatial tractrix which is the extension of classic tractrix is presented to establish the geometry model of the tool orientation kinematics in the part coordinate system. The proposed model describes the relations between the tilt angle and the variation of ball-end tool orientation. Two spatial tractrix-based methods, synchronizing tractrix-based method and equilibrating tractrix-based method, are developed to minimize the variation of tool orientation by controlling the variation of tilt angle. These methods are used to plan the tool orientation on a part surface modeled by a bicubic spline surface. The performance evaluation carried by intense simulations demonstrates the equilibrating tractrix-based method provide the best results in most cases compared with the existing differential geometry-based methods such as the tractrix-based method and parallel transport method. The synchronizing tractrix-based method works well in some special cases.

Research Area(s)

  • Ball-end tool orientation, Five-axis machining, Spatial tractrix, Tool orientation optimization

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