a geometric algorithm for finding the largest milling cutternau/papers/yao2001geometric.pdf · mine...

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AbstractThis paper describes a new geometric algorithm to deter-

mine the largest feasible cutter size for 2-D milling operationsto be performed using a single cutter. First is given a generaldefinition of the problem as the task of covering a targetregion without interfering with an obstruction region. This def-inition encompasses the task of milling a general 2-D profilethat includes both open and closed edges. Discussed nextare three alternative definitions of what it means for a cutterto be feasible, with explanations of which of these definitionsis most appropriate for the above problem. Then, a geometricalgorithm is presented for finding the maximal cutter for 2-Dmilling operations, and the algorithm is shown to be correct.

Keywords: Computer-Aided Manufacturing, ProcessPlanning, Cutter Selection for Milling

1. IntroductionNumerical control (NC) machining is being used

to create increasingly complex shapes. These com-plex shapes are used in a variety of defense, aero-space, and automotive applications to (1) provideperformance improvements and (2) create high-per-formance tooling (such as molds for injection mold-ing). The importance of the machining process isincreasing due to the latest advances in high-speedmachining that allows machining to create evenmore complex shapes. Complex machined partsrequire several roughing and finishing passes.Selection of the right sets of tools and the right typeof cutter trajectories is extremely important in ensur-ing high production rates and meeting the requiredquality level. It is difficult for human planners toselect the optimal or near-optimal machining strate-gies due to complex interactions among tool size,part shapes, and tool trajectories.

Although many researchers have studied cutterselection problems for milling processes, there stillexist significant problems to be solved. Below aretwo examples:

• Most existing algorithms only work on 2-Dclosed pockets (that is, pockets that have noopen edges), despite the fact that open edges arevery important in 2-D milling operations.

• Because there are several different definitions ofwhat it means for a cutter to be feasible for aregion, different algorithms that purport to findthe largest cutter may in fact find cutters of dif-ferent sizes.

This paper presents an algorithm for finding themaximal cutter for general 2-D milling operations tobe performed using a single cutter.

• The general 2-D milling problem is formulatedin terms of a target region and an obstructionregion. This problem formulation encompassesthe general problem of how to mill a 2-D regionthat has both open edges (edges that don’t touchthe obstruction region) and closed edges (edgesthat touch the obstruction region) (see Section 3for definitions). The formulation allows arbitrar-ily complex starting stocks. Therefore, it can beused to model machining of geometrically com-plex castings such as engine blocks.

• Three different definitions are given of what itmeans for a cutter to be feasible, and it isexplained why one of these definitions is moreappropriate than the others.

• A new “region covering” algorithm is describedthat finds the largest cutter that can cover the tar-get region without interfering with the obstruc-tion region, and the correctness proof is givenfor the algorithm.

In practice, quite often multiple cutters are usedto machine a complex milling feature. The region-covering algorithm is also useful as one of the stagesin finding an optimal sequence of cutters for

Journal of Manufacturing ProcessesVol. 3/No. 1

2001

A Geometric Algorithmfor Finding the Largest Milling CutterZhiyang Yao ([email protected]) and Satyandra K. Gupta ([email protected]), MechanicalEngineering Dept. and Institute for Systems Research, and Dana S. Nau ([email protected]), Computer ScienceDept. and Institute for Systems Research, University of Maryland, College Park, Maryland, USA

machining a complex feature (for subsequent workon this subject, please see Yao, Gupta, and Nau1).

2. Related WorkBecause of the wide range of the complexity of

products, requirements for machine accuracy, differ-ent machining stages, selecting optimal cutter size isan active research area. Below is a summary of pre-vious research in the area of milling cutter selection.

Bala and Chang presented an algorithm to selectcutters for roughing and finishing milling opera-tions.2 Their work encompasses almost all of the fea-tures found on prismatic parts, such as slots andsteps. Their algorithm is based on geometric con-straints. The basic idea is trying to fit the possiblelarge circle into contours to select possible large cut-ters to save processing time. Bala and Chang takeboth cutter change time and geometric constraintsinto consideration. They state the problem as fol-lows. If there exists a set of cutters and, aftermachining with these cutters, only finish machiningis needed for fillet radii corners, the problem is todetermine the cutter with the largest radius in thisset. The main concern is to make sure that the mate-rial left behind by the cutter at each of the convexvertices can be removed by one pass along theboundary of the finishing cutter. A convex vertex isdefined as a vertex at which the interior angle is lessthan 180°. For each convex vertex, the radius of thecircle touching the edges forming the vertex as wellas the fillet circle is found. Then Bala and Changcheck to see if this circle intersects any of the edgesof the bounding polygon or any of the islands. If anedge that violates the circle is found, the circle hasto be modified or the radius has to be reduced toremove this violation. The aim is to make the circletangential to this edge. After that, the reflex verticeshave to be handled. Reflex vertices are those ver-tices at which the interior angles are greater than180°. To solve the problem caused by edges, per-pendiculars are drawn from the reflex vertex to eachof the edges. First, Bala and Chang check to see ifthe perpendicular hits the edge or not. If not, noaction is taken. Otherwise, they have to check to seeif that particular edge is causing a valid constraint ornot. To resolve the constraint caused by other reflexvertices, the distances to all the valid reflex verticesare determined. The smallest of all these distancesgives the smallest constriction within the pocket.

After determining the cutter size, the feasible cuttermotion region can be identified and the cutter move-ment within the region can be optimized. By usingthe Bala and Chang algorithm, the maximal cutterthat can cover the target region will be the one thatis based on Alternative 1 of cutter feasible definition(see Section 3.1 for details).

Veeramani and Gau have developed a two-phasemethod to select a set of cutters.3 In the first phase,a concept called the Voronoi mountain is employedto calculate the material volume that can be removedby a specific cutting tool size, the material volumeremaining to be machined subsequently, and the cut-ter paths for each cutting tool. In the second phase,a dynamic programming approach is applied foroptimal selection of cutting tool sizes on the basis ofthe processing time. This algorithm considers geo-metric constrains as well as total processing time. Itis possible to save processing and machining costcompared to using a single cutter to machine theentire pocket. However, the algorithm of using theVoronoi mountain can only handle problems withoutopen edges.

Yang and Han presented a systematic tool pathgeneration methodology in which they incorporateinterference detection and optimal tool selection formachining free-form surfaces on three-axis CNCmachines using ball-end cutters.4 To find the optimaltools, a comparison of all possible combinations oftools is performed. The maximum number of select-ed cutters is restricted to three. This optimal toolselection method is designed for any type of para-metric surfaces to be machined. This algorithm hasthe following limitations. First, because the algo-rithms are grid-line based, if very fine resolution ofthe grid is imposed, high computational power isdemanded to implement the algorithm. Second, ifthe number of available tools is large, the compari-son of all possible combination of tools could betime consuming.

Mahadevan, Putta, and Sarma have developed afeature-free approach to automatic CAD/CAM inte-gration for three-axis machining.5 The algorithm isbased on the Voronoi diagram. The objective is toselect tools for global roughing and generate toolpaths directly from the shape of the workpiece. First,slices are generated as sequences of closed contours.Then a Voronoi diagram is employed to generate thepath of the centerline of the tool and calculate theaccessibility region on each slice. The criterion to

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select a cutter in this algorithm is to select the cutterthat can sweep much of the region of the sliceinstead of selecting a largest possible cutter. Becauselarge tools have a less reachable region than smalltools, they incur the penalty of tool changes, and sothe optimal tool sequence should be selected basedon the total time, which depends on the region thateach tool can access in each slice. To calculate theaccessible region, the overall geometry of the toolassembly including the tool holder and the spindle isconsidered in this algorithm.

Other research in the area of cutter selectionincludes work by Arya, Cheng, and Mount6 on mul-tiple tool selection; Dong, Li, and Vicker7 and Li,Dong, and Vicker8 on rough machining of sculp-tured parts; Lee and Chang9,10 on 2.5-D and 3-D NCsurface machining and five-axis sculptured surfacemachining; Lim, Corney, and Clark11 on tool sizing

for feature accessibility; Sun et al.12 on operationdecomposition for freeform surface features; andYou and Chu13 on NC rough cut machining. Theseworks are related to cutter selection but are signifi-cantly different in scope from the problem beingaddressed in this paper.

3. Definitions

3.1 BasicsThe most common milling problem is the problem

of cutting a given 2-D region at some constant depthusing one or more milling tools. In addition to theregion to be machined (called the target region, T),there is also an obstruction region, O, a region that thetool should not cut during machining. An example isshown in Figure 1. The target region and the obstruc-

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Figure 1Examples of Stock, Final Part, Target Region, and Obstruction Region

(a) Stock (b) Final part

Milling cutter

(c) Target and obstruction regions

Target boundary

Target region, T

Obstruction boundary

Obstruction region, O

tion region must both be regular sets but may eachconsist of a number of non-adjacent subregions:

T = T1 ∪ ... ∪ Tj

O = O1 ∪ ... ∪ Ok

It is assumed that the boundary of each subregion con-sists only of line segments and segments of circles.

As shown in Figure 1c, the target boundary, BT,is the boundary of the target region, and theobstruction boundary, BO, is the boundary of theobstruction region. The edges of the obstructionboundary are called obstruction edges. An edge ofthe target boundary is called a closed edge if itcoexists with an obstruction edge; otherwise, it iscalled an open edge. (Note that 2-D closed pocketsdo not have any open edges.) Dashed lines repre-sent open edges, solid lines represent closededges, and diagonal stripes represent the obstruc-tion region.

Figure 2 shows examples of open and closededges. Each closed edge, e, separates the material(that is, the obstruction region) from part of the tar-get region. The side of e on which the material liesis called e’s material side, and the other side is callede’s non-material side.

Let p = (x,y) be a point, and r ≥ 0. Then p’s r-offsetregion is the set of all points within distance r of p:

If S is a set of points, then its r-offset region is theunion of the r-offset regions of the individual points:

Intuitively, a point p is r-safe if a cutting tool ofradius r can be placed at p without intersecting theobstruction (an example is shown in Figure 3).Mathematically, this is equivalent to any of the fol-lowing two statements:

• p is r-safe if the distance between p and everypoint in the obstruction region is at least r;

• p is r-safe if offset(p,r) ∩* O = Ø.

where ∩* is regularized intersection. Similarly, –*denotes regularized difference and ∪ * denotes regu-larized union.

A set of points S is r-safe if all of the individualpoints are r-safe or, equivalently, if O ∩* offset(S,r)= Ø. Safe(S,r) is defined as the r-safe subset of a setS. Therefore,

safe(S,r) = S –* offset(O,r)

Figure 4 shows an example of safe(T,r).A point p is r-cuttable if there is an r-safe point q

such that p ∈ offset(q,r). Intuitively, this means p isr-cuttable if a cutting tool of radius r can be posi-tioned to cover p without intersecting the obstruc-tion region. A set of points S is r-cuttable if everypoint of S is r-cuttable.

Lemma 1 (Cuttability of all safe points). Any r-safepoint is also r-cuttable.

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( ) ( ) ( ) ( ){ }2 2offset , , :p r u v u x v y r= − + − ≤

( ) ( ) ( ) ( ){ }( )

2 2

,

offset , , :x y S

S r u v u x v y r∈

= − + − ≤∪

Figure 2Examples of Open, Closed, and Obstruction Edges

Non-material side of eMaterial side of eA closed edge, e

Obstructionedges

An open edge

Figure 3An r-offset Region, and Locations that are r-safe and Not r-safe

(x2,y2) is r-safe

offset(x,y),r)

Targetregion

Obstructionregion

(x1,y1) isnot r-safe

(x1,y1) (x2,y2)

(x,y)r

Proof. Let p be any r-safe point. Then p ∈offset(p,r), so p is in the r-offset region of an r-safepoint. □□

Lemma 2 (Non-cuttability of obstruction points).No point in the interior of O is r-cuttable.

Proof. Suppose p is in the interior of O. Let q be anypoint such that p ∈ offset(q,r). It suffices to showthat q cannot be r-safe. To show this, it is first notedthat the distance between p and q is ≤ r. Since p is inthe interior of O, it is easy to construct another pointp′ in the interior of O such that the distance betweenp and q is < r. Thus q is not r-safe. □□

If it is obvious what r is, then “offset” is used for “r-offset,” “cuttable” is used for “r-cuttable,” and so forth.

3.2 Cutter FeasibilityMost existing algorithms for cutter tool size

selection just find the “largest feasible cutter” with-out clearly stating what it means for a cutter to befeasible. There are at least three different possibledefinitions, based on three different criteria for whatkind of cutter path is acceptable.

Alternative 1: cutter feasibility based onVoronoi diagrams. It is easy to think of definingthe largest feasible cutter to be the largest cutterthat can go through all “bottlenecks” in the targetregion, as shown in Figure 5a. This is equivalent tosaying that a cutting tool C of radius r is feasibleif T’s Voronoi diagram6 is r-safe. In Figure 5a, C1

is an example of the largest feasible cutter basedon this definition.

Alternative 2: cutter feasibility based on a con-tinuous tool path. Rather than forcing the cutter togo through every bottleneck, the cutter can beallowed to go around some of them instead (Figure5b). Thus, one might want to say that a cutting toolof radius r is feasible if there is a continuous toolpath H that is r-safe and whose r-offset region con-tains T. Intuitively, this means that C can machineall of T in one continuous pass. In Figure 5b, C2 isan example of the largest feasible cutter based onthis definition.

Alternative 3: cutter feasibility based on cuttabil-ity. If the machining process can be interruptedbriefly, each bottleneck can be jumped over by liftingthe cutter up and putting it down again on the otherside of the bottleneck (Figure 5c). Thus, it might besaid that a cutting tool C of radius r is feasible if thereis an r-safe set of points S whose r-offset region con-tains T (or equivalently, if T is r-cuttable). Intuitively,this means that C can machine T using one or morepasses. In Figure 5c, C3 is an example of the maxi-mal cutter based on this definition.

For simple cases such as the situation shown inFigure 6, all three alternatives will give the sameanswer. However, in more complicated situations,such as the situation shown in Figure 5, Alternative3 will give the largest cutter size and Alternative 1will give the smallest cutter size.

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Figure 4Example of safe(T,r)

(a) Target and obstruction regions (b) Cutter and corresponding safe(T,r)

safe(T,r)

offset(O,r)

r

Cutter

T

O

The main goal of finding the maximal cutter is toreduce the manufacturing time and thereby reducethe manufacturing cost. Generally, using a small cut-ter requires much more time than would be neededby a larger cutter, even if the larger cutter needs tobe lifted up and set down again. Thus, sinceAlternative 3 gives the largest cutter, it is the defin-ition of cutter feasibility that is preferable in manymachining problems. Thus, the definition is what isused in this paper.

Problem Formulation: With the above definitionin mind, the cutter selection problem is defined as fol-lows: given a target region T and an obstruction regionO, find r* = max {r : a cutter of radius r is feasible}where feasibility is as defined in Alternative 3.

Note that if any two closed edges meet at a con-vex corner (see Figure 7 for an example), then thecorner point is not r-cuttable for any r > 0. In thiscase, it is said that the cutter selection problem isunsolvable. In all other cases, it is solvable.

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Figure 5Different Largest Feasible Cutters Resulting from Different Definitions of Cutter Feasibility

(a) Largest feasible cutter based on Alternative 1

(b) Largest feasible cutter based on Alternative 2

(c) Largest feasible cutter based on Alternative 3

C1 has to go through allbottleneck segments

A bottlenecksegment

C1

C2 can cut thisbottleneck segmentby an alternative path

C2 C2 has to go throughthis bottleneck to geta continuous path

C3C3 can jump overthis bottleneck,and thus can coverthe bottleneckshape by cutting infrom both sides

3.3 Critical and Non-Critical PointsIntuitively, the edge region of radius r for a

closed edge e is the region E(e,r) formed bysweeping a cutter of radius r along the non-mater-ial side of e. Mathematically speaking, a point p isin E(e,r) if p lies within a circle of radius r that istangent to ei on the non-material side of e. Figure8c shows an example of an edge region. The cumu-lative edge region E(r) is the union of the edgeregions of all closed edges. For an example, seeFigure 8c. From this definition, the followinglemma follows immediately:

Lemma 3.

max {r : E(r) ∩* O = Ø} = mine max{r : E(e,r)∩* O = Ø},

where the minimum is taken over all closed edges e.A point in T is r-critical if it is neither r-safe nor

in an edge region of radius r. The r-critical region isthe set K(r) of all r-critical points of T. Note that

K(r) = T –* safe(T,r) –* E(r) = (T –* E(r)) ∩* off-set(O,r).

The r-critical region may consist of several non-adjacent subregions:

K(r) = K1(r) ∪ ... ∪ Kk(r)

Below are several examples of what these subre-gions can look like:

• One kind of critical subregion can occur whentwo closed edges meet, as shown in Figure 9.However, this kind of critical region will notoccur if the angle between the two closed edgesis greater than 60°.

• Another kind of critical subregion can occurwhen an edge of an obstruction region occursslightly outside the target region, as shown inFigure 9. However, this will not happen if thedistance between the edge and the target regionis greater than the cutting tool radius r.

The following theorem says that a cutting toolcan cut every non-critical point if and only if thecutting tool’s radius is small enough that none ofthe edge regions intersect the obstruction region.Because the above examples suggest that mostdesigns are unlikely to contain critical points, thismeans that in most cases it is easy to compute themaximal cutting tool radius: just find the largestradius for which no edge region intersects theobstruction region.

Theorem 1: E(r) ∩* O = Ø if and only if every pointin T –* K(r) is r-cuttable.

Proof. It will first be proved that if E(r) ∩* O =Ø then every point in T –* K(r) is r-cuttable. Letp be any non-critical target point, that is, anypoint in T –* K(r). From the definition of K(r),there are two cases.

Case 1: p is r-safe. Then from Lemma 1, p is r-cuttable.

Case 2: p is in some edge region E(e,r). Then p iscontained in a circular region R of radius r that istangent to e on e’s non-material side. Let q be thecenter of R. Then R = offset(q,r). Every point of off-set(q,r) is in E(r), so offset(q,r) ∩* O = Ø, whenceq is r-safe. Thus p is in the r-offset region of an r-safe point, so p is r-cuttable.

Now, it will be proved that if E(r) ∩* O ≠ Ø then somepoint in T –* K(r) is not r-cuttable. Suppose E(r) ∩*

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Figure 6Simple Example of Maximal Cutting Tool

This cuttercan cover thewhole targetregion, but itis too small

This is the largestfeasible cutter

This cutter is toolarge because it willintersect with theobstruction region

Target region

Obstructionregion

Figure 7Example of Unsolvable Cutter Selection Problem

No non-zerocutter cancover thisconvex corner

Target region

Obstructionregion

O ≠ Ø. Then for some closed edge e, E(e,r) ∩* O ≠ Ø,so there is a point p ∈ E(e,r) such that p ∈ O. Fromthis it is easy to construct a point p′ ∈ E(e,r) such thatp′ is in the interior of O. Let q be the point of e that isclosest to p′. The only location where the cutter can cutq is the point c for which offset(c,r) is tangent to e atq. It is easy to show that offset(c,r) also contains p′.Thus c is not r-safe, so q is not r-cuttable. An exampleis shown in Figure 10. □□

The above theorem says nothing about whetherp is r-cuttable if p is in the critical region K(r). Insuch cases, p may or may not be r-cuttable. If p ∈K(r), then p ∈ offset(Oj,r) for some subregion Oj

of O. If Oj is convex and if p ∉ offset(Ok,r) for allk ≠ j, then p is r-cuttable (see Figure 11 for anexample). However, if Oj is not convex or if p ∈offset(Ok,r) for some k ≠ j, then sometimes p is r-cuttable and sometimes it is not.

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Figure 8Examples of Edge Region, Safe Region, and Critical Region

(f) After obtaining the safe region,the critical region can be found

Critical region, K(r)

(e) Safe(T,r) can be found after the offset regionand cumulative edge region are found

r-safe subset of T,safe(T,r) E(r)

offset(O,r)

(d) Given a radius r, the offset of theobstruction region can be found

(c) For closed edges, the edge region for each edgecan be found, thus finding the cumulative edge region

offset(O,r)Edge region, E(e,r) Cumulative edgeregion, E(r)

(b) For closed edges, the maximal edgeregion radius can be found

(a) Obstruction and target regions

Closed edge, e

Target region, TObstruction

subregion, Oj

Maximal edge regionradius, rmax(e) Radius of this cutter

is too big

Radius of this cutteris too small

4. Algorithm for Findingthe Maximal Cutter

The main algorithm for the maximal cutter selec-tion problem is called Find_Maximal_Cutter_Ra-dius (FMCR for short). For every closed edge a, thisalgorithm calls the subroutine Maximal_Edge_Re-gion_Radius to find the maximal edge-region radiusfor a. Then it uses the smallest of those radii (denot-ed by rE) to compute the cumulative edge regionE(rE), the safe region S(T,rE), and the critical regionK(rE). The algorithm then calls the subroutineMaximal_Critical_Region_Radius to find the largestr (denoted by rK) such that K(rE) is r-cuttable. Thefinal result is the minimum of that radius rE and rK.

Procedure Find_Maximal_Cutter_Radius(T,O)//T is the target region, and O is the obstruction region.

1. rE = ∞, rK = ∞;2. For each closed edge a and obstruction edge b, do

• r = Maximal_Edge_Region_Radius(a,b)//this subroutine is described in Section 5

• rE = min{r,rE}; //rE is now the largest radiusfor which no edge region intersects the obstruction

3. E = E(rE); //the cumulative edge region4. F = offset(O,rE); //the offset of the obstruc-

tion region5. S = T –* F; //the “safe” region6. K = T –* S –* E //the critical region7. If K is nonempty then

• rK =Maximal_Critical_Region_Radius(T,O,K,rE)//this subroutine is described in Section 6//rK is now the largest r such that K is r-cuttable

• Return r = min{ rE, rK }

8. Else return r = rE;

5. Finding the Maximal Swept Cutterfor a Closed Edge

In the algorithm Find_Maximal_Cutter_Radiusdescribed in Section 4, the purpose of the subroutineMaximal_Edge_Region_Radius is to solve the fol-lowing problem: given a closed edge a and anobstruction edge b, find the largest r such that theregion E(a,r) does not intersect the edge b.

If the closed edge is an arc segment and its angleis greater than 180°, then this arc is split into two arcsegments such that each arc segment’s angle is lessthan or equal to 180°. For each closed edge a, thefollowing is done:

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Figure 9Some Examples of Critical Regions

Obstructionregion

Critical subregionsthat occur when anedge of an obstructionregion occurs slightlyoutside the targetregion

Target region

Figure 10In this example, if E(r) ∩∩* O ≠ Ø,

then some points in T –* K(r) are not r-cuttable

q

Offsetregion Critical subregions

that occur whentwo closed edgesmeet

Edge region

p´

Figure 11Example of a Critical Subregion

That Can Be Covered Without Reducing the Tool Radius

Obstruction region, O

Target region, T

Offset(O,r)

The criticalsubregion isr-cuttable

CriticalsubregionOne end ofan edgeregion ofradius r

a. Extend closed edge a in each direction at its endpoints to infinity using rays that are tangent toend points and going away from the edge.

b. The extended edge divides the space into twohalf-spaces, Hm and Hn. Hm is the half-space thatcontains the material side of a. Hn is half-spacethat contains the non-material side of a.

c. Use perpendicular lines at end points to splitHm into the following three regions: one mid-dle region of a and two end-regions of a.Figure 12a shows examples of these regionswhen a is a line segment, and Figure 13ashows examples of these regions when a is anarc segment.

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Figure 12Finding Maximal Cutter Radius for a Linear Closed Edge in Presence of Different Types of Obstruction Edges

End region

b

(d) Another example where the obstructionedges are located in the end regions

(c) An example where the obstructionedges are located in the end regions

Yr

a

Yr

b

b Y

ra

Y r

b

(b) A case where the obstruction edgesare located in the middle region

(a) Closed edge a is a line segment

b

Yr

a

b

Hm

p1

l1 l2Hn

p2

Middle region End region

Closed edge a

Figure 13Finding Maximal Cutter Radius for Circular Closed Edge in Presence of Different Types of Obstruction Edges

End region

b

(d) Another example where the obstructionedges are located in the end regions

(c) An example where the obstructionedges are located in the end regions

Y

r

aY

r

bb′

Y

r

a Y

r

b

(b) A case where the obstructionedges are located in the middle region

(a) Closed edge a is an arc segment

b

Yr

a

b

Hm

Hn

Middle region

End region

b

a

b′

a

a

bb

a

a

Procedure Maximal_Edge_Region_Radius (a, b)1. Split b into at most two segments such that each

segment is completely contained in Hm or Hn.2. r = ∞;3. for every segment b′ of b that is in Hn, do the fol-

lowing:r′ =

Max_edge_region_radius_for_closed_edges(a, b′);r = min{r, r′};

4. return r.

Procedure Max_edge_region_radius_for_closed_edges(a, b)

1. Split b into at most three segments such thateach segment is in the middle region of a or inone of the two end regions of a.

2. If b is in the middle region of a, thenReturn 1/2 of the distance between a and b.//This is the same as the maximum diameter of

any circle that touches b and is tangent to a.Examples are shown in Figures 12b and 13b.

3. If b is in the end region of a, then• Let e be whichever end point of a is closest

to b;• Let p be the point in b that is closest to e;• Let Y be the circle that is tangent to a at e

and contains p;• Return the radius of Y.//In practice, if a circle can be found that is tan-gent to both a and b at p, then Y is that circle.Otherwise, Y is found by finding the minimalcircle that is tangent to a at p and passes throughone end point of b. Examples are shown inFigures 12c and 12d and Figures 13c and 13d.

6. Finding the Maximal Cutter forCritical Region

In Section 4, if the critical region in Step 6 ofFind_Maximal_Cutter_Radius is not empty, thena cutter of radius rE may be too large to cover all ofthe critical region. In this case, the subroutineMaximal_Critical_Region_Radius, described be-low, will find a maximal cutter radius that cancover the critical region. In this subroutine, thenumber ri is a constant set by the user. It shouldcorrespond to the increment in tool sizes that isavailable on a shop floor.

Procedure Maximal_Critical_Region_Radius(T,O,K,rE)//T is the target region, O is the obstruction region,K is the critical region, and rE is the maximal edge-region radius.

Let rK = rE.1. Let U be a finite region that encloses the

obstruction and target regions.//In practice U is computed by computing thebounding box of T ∪ * O.

2. loop• P = U —* offset(O,rE).

//P is = {safe locations for a cutter of radius rE}

• S = offset(U,rE)//S = {points cuttable by a cutter of radius rE}

• if K ⊆ S, then return rK.• else rK = rK – ri

//the constant ri is described in the text.3. repeat

//Figure 14 shows an example of how this procedureworks. In Figures 14c and 14d, rK = rE, and K ⊄ S.After one or more iterations, rK = rE – crI, where c is aconstant, then K ⊆ S as shown in Figures 14e and 14f.

7. Discussion of Correctness ofAlgorithm

For a solvable problem, the algorithm exhibits thefollowing two properties:

Property 1. If the critical region in Step 6 of FMCRis empty, then FMCR returns r* = max {r : a cutterof radius r is feasible}.

Proof. Suppose there is no critical region, and let rbe the number returned by FMCR. Here r = rE. ThenrE is the minimum, over all closed edges e, of max{r : E(e,r) ∩* O = Ø}. Thus from Lemma 3, rE is thelargest number such that E(rE) ∩* O = Ø. Therefore,from Theorem 1, a cutter of radius rE will be able tocut all of T. For any r > rE, E(r) ∩* O ≠ Ø. Thus fromTheorem 1, T would not be r-cuttable. □□

Property 2. If the leftover region in Step 6 of FMCRis not empty, then FMCR returns a number r such thata cutter of radius r is feasible, and r* – r < ri (wherer* is the maximum feasible cutter radius).

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Proof. Let r be the number returned by FMCR. Here r= min{ rE, rK }. Then E(r) ∩* O = Ø, so from Theorem1 it is known that T – K(r) is r-cuttable. ProcedureMaximal_Critical_Region_Radius only allows cuttersthat can cover K. Therefore r can cover the target regionand is feasible. If rK is equal to rE, then r is the exact solu-tion and there is no difference between the theoreticalanswer and the result found by FMCR. If rK < rE, then r+ ri cannot be a feasible solution. Otherwise, FMCR

would have returned this solution. Therefore, in this case,theoretically maximal radius r* < r + ri. Therefore thedifference between the theoretically maximal diameterand result returned by FMCR is smaller than ri.

8. Implementation and ExamplesThe example shown in the Figure 15 is used to

illustrate the operation of the algorithm. The target

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Figure 14Using Approximation Algorithm to Find Near Maximal Cutter for Critical Region

(f) After offset P(r2) by using radius r2, the whole target regionis safe; therefore, r2 is the approximate maximal cutter

Part of K is not in S(r1)

(e) Reduce the radius from r1 to r2 and offsetobstruction region by using radius r2

offset(O,r), r2 = r1 - r´, r´ = crI

(d) After offset P(r1) by using radius r1,there exists some region that is not safe

(c) Offset obstruction region using radius r1

P(r1)

Edge region, E(e,r)offset(O,r)

(b) Critical regions(a) Obstruction and target regions

O

T

K

S(r1)

P(r2)

S(r2)

region and obstruction regions are shown in Figure15a. The details are as follows: The main algorithm forthe maximal cutter selection problem is calledFind_Maximal_Cutter_Radius (FMCR for short).For every closed edge a, this algorithm calls the sub-routine Maximal_Edge_Region_Radius to find themaximal edge-region radius for a. Then it uses thesmallest of those radii (denoted by rE) to compute thecumulative edge region E(rE), the safe region S(T, rE),and the critical region K(rE). The algorithm then callsthe subroutine Maximal_Critical_Region_Radius tofind the largest r (denoted by rK) such that K(rE) is r-cuttable. The final result is the minimum of that radiusrE and rK.

• First, for every closed edge a is found the maximaledge-region radius for a (shown in Figure 15b).

• The smallest of those radii, rE, is used to com-pute the cumulative edge-region E(rE) and saferegion S (shown in Figure 15c).

• Then the critical region K is obtained. For the crit-ical region is found the maximal cutter with radiusrK that covers it and does not interfere with theobstruction region, as shown in Figure 15d.

• The maximal cutter that covers the target regionwithout interfering with the obstruction regionwill be the minimal one of rE and rK.

The algorithm has been implemented to find themaximal cutter for 2-D milling operations. The coreprogramming work is done by using C++ on a UNIXsystem. Meanwhile, the core code has been linkedwith the ACIS Toolkit® and JAVA 3D® such that byinputting a solid model of a milling problem, theprofile of target and obstruction regions can beextracted and then the core code can be executed toget the maximal cutter. Finally, the result can beshown in a 3-D version.

Figures 16-19 show the results of the maximalcutters selected by the algorithm. The algorithm

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Figure 15Example of Operation of Algorithm

Obstruction region

(a) Problem illustration (b) Maximal edge region radius

(c) Edge region, safe region, and critical region (d) Maximal cutter that can cover critical region

Target region

Closed edge

Maximal edgeregion radius, rE

Edge region

Safe region

Critical region

Maximal circlewith radius rKthat can covercritical region

solved every one of those examples in less than onesecond on an Ultra 10 computer.

9. Conclusion and DiscussionThis paper presented a geometric algorithm for

finding the maximal cutter size for a 2-D millingprocess. The algorithm has the following properties:

1. It finds the largest cutter that can cover theregion to be machined without interfering withthe obstruction region.

2. In addition to solving traditional pocket-millingproblems, the algorithm can solve a wide varietyof milling problems that involve open edges.Consideration of open edges is extremely impor-tant when near net shape castings are used asstarting stocks.

3. The algorithm uses a cutter feasibility definitionbased on the cutter’s ability to cover the targetregion. Therefore, it can find larger cutters thanthe ones found by algorithms that are based on

alternative definitions of feasibility (for example,either based on covering every bottleneck in thetarget region or existence of a continuous pathbetween every pair of points in the target region).

The maximal cutter found is based on the geo-metric constraints. In actual machining, it is neces-sary to consider several other cutting constraints.Here are some examples:

• There are several cutting parameters that may influ-ence the selection of cutter size. Cutters that conflictwith the machining constraints cannot be used. Forexample, it is known that the material removal rateis proportional to the diameter of the cutter inmilling operations. As a result, if a bigger cutter isused, there can be a higher material removal rate,thus saving cutting time. On the other hand, themaximum power of a machine is constant. Therequired cutting power is proportional to the metalremoval rate. Therefore, the maximal diameter isactually limited by the maximum machine power.

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Figure 16Example 1

(a) Starting stock (c) Profile of final part along withmaximal cutter found by algorithm

(b) Final part

Figure 17Example 2


(b) Final part

• Sometimes the cutter selected by the region-cov-ering idea may not be the best one for manufac-turing. For example, Figure 20 shows an exam-ple in which, if the maximal cutter selected bythe algorithm is used, it will have to be lifted upand put down several times. In this particularcase, the maximal cutter selected may not savetotal cutting time and may result in a bad manu-facturing surface. Besides the geometric con-straints, manufacturing knowledge is also need-ed to help decide which is the best cutter size.

• In addition to geometric considerationsdescribed in this paper, several other machiningconsiderations, such as available fixturingoptions, surface finish requirements, availablecutting tool geometries, and the resulting cuttingforces, play a role in cutter selection and shouldbe considered in selecting milling cutters.

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Figure 18Example 3


(b) Final part

Figure 19Example 4


(b) Final part

Figure 20Manufacturing Consideration in Choosing Maximal Cutter

The algorithm is currently being extended to per-form cutter selection optimization by consideringmultiple cutters. Preliminary results are described inYao, Gupta, and Nau.1

AcknowledgmentThis research has been supported by NSF grants

DMI9896255, DMI9713718, and EIA-9729827.Opinions expressed in this paper are those of theauthors and do not necessarily reflect the opinion ofthe National Science Foundation.

References1. Z. Yao, S.K. Gupta, and D.S. Nau, “Selecting Flat End Mills for 2-

1/2D Milling Operations,” ISR Technical Report, TR 2000-41 (CollegePark, MD: Univ. of Maryland, 2000).

2. M. Bala and T.C. Chang, “Automatic Cutter Selection and OptimalCutter-path Generation for Prismatic Parts,” Int’l Journal of ProductionResearch (v29, n11, 1991), pp2163-2176.

3. D. Veeramani and Y.S. Gau, “Selection of an Optimal Set of Cutting-tool Sizes for 2.5D Pocket Machining,” Computer Aided Design (v29, n12,1997), pp869-877.

4. D.C.H. Yang and Z. Han, “Interference Detection and Optimal ToolSelection in 3-axis NC Machining of Free-form Surface,” Computer AidedDesign (v31, n5, 1999), pp303-315.

5. B. Mahadevan, L. Putta, and S. Sarma, “A Feature Free Approach toTool Selection and Path Planning in 3-axis Rough Cutting,” Proc. of 1stInt’l Conf. on Responsive Mfg., Nottingham, UK, Sept. 1997, pp47-60.

6. S. Arya, S.W. Cheng, and D.M. Mount, “Approximation Algorithmfor Multiple-tool Milling,” Proc. of 14th Annual ACM Symp. onComputational Geometry, 1998, pp297-306.

7. Z. Dong, H. Li, and G.W. Vicker, “Optimal Rough Machining ofSculptured Parts on a CNC Milling Machine,” Trans. of ASME, Journal ofEngg. for Industry (v115, n64, 1993), pp424-431.

8. H. Li, Z. Dong, and G.W. Vicker, “Optimal Toolpath Pattern Identificationfor Single Island, Sculptured Part Rough Machining Using Fuzzy PatternAnalysis,” Computer Aided Design (v26, n11, 1994), pp787-795.

9. Y.S. Lee and T.C. Chang, “Application of Computational Geometry inOptimization 2.5D and 3D NC Surface Machining,” Computers in Industry(v26, n1, 1995), pp41-59.10. Y.S. Lee and T.C. Chang, “Automatic Cutter Selection for 5-axisSculptured Surface Machining,” Int’l Journal of Production Research (v34,n4, 1996), pp977-998.11. T. Lim, J. Corney, and D.E.R. Clark, “Exact Tool Sizing for FeatureAccessibility,” Int’l Journal of Advanced Mfg. Technology (v16, 2000),pp791-802.12. G. Sun, F. Wang, P. Wright, and C. Sequin, “Operation Decompositionfor Freeform Surface Features in Process Planning,” Proc. of DETC 1999:1999 ASME Design Engg. Technical Conf., Las Vegas, NV, Sept. 12-15,1999.

13. C.F. You and C.H. Chu, “An Automatic Path Generation Method ofNC Rough Cut Machining from Solid Models,” Computers in Industry(v26, n1, 1995), pp161-173.

Authors’ BiographiesZhiyang Yao is a PhD student in the mechanical engineering depart-

ment at the University of Maryland, College Park. He received a BS in1995 and an MS in 1998 in mechanical engineering from TsinghuaUniversity, P.R. China. His research interests are computer-aideddesign and manufacturing, concurrent engineering, and geometric rea-soning. Currently, he is working on constructing innovative processplans that can significantly reduce time to market and enable cost-effective small-batch manufacturing. He has authored or coauthored 11articles in journals, conference proceedings, and technical reports. Heis a student member of ASME.

Satyandra K. Gupta is an assistant professor at the University ofMaryland in the mechanical engineering department and the Institute forSystems Research (ISR). He received a PhD in mechanical engineeringfrom the University of Maryland at College Park in 1994. Prior to joiningthe University of Maryland, he was a research scientist in the RoboticsInstitute and an adjunct assistant professor of manufacturing in theGraduate School of Industrial Administration at Carnegie MellonUniversity. The objective of Dr. Gupta’s research is to provide innovativeproduct realization methodologies that can significantly reduce time tomarket and enable cost-effective small-batch manufacturing. Over the last10 years, Dr. Gupta has participated in a number of design and manufac-turing automation research projects. Representative projects include gen-erative process planning for machining, automated manufacturabilityanalysis, automated redesign, generative process planning for sheet metalbending, assembly planning and simulation, extraction of lumped para-meter simulation models for microelectromechanical systems, distributeddesign and manufacturing for solid freeform fabrication, and automatedmold design and fabrication. He has authored or coauthored more than 70articles in journals, conference proceedings, and book chapters. He hasorganized several conference sessions on computer-aided design andmanufacturing areas. Dr. Gupta has won many honors and awards for hisacademic excellence and his research contribution to design and manu-facturing automation area.

Dana S. Nau is a professor at the University of Maryland in theDepartment of Computer Science and the Institute for Systems Research(ISR). He also has affiliate appointments with the Institute for AdvancedComputer Studies and the Department of Mechanical Engineering. Dr. Naureceived a BS in applied mathematics from the University of Missouri atRolla in 1974. He received an AM (in 1976) and PhD (in 1979) in comput-er science from Duke University, where he was an NSF graduate fellow anda James B. Duke graduate fellow. He has had summer and/or sabbaticalappointments at IBM Research, NIST, the University of Rochester, andGeneral Motors Research Laboratories. Dr. Nau’s research interests includeAI planning and searching techniques and computer-integrated design andmanufacturing. He has co-edited two books and has published more than200 refereed technical papers. Copies of recent papers and summaries ofcurrent research projects are available at http://www.cs.umd.edu/users/nau.

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