KINEMATIC APPROXIMATION OF RULED SURFACES USING NURBS MOTIONS OF A CYLINDRICAL CUTTER

J. XiaDepartment of Mechanical EngineeringState University of New YorkStony Brook, New York 11794-2300xia@design.eng.sunysb.eduQ. J. GeDepartment of Mechanical EngineeringState University of New YorkStony Brook, New York 11794-2300qge@notes.cc.sunysb.eduAbstractThis paper deals with the problem of 5-axismachining ofruled surfaces. It is well known that a ruled surface may be effi-ciently machined with a cylindrical cutter using the side millingprocess. This paper studies the problem of planning the tool mo-tion for side milling of ruled surfaces as a kinematic approxima-tion problem. The goal is to develop a new method for synthesiz-ing a Non-UniformRational B-Spline(or NURBS) motion suchthat the swept surface of such a motion with a cylindrical cutterclosely approximates a given ruled surface. First a NURBS mo-tion is used to interpolate or approximate a set of discrete cutterlocations for the side milling of a given ruled surface. Then thedeBool control positions of the NURBS motion is fine-tuned tominimize the error between the swept surface of the cylindricalcutter under the NURBS motion and the desired ruled surface.In this way, the tool motion is represented by the set of deBoorcontrol positions and associated knot sequence as opposed to ahuge set of discrete cutter locations.1 IntroductionTo achieve the high precision and high productivity requiredfor new products manufactured by CNC machining processes,new real-timeCNC servo-controllershave been proposed anddeveloped (see, for example, Koren, 1997). A real-time5-axisinterpolator is capable of producing more accurate and smoothersurfaces than the off-lineinterpolator since the position and ori-entation of a cutter are accurately interpolated along the surfacein real-time. The conventional machine tools use off-linein-terpolation method, which adopts a constant orientation or lin-early interpolates two orientations between two approximatedend-points. This results in abrupt change in cutter orientationaround the end-pointsand thus produces roughness around theend-points. The off-linemethod produces also large change inacceleration and deceleration at each interval, which results inincreased surface non-uniformityand machining time. The cur-rent approach to the problem of tool path planning has focusedon obtaining a discrete representation of the tool path. In orderto realize the full potential of a real-timeservo-controller, an an-alytic description of cutter tool motion is needed to provide thecontroller with continuous tool locations as well as velocity andacceleration information.The goal of this research is to explore methods for planningtool paths of a CNC machine tool using Non-UniformRationalB-spline(or NURBS) motions. The focus of the present paperis on 5-axismilling of a ruled surface using the cylindrical cut-ter’s side edge, or side milling process. Compared with bottom-edge milling process, the side milling process is highly advan-tageous as it allows for significant reduction in machining time,and therefore the cost.Sincethe location (or the positionand orientation) of a cylin-drical cutter can be defined by the axis of the cylinder, the ruledsurface swept by the axis of the cylinder defines the cutter pathfor side milling of the ruled surface. Thus the problem of plan-ning the tool path for the 5-axissiding milling of a ruled sur-face may be studied as that of obtaining the “offset” surface ofthe original ruled surface. Considering a given ruled surface as atwo-parameterfamily of points, then its normal offset is a surface1 Copyright 2000 by ASME