recent development in cnc machining of freeform surfaces: a state

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Computer-Aided Design 42 (2010) 641–654

Contents lists available at ScienceDirect

Computer-Aided Design

journal homepage: www.elsevier.com/locate/cad

Review

Recent development in CNC machining of freeform surfaces:A state-of-the-art reviewAli Lasemi, Deyi Xue ∗, Peihua GuDepartment of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Alberta, Canada, T2N 1N4

a r t i c l e i n f o

Article history:Received 28 July 2009Accepted 6 April 2010

Keywords:Freeform surface5-axis CNC machiningTool pathTool orientationTool parameters

a b s t r a c t

Freeform surfaces, also called sculptured surfaces, have been widely used in various engineeringapplications. Freeform surfaces are primarily manufactured by CNC machining, especially 5-axis CNCmachining. Various methodologies and computer tools have been developed in the past to improveefficiency and quality of freeform surface machining. This paper aims at providing a state-of-the-art review on recent research development in CNC machining of freeform surfaces. This reviewprimarily focuses on three aspects in freeform surface machining: tool path generation, tool orientationidentification, and tool geometry selection. For each aspect, first concepts, requirements and fundamentalresearchmethods are briefly introduced. Themajor researchmethodologies developed in the past decadein each aspect are presented with details. Problems and future research directions are also discussed.

© 2010 Elsevier Ltd. All rights reserved.

Contents

1. Introduction........................................................................................................................................................................................................................6422. Tool path generation..........................................................................................................................................................................................................642

2.1. Overview ................................................................................................................................................................................................................6422.1.1. Path topology and parameters...............................................................................................................................................................6432.1.2. Requirements..........................................................................................................................................................................................6442.1.3. Traditional methods ...............................................................................................................................................................................644

2.2. Recent developments ............................................................................................................................................................................................6452.2.1. Curvature matched machining ..............................................................................................................................................................6452.2.2. Isophote based methods ........................................................................................................................................................................6452.2.3. Configuration space methods (C-space methods) ................................................................................................................................6462.2.4. Methods for polyhedral models and cloud of points ...........................................................................................................................6462.2.5. Region based tool path generation ........................................................................................................................................................6482.2.6. Compound surface machining ...............................................................................................................................................................648

3. Tool orientation identification ..........................................................................................................................................................................................6483.1. Overview ................................................................................................................................................................................................................648

3.1.1. Requirements..........................................................................................................................................................................................6483.1.2. Traditional methods ...............................................................................................................................................................................648

3.2. Recent development ..............................................................................................................................................................................................6493.2.1. C-space based tool orientation methods...............................................................................................................................................6493.2.2. RBM and AIM ..........................................................................................................................................................................................6493.2.3. Tool orientation smoothing ...................................................................................................................................................................6503.2.4. Other methods ........................................................................................................................................................................................650

4. Tool geometry selection ....................................................................................................................................................................................................6504.1. Overview ................................................................................................................................................................................................................6504.2. Recent development ..............................................................................................................................................................................................651

4.2.1. Tool selection for the roughing stage ....................................................................................................................................................6514.2.2. Tool selection for finishing stage ...........................................................................................................................................................6514.2.3. Tool selection for semi-finish and clean-up stages ..............................................................................................................................652

∗ Corresponding author. Tel.: +1 403 220 4168; fax: +1 403 282 8406.E-mail address: [email protected] (D. Xue).

0010-4485/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2010.04.002


642 A. Lasemi et al. / Computer-Aided Design 42 (2010) 641–654

5. Summary and discussions .................................................................................................................................................................................................6525.1. Summary ................................................................................................................................................................................................................6525.2. Discussions .............................................................................................................................................................................................................652Acknowledgement .............................................................................................................................................................................................................652References...........................................................................................................................................................................................................................652

1. Introduction

Freeform surfaces, also called sculptured surfaces, have beenwidely used in aerospace, automobile, consumer products andthe die/mold industry. Freeform surfaces are usually designed tomeet or improve an aesthetic and/or functional requirement. Thedefinitions of freeform or sculptured surfaces are intuitive ratherthan formal [1]. Often they are defined as surfaces containing oneor more non-planar non-quadratic surfaces generally representedby parametric and/or tessellated models.3- and 5-axis CNC machines have been most widely used in

machining freeform surfaces. Five motions can be continuouslyand simultaneously controlled in a 5-axis machine. Translationalmotions in the X, Y and Z directions and two rotational motionsare either applied to the tool holder or the machine table or both[2]. Compared to 3-axis machines, 5-axis machines can producecomplex surfaces with better quality and efficiency. 3 12

12 machines

have also been used due to a lower initial and operational cost.They have better stiffness compared to 5-axis machines becausethe rotary axes are locked during the cutting movement of the tool[3]. However unlike 5-axis machines, in 3 12

12 machines orientation

cannot be continuously adjusted during cutting process [2], thusrequiring longer machining time.Stages to complete freeform surface machining are usually

classified into rough, semi-finish, finish, clean-up and finalpolishing and treatment. In rough cutting, most of the material isremoved from the surface to generate an approximate shape of thesurface. Shoulders left from the roughing stage by large machinetools are removed in semi-finishing to yield a continuous offsetsurface for finishing [4]. At the finishing stage, the rough surfaceis transformed into the exact shape. Another method classified thestages as rough, finish and clean-up [5]. In this review, we use the3-stage scheme throughout the paper. During clean-upmachining,the uncut volumes that have not been machined in the finishingstage due to the use of larger cutters are removed. Thus clean-upregions play an important role in reducing the machining time ofcomplex surfaces [5].The concept of tolerance is used to measure quality of freeform

surface machining. An upper bound and a lower bound should beconsidered for a designed freeform surface. The former controls themaximum scallop height while the latter corresponds to gouging.For a surface to be machined within the designed tolerance, thescallop height should not exceed the maximum allowed toleranceand the surface should be gouge free.Gouges are classified into 3 categories: local, rear and global

gouges as shown in Fig. 1. A local gouge occurs when the effectiveradius of curvature of the tool in the cutter contact (CC) point islarger than that of the surface. A rear gouge happens when thebottom of the tool interferes with the surface in the points otherthan the CC point. Local and rear gouging happens in saddle andconcave surfaces [6]. Global gouging (or collision) results from theinterference between the part’s surface and the non-cutting areasof the tool such as the tool shaft or tool holder. In the presenceof a gouge, the surface accuracy and texture specifications are notsatisfied and/or serious damage may happen to the part’s surfaceand machine tool.Several techniques have been developed for gouging avoidance.

For local gouging, methods based on matching the curvature ofthe surface at the CC point and the effective tool curvature have

beenmostwidely used [7–13]. Methods for proper tool orientation[14–18] have mostly addressed rear gouge avoidance. Among themethods for the identification of global gouging (or collision),feasibility cone checking method [19] and configuration spacemethod [18,20–22] demonstrated effectiveness. A comprehensivereviewonmost recent gouge detection and avoidancemethods canbe found in [23].To machine a freeform surface with a 5-axis CNC machine,

cutter location (CL) points generated by tool path planning, toolorientation at each point, tool shape and size, spindle speeds,and the traveling velocity of the CL points need to be considered.This state-of-the-art review focuses on the three most importantissues: tool path generation, tool orientation identification, andtool geometry selection. These issues are related to each other [24].However in 5-axismachining, because of the complex nature of theproblem, it is difficult to solve the whole problem considering theoptimality of all relevant aspects such as path pattern, path length,path parameters, tool orientation, smooth orientation changes,tool size, scheduling of the feedrates as well as other objectives toavoid gouging [15]. As a result, many researchers try to solve theseproblems separately.It should be noted that three major reviews on freeform

machining have been observed by the authors [25–27] that coverthe research issues up to 1997. The review of the present paper,however, focuses on recent developments from 1997 up to 2008.The rest of this paper is organized as follows. The three main

aspects in freeform machining, including tool path generation,tool orientation identification, and tool geometry selection, arediscussed in Sections 2–4, respectively. For each aspect, firstconcepts, requirements and fundamental research methods arebriefly introduced. The major research methodologies developedin the past decade in each aspect are presented with details.Problems and future research directions are also briefly providedin each section.

2. Tool path generation

2.1. Overview

Tool path planning is a critical task in themachining of freeformsurfaces. Specific constraints are applied in path planning fordifferentmachining stages to achieve the optimal time and quality.For example in finish machining, the machining time should beminimized while the scallop height must be maintained belowthe specified level. An ideal tool path should generate uniformlydistributed scallops across the whole surface [28]. Smaller scallopsize does not necessarily mean a better tool path, since it isachieved at the cost of increased machining time. On the otherhand, the minimum machining time will be achieved when thescallop height is set to the maximum allowable measure. Toolpath planning is composed of 2 aspects: path topology and pathparameters. The former is defined by the pattern that the cuttermoves to produce the surface, and the latter is modeled by thetool side step between successive paths and the tool’s forwardstep in each path. Many researches have been carried out on theoptimization of the tool path in these two areas.Hence the tool path generation problem will be converted into

the following sub-problems:with a defined cutting tool, (1) specifypath pattern and the linking strategy (path direction), (2) specifypoints on the surface that the tool should track, and, (3) check toollocal and global interferences.


A. Lasemi et al. / Computer-Aided Design 42 (2010) 641–654 643

a b c

Fig. 1. Three types of gouging in 5-axis machining. (a) Local gouging. (b) Rear gouging. (c) Global gouging (collision).

2.1.1. Path topology and parametersEach tool path generation is conducted by selection of path

topology and path parameters [29]. For machining, the cutter tracesa sequence of CC points to approximate the freeform surface, andthe pattern of tracing is called tool path topology. Path topologyand the method of linking the generated paths directly affect themachining time [30]. A proper topology can result in theminimumpath length, the minimum number of tool retractions, and theflexibility of being locally refined to match the surface’s geometricproperties. Many researches have focused on minimizing the totalpath length and the number of tool retractions [30–33]. Formillingof freeform areas, contour parallel and direction parallel paths aremostwidely used. Strip normal and parallel paths are suitable in themachining of clean-up areas [5,34]. It has been shown that a stripparallel pattern generates tool paths with theminimumnumber ofretractions for clean-up machining [34].In a direction parallel path, path segments are parallel with

a predefined line (Fig. 2(a)). This line could be parallel with ornormal to the surface boundary or parallel with the axis of aspecified coordinate system. Proper selection of the reference linedirectly affects the generated path length [36]. Specifically theoptimum path direction will result in longer individual pathsand the minimum non-cutting movements of the cutting tool. Aspecific case of direction parallel path, zigzag path, is commonlyused in commercial CAM systems for roughing [8,37].A contour parallel path is constructed by the boundary curves

of the surface [38]. Each path is an offset of the boundaryof the surface. Voronoi diagram, pair-wise offsetting and pixelbased approaches are used to generate the 2D offsets in contourparallel paths [31,33,39]. These methods may be computationallyexpensive [40]. Thepaths in thepattern could be spirally connectedor each path could be independently an offset of the boundary(Fig. 2(b)). Kim and Choi [30] and El-Midany et al. [41] comparedthe machining time for direction- and contour-parallel pathsby considering the feedrate acceleration and deceleration. Inboth researches, the machining times for different direction- andcontour-parallel paths were compared by using a linear feedrateacceleration/deceleration model. El-Midany et al. [41] concludedthat the selection of the optimal path topology depends on thegeometry of the surface’s boundary and the cutting conditions.Kim and Choi [30] mentioned that although pure zigzag path (withsharp corners) results in a longer machining time compared tocontour parallel path, it is practically preferred to contour parallelin die and mold machining because of enhanced constant cuttingloads. An investigation of optimal path pattern selection for layer-by-layer rough cutting has been carried out by Li et al. [42]. Inthis method, the optimal path topology for each layer was selectedby fuzzy pattern analysis among a variety of topologies in a pathtopology database. So for each roughing process, more than onepattern could be selected to machine all layers based on thegeometry of these layers.Cox et al. [43] introduced the space filling curves (SFCs) for finish

tool path generation of freeform surfaces considering that parallel

or spiral line pattern is not the best way of traversing the toolin an area. They used truncated Palmer’s and Moore’s curves toavoid overlapping and crossing points in the generated tool paths.However this truncationmethod doubles the number of lines to betraversed and sharp edges still remain in the tool paths. Marshalland Griffiths [38] mentioned that extensive changes in pathdirection decrease a tool’s lifewhen using space filling curves. Theyused Hilbert’s curve to avoid this problem. The order of the curvecan be locally adjusted for the areas that need finer cuts. Howeverthis increases the number of lines in the whole refined areasby a factor of 2 that causes the efficiency problem. Furthermoresharp corners are rounded by Bezier curves, thus improving themachine’s dynamic behavior [37]. SFCs method is not popular in5-axis machining due to the path convolution, frequent changes inpath direction and the computational difficulties.SFCs have also been generated based on local optimal cutting

direction to produce a shorter tool path [35,44]. The process iscomposed of three steps: grid construction, SFCs generation, andcorrection of generated SFCs. The curvilinear space filling curvesintroduced by Anotaipaiboon and Makhanov [35] can improvelocal adaptation and decrease frequent changes in machiningdirections usually observed in SFCs. In this work, the space fillingcurve was developed based on the rectangular grid generatedfrom locally adaptable parametric curves in 2 directions (Fig. 2(c)).The rectangular grid is the result of an expensive numericaloptimization with tolerance constraints. Optimal grid connection(path linking) is modeled as a traveling salesman problem. Thegenerated tool path is locally adjustable, and the optimization triesto find the minimum tool path length, with the minimum toolretractions and sharp corners.Based on the strengths and limitations of different path patterns

investigated above, the direction parallel and contour parallelpaths are considered the most widely used ones due to theirsimplicity and adaptability in engineering applications. Althoughspace filling curves have found different applications such as inautomatic polishing [45] and shape representation, they are notpopular in 5-axismachining due to the path’s complexity, frequentchanges in cutting direction, and the machine tool’s dynamicproblems [46].In addition to the path pattern, the tool path should includepath

parameters which are defined by side step and forward step. Theresultingmachining error is closely related to these parameters. Asmentioned above, the cutter traces a sequence of CC points alongthe path pattern to machine the surface. The distance betweenconsecutive CC points is called forward step denoted by f in Fig. 2.The distance between 2 adjacent paths is called path interval (alsoside step or stepover) denoted byw in Fig. 2.Two methods are often used to calculate the forward step

parameters: circular arc approximation [8,47] andmaximumchordaldeviation [6,8,48]. Although in most of the tool path generationmethods, line segments are used to define the lengths of theforward steps, interpolation of the generated tool paths withpolynomial curves has also been carried out [49–52]. This will



1

0.8

0.6

0.4

0.2

00 0.2 0.4 0.6 0.8 1

u

v

a b c

Fig. 2. Different path topologies. (a) Direction parallel. (b) Contour parallel. (c) Adaptive Curvilinear space filling curve [35].

reduce machining time when the tool path is transformed tomachine axis movements by the NC unit, and decrease the sizeof memory required inside the CNC controller (which is largewhen the tool path is represented by line segments). Path intervalis determined by scallop height, cutter geometry and surfacegeometry information. The number of CC points is related to sideand forward steps. In general with the larger numbers of CC points,the surface will appearmore precise andwith smaller scallops. Buton the other hand, CPU time, memory usage and the machiningtime will increase with the large numbers of CC points.

2.1.2. RequirementsBased on the above discussions, a tool path is developed by

selecting the path’s topology and parameters considering theaccuracy and time constraints. Hence a generated tool path can beevaluated by the following 3 criteria [38,53]:

• Quality: The generated tool path must be gouge free and thescallop height should be within the specified tolerance. Theserequirements are controlled by path parameters.• Efficiency: Two types of efficiency measures need to beconsidered: (1) efficiency in simulation based on CPU timeand memory usage, and (2) efficiency in the actual machiningtime. Efficiency is achieved by the system through generatingand simulating different path topologies and parameters andidentifying the optimal one.• Robustness: We consider robustness as an adaptation capabilitywith the different surfaces and machines. A robust systemshould be able to work with multi-patch surfaces and theircontinuity conditions. It should work for both parametricmodels and tessellated models. A robust system shouldallow for selecting among a variety of tool paths withdifferent topologies and parameters. The machine’s kinematiclimitations should also be taken into consideration.

2.1.3. Traditional methodsTraditionally iso-parametric and iso-planar methods have

been used for the tool path generation [10,48,54–56]. The iso-parametric paths have first been addressed by Loney and Ozsoy[57]. In this method by keeping one of the two parametersconstant, CC points are generated along the other parameter of aparametric surface S(u, v). The iso-parametric method is popularfor freeform surface machining, because the surface data aredirectly utilized in the tool path generation [48]. However thepath interval (i.e., the constant parameter increment) is controlledby the scallop height constraint. Since a line in the parametricdomain is usually mapped nonlinearly to the Euclidean space, aconstant step in the parametric domain results in unequal path

intervals between adjacent paths [8]. This leads to non-uniformdistribution of scallops across the surface and affects machiningefficiency [53,58,59]. Besides it is difficult to generate tool pathsby the iso-parametric method in surfaces consisting of severaltrimmed surfaces [60].An iso-planar tool path is developed by intersecting the surface

with parallel planes in Cartesian space. The side step (i.e., thedistance between the parallel planes) is decided based on thescallop height constraint. This method is very robust and widelyused in commercial CAM systems [46]. Unlike the iso-parametricmethod, the iso-planar method can be used for compound andtrimmed surfaces [61] and triangular meshed models. Howeverproper selection of the intersecting planes in this method greatlyaffects the path length and machining time.It is evident that both the iso-parametric method and the iso-

planar method lead to conservative path intervals in an attemptto control the scallop height [10,58,61]. Only at certain pointsof the surface, scallop heights are close to the design constraint,and in other areas unnecessary high surface quality is achievedwhich leads to non-optimalmachining time. Iso-scallopmachiningwas introduced by Suresh and Yang [47] and Lin and Koren [62].The iso-scallop height method is an improved version of the iso-parametric and iso-planar methods. Based on this method, scallopheights are the same everywhere in the surface. Most of the recentmethods have focused on the development of iso-scallop toolpaths.In iso-scallop tool paths, a master path can be selected from

one of the surface boundary curves from which the other pathsare constructed [7,47,62,63]. Each CC point in the next cutter path(Pi+1,j) is calculated from the CC point of the current path (Pi,j),so that the scallop height remains the same (or the maximumallowable tolerance) all over the surface. The direction of side stepat each CC point is normal to the cutting direction at that CCpoint. The newly generated CC points form the new cutter pathwhich can be fitted to a cubic spline on which new cutter pointsare determined by forward stepping. This secondary path is thenused as a master path for generating another path. Although thismethod works for every kind of surface and significantly reducesthe number of CC points and the path length, it suffers from itscomputational complexity. Another issue is the accumulation oferrors due to the numerical fitting of cubic splines from themasterpath to the last one.With the same scallop height, the path interval in concave

areas was found to be larger than that of convex areas [62]. Byutilizing this result, Giri et al. [63] suggested that the direction ofthemaximum convex (or theminimum concave) curvature shouldbe selected as the initial cutter path for iso-scallop machining. Themachining potential field approach developed by Chiou and Lee [64]



Fig. 3. Coordinate systems for tool path generation.

is anothermethod for the selection of themaster cutter path basedon the maximum material removal rate in the first path. Kim andSarma [65,66] suggested the selection of the maximum kinematicperformance (i.e., fastest path according to machine’s structure)along with the maximum material removal rate for the first cutpath. They defined the machining time as a function of cuttingspeed, cutting direction and side step which should be minimizedin presence of a fewmachine and surface constraints. However thedeveloped method is not generic and needs structural informationof the specific machine tool [67].

2.2. Recent developments

Many new tool path generation techniques have been devel-oped in recent years to solve different problems in 5-axis ma-chining. Some of the typical problems include how to reducesimulation and actual machining time, machining of compoundsurfaces and non-parametric surfaces, innovative techniquesleading to decreasedmachining and investment cost, etc. Our clas-sification is based on an extensive study of the large body of litera-ture on tool path generation techniques. For most of the methods,path topology and parameters, gouge detection techniques, as wellas the improvement compared to the traditional methods, will bediscussed.

2.2.1. Curvature matched machiningCurvaturematchedmachining is a tool path generation and tool

positioningmethod based onmatching the curvatures between thecutting tool and the workpiece surface at the cutting point. Iso-scallop gouge free tool paths can be generated for parametric ortriangular meshed surfaces. Based on this method, the effectivecurvature of the tool should be smaller than or equal to that of thesurface at the CC point. As a result of this assumption, local gougingis automatically avoided in the generated tool path. Two methodshave been presented by Jensen [68] for curvature matching: theinstantaneous approach and the swept silhouette approach. Theinstantaneous approach analyzes the cutter’smovement at each CCpoint, while the silhouette approach considers the volume sweptby the cutter during movement between two CC points.In a 5-axis machine, the cutter can be tilted towards the

feed direction to match the surface curvature at CC point. Fig. 3shows tool coordinate system (xt yt zt), surface coordinate system(xs ys zs), local coordinate system (xL yL zL), and the global machinecoordinate system (xG yG zG). z axis is along the direction ofsurface normal at the CC point in the 3 coordinate systems. xand y in the surface and tool coordinate systems are along theprincipal directions, and yL is the feed direction at CC point. xLis the cross product of yL and zL. α is the tilt angle, and λ andθ are the angles between xL and the surface and tool principal

Fig. 4. Machining strip width for scallop height h. wL and wR denote left and rightstrip widths respectively.

direction respectively. When θ = 0, the cutting direction alignswith the tool’s principal direction and the tool has the largesteffective curvature. When λ and β are equal to zeros, tool andsurface principal directions overlap each other and with xL. In thissituation, the maximum machining strip width and path intervalare achievable. However to avoid gouges, especially to avoid globalgouges, it is not possible to set θ equal to zero. It should be notedthat throughout this paper, ω (tool rotation about surface normalzL) is considered as inclination angle. The fillet end tool surface atthe CC point in the tool’s coordinate system is given by [8]:

zt =12(κt1x2t + κt2y

2t ) (1)

where κt1 and κt2 are principal curvatures of the cutting tool. Byusing Eq. (1) and the transformation, the tool projection in theplane with the same normal to feed direction (i.e., xLzL) in thelocal coordinate system can be derived. The same procedure canbe applied for the part’s surface. The effective tool and surfaceshape are used to estimate the machining strip width and toolpath interval for iso-scallop tool paths (Fig. 4). It should be notedthat these calculations for matching the tool and surface projectedcurvatures are only required for concave and saddle areas. Forconvex or flat areas, the tilt angle (α) should be selected as zeroor near zero [69]. The projected shape for a generalized cutter hasbeen studied by Chiou and Lee [9,64]. The effective curvatures ofthe cutter and workpiece surfaces can be calculated and comparedin the local coordinate system.In a similar manner, Lee and Ji [70], and Lee [7] used the

effective cutting shape in the instantaneous cutting plane to findits intersection with the projected surface offset. The left and rightmachining strip widths found in each CC point are then usedto calculate the path interval for the points in adjacent paths.However in both methods, the numbers of CC points in all theconsecutive paths are the same. To overcome this problem, Lee[7] suggested selecting the longest surface of the boundary as theinitial path and checking and bisecting the path segments thatare out of the tolerance zone. Another method is to pass a splinethrough a set of all the newly generated points and find forwardsteps for the new curve [63]. Yoon et al. [12] developed an accuratemathematical solution to calculate the machining strip width andinteraction between the part’s surface and cutter at CC point basedon the curvature matching.Master cutter path for iso-scallopmachining could be identified

by curvature matching concept.Machining potential field approachdeveloped by Chiou and Lee [64] defines the path with themaximum average machining strip width as the starting pathamongst the potential paths across the surface. Maximumaveraged machining strip width is the total swept area of a cuttingtool divided by the total path length.

2.2.2. Isophote based methodsIsophotes are points on a surface with the same light intensities

[46]. An isophote is a region on the surface where the normal



n

nV

VV

Fig. 5. Isophotes in a freeform surface.

vectors n in all the points of the region are equal to a referencevector V (usually z axis) by a predefined tolerance called theinclination range (Fig. 5). In isophote partitioning, the surfaceis segmented into regions which have rather the same normal,and hence the same path interval is required to machine thepoints on the surface. This method has been mainly applied in 3-axis machining. The isophote angle θ for a point P(ui, vj) on theparametric surface with reference vector V is given by:

θ = cos−1

∂P∂u

∣∣ui,vj×

∂P∂v

∣∣ui,vj∣∣∣ ∂P∂u ∣∣ui,vj × ∂P

∂v

∣∣ui,vj

∣∣∣ • V

. (2)

In the method developed by Han and Yang [46], the free formsurface is first segmented into isophote regions. Then for toolpath generation after partitioning, the iso-inclination curves of theboundaries are projected to a plane perpendicular to the referencevector. Paths are generated in this plane and then projected backto the 3D surface. The path interval is considered to be the same allover the projected 2D plane and calculated as follows:

w = D cos(θi+1) (3)

where D is the approximation of path interval in 3D space forthe iso-inclination curve with the minimum inclination angle, andθi+1 is the upper boundary of the inclination range as shown inFig. 5. This ensures that the design tolerance is satisfied. Howeverthe scallops are not constant in the isophote anymore. To addressthis problem, an allowable variation for scallop heights can bespecified with which the inclination range is defined for thesurface’s partitioning.In the research by Han et al. [6], a freeform surface was

approximated by a ruled surface based on the isophote method.Iso-inclination curveswere used as generators for the ruled surfaceand also as the boundary curves for the tool path generation. Anadaptive iso-planarmethodwas used by Ding et al. [61] for the toolpath generation after isophote partitioning of freeform surfaces. Itshould be noted that in surfaces connected with C0 continuity, iso-inclination curves break at the edge of the junction and do not forma closed loop [61]. Decision on uniting or separating these regionsis not an obvious choice. In the global-local path planning strategyby Ding et al. [61], first a global iso-planar path is developed forall the surface regions. Then the path is locally adjusted for theregions with higher inclination ranges to satisfy the scallop heightconstraints. Yang and Han [71], Yin [72], and Yin and Jiang [73]have also reported applications of this method in generating toolpaths.Currently developed tool path generation methods based on

isophote partitioning are fairly accurate, simple, and computation-ally inexpensive. However further improvements still need to becarried out. More accurate calculation of path interval with iso-scallop point-by-point strategy for each isophote region could be

conducted. Since the shortest path length and the minimum num-ber of tool retractions are often used as criteria to evaluate cuttingpaths, the optimum path linking strategy could be investigated forlinking the generated paths for the regions.

2.2.3. Configuration space methods (C-space methods)Choi et al. [20] and Morishige et al. [21] employed the

configuration space (C-space) technique in tool path generation.The configuration space (C-space) of a rigid object with a certaindegrees of freedom is defined as the space of the parameterscorresponding to the degrees of freedom. Each point in the C-spacerepresents one configuration. By considering the design toleranceand mapping the obstacles corresponding to the local and globalgouges in the C-space, the problem of tool path planning can beconverted into the problem of planning the movements of a pointin the C-space [20,22]. The 2D C-space of the tool at the CC pointis constructed by only considering two orientation parametersof the tool (α and ω in Fig. 2) and removing the configurationswith collision from the available space [21]. Then the optimalparameters are selected in the C-space based on the maximummachining efficiency.The 3D C-space is constructed by adding the tool motion

parameters to the 2D C-space. Morishige et al. [22] and Lu et al.[18] developed different approaches for building 3D C-spaces. Inthe research by Morishige et al. [22], the aim is to find a 3Dcurve that describes the evolution of tool postures through CCpoints. The method by Lu et al. [18] first constructs the C-spaceby finding the upper and lower surface boundaries correspondingto scallop and gouge in each orientation set at the CC point. Thenthe optimal 3D set for a tool path is constructed by minimizingthe motion distance between 2 neighbor C-space sets. Otherobjectives can be used as well. However searching for all thefeasible configurations to identify the optimal tool trajectory inboth 2D and 3D C-spaces can lead to computational perplexingproblems. In this case, simplifying assumptions, such as neglectingthe axis acceleration/deceleration [18], larger search increments,avoiding detailed kinematic analysis, or removing the unnecessarysearch space such as limiting the tilt angle range, may reduce thecomputational time.

2.2.4. Methods for polyhedral models and cloud of pointsPolyhedral models (also called tessellated, faceted or triangular

meshmodels) have become popular in CAD/CAM systems. Becauseof the simplicity for data exchange and geometric computation[5,60], they are used as representationmodels for CAMand processplanning purposes. Thesemodels are created either from a cloud ofpoints [74,75] or fromaparametric surface [53]. Polyhedralmodelsfacilitate the tool path generation for surfaces with multipleand large number of patches [16]. Apart from this, sometimesmachining of non-parametric or non-implicit surfaces is inevitable,for example in cases that the design surface is created from a stylistprototype in an intermediate medium such as clay and convertedinto CAD data using contact or non-contact digitizers [55,76]. Fig. 6shows the different input models and procedure for tool pathgeneration.The tessellation error during the generation of a polyhedral

model may greatly affect the quality of the resulting tool paths[5,55,77]. Thus, grid density for sampling and triangulation shouldbe selected with respect to the required accuracy and timeefficiency. Estimation of differential geometric properties suchas normal vector and curvature may also add another source oferrors whichmakes it difficult to evaluate the generated tool pathsagainst the surface finish requirements.Based on this discussion, two newdifferent trends are emerging

in this area:



Fig. 6. Possible freeform inputs for tool path generation.

(a) Machining of a triangular polyhedral surface generated eitherfrom a parametric surface or a cloud of points

(b) Direct machining of cloud of points in reverse engineeringapplications to remove the time consuming and error pronetask of surface fitting or triangulation

Issues in 5-axis machining of tessellated models are slightlydifferent. It is not easy to generate iso-scallop tool paths as thegeometric characteristics of the surface are not readily available.Furthermore 5 different situations of gougemay occur between thetool and the edges, vertices and faces of the faceted model whichrequires specific detection techniques [18,77].The iso-planar method has been used for machining of poly-

hedral surfaces. Park [78,79] introduced algorithms for iso-planarcutter location (CL) based tool path generation for 3-axis machin-ing of polyhedral surfaces. This problem can be stated as findingthe intersections of lines and curves with an offset triangular meshmodel. The core of the method is to select proper segments ofthe line-surface intersection and removing the unwanted intersec-tion lines by the polygonal chain intersection algorithm. The curve-surface intersection problem can be approximately solved by theline intersection method. The line projection problem can be usedfor the generation of direction parallel path patterns and the curveintersection problemcan be used for the generation of contour par-allel and SFC path patterns.A few methods have been reported for the 5-axis machining of

polyhedral models with flat- and fillet end cutters. Xu et al. [76]developed a method for 5-axis machining. In this method, first a2D C-space for gouge and collision free tool orientation parametersis constructed considering the machine’s kinematic limits. Thetool path generation is based on the iso-planar method with theminimal orientation changes (also called orientation smoothing)between successive CC points.Lauwers et al. [69] developed a method for the calculation of

cutter tilt angle based on the curvature matching. At the CC pointnormal to instantaneous feed direction, a B-spline curve is passedthrough the intersection points of a cutting plane and the facetedmodel. This cutting plane contains the surface normal with itsnormal parallel with the feed direction. Using this intersectioncurve and the cutter projected shape, the cutter inclination anglecould be calculated to result in the maximum material removalrate. Then the rear gouge is detected and avoided by a combinationof tilt angle adjustment and tool retractions. Still the method

is iso-planar with variable surface finish across the surface. Luet al. [18] improved this method to calculate path intervals forthe generation of iso-scallop tool paths. However they selectedrather conservative path intervals due to the unknown inclinationangle at the stage of path calculation. The latter two techniques arecomplete in the sense that they consider all the 3 types of gougeavoidance. These methods also benefit from 2 additional degreesof freedom in 5-axis machining and path parameters are selectedin the way that the maximummachining efficiency is achieved.A method for the identification and machining of clean-up

regions for tessellated surfaces, called the contraction tool method,has been developed byRen et al. [5]. After identification of clean-upregions using gouging boundaries of the cutter on the tessellatedmodel, virtual intermediate cutters, ranging from the finishing toolto a pencil cut in size, are calculated to trace the clean-up band ina strip parallel path pattern. For variable clean-up bands, the pathinterval and intermediate cutters are selected to satisfy the scallopconstraint for the worst situation. Furthermore, the generation ofpencil-cut and fillet-cut tool paths for polyhedral models has beenintroduced by Kim et al. [80].For direct machining of cloud of points, the earlier methods

rely on arrangement of a point pattern of the initial point cloudinto a rectangular grid (Z-map model) and then selecting the linesegments [81] or a cubic spline passing through the line segmentsof pattern rows [82] as the tool path. In the method developedby Lin and Liu [81], rough and finish paths were developed byslice-by-slice and height correction methods (to avoid gouging)respectively. A study on recent Z-map model slicing methods canbe found in [78].Methods based on construction of a parametric surface from

measured data points such as those given by Yin [72], Yin andJiang [73] and Sun et al. [83] are advantageous in the sensethat machining of a parametric surface is straightforward and/orin some cases mechanical behaviors resulting from boundaryconformed iso-parametric paths are improved. However theseprocesses are less efficient because of the added parameterizationerrors and the inherent drawbacks of the iso-parametric methodspresented in Section 2.2.In a similar way, Chui et al. [84] developed a method for 5-axis

machining of cloud of points. CL data are approximated by 3D biarcfitting and the cutter’s orientation is calculated by the informationof triangulated grid of points. However in all themethods discussedabove, path parameters and the grid resolution have not beenestablished based on the design tolerance. To ensure the finishedquality, conservative path intervals and forward steps are oftenselected, thus resulting in significantly increased machining time.In the method developed by Feng and Teng [55], side step andforward step are adaptively calculated by construction of theprojected cutter location net (so called CL-net) using least squaresplane fitting. The concept is similar to previous methods, exceptthat machining error and maximum scallop height are consideredin the grid spacing and selection of CL points in both directionsof the grid. Although side and forward steps are adaptive to thegeometry of point cloud, unnecessary dense path intervals stillexist in the final tool path.Rough and finish tool paths have also been generated from

the measured data in the form of point sequence curves (PSC-map model) where each curve of the point sequence exists ina measuring plane of the physical object [40]. PSC-map modelis basically a result at the data acquisition stage and eliminatesthe need for further processing of the point cloud for tool pathgeneration.From the above study of the literature on machining of

polyhedral models and the cloud of points, it is evident thatmore research is still needed to improve machining qualityand efficiency for producing these surfaces. Especially very little



work has been carried out in 5-axis machining of a cloud ofpoints. The areas to be improved include the tessellation method,approximation of geometric characteristics, and exploration ofadvantages of 5-axis machining in tool path generation.

2.2.5. Region based tool path generationRegion based tool path generation methods are based on di-

viding the freeform surface into regions by identifying meaningfulfeatures. This process is also called segmentation, subdivision, par-titioning, or feature extraction.The isophote concept has been used for surface segmentation

in the method developed by Han and Yang [46] and Han et al.[6]. The idea is to segment surface into regions based on similarnormal vectors. Applications of isophote partitioning in tool pathgeneration have been reported in [61,72,73]. Elber [85] divided thesurface into 3-axis regions and 5-axis regions based on the surfacecurvatures to improve the machining productivity. Lee and Ji [70]used first and second fundamental forms of the surface [86] toextract different regions by classification of differential geometriccharacteristics of the surface. In the method by Chen et al. [2],surface segmentation was used to enhance 3 12

12 machining. First

the freeform surface is divided into a number of regions, andthen the part set-up is adjusted for machining each region by atilt/rotary table. Hence each region is machined by 3-axis CNCtool paths. Although machining time is increased by using severalsetups, the machining cost is lower than 5-axis machining dueto lower investment cost. Surface segmentation has been carriedout based on surface shape category and machinability by ahierarchical data structure. Shape categories are determined by thesigns of Gaussian andmean curvatures, and for machinability localand global gouging has been considered. To define the boundariesof the surface patches, a Voronoi diagram that is capable of sortingthe discrete points based on their similarities has been used.Tool paths can be generated for each region based on their

similarities. This may lead to reduced tool path computation timeand improved machine axis movement and machining efficiency.Machining cost will increase in region-by-region machining ifonly path length and machining time are considered as the costfactors. However the possibility of machining complex surfaceswith 3- and 3 12

12 -axis machines and decreased rate of orientation

changes during 5-axis machining could be considered as some ofthe advantages of region based machining.

2.2.6. Compound surface machiningCompound surfaces are used in many products due to the

complexity of the required shape. A compound surface maycontain several distinct surface patches such as Coon’s patches,ruled surface patches, Bezier patches, or NURBS patches connectedwith C0 or higher continuity [87]. Depending on the situation,patch-by-patch strategy or the whole surface machining approachis applied for tool path generation. The iso-parametric patch-by-patch method is useful when different tolerances are required fordifferent patches. It allows the selection of the optimal parametersfor each individual patch. The iso-planar method can be appliedto generate the tool path for the whole surface [48]. Comparedto patch-by-patch machining, this method may result in a shorterpath, but it takes longer to solve the nonlinear sets of equations forthe plane intersection.An issue in patch-by-patch machining is the proper linking of

the paths between patches. Selection of the path start and endpoints in each patch and connecting the paths of different patcheswith each other are two coupled problems and affect the totalmachining time [48,88]. Veeramani and Gau [89] addressed theproblem of optimal path linking in patch-by-patch machining. Theproblemwas transformed into traveling salesman problem by graphmodels.

Inter-patch gouging may occur for CL based tool path generationwhen the tool removes material from the neighboring patch whilecutting near the boundary of a patch. In this case, tool orientationadjustment along with proper trimming of the path (for pathsnormal to the boundary) and path interval adjustment (for pathsparallel with the boundary) can be utilized to avoid gouges [87].Generation of an iso-scallop path over the whole parametric

patches has been developed by Sarma and Dutta [90]. Specificallythey addressed identification of the CC points of the adjacent iso-scallop path in the situation where the current tool path passesthrough multiple parametric patches. The points of the adjacentpaths were interpolated by a Hermite curve in the domains of eachof the underlying patches. Then heuristics were employed for theconnection of Hermite curves in the parametric surface to eachother based on the type of discontinuity.Tool path generation for multi-patch trimmed parametric

surfaces is often required in aerospace and automotive industries.In these situations, the tool path should adapt to the boundariesof the surface for superior aerodynamic properties. Various re-parameterization methods have been introduced by Yang et al.[91,92] that can be used to generate a new parametric surfaceand hence iso-parametric paths that conform to the boundary.These methods have the potential to be used for multiple trimmedsurfaces. The geometric bisectionmethod given by Li [93]works bysuccessive partitioning of the surface patch by using its boundaries.This method has been used to generate boundary conformed toolpaths for compound surfaces.

3. Tool orientation identification

3.1. Overview

As discussed in Section 2, in 5-axis machining the tool axes canhave two rotational movements in addition to three translationalmovements. These two rotations are recognized by a tilt angleα about the xL axis and an inclination angle ω about the zL axis(α, ω) : −π/2 ≤ α ≤ π/2, 0 ≤ ω ≤ 2π , as shown in Fig. 3. Theserotations allow for the machining of areas where are inaccessiblefor 3-axis machining. Compared to ball end milling, by using fillet-or flat endmill, increased material removal rates are obtained [94]and lower spindle speeds are required. However the problem offinding the optimal tool orientation for 5-axis machining is muchmore complicated than the typical problems in 3-axis machining.The surface should be gouge free while the material removal rateis the maximal, and the orientation should change smoothly insuccessive CC points.

3.1.1. RequirementsThe main objective in tool orientation identification is to select

the orientation parameters at each point such that the minimummachining time (the maximum material removal rate) can beachieved while the generated surface is gouge-free and withinthe profile tolerance with even quality across the surface. Somemethods conduct extensive searching of the orientation area (C-space) to find the optimal smoothly varying gouge-free orientationwhile the others try to incrementally change the orientation to findthe feasible one. Both methods have advantages and limitationswhich will be discussed later.

3.1.2. Traditional methodsThree methods have been widely used in tool orientation

identification. The initial method for tool orientation was toconsider a fixed tilt angle (i.e., Sturz angle) in the plane containingthe feed direction and surface normal at CC point [14,95,96]. Thetilt angle was chosen between 3°–10° [95]. This method obviouslycan improve material removal rate compared to 3-axis machining.



a b

Fig. 7. C-space for orientation parameters. (a) Discretized 2-D orientation space (white area shows safe orientation space). (b) 3D C-space for one tool path [102].

However a trial and error process was required to select theoptimal angle. Smaller angles could result in rear gouging whilelarger angles reduced the material removal rate.Principal axis method (PAM) [14,96,97] borrowed the concept of

curvaturematchedmachining. Ideally the tool is tilted towards thefeed direction so that the minimum effective curvature of the toolis equal to the maximum effective curvature of the surface at theCC point. However practically this may not be possible becausethe cutting direction in iso-scallop machining cannot be alwaysthe same as the direction of the minimum principal curvature ofthe surface. Moreover to avoid rear and global gouging in 5-axismachining, the inclination angle ω may not be always kept equalto zero at each CC point. However it should be noted that PAM onlyavoids rear gouging by incrementally tilting the tool away fromthe ideal situation, thus it is suitable for open face freeform areas[14,97].Another method called multi point machining (MPM) [96,

98,99] searches for the second CC point while keeping thetool in contact at the first CC point. These two points areapproximately symmetrical about the direction of the minimumsurface curvature. This results in increased material removal rateand a controllable scallop shape. MPM has not been popular due tothe mathematical complexity in finding the second CC point andthe possibility of not converging to a solution [14].

3.2. Recent development

For optimization of tool orientation, many recent investigationsfocus on an improvement of the computational efficiency of findinga gouge-free orientation [14,17,23],whilemany otherswork on theimprovement of machining quality and efficiency [15,18,67,100–102]. Classification of thesemethods in this review is based on theircharacteristics and algorithms to achieve the required objectives.

3.2.1. C-space based tool orientation methodsAs discussed in Section 2, the C-space method can be utilized

for both tool path generation and tool orientation [18,20–22].In tool orientation identification, the C-space will be the tooltilting and inclination parameter areas excluding the gouge proneorientations (Fig. 7). The sampling approach and the resolutionof rotation parameters for sampling are critical for C-spacecomputation efficiency and accuracy. After construction of the C-space, selection of the smaller tilt angles as well as the minimumchanges of orientations in successive CC points can be objectivesof the optimization process. As a result the optimal solution liesclose to the boundaries of the C-space [15]. Morishige et al. [21]developed a collision free C-space construction method. Lee [95]

introduced the domain of admissible orientation for avoiding localand rear gouges. Jun et al. [15] introduced a tool orientationmethod based on local, rear and global gouge avoidance. Theoptimization goal is to maximize scallop height and minimize tiltand inclination angles. A complete gouge free 3D C-space methodwas developed by Lu et al. [18] based onminimizing the time traveldistance to smooth the tool orientation changes. Although C-spaceis effective to monitor all the possible orientations, exhaustivecomputationmay be required to reach the optimal solution inmostof the above mentioned methods.

3.2.2. RBM and AIMRolling ball method (RBM) and arc intersect method (AIM) are

two powerful tool orientation techniques for local and rear gougeavoidance. Local gouging is avoided by the curvature matchingconcept. RBM takes into account the approximate curvature ofthe surface in the vicinity of the CC point. Generated tool pathsare free of rear gouges. A rolling sphere is constructed based onthe characteristics of the surface under the arbitrary tool shadow[14]. This shadow, called the shadow checking area, is divided intoconcentric grid of points and the pseudo radius of curvature iscalculated at each grid point under the tool shadow as shown inFig. 8. Calculation of the pseudo radius of curvature is similar to thecalculation of an osculating plane for a point on 3D curves [103].The tool should be positioned in the sphere constructed by themost concave radius of curvature such that it makes a circular lineof contact with the sphere.RBM only requires the surface normal at the CC point and

the positions of grid points. However the equation of the surfaceis required to calculate the shadow grid area and points. Thisproblem was addressed by graphic-assisted RBM [104] where thecomputer’s graphic hardwarewas used to enhance tool orientationcomputation for polyhedral models. In RBM the shadow checkingarea beneath the tool is oversized which may result in tilt angleslarger than required and over computation.In AIM the same shadow checking area and grid points in RBM

are used, however instead of constructing rolling spheres for everygrid point at each CC point, the surface under the tool shadow isrotated to intersect the surface [3,16]. The smallest resulting angle(equivalent to the largest tilt angle) is selected as the gouge freeorientation. It should be noted that as a single independent toolorientationmethod, AIMandRBMare only applicable tomachiningof open faced freeform surfaces (which are assumed to be collisionfree). The methods can be used for both multi-patch triangularand parametric surfaces; however it is more complicated to checkthe points of adjacent patches. Like every other method used forpolyhedral models, their accuracy is greatly related to the grid size



Fig. 8. Shadow checking area and pseudo radius of curvature for a grid point.

of triangulation. The computational efficiency of AIM is reported tobe low due to the calculation of exact rotation angle at every gridpoint [17,105].

3.2.3. Tool orientation smoothingDramatic changes in tool orientation from point to point may

increase machining time [76] and decrease its quality by leavingtool marks on the machined surface [67]. As a result recentlyconsiderable researches have focused on development of methodsfor smoothly varying tool orientation. Jun et al. [15] developed atool orientation smoothing method by minimizing the C-distancebetween successive tool orientations. The forward scheme for C-distance calculation between (αi−1, ωi−1) and (αi, ωi) is calculatedas follows:

(C_distance)i = Min(√(αi−1 − αi)2 + (ωi−1 − ωi)2). (4)

Both Morishige et al. [22] and Jun et al. [15] used forwardand backward smoothing to reach the optimal solution. Lauwersand Lefebvre [106] applied a two-step iterative procedure forsmoothing the variations of tilt angle. Tilt angles are smoothedby removal of one or two intermediate points that form a valley,while gouging is avoided by not allowing the tilt angle to decreaseduring the smoothing process. In the method by Ho et al. [100],first critical areas were manually assigned orientations. This cangreatly improve the complexity and reduce computational time.Then smooth tool orientations were generated by using a vectorinterpolation algorithm called quaternion interpolation. Furthercollision checking was conducted after interpolation and theoverall process was iterative. Wang and Tang [102] used theconcept of discretized visibility map (Vmap), whichwas introducedby Balasubramaniam et al. [107], to identify the set of validorientations by inspecting the valid area with all the gougingconstraints. Then a feasibility map (Fmap) was defined at each CCpoint. The first Fmap is a single point in Vmap(CC1). Then thesuccessive Fmaps for i > 1 are defined recursively as follows:

Fmap(CCi) = Vmap(CCi) ∩ Fmap(CCi−1). (5)

Please note that the Fmap(CCi) for i > 1 spreads in the discreteorientation domain and does not remain a single point. Then usingthe combination of Fmaps and an angular velocity change limitoperator, they solved the problem of smooth orientation in aforward scheme. Castagnetti et al. [67] first developed the domainof admissible orientation [95] in the part coordinate system andtransformed it to the machine’s coordinate system. Then smoothlyvarying orientationswere generated byminimizing twomeasures:(1) the angular difference between two successive CC points, and,(2) the curvature of tool axis orientation evolution. Their resultsshowed that the optimization of tool orientation in the machine’scoordinate system could reduce machining time. The iso-conicpartitioning method developed by Wang and Tang [23] integratedoptimal path generation with tool orientation smoothing. As canbe seen these methods are similar in nature. The main objective intool orientation smoothing is to limit the solution bound to reducethe variation level of tool orientation in successive points and alsooverhaul the large body of required computation.

3.2.4. Other methodsHosseinkhani et al. [17] developed a method called the

penetration–elimination method (PEM) for rear gouging avoidance.Similar as RBM and AIM, the PEM is only applicable to open facecollision-free areas. A new concept for gouge quantification wasintroduced based on radial and axial depths of gouge. If a point isinside the tool area, it is assigned a negative radial depth of gouge.The gouging intensity function is defined as the multiplication ofradial and axial depths of gouge. Hence, a point in the gouge areahas a negative gouge intensity value. The penetration–eliminationis conducted by incremental checking of the tool shadow areaand adjusting the tilt angle, while each time the points calculatedwith positive gouge intensity values are deleted from the checkingseed points. Together with an optimized root finding method,this method greatly reduces the calculation time by dynamicallyremoving the safe areas out of the calculation loop.Optimization of tool orientation has also been conducted to

reduce the kinematic error of 5-axis machining [108]. Analysis ofkinematic error for 5-axis machining can be found in [50,109–111]. The errors addressed by Ye and Xiong [108] are the onesresulting from postprocessor of the CAM software system duringthe linearization process when it transforms the tool orientationdata (and the CC points) into the machine axis movements. Whenthe postprocessor controls the deviation of the generated axismovement with the original CC points, it transforms back themachine axismovements into CC points. Thismay result in over- orunder-estimation of kinematic errors. Ye and Xiong [108] showedthat the proper selection of tool orientation can reduce this type oferror.Approximation of rear gouging area by a quadric surface has

been used by Fan and Ball [105] for the tool orientation in 5-axismachining. A major advantage is the computational efficiency offitting and distance calculation and the applicability to implicit,polyhedral and point based surfaces. However the required samplepoint patterns exaggerate the gouge prone area under the toolshadow, and hence increase the time of gouge checking. Anothergroup of methods tends to locate the workpiece in a few setupsby using tilt/rotary tables [2,112,113]. Note that this is differentfrom 5-axis machining when the tool rotations are controlled bythemachine table. In thesemethods, setups are generated based onpartitioning the surfaces into a few areas and then machining by amulti-axis machine. This partitioning process has been conductedby the shape classification and machinability in [2], accessibilitydomain analysis in [112] and Gmap (Gaussian map) based areanormal matching in [113]. Then tool orientation is adjusted inevery setup by tilt/rotary table to machine with 3, 3 12

12 , or 4 axis

NC machine.

4. Tool geometry selection

4.1. Overview

The main objective for tool geometry selection in 5-axismachining is to reduce the machining cost. Machining cost isdetermined by the machining time and tool cost. Lee et al. [4]showed that the best way tominimizemachining time is to choosethe largest possible tool along with the minimum number oftool changes. Also a number of geometric constraints should besatisfied. The surface should be gouge free and within the boundsof assigned tolerances. The cutter should be selected for differentstages of machining, i.e., roughing, semi-finishing, finishing andclean-ups. It should be noted that proper assignment of clean-upregions by selection of larger tools during the finishing process andthen cleaning of sharp filleted areas may be a good strategy forreducing the total machining time.



Fig. 9. Generalized cutter shape based on APT definition.

Fig. 10. Layer-by-layer machining for roughing.

Tool geometry is defined by tool type and size parameters.Selection of tool type to a great extent depends on the geometryof the surface and the process planner’s experience. Heuristicmethods have been used for the identification of the tool type[4,114]. Generally endmills (ball-, flat-, and fillet endmills) aremost widely used in 5-axis machining of freeform surfaces. Themathematical descriptions and the formulation of the interactionwith workpiece surface for three types of endmills have beenprovided in [115]. A tool with generic shape based on APTdescription is shown in Fig. 9. The three types of endmills arespecial cases of the generic tool. Selection of tool size includes theselection of different tool parameters shown in Fig. 9.

4.2. Recent development

4.2.1. Tool selection for the roughing stageAt the roughing stage, the excess material should be removed

as quickly as possible. Tool selection methods for 3- and 5-axismachining are similar. A common method for roughing freeformsurfaces is the layer-by-layer machining approach (Fig. 10) [4,42,114,116,117]. Tool changes occur in cutting planes while the partis machined at each hunting plane. Of particular importance is theselection of largest possible tool for each layer (hunting plane) aswell as selection of the optimal set of tools for the whole cuttinglayers. Flat endmills are most widely used for rough machining[4,114,118]. The cutter size for each boundary is restricted by thegeometry of the inner and outer boundaries of that hunting plane[27,119]. The total machining time is the sum of machining timesfor all slices plus the time required for tool changes. Althoughselection of the largest possible tool is preferred, it should be notedthat in the problem of optimal tool sequence selection it does notnecessarily yield the minimummachining time [116].Selection of the optimal tool sequence in layer-by-layer

machining is a complex problem. A method for merging selectedcutters in different layers by analyzing the relative machiningtime has been developed by Lee et al. [4]. Balasubramaniam et al.[116] used a flow network model to formulate the tool sequenceselection problem considering the accessibility limits of the tool. In[114], a full description of the hunting layer intersection problemandmergingmethod have been discussed. Chen et al. [119] applieddynamic programming for the simultaneous selection of huntingplanes and the cutter to minimize the machining time. A generic

formulation of dynamic programming for selection of the optimalset of cutters with applications in rough cutting modes has alsobeen developed by Veeramani and Gau [120].Octree volume decomposition has been used to divide each

layer into areas with simple and complex geometries [118,121].In the method developed by Lee et al. [118], simpler portions(full octants) in all layers are cut by larger cutters first, and thenthe complex portions (partial octants) of the layer boundariesare removed by subdivision and matching with smaller tools.Evaluation of the machining time improvement compared tohunting–merging methods was not provided in this paper.Meanwhile thepotential collisiondue to theundercuts in the initialmachining of full octants was not discussed.

4.2.2. Tool selection for finishing stageThe same as roughing, themain objective for the finishing stage

is to minimize the machining time. However the surface should begouge-free and within tolerance limits. In 5-axis finish machining,cutter selection is closely related to the tool’s orientation, tool pathtopology and tool path parameters. The problem can be stated asfollows:

Given a set of cutting tools, select the best cutter/cutter set thatcan traverse the entire surface in the minimum time withoutcausing the three types of gauges and within the tolerances.

This formulation requires optimization of the tool orientation,tool path, and the tool geometry at the same time. Most of thecurrent methods avoid modeling of this problem and assume thatthe tool has been selected before tool path generation [122,123].For 3-axis machining, a big part of the problem, i.e., optimizationof tool orientation, will not be considered. In this case, the toolselection is basically to calculate the minimum radius of curvaturefor the surface and match the largest possible tool with that[71,87,114,124]. However in 5-axis machining due to the variableorientation parameters, it is possible to select a tool with a radiuslarger than the smallest radius of curvature of the surface.Lee and Chang [19] introduced a method for the identification

of a feasible cutter at a CC point. At first the collision-free tiltand inclination angle limit at each CC point are determined by afeasibility cone. Then in samples of tilt and inclination angles, theeffective radius of the tool and that of the surface are evaluated.To avoid local gouging, the effective radius of the tool should besmaller than that of the surface in different samples of tilt andinclination angles. The effective curvatures of tool and surface ata specific direction (yL) are given as follows (see Figs. 3 and 4):

κtL = κt1 cos2(θ)+ κt2 sin2(θ) (6)

κsL = κs1 cos2(λ)+ κs2 sin2(λ) (7)

where κs1 and κs2 are surface minimum andmaximum curvatures,and κt1 and κt2 are those of the tool defined by:

κt1 =sin(α)

r1 + rf sin(α); κt2 =

1rf. (8)

Thus all the possible cutter orientations can be checked for localgouging at different feed directions. Jensen et al. [125] mentionedthat because of the convex hull property, it is not required to checkall the feasible orientation areas for tool selection. As mentionedin Section 3, the optimal orientation must lie in the boundariesof the feasible orientation area. Jensen et al. [125] proposedthe relative optimal cutter selection method to initialize with thelargest available tool in the tool library and the following tilt andinclination angles:

α = sin−1(

rf1κs1− r1

); ω = 0 (9)



where rf and r1 are the cutter fillet radius and the cutter radius,and κs1 is the surface minimum principal curvature. Cutter sizeparameters (e.g., corner radius and shank radius) will change incases where the interference constraint is not satisfied. Comparedto the method by Lee and Chang [19], the method by Jensen et al.[125] requires less computation effort. A few good examples ofdifferent cutter parameter selection and the resulting machiningtime can be found in [125]. Tool selection developed by Li andZhang [123] is based on accessibility analysis before the generationof tool path. Accessibility of a cutter has been defined as beingoriented at a point without causing any of the three types ofgouges. The largest cutter which is accessible all over the surfaceis then selected as the feasible cutter. Similarly in the research byLi and Zhang [122], it has been assumed that tool path patternand direction are not decided at the cutter selection stage. Thesurface is decomposed into convex, concave and saddle regions andthe sampling points for accessibility checking are only consideredin concave and saddle regions. Although the methods by Li andZhang [122,123] give flexibility for the selection of anypathpatternand direction for the selected cutter, the results are relativelyconservative and computationally expensive compared with themethods by Lee and Chang [19] and Jensen et al. [125].

4.2.3. Tool selection for semi-finish and clean-up stagesPencil-cut and fillet-cut methods are used for clean-up machin-

ing [80]. Few works have been carried out on the selection of ap-propriate clean-up and semi-finish cutters. In the method by Renet al. [5], the clean-up area has been approximated by a V-shapeto generate a tool path by the pencil-cut method. Then the num-ber and size of ball endmills to clean the sharp filleted areas arecalculated by considering intermediate virtual cutters ranging fromthe finishing tool to the clean-up tool. For semi-finishing, the cut-ter selection should be based on the geometric constraints and thethickness of the shoulders left from the roughing process [4,114].

5. Summary and discussions

5.1. Summary

A review of the fundamental issues and new developmentsin CNC machining of freeform surfaces has been carried out inthis work. Three major issues, tool path, tool orientation, andtool geometry, have been considered. Numerous researches inthe last two decades have resulted in significant improvement inthese aspects. Among different evaluation measures, quality andefficiency of machining are primarily used to study and comparethese different developed methods.

• In tool path generation, iso-scallop tool paths along withthe curvature matching method have significantly lead tothe improved surface quality and reduced machining time.Many achievements have also been observed in the machiningof polyhedral surfaces, point clouds and compound surfaces.Various surface segmentation techniques such as the isophotepartitioning method, which leads to decreased machining timeand/or reduced investment cost, have also been studied.• Development of optimal tool orientation techniques for 5 axismachining has significantly improved machining productivityand quality. Methods for the tool orientation identificationfocus on either improvement of the computational efficiencyof the existing methods or development of new methods. Toolorientation smoothing plays an important role to achieve a highquality of freeform surface machining.

• Tool selection methods for freeform surface machining havebeen developed for roughing, semi-finishing, clean-up andfinishing stages. One of the critical issues is the selection ofthe optimal tool sequence for roughing stage in layer-by-layermachining. For 5 axis machining, the tool selection for finishingstage is closely related to tool orientation, tool path topologyand path parameters. The majority of the developed methodscombine different techniques with the curvature matchingprinciple to identify the largest possible cutter for finishing ofa freeform area.

5.2. Discussions

Some studies show that introduction of the freeform surfacemachining techniques has considerably reduced the time and costin industry [28,64]. Despite the progress, many challenges stillneed to be further addressed to improve the quality and efficiencyof the freeform surface machining methods.Computation time and machining time are the two major

issues for further improvement of efficiency in freeform surfacesmachining. As a core method, curvature matched machininghas largely affected the recently developed techniques in toolpath generation, tool orientation identification and tool selection.Employment of the curvature matched machining approachtogether with the iso-scallop method in these three areas cansignificantly reduce actual machining time while ensuring thequality of the final surface at the cost of increased computationtime. However in some of the recently developed methods, it hasbeen observed that surface quality has been overlooked in orderto decrease either the machining time or the simulation time. Theiso-scallop rule no longer applies to the resulting tool paths in thesemethods.The quality issue needs to be further investigated in the

machining of polyhedral surfaces, cloud-of-points surfaces, andcompound and trimmed surfaces.Many new techniques have beendeveloped for the local and global gouge detection in machiningof polyhedral surfaces which was previously assumed to be anerror prone, time consuming and approximation based task. Formachining of cloud-of-points surfaces, the developed methodsare conservative for achieving the required quality. This is dueto lack of a precise measure for determining path parametersfor machining these surfaces as well as compensation for theapproximation errors introduced by geometric modeling. Forcompound surfaces, more efficient methods for rear and globalgouge detection in the boundary areas need to be developed.In addition to the three major issues studied in this review

considering the geometric aspect of freeform machining, otherissues such as machining feedrate scheduling [126–129] andkinematics and dynamics of CNCmechanisms [130,131] also affectthe quality and efficiency of freeform surface machining. All theseproblems need to be further addressed to apply the developedfreeform surface machining methods effectively in industry.

Acknowledgement

The authorswish to thank the Natural Sciences and EngineeringResearchCouncil (NSERC) of Canada for providing financial supportto this research.

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