Five-axis tool path generation for a flat-end tool based on iso-conic partitioning

Computer-Aided Design 40 (2008) 1067–1079

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Computer-Aided Design

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Five-axis tool path generation for a flat-end tool based on iso-conic partitioningNan Wang, Kai Tang ∗Department of Mechanical Engineering, Hong Kong University of Science and Technology, Hong Kong, China

a r t i c l e i n f o

Article history:Received 17 March 2008Accepted 26 September 2008

Keywords:5-axis NC machiningTool orientationTool path planningKinematic performanceLocal gouging

a b s t r a c t

Traditionally, for the flat-end tool, due to the intertwined dependence relationship between its axis andreference point, most 5-axis tool-path generation algorithms take a decoupled two-stage strategy: first,the so-called cutter contact (CC) curves are placed on the part surface; then, for each CC curve, toolorientations are decided that will accommodate local and/or global constraints such as minimum localgouging and global collision avoidance. For the former stage, usually simplistic ‘‘offset’’ methods areadopted to determine the cutter contact curves, such as the iso-parametric or iso-planemethod; whereasfor the latter, a common practice is to assign fixed tilt and yaw angle to the tool axis regardless the localcurvature information and, in the case of considering global interference, the tool orientation is decidedsolely based on avoiding global collision but ignoring important local machining efficiency issues. Thisindependence between the placement of CC curves and the determination of tool orientations, as wellas the rigid way in which the tilt and yaw angle get assigned, incurs many undesired problems, suchas the abrupt change of tool orientations, the reduced efficiency in machining, the reduced finishingsurface quality, the unnecessary dynamic loading on the machine, etc. In this paper, we present a 5-axis tool-path generation algorithm that aims at alleviating these problems and thus improving themachining efficiency and accuracy. In our algorithm, the CC curves are contour lines on the part surfacethat satisfy the iso-conic property — the surface normal vectors on each CC curve fall on a right smallcircle on the Gaussian sphere, and the tool orientations associated to a CC curve are determined by theprinciple of minimum tilt (also sometimes called lead) angle that seeks fastest cutting rate without localgouging. Together with an elaborate scheme for determining the step-over distance between adjacent CCcurves that seeks maximummaterial removal, the presented algorithm offers some plausible advantagesover most existing 5-axis tool-path generation algorithms, particularly in terms of reducing the angularvelocity and acceleration of the rotary axes of the machine. The simulation experiments of the proposedalgorithm and their comparison with a leading commercial CAM software toolbox are also provided thatdemonstrate the claimed advantages.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

5-axis Numerical Controlled (NC) machining produces betterquality surfaces and is more efficient than 3-axis NC machining.However, on the other hand, the two rotational axes also make itscontrol and planningmuchmore difficult. Over the past 15 years orso, more and more researchers have focused their study on 5-axismachining. Among a number of issues, the following are the mostfocused.

1.1. Local gouging prevention

On local gouging detection and prevention, several tech-niques [1–5] have been developed that are based on comparing

∗ Corresponding author. Tel.: +852 2358 8656; fax: +852 2358 1543.E-mail address:[email protected] (K. Tang).

0010-4485/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2008.09.005

the effective cutting curvature of the tool’s swept surface withthe normal curvature of the part surface at the contact point. Andothers, such as the Rolling Ball Method [6], the Arc-intersectionMethod [7], the Penetration–elimination Method [8] and the al-gorithm proposed by Xu [9], are area-based methods. Yoon [10]and Jensen [11] not only considered local gouging but also animprovement of the process efficiency by selecting the best toolorientations.

1.2. Global interference avoidance

Global interference detection and avoidance has remained tobe a major challenge. Lee [12] used quick feasibility checkingand detailed feasibility checking to judge 2D and 3D collisionbetween the cutter and the sculpture surface. Morishige [13]used the 2D configuration space to describe the relationshipbetween postures of a ball-end tool and their collision with theenvironment. Vafaeesefa [14] projected infeasible domains to aunit sphere centered at the cutter contact point to find the feasible

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1068 N. Wang, K. Tang / Computer-Aided Design 40 (2008) 1067–1079

tool orientations. Ho [15] handled the real-time collision detectionproblemwith haptic rendering. Ilushin [16] proposed an approachthat uses space subdivision techniques and ray-tracing algorithmsto derive a surface-tool intersection algorithm to find betterresults. To reduce the calculation time, Umehara [17] proposedthe idea of expanding a group of interference-free tool postures tofill the machined surface before defining the cutter contact points.Besides these, Lauwers [18,19] also considered collisions betweenthe machine and part or the tool.

1.3. Tool path planning

How to plan a 5-axis tool path with both local and globalgouging considered has remained the most difficult problem. Thetool path planning strategies can be classified into five groups:the iso-parametric [5], the iso-offset [20], the iso-planar [21–24], the iso-scallop [25–27] and the iso-phote [28]. Most toolpaths are generated in the iso-parametric direction due to itsstraightforwardness. The iso-planar method is often preferred forthree-axis machining and is capable of dealing with trimmedsurfaces, despite its heavy computing cost due to the need ofsurface intersection. In the iso-scallop height approach, the initialtool path is the center line or the boundary curve of the part, andthe following paths are offsetting lines of the current tool path. Theiso-phote method and its variances are only suitable for three-axismachining, in which some inclination curves are defined as theinitial paths, and other tool paths are generated by offsetting theinclination curves.Otherwise, there have been some new developments in 5-

axis tool path generation, apart from the aforementioned. Theyare the principal axis method [3], the fractal method [29], theguide surface method [30], the maximum kinematics performancebasedmethod [31], the contour-parallel offsetmethod [32–34], theboundary conforming method [35], the spiral cutting method [36,37], etc. In [38], Chiou and Lee introduced the machining potentialfield method that tries to define a series of contact points thatshould obtain the largest feasible machining strip width so as toreduce the machining time and improve the surface quality.

1.4. Tool orientation optimization

Traditionally, tool orientation determination was studiedmainly concerning global collision avoidance but without consid-ering other due factors such as kinematic or dynamic performance.Morishige [39] improved his original method in [13] and extendedit to 3D configuration space, which is able to precisely describe therelationship between ball-end tool postures and the existence ofcollision. Jun et al. [40] proposed a method, which is also C-spacebased, to find better tool orientations with both local gouging andglobal interference considered. In theirmethod, theminimumcuspheight is used as the objective function for determining tool orien-tations, in addition to the interference-free constraint. Chiou [41]proposed a swept envelope approach to finding optimal tool ori-entations with the considerations of gouge and collisions. Balasub-ramaniam [42] used the concept of visibility to generate globallycollision-free 5-axis tool postures. Gian [43] used open regions andvector fields to generate the tool orientation in 5-axis NC machin-ing of cavity regions. If the orientation of the tool is predefined, theCL surface deformation approach introduced by Kim [44] could beused. Hsueh [45] proposed a method to generate the tool orienta-tion automatically. First, the tilting collision-free angle range in theplane normal to the tool path is determined. Then, the correspond-ing collision-free yaw angle range is formed by intersecting theneighboring surfaces and the cone generated by the tilt collision-free angle range.

There is a number of limitations in the above mentionedmethods for the determination of tool orientations. First, thetool is almost always assumed to be of the ball-end type.Second, and more critically, a series of tool postures computedby these algorithms often requires a drastic change in tool’sorientation between neighboring tool contact points. Such anextreme change in orientation can never be feasible in realmachining due to the physical limits of the angular velocityand acceleration of the rotary motions of the machine tool.Meanwhile, the dynamics of machining tool will increase surfacedeviation (cf. [46]). As a remedy, a smoothing technique is usuallyadopted to lessen the angular change (cf. [39,40,47]). Recently, wepresented a method [48] that uses VMap and FMap to determinethe tool orientations that meet the angular velocity constraints.Nevertheless, surface properties are never considered in theseremedy solutions.When planning the 5-axis tool paths for a flat-end tool, which is

preferred over ball-end type tools but whose computation is muchmore difficult than that of the ball-end tool due to the dependencyof the reference point of the tool on its orientation, a particulartool posture is represented by the cutter contact (CC) point and theorientation of the tool. Except for very few works (cf. [38,50]), thedetermination of the CC points has universally been in the offset-fashion: an initial CC curve is first decided (usually the boundary ofthe part surface), and subsequent CC curves are just offset curvesof the previous ones with a constant step-over distance measuredeither in the parametric domain or geodesically on the part surface.Moreover, the determination of the CC points and tool orientationsare decoupled — a CC point is established without concerning theeventual orientation of the tool at the point.In this paper, we present an implemented work in automatic

generation of 5-axis tool paths for a flat-end tool that seeks bettercorrelation between CC points and tool orientations and also theoverall planning/ordering of the CC points on the part surface.We introduce a concept called iso-conic that retrains the toolorientations along a CC curve to a right small circle on the Gaussiansphere. This constraint, together with the minimum inclinationangle of the tool axis defined at a CCpoint, uniquely defines the toolaxis at the CC point. The resultant tool postures along the CC curveenjoy the good characteristic of small change in tool orientationswhile at the same time achieving a faster material removal rate.And, by orchestrating a carefully designed propagation scheme forneighboring CC curves that seeks maximum machining strips, ouralgorithm outputs 5-axis tool paths that are believed to performbetter in terms of the dynamics of the machine (i.e., the slowerangular velocity and acceleration of the machine rotary axes,though not the overshoot and other issues due to potential largegeodesic curvatures on the CC curves) and themachining efficiency(i.e., faster material removal rate), as well as the surface quality ofthe finished part.

2. Preliminaries

Let our part surface S be a C2 surface with a parameter u–vdomain. A flat-end cutter could be fixed at any CC point on S withsurface normal Np and tool axis T, as shown in Fig. 1. Assumingthat T is not parallel with Np, the Cutter Location (CL) point couldbe calculated using Eq. (1), where r is the radius of the cutter.

CL = CC+ rNp − (T · Np)Np∥∥Np − (T · Np)Np∥∥ . (1)

There are two spherical coordinate systems shown in Fig. 1(a) andFig. 1(b) — the Machine Coordinate System (MCS) and the LocalCoordinate System (LCS). The LCS is expressed by F–Np–K, where F

N. Wang, K. Tang / Computer-Aided Design 40 (2008) 1067–1079 1069

Fig. 1. Local coordinate and machine coordinate system.

is the feed direction andK the cross productNp×F. In LCS, the unitvector T is defined by two angles, the tilt angle α and yaw angle β ,whereas in MCS T is expressed in terms of one of (A, B), (B, C) or (A,C), depending on the specific machine type and also which two ofthe machine rotary axes A, B, and C are used. We emphasize that,to facilitate our discussion, the spindle-tilting machine as shown inFig. 1(c)will be chosen as ourmachine throughout the entire paper.In addition, (A, C) are chosen.The effective cutting ellipse (ECE) is the projection of the cutter

bottom on the Np–K plane in LCS, as shown in Fig. 2. In the figure,surface S+ is the offset of part surface Swith an offset hwhich is themaximum scallop height allowed. The points on the cutting edgecould be expressed using Eq. (2) (where αL and βL are the tilt andyaw angle of the tool axis and ϕ is the parameter for the points on

Fig. 2. Effective cutting ellipse.

ECE). The ECE is mainly used for the two purposes as given below.

P(ϕ) =

r cosαL cosβL sinϕ − r cosαL cosβL − r sinβL cosϕ−r sinαL sinϕ + r sinαLr cosαL sinβL sinϕ − r cosαL sinβL + r cosβL cosϕ

. (2)(i) Local gouging test. The effective cutter curvature, keff, of the

cutter is the curvature of the effective cutting ellipse at the CCpoint, and can be calculated using Eq. (3). The traditional localgouging analysis is to compare keff with the normal curvature ofthe surface at the CC point in direction K, i.e., the κ in Fig. 2.Local gouging is prevented by determining the smallest tilt

angle α to achieve the fastest material removal. It is calculatedby equating keff to the maximum principal curvature kmax of thesurface, expressed by Eqs. (3) and (4). Here, β is equal to 0.

keff = sin(α)/(r∗ cos2 β) (3)

αmin = sin−1(r∗kmax). (4)

(ii) Maximal machining strip. Lee [49] suggested a method thatuses ECE to evaluate the machining strip. Given the machiningtolerance h and tool orientation (αL, βL), ϕa and ϕb could becalculated by solving Eq. (5) for ϕ.

D1 sin2 ϕ + D2 sinϕ + D3 sinϕ cosϕ − D3 cosϕ + D4 = 0 (5)

where

D1 =r2 cos2 αL sin2 βLK

2D2 = r sinαL cosβL sinβLκ

D3 = r2 cosαL cosβL sinβLκD4 = h− r sinαL + D1.

(6)

By finding ϕa and ϕb, the coordinates of Ca and Cb could be readilycalculated using Eq. (2).Assuming that Np and the surface curvature (at least the two

principle curvatures) are known, our task is to:(a) Determine a series of non-iso-parametric CC curves on

the part surface S – hereafter simply to be called tool paths –{L1, L2, . . . , LN}. The neighboring tool paths are related to eachother by the constraint of maximal machining strip.(b) Determine the tool postures at the tool paths obtained from

(a). These tool postureswill ensure a fastermaterial removal underthe requirement of local gouging prevention. At the same time,they are required to vary slowly and smoothly in MCS.(c) Finally, develop a real computer system that implements (a)

and (b).


Fig. 3. Local gouging constraint cone (LGCC).

3. Tool posture determination

For the time being, suppose that a list of CC points, {CC1,CC2, . . . , CCn} that comprise a tool path, is already available. (Howit will be generated will be discussed in the next section.) We areto determine the corresponding tilt angles {α1α2, . . . , αn} and yawangles {β1β2, . . . , βn}.At a CC point CCi, if the tilt angle αi is larger than someminimal

angleαmin, local gougingwill not occur.We define the local gougingconstraint cone (LGCC) at CCi to be the cone with its apex on CCi, itssymmetry axis the normal vector Ni to the surface S at CCi, and aspan angle of αmin. For example, in Fig. 3, Cone 1 (CC1, N1, αmin)is the LGCC of CC1 – the tool axes that lie outside Cone 1 will notincur local gouging at CC1, with the cone itself as the permittedminimal tilt angles, which though have a faster material removalrate. Note that these cones are not congruent to each other – theminimum tilt angle αmin depends on the local geometry (such asthe principal curvatures and the feed direction) and is different atdifferent CC points. On the other hand, conceivably, the change oftheseminimum tilt angles is small or gradual, since the part surfaceis assumed to be smooth.Each CC point has one such corresponding cone, and thus there

is a list of n cones, {Cone1, Cone2, . . . , Conen}, associated with{CC1, CC2, . . . , CCn}. Our task is to determine a list of tool axes,{T1, T2, . . . , Tn}, at these CC points. To satisfy the (local) gouge-freeconstraint, Ti must be outside Conei. At the same time, the changeof these axes should be gradual and small in MCS.Fig. 4 gives a pictorial illustration of our tool posture

determination method. The key to our method is the pre-assumption that the surface normal vectors at the n CC points CCi,{N1,N2, . . . ,Nn}, satisfy the iso-conic condition, that is, they all fallon a conewith its apex at the origin.Wewill discuss in detail in thenext section on how this condition can be upheld. Let one of therotary axes of the MCS, e.g.,+Z (the C axis), be the symmetry axisof this cone, e.g., Cone-G in Fig. 4. Refer to Fig. 4. After translatedto the origin of the MCS, each Conei intersects Cone-G at two linesPi1 and Pi2, e.g., P11 and P12. Since the tool path is already given,and hence the feed direction, we just take the one of Pi1 and Pi2,whichever is closer to the feed direction at CCi.As a result, we have obtained nTi = [A

pi C

pi ]T, where Api and C

pi

are the two rotary angles of Ti in MCS, which can be expressed bythe following equation (xi, yi, and zi are respectively the projectionof Ti on the X-, Y -, and Z-axis):

Ti =

{Api = arctg(yi/zi)

Cpi = arctg(yi/xi)i = 1, 2, . . . , n. (7)

Fig. 4. Tool posture determination.

Fig. 5. Rotation of T in MCS.

Analyzing the geometric relationship between Ti and Ti+1(see Fig. 5), one finds that in MCS the angle changes betweenneighboring tool axes are

Ti+1 − Ti ={1A = Amid − Ai1C = Ci+1 − Cmid

i = 1, 2, . . . , n− 1 (8)

where

Tmid =

Amid = ±arctg(√1− x2i −z

2i+1 /z i+1

)

Cmid =

arctg

(√1− x 2i− z

2i+1/xi

)π − arctg

(√1− x2i − z

2i+1/xi

)In our iso-conic method, since zi = zi+1 throughout the entire

path, then,

Ti+1− Ti ={1A = Amid − Ai = arctg(yi /z i)− arctg(yi/zi) = 01C = Ci+1 − Cmid = arctg(yi+1/xi+1)− arctg(yi/xi).

Because CCi andCCi+1 are close, their surface normal changewill besmall, so is the difference (αmini+1 − α

mini ) (as the surface is C

2), and1C . Therefore, one can conclude that the thus defined tool axes{T1, T2, . . . , Tn} have the following desirable properties:• There is no local gouging at each CCi.• Along the entire tool path, the tilt angle (measured in LCS)varies, but smoothly and slowly, so as to adapt to the localcurvature and geometry to attain better finishing surfacequality.


(a) Part surface S. (b) φ function.

(c) Equi-space contour lines in u–v domain. (d) Equi-space contour lines on surface S.

Fig. 6. Part surface (a), the φ function (b), the equi-spacing contours of φ in the u–v domain (c), and the contours on S (d).

Fig. 7. Finding the next path from the current: initial guessing points.

• Overall, the difference ||Ti+1 − Ti|| along two rotation axes willbe small and smooth, even zero for one axis, when measured inMCS.

4. Tool path determination

In the previous section, there is a pre-assumption that thesurface normal vectors at the tool path {CC1, CC2, . . . , CCn} all fallon some cone (the iso-conic property). This requirement is actuallythe central point in our overall planning of tool paths. Let the partsurface be given as a parametric surface, i.e.,

S = (X(u, v), Y (u, v), Z(u, v)), (u, v) ∈ Φ ∈ [0, 1]2 (9)

Fig. 8. Checking the relationship between {A, B} and {At , Bt }.

where Φ is some connected region in the u–v plane. To facilitateour task, we define a scalar function φ(u, v), called the inclinationfunction, in the u–v domain:

φ(u, v) = ACOS(N(u, v) · [0, 0, 1]) (10)

whereN(u, v) is the surface normal vector at S(u, v). Obviously, allthe contour lines of φ(u, v) are iso-conic, i.e., their correspondingembedded curves on S are iso-conic. In the example shown in Fig. 6,(a) is the part surface S, (b) is the corresponding φ function, (c) and(d) are the equi-spacing contour lines of the φ function in the u–vdomain and on the S respectively.


Fig. 9. New clusters start due to the change of topology of paths.

Fig. 10. Difference between the u–v measure and φ measure.

The equi-spacing contour lines provide an approximatedistribu-tion of tool paths, but may not be the tool paths themselves. Whendetermining the tool paths, {L1, L2, . . . , LN} for someN , we need toaddress the following critical questions:

• What is the initial (first) tool path?• After the initial tool path, how to determine the next?• How to guarantee that the tool paths {L1, L2, . . . , LN}will coverthe entire S? That is, the resultant cut surface will be identicalto S under the given maximum scallop-height requirement?

In the following, we lay out our tool path generation algorithmcentered around answering the above questions. Before thealgorithm, several things need to be clarified or defined. First, acontour line φ(u, v) = c may consist of several disjoint pieces,each of which is either a closed loop or an open curve with itstwo ends on the boundary of the u–v domain. Hereinafter, theterm ‘‘contour’’ or ‘‘contour line’’ will always mean a single piece— exactly which piece will depend on the context. Second, we

Fig. 11. Partitioning of the u–v domain of the part of Fig. 6(a).

Fig. 12. Determination of tool axis with the VMap.

will use Strip (L1, L2) to indicate the strip-like region between twocontour lines L1 and L2 in the u–v domain. Third, let us recall thesingularity of a multi-variate function F : it is where the gradient∇F vanishes. φ thus can also have singularity points, e.g., the threemarked points in Fig. 6(c), which will be used in the algorithm. Atlast, let dP(C, p) denote the Euclidean distance between a curve Cand a point p in the plane, i.e., it is the Euclidean distance betweenp and its closest point on C .

4.1. Single expansion

The tool paths are computed in a clustering fashion and eachcluster is computed individually. The core of our program forgenerating a cluster is a procedure called EXPANSION. Startingfrom some initial path, and given an incremental direction forφ, this procedure computes the rest of the paths in the clustersequentially that strictly follow the given incremental direction ofthe φ. Thus, by invoking EXPANSION twice on an initial path, oneon increasing and one on decreasing of the φ, the entire clustercan be generated. For the codes of EXPANSION given below, theincremental direction ofφ is ‘‘decreasing’’; that for the ‘‘increasing’’direction can be derived similarly. The input to EXPANSION is thestarting path L = {(ui, vi) : 1 ≤ i ≤ n}, its φ value φc , and itsCa-curve A and Cb-curve B (see Fig. 2).


Procedure EXPANSION (L, φc, A, B)BeginStep 1. puv ← the closest singularity point to L.Step 2. If dP(L, puv) < δuv , then exit.Step 3. Refer to Fig. 7. For every point CCi = S(ui, vi) find theprojection point CC∗i on surface S of the point CCi + K · ||Cb − Ca||,where K, Ca, and Cb are the entities as given in Fig. 2 for CCi. Letφ∗c be the largest among the φ values of the n points CC

∗

i , that is,it has the smallest difference from φc .Step 4. Find those singularity points that lie inside the regionStrip (φ(u, v) = φc, φ(u, v) = φ∗c ); if none is found, go to Step 6.Step 5. φ∗c ← max{φ∗c , φs}, where φs is the largest of the φ valuesof the singularity points found in Step 4.Step 6. φ1 ← φc;φ2 ⇓ φ

∗c .

Step 7. φt ← (φ1 + φ2)/2.Step 8. Compute the contour line φ(u, v) = φt . If this contour isabnormal, exit. Otherwise, let Lt be this contour line.Step 9. Find all the tool postures on Lt , using the algorithm fromSection 3.Step 10. Compute the Ca-curve At and the Cb-curve Bt of Lt usingthe tool postures from Step 9.Step 11. Check the relationship among A, B, At , and Bt on surfaceS+. There could be three distinct cases and they are treatedaccordingly as follows.Case 1. (Fig. 8(a)) At is completely inside the other side of Bw.r.t.Bt , but the minimum distance between At and B is below a giventhreshold µ. Lt then is taken as the current path, i.e.,{L, φc, A, B} ← {Lt , φt , At , Bt}, and we go to Step 1 to find thenext path.Case 2. (Fig. 8(b)) At is completely inside the other side of Bw.r.t.Bt , and the minimum distance between At and B is larger than µ.We then expand the φ interval by setting φ1 ⇓ φt , and go toStep 7.Case 3. (Fig. 8(c) or (d)) Otherwise, we shrink the φ interval bysetting φ2 ← φt , and go to Step 7.End.

In the above iterative procedure, there are two places wherethe iteration can or will terminate — Steps 2 and 8. At Step 2, wecheck if the current path L passes through (i.e., numerically, veryclose to) a singular point. When this occurs, the current expansionmust stop, since, at a singular point, the gradient ∇φ is 0 and thecondition of the monotone decreasing φ can no longer hold whencrossing the point. In fact, a closer look into a singular point revealstwo distinct scenarios: it is either elliptic (e.g., the sole circularpoint in Fig. 6(c)) in which case the current path L has alreadyshrunk to a tiny loop, or hyperbolic (e.g., the two square points inFig. 6(c)) inwhich case the nearby contours fall into a unique groupof clusters around the point. Either way, the current expansion hasno where to go but stop.The other possible terminal condition is when the contour

line φ(u, v) = φt found in Step 8 is abnormal. Here the word‘‘abnormal’’ means two possible situations. One is when thecontour φ(u, v) = φt is entirely outside the u–v domain Φ . Sincethe part surface S is C2 continuous, this could happen only whenthe current path L is already very close to the boundary ofΦ and theexpansion should stop there. The other is when the topology of thenewcontour changes from that of L—either it is openbut L is closedor vice versa, as illustrated in Fig. 9.When this happens, the currentexpansion stops and new cluster(s) will emerge at the new contourline(s) of φ(u, v) = φt , e.g., in Fig. 9, four new expansions will beinitiated at the four open paths σ1, σ2, σ3, and σ4, independently.The φ value φ∗c found in Step 3 indicates the largest deviation

of φ from φc for the next path before the scallop-height constraintwould be violated. Additionally,φ∗c should be subject to our secondconstraint that a cluster be free of any singular points, and this

is enforced by Step 4 and 5. It is imperative to point out thedifference between the Euclidean distance in u–v and that in φ —the contributor to φs in Step 5 could be some one other than puv(see Step 1), as exemplified in Fig. 10, in which, when compared topoint q, point p is closer to contour L in u–v but farther in termsof φ.Another item of EXPANSION that needs elaboration is Step 11:

how to check the inclusion relationship between two machiningstrips on surface S+. Theoretically, this requires finding theintersection between two regions on S+, which nevertheless is notrecommended, if not absolutely impossible, because of the usualhigh algebraic complexity of S+ which is an offset surface. In ourimplementation, the machining strip between the two curves Aand B is approximated by a ruled surface (with A and B as therails), and the inclusion checking is achieved by first projecting (thesample points of) At onto this ruled surface and then examiningthe parameters of the projected points. Since the Ca-curve andCb-curve are approximations anyway (see Fig. 2), we found thatthis approximation (by a ruled surface) performs well in ourexperiments.

5. Overall partitioning algorithm

Based on the single cluster generation procedure EXPANSION,the overall algorithm for generating all the clusters (the overallpartitioning algorithm) is readily available. First of all, however,it is beneficial to note this observation: if a cluster does notcontain an elliptic singular point, it must end on the boundaryof the u–v domain. For example, Fig. 11 depicts the partitioningof the part surface shown in Fig. 6(a) (also referring to Fig. 6(c)),in which all the clusters end at the boundary, even for the onethat contains the lonely elliptic singularity point (cluster 9). Thisplausible property suggests a succinct strategy of controlling theflow of the algorithm. Let ∂S denote the boundary curve of S in theu–v domain (e.g., in the example of Fig. 6, it is made of four iso-curves u = 0, 1 and v = 0, 1). In ∂S, we keep a listΛ of uncoveredsegments — those on ∂S that have not been covered by the clustersthat have been so far generated; initially,Λ has only one segment,the ∂S itself. An execution of procedure EXPANSION will not onlygenerate paths in a cluster, but also remove certain segments fromΛ that are covered by the generated cluster.

Program PARTITIONINGBeginΛ← ∂S;WhileΛ 6= Nil Do {p← one point onΛ;φc ← φ(p);L← the piece of the contour φ(u, v) = φc that

contains p;Compute the Ca–curve A and Cb–curve B of L using the

toolposture algorithm from Section 3;Call EXPANSION (L, φc, A, B) in the ‘‘increasing’’

direction ofφ;Call EXPANSION(L, φc, A, B) in the ‘‘decreasing’’

direction ofφ;

}End

Fig. 11 shows the result of algorithm PARTITIONING whenapplied to the part of Fig. 6(a), where the ID numbers of the clustersalso correspond to the order of their creation in the partitioningprocess. In this example (and actually inmost of our test examples)


Fig. 13. Test example 1.

we found that the u–v domain is completely covered by thegenerated clusters from PARTITIONING. In case – though very rare– if there is some region in the u–v domain that is left uncoveredafter PARTITIONING, we can pick from this region a singular point(elliptic) as the starting point – notice that whose contour is thepoint itself – and call EXPANSIONwith the appropriateφ direction,and repeat this process until the u–v domain is completelycovered.

6. Consideration of global collision avoidance

So far in our work, global interference has not been considered.In the presence of global obstacles (and the part itself), the iso-conic condition imposed on a path some times might not hold. Inpractice, usually local machining quality and global interferenceare considered independently. This in large part is due to the factthat for many kinds of parts in which local quality is the utmost


(a) Tool paths generated by our method. (b) The iso-parametric tool paths.

Fig. 14. Test example 2.

Fig. 15. Some tool paths on the part surface of Fig. 6(a).

requirement, such as optical lenses, high quality watch frames,high precision instruments, etc., there is no global interference atall or it can be safely ignored. Nevertheless, a sound solution to thisproblem is still very important. Here we propose our general ideafor solving the problem.

In our modified algorithm, there will be no change to theway when CC curves are computed; in other words, the entireSection 4 remains the same. However, themethod for determiningthe tool postures (Section 3) must be amended. Fig. 12 illustratesthe needed amendment. Recall that the VMap (CCi) [48] of contactpoint CCi is a connected region on the Gaussian sphere thatrepresents all the gouge-free (both local and global) tool axes atCCi. In Fig. 12, p1 and p2 are the intersections between Cone G andConei (see Fig. 4 also). If VMap (CCi) covers p1 or p2, there will beno change to the original way, i.e., one of themwill be taken as thetool axis, depending on whichever is covered and closer to the+Zaxis. Otherwise, we find the point in VMap (CCi) that is closest top1 or p2 (e.g., q in Fig. 12) and use it as the tool axis.The tool postures thus computed can incur abrupt changes

in their tilt or yaw angle that may violate the dynamic con-straints on the machine. As a solution, we can use the algorithmin [48] to smooth them and enforce them to meet the angu-lar–velocity–compliance requirement. For details on this smooth-ing, please refer to [48].

7. Experimental results and discussions

We have implemented our algorithm and applied to severalexamples, and compared the simulated cutting results to that ofa leading commercial CAM software package.The first test and comparison are shown in Fig. 13, in which

the part surface is the one shown in Fig. 6(a). Fig. 13(a1) depictsthe tool paths (including the tool retractions) generated by ouralgorithm, (a2) is the simulated machined surface, and (a3) shows

Fig. 16. Tilt angle and yaw angle in LCS of tool paths from Fig. 15.


Fig. 17. Comparison of angular velocity between iso-conic and zigzag method.

the tool orientations on some tool paths. Fig. 13(b1)–(b3) showthe corresponding results of the chosen commercial CAM software,in which the iso-parametric strategy was used and the tilt andyaw angle are fixed at 4◦ and 0◦, respectively. The flat-end toolhas a radius of 1 cm, and the scallop-height tolerance h is 1 mm.Comparing (a2) with (b2) in Fig. 13, one clearly sees a bettersurface quality achieved by the proposed algorithm. Meanwhile,when compared with (b3), in (a3) the change of tool orientationsgenerated by our method can be seen to be small, continuous and

smooth. The second test and comparison, inwhich the same settingof the tool size and tilt and yaw angle are used, are shown in Fig. 14.In both these test examples, no global obstacles are present, andthus the tool postureswere computed exactly by themethod givenin Section 3.To analyze the behavior of the tilt and yaw angle under

the proposed algorithm, for the part surface of Fig. 6(a), weselect from the clusters some computed individual tool paths andtheir corresponding tool postures, displayed in Fig. 15, and show


Fig. 18. Comparison of angular acceleration between iso-conic and zigzag method.

respectively in Fig. 16(a) and (b) the corresponding tilt and yawangle in LCS. As evidently revealed in Fig. 16(a), the tilt anglecontinuously changes throughout the machining process, so as toseek the best machined surface quality.For each tool path in Fig. 15, when measured globally in MCS,

its angular velocity and acceleration about the rotary axis A, asshown in Figs. 17 and 18, remain a constant zero, while that aboutthe C-axis are of small magnitude and smooth, as also shown inFigs. 17 and 18. Evidently, our results are quite different frommost conventional methods, for example, the zigzag method. In

those methods, the local tilt and yaw angle are constant regardlessof the local geometry of the CC point, and, to accommodatethem, the machine’s rotary axes sometimes have to undergodrastic motions, causing large angular velocity and acceleration.For comparison purpose, the angular velocity and accelerationof the A- and C-axis for three zigzag tool paths (which arearbitrarily chosen from Fig. 13(b1)) are given in the middle row ofFigs. 17 and 18, respectively.When calculating the angular velocityand acceleration, for simplicity, we assume the tool maintains aconstant feed speed and hence the differentiation is carried out


Fig. 19. The computed tool postures in the presence of global obstacles.

with respect to the arc length of the tool path. In addition, theangular velocity and acceleration are normalized in the figure,since, for comparison only, their exact values are not important.(Note: the jagged curves in the right of the top row of Fig. 18 are

due to round-off errors in the numerical calculation — the valuesare 2–3 orders smaller in magnitude than their counter parts ofzigzag method and we used a very simple centered difference tocalculate them.)Additionally, as a more convincing comparison, in the bottom

row of Figs. 17 and 18 respectively, we show the percentagehistograms of the A- and C-axis angular velocity and accelerationof the tool paths from Fig. 15 (ourmethod) and three arbitrary onesfrom Fig. 13(b1) (the zigzag method). These histograms clearlyconfirm the achievement by our method of the set objective: togreatly reduce the dynamic loading, in terms of the A- and C-axisangular velocity and acceleration, on the machine.Finally, to evaluate our method in the presence of global

interferences, we ran our algorithm again on the part surface ofFig. 6(a), which though this time is subject to a global obstacle,and the same group of the computed tool paths as that in Fig. 15along with their tool postures are shown in Fig. 19. Apparently, theconsideration of global interference avoidance does not affect thetool paths; but it does affect the tool postures. In Fig. 19, path 4 isaffected by the obstacle (the sphere) and, to avoid this obstacle, thecomputed tool postures on the path near the sphere are differentfrom their counter parts in Fig. 15.

8. Conclusions

In this paper, we have presented a 5-axis tool-path generationalgorithm for a flat-end tool. The primary goal of this work isto address some lingering and serious problems associated withmost existing 5-axis tool-path generation algorithms. Specifically,the proposed algorithm aims at finding a better placement of CCcurves on the part surface such that the tilt and yaw angle of thetool change adaptively according to the local geometry of the CCpoint so as to avoid or minimize the local gouging, while alongeach CC curve the rate of change of the tool orientations in themachine coordinate system is kept small and smooth. Toward thisobjective, we proposed the concept of iso-conic property – under

which the part surface normal vectors along each CC curve areall restrained to some right small circle on the Gaussian sphere –and the idea of minimum tilt angle for determining the tool axis.Together with an elaborate partitioning scheme that seeks fastermaterial removal, the proposed algorithm aims at performingbetter than many existing algorithms in terms of machiningefficiency, kinematics and dynamics performance of the machine,and finished surface quality. Our initial simulation experimentalresults and their comparison with the popular Zigzag methodindicate certain promising potential of the proposed algorithm.However, more simulation experiments and comparisons with

other existing 5-axis tool path generation approaches are neededin order to assess the behavior and efficacy of the proposedalgorithm. In addition, although our formulation and the algorithmitself apply to any types of 5-axismachine, the analysis, simulation,and comparison in the paper are all based on a specific typeof 5-axis machine, the spindle-tilting machine. It is necessary toappraise how the proposedmethodology can be extended to othertypes of 5-axismachines. One particular entry point is the selectionof the axis of iso-cones. For the spindle-tilting machine, this axisis static and naturally coincides with the +Z axis since C-axis isthe +Z axis and all the A-, B-, C-axis, and the tool axis itself passthrough a common point, i.e., the origin of the global coordinatesystem XYZ. Other types of 5-axis machine however do not havethis luxury — the relationship between the change of the tool axisand the rotation about the A-, B-, or C-axis also depends on thelocation of the CL point in space and hence cannot be put into asimple equation such as Eq. (8). One plausible solution is to finda few dynamic axes of iso-cones for different regions of the partsurface, now that the rotation about the A-, B-, or C-axis dependson the CC point.

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Five-axis tool path generation for a flat-end tool based on iso-conic partitioning