Mahmoud M. RababahDepartment of Mechanical Engineering,

The Hashemite University,

Zarqa 13115, Jordan

Zezhong C. Chen1

Department of Mechanical and

Industrial Engineering,

Concordia University,

Montreal, QC, H3G 1M8, Canada

e-mail: zezhong.chen@concordia.ca

An Automated and AccurateCNC Programming Approachto Five-Axis Flute Grindingof Cylindrical End-MillsUsing the Direct MethodIn solid carbide end-mills, the flutes significantly affect the tool’s cutting performanceand life, and the core radius mainly affects the tool’s rigidity. The current CNC program-ming techniques can correctly determine the orientation of the wheel so that it grinds therake face with the specified rake angle; however, it cannot accurately determine thewheel location for the direct method and, consequently, the desired core radius is notguaranteed. To address this problem, a new CNC programming approach is proposed toaccurately calculate the wheel orientation and location (WOL) in 5-axis grinding of thecylindrical end-mill flutes. In this work, a new concept of 5-axis CNC grinding—effectivegrinding edge (EGE)—is first proposed to represent the instantaneous grinding edge ofthe wheel, and the parametric equations of the effective grinding edge are formulated.The wheel orientation and location in 5-axis flute grinding are calculated automaticallyand accurately so that the rake angle of the rake face and the core radius are ensured.The new approach is verified with several examples in this work. Therefore, it canimprove the end-mill quality and lays a good foundation for the computer-aided design/computer-aided engineering/computer-aided manufacturing (CAD/CAE/CAM) of end-mills. [DOI: 10.1115/1.4023271]

Keywords: effective grinding edge, wheel orientation and location, CNC programmingfor grinding, end-mill flute grinding, 5-axis tool grinding

1 Introduction

For an end-mill, its flute geometries substantially affect the toolrigidity and the cutting forces in machining. By definition, a fluteincludes a flute surface and a rake face along a helical side cuttingedge; the two surfaces are next to each other. The rake angle ofthe rake face determines the cutting forces. The flute surfacedetermines the core radius and the tool moment of inertia; it alsoprovides space for chip evacuation. Thus, the flutes of a tool affectits cutting performance and life. Therefore, many researchershave conducted a large amount of work on flutes grinding of solidcarbide end-mills in the past years; their research is normally clas-sified into two categories: the direct and the inverse methods. Inthe direct method, a standard grinding-wheel is subjectivelyselected, the WOL is approximately calculated, and then it grindsthe flute along the helical cutting edge to generate the rake faceand the flute surface simultaneously. The flutes cut with thismethod can be modeled using the envelop theory [1]. This processcan make the rake face with the specified rake angle; however, thedesired core radius is not guaranteed, and the machining error canreduce the tool rigidity.

The inverse method is more complicated in computing thegrinding-wheel profile with the following principle. In 2-axis flutegrinding, the grinding-wheel and the end-mill axes are skew lines,and geometrically, at any WOL of the wheel path, the grinding-wheel contacts the flute of the end-mill at a curve. Thus, this curveis obtained based on principle that is the normal vector of the flutesurface or the rake face at any point on the curve should pass

through the grinding-wheel axis. Tsai and Hsieh [2] developed aninverse method of finding a grinding-wheel profile for makingball end-mills. Similarly, Ren et al. [3] used the inverse method tocalculate a special grinding-wheel profile and grind tapered filletend-mills. Moreover, Chen et al. [4] generated the profile of anonstandard grinding-wheel for making concave cone end-mills,and Wu and Chen [5] found the profile of a grinding-wheel formaking cutters with a circular arc generatrix. Besides, severalresearchers [6–10] applied this principle to calculate the profilesof special grinding-wheels for making new types of end-mills.

Since the nonstandard grinding-wheels aforementioned canfacilitate grinding a prespecified flute surface of an end-mill in a2-axis path, the normal rake angle fluctuates along the cuttingedge, which can cause different cutting forces along the end-millaxis, resulting vibration in machining (the helical angle of themachined flute is not precise). Besides, the profiles of thegrinding-wheels are not presented with explicit equations. Toaddress the problems, Chen and Bin [11] and Feng and Bin [12]proposed an approach. In the approach, the rake face is cut on amulti-axis grinding machine tool with standard grinding-wheels.Since simple standard grinding-wheels are used, more relative motionsare required to ensure a constant normal rake angle along the cuttingedge. Nevertheless, the approach is only used for rake face grinding,inappropriate for flute surface grinding. Moreover, it is only designedto use special grinding-wheel shapes (torus and spherical).

To eliminate the existent problems of making flutes of solidcarbide end-mills using the direct method, our innovative workestablishes a NC programming theory for 5-axis grinding of cylin-drical end-mill flutes and constructs genuine geometric models ofmachined flutes, which are necessary for finite element analysiswith high fidelity. In this work, first, the generic mathematicalequations of grinding-wheel locations and orientations in 5-axisgrinding of cylindrical end-mill flutes with standard profile

1Corresponding author.Contributed by the Manufacturing Engineering Division of ASME for publication

in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedSeptember 30, 2012; final manuscript received November 13, 2012; published onlineJanuary 22, 2013. Assoc. Editor: Allen Y. Yi.

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grinding-wheels are derived as a new multi-axis CNC programmingtheory. Then, the new theory is applied to a cylindrical end-millflute for its 5-axis grinding-wheel paths. Finally, the method isverified using grinding simulation.

2 Geometric Fundamentals of the Cylindrical

End-Mill Flutes

A solid carbide end-mill mainly includes several geometric fea-tures, such as several helical flutes and the corresponding side andbottom cutting edges, the first and the second flank faces of eachcutting edge, a gash (or a split) on the bottom, and the centralcore. In this work, a helical flute is defined with a flute surfaceand a rake face along the side cutting edge; the reason for defininga flute in such a way is that a flute is machined with a standardgrinding-wheel along a specified path in the direct method. Sincethe flutes of a solid carbide end-mill are important to the tool per-formance and life, our research is focused on accurate machiningof the flutes. Due to the complex shape of the flutes, they are oftenmachined with the 5-axis CNC grinding; unfortunately, the CNCprogramming technique for 5-axis flute grinding has not fullyestablished and the flute cannot be accurately machined. To de-velop an advanced programming approach, the basics of the flutegeometry of a cylindrical end-mill are introduced in this section.

2.1 Parameters of the Flute Cross-Section Profile.Theoretically, a solid carbide cylindrical end-mill can be firstdesigned with a commercial CAD/CAM software system, and then itcan be made on a 5-axis CNC grinding machine. This is called theinverse flute machining method. As an important feature of thetool, the flute is designed in a way that the flute cross-section pro-file is defined and swept along the helical side cutting edge. Forthe illustration purpose, a conventional design of the flute profileis provided and plotted in Fig. 1. In this design, a tool coordinatesystem <T ¼: o; x; y; zð Þ, in which the axes are denoted with smallitalic letters, is established in the following process: (a) the origino is at the tool tip, (b) the z-axis is aligned with the tool axis andpointing inside the tool, and (c) the principle plane xoyð Þ is per-pendicular to the tool axis representing its cross sections, and thex-axis passes the first flute point on the cutting edge. This diagramshows a flute profile on the bottom of a four-flute end-mill and thetool and the core circles with the tool radius rT and the core radi-usrC, respectively. In detail, the flute includes three segments: (1)line segment f0f1 forms the rake face with radial rake angle aR,

which is the angle between f0f1 and the x-axis; (2) circular arc_f1f2of radius r1 is tangent to both f0f1 and the core circle of radius rC;(3) circular arc _f2f3 of radius r2 is tangent to_f1f2, and the two cir-cular arcs form the flute surface. Besides, this diagram shows thetwo relief faces: line segment f3f4 tangent to_f2f3 generates the sec-ondary relief surface with relief angle cS, and line segment f4f5generates the primary relief surface with relief angle cP. For dif-ferent end-mills, the flute profile can be varied. Unfortunately, theinverse flute machining method is not popular in industry.

To machine the flutes, compared to the aforementioned inversemethod, the direct method is more popular in the tool manufactur-ing companies. In this method, a standard grinding-wheel isselected subjectively, and then the grinding-wheel moves along itspath in the 5-axis CNC grinding. Since the key parameters of theflutes are the rake angle and the core radius, their accuracy shouldbe ensured during machining. Regarding the flute shape, it iswidely accepted that the flute shape is less important to the toolperformance and life; thus, it is more flexible without any machin-ing tolerance specified, and the actual shape is mainly dependenton the shape of the selected grinding-wheel. Therefore, the objec-tive of this work is to establish a new CNC programming tech-nique for the 5-axis flute grinding with a variety of standardgrinding-wheels. More specifically, based on a selected grinding-wheel, its orientation and location in the 5-axis grinding should bedetermined automatically and accurately so that the specified rakeangle and the core radius are guaranteed. Due to the large flexibil-ity of the flute shape, the way of grinding-wheel selection is notunder investigation in this work.

2.2 Parametric Representation of the Side CuttingEdge. As guides of the flutes, the side cutting edges of an end-mill are crucial to the 5-axis flute grinding; by natural, they arethe paths of wheel location in the 5-axis CNC program. In thiswork, the side cutting edges are defined with the parametric repre-sentation in the aforesaid tool coordinate system <T. Generally,each side cutting edge is a helix on the envelope of the tool, andall of them are evenly distributed around the tool axis. The toolenvelope t is a revolving surface, and a longitude of this surface ishere called a generatrix. Hence, as a mathematical model of theside cutting edge, the helix has a constant helical angle w at anypoint on the side cutting edge, which is the angle between the cor-responding tangent vector of the cutting edge and the generatrix.To define the side cutting edge in the tool coordinate system, theend-mill envelope is represented with two parameters: the z

Fig. 1 The plot of a basic flute profile of a four flute end-mill in the tool coordinate system

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coordinate and the rotation angle h about the z-axis. The paramet-ric equation of the envelope is

t z; hð Þ ¼r zð Þ � cos h

r zð Þ � sin h

z

264

375 (1)

where r zð Þ is the radius of the corresponding circle on the planewith coordinate z. Figure 2 shows the envelope and the circle inthe tool coordinate system. Based on Eq. (1), the first partial deriv-atives of the envelope t z; hð Þ in terms of the parameters arederived as

@t

@z¼

@r zð Þ@z� cos h

@r zð Þ@z� sin h

1

266664

377775 (2a)

and

@t

@h¼�r zð Þ � sin h

r zð Þ � cos h

0

264

375 (2b)

In this work, the differential 1-forms of the tangent vectors ofthe helical side cutting edge and the generatrix are denoted as dtand dt, respectively. They can be found with the followingequations:

dt ¼:@t

@z� dzþ @t

@h� dh; and dt ¼:

@t

@z� dz (3)

Thus, the relationship between the helical angle w and thesetwo vectors can be formulated as

cos w ¼ dt � dt

dtk k � dtk k (4)

By substituting Eq. (3) into (4), the relationship between the dif-ferential 1-forms of the parameters, z and h, can be formulated as

dzð Þ2¼

@t

@h

� �0� @t

@h

@t

@z

� �0� @t

@z

� cot wð Þ2� dhð Þ2 (5)

By integrating both sides of Eq. (5), the explicit function of theparameter z hð Þ in terms of h can be established. Thus, the para-metric representation of the side cutting edge c is

c hð Þ ¼:

r z hð Þð Þ � cos h

r z hð Þð Þ � sin h

zðhÞ

264

375 (6)

Therefore, the helical side cutting edge of a cylindrical end-millof radius rT can be expressed as

c hð Þ ¼

rT � cos h

rT � sin hrT

tan w� h

2664

3775 (7)

2.3 Unit Normal Vector of the Rake Face. It is well-knownthat the rake angle of the rake face along the side cutting edge is acritical parameter that determines the cutting forces and the cut-ting temperature. For clarity, the rake angle is named more specif-ically, according to its position; the radial rake angle aR is therake angle on a plane perpendicular to the tool axis (see Fig. 1),and the normal rake angle an is on a plane perpendicular to thetangent vector of the side cutting edge (see Figs. 1 and 3). Inindustry, the rake angle often refers to the normal rake angle. Togrind a flute with the specified (normal) rake angle, the grinding-wheel should be properly oriented during machining. Hence, thewheel orientation should be accurately calculated in the 5-axisCNC programming.

According to the geometric model of the 5-axis fluting, a lateralface of the grinding-wheel should be aligned with the rake face,which means the normal vectors of the two faces are in-lined.Therefore, it is necessary to find out the unit normal vector of therake face in the 5-axis CNC programming. To formulate this nor-mal vector, we start with the definition of the normal rake angle.The following diagram, Fig. 3, illustrates a normal rake angle at apoint on the side cutting edge and the angle-related geometries.The procedure of defining the normal rake angle of a rake face ata cutting edge point p is (1) to construct a plane C perpendicularto the cutting edge; and (2) to construct a plane P perpendicular

Fig. 2 Illustration of the parameters of the cylindrical end-millenvelope in the tool coordinate system

Fig. 3 Illustration of constructing the unit normal of the rakeface at a side cutting edge point

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to the cutting velocity. Here, the plane C is perpendicular to thetangent vector s of the helical cutting edge c at the point p, andthe plane P passes through the tool axis (or the z-axis) and thepoint p. The intersection between these planes is line i, and theintersection between the plane C and the rake face (not shown inFig. 3) is line r. Thus, the angle between the lines i and r is thenormal rake angle an. Since the rake face is spanned by r and s,the normal of this face can be found as the cross product of r ands. Based on the definition of the normal rake angle, its equationcan be formulated in the tool coordinate system <T.

Suppose the coordinate of the point p is ½px; py; pz�0

in the toolcoordinate system <T. According to the equation of the side cut-ting edge, its unit tangent vector s at the point p is

s ¼sx

sy

sz

24

35 ¼

dc

dhdc

dh

��������

(8)

The plane C is perpendicular to s. When the tool self-rotates, thecutting speed v at the point p is normal to the tool axis and is tan-gent to the tool envelope. The plane P is perpendicular to the cut-ting speed v at the point p, which can be simply represented as

v ¼�py

px

0

264

375 (9)

Since line i is the intersection of the planes, C and P, the line vec-tor i can be found as the cross product of v and s. The equation ofthe line vector i is

i ¼ix

iy

iz

264

375 ¼ v� s

v� sk k (10)

To represent the line r in <T, a coordinate system <P at the pointp is constructed with vectors s, w, and i, and the vector w is thecross-product of i and s. Based on Fig. 3, the representation ofw is

w ¼wx

wy

wz

264

375 ¼ i� s (11)

Since the normal rake angle an is the angle between r and i, theline vector r can be easily represented in the coordinate system<P as

r<P ¼

rS

rW

rI

264

375 ¼

0

sin anð Þcos anð Þ

264

375 (12)

According to the relationship between the two coordinate systems,<P and <T, the mapping matrix M<P!<T

from <P to <T is

M<P!<T ¼

sx wx ix 0

sy wy iy 0

sz wz iz 0

0 0 0 1

26664

37775 (13)

Therefore, the equation of vector r in <Tcan be found as

r

1

� �¼

rx

ry

rz

1

26664

37775 ¼M<P!<T � r<

P

1

" #

¼

sx wx ix 0

sy wy iy 0

sz wz iz 0

0 0 0 1

2664

3775 �

0

sin anð Þcos anð Þ

1

2664

3775 (14)

Finally, the unit normal vector n of the rake face can be expressedas

n hð Þ ¼ r� s (15)

In the 5-axis flute grinding, the grinding face should be alignedwith the rake face of the tool, therefore, the normal vector of therake face is used to determine the grinding-wheel orientation inthe 5-axis CNC programming.

3 Five-Axis CNC Grinding of the End-Mill Flutes

The solid carbide end-mills are complex in shape, especially,their flutes; and their two important parameters are the normalrake angle and the core radius. To grind the end-mill flutes withthe specified core radius and normal rake angle, the selectedgrinding-wheel should be properly located and oriented (in termsof the end-mill) on the 5-axis CNC tool grinding machine. How-ever, in current techniques, the wheel location is approximated,resulting in large deviation of the core radius. Therefore, it is neces-sary to accurately model the volume swept by the grinding-wheelin its 5-axis motions; and, with this model, the wheel location canbe calculated to ensure the specified core radius after grinding.

3.1 Five-Axis Tool Grinding Machine and StandardGrinding-Wheels. With development of 5-axis tool grindingmachines, such as 5-axis WALTER and ANCA machines, solidcarbide end-mills can be ground with higher accuracy and quality,compared to the conventional method of grinding tools. Particu-larly, the end-mill flutes have to be machined with the 5-axis CNCgrinding since their geometry is quite complicated. Figure 4

Fig. 4 The kinematics chain of a 5-axis tool grinding machine

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shows the configuration of a general 5-axis tool grinding machine,and the machine coordinate system <M includes the �X, �Y, �Z, �A,and �B axes. On this machine, the grinding-wheel can translatealong the �X, �Y, and �Z axes and rotate about the �A and �B axessimultaneously in order to machine complex geometries. In indus-try, there are many types of standard grinding-wheels, and theyare often used in end-mill production. Figure 5 displays fourstandard grinding-wheels with their profiles. The selection of theproper grinding-wheel is based on experience.

3.2 Effective Grinding Edge of a Grinding-Wheel in Five-Axis CNC Grinding. The main objective of CNC programmingfor the 5-axis flute grinding is to calculate the WOL in terms ofthe end-mill in the tool coordinate system. A kernel technique ofthe programming is to accurately and efficiently represent the ge-ometry of the wheel swept volume in the 5-axis CNC grinding.Since the kinematics of the 5-axis grinding and milling are similarand the 5-axis milling has been under extensive research, it wouldbe easier to describe the 5-axis flute grinding in comparison withthe 5-axis milling. For the 5-axis milling, the orientation of thecutting tool can be simultaneously changed while the tool movingalong preplanned paths in order to achieve better geometric matchbetween the tool and the part local surface without local gougingand global interference. From the geometric point of view, thetool sweeps a complex volume, in which the stock material isremoved; and the exterior surface of this volume, mathematically,is the envelope of the tool revolving surface at different locationsin the 5-axis milling process. This envelope is called cutter sweptsurface. Actually, the cutter swept surface is composed of theeffective cutting edges of the tool at different tool locations. Theeffective cutting edge of the tool at a location is defined as the sil-houette boundary of the tool revolving surface in the tool velocitydirection at this location (see Fig. 6). The effective cutting edge isdifferent if the tool feed direction is changed. With the effectivecutting edge, the cutter swept surface can be easily constructed,which is used to evaluate the 5-axis CNC tool paths. Basically,the geometric feature of the 5-axis flute grinding is similar to thatof the 5-axis milling.

To ensure the specified core radius and rake angle of a flute, theflute has to be cut on a 5-axis tool grinding machine, and a stand-ard grinding-wheel should be properly selected and then locatedand oriented. During machining, the grinding-wheel moves alongthe helical side cutting edge of the flute, sweeping an imaginaryvolume. In this work, the volume is called wheel swept volume,and any workpiece material inside the volume is removed. The ge-ometry of the wheel swept volume can be represented by the en-velope of the wheel surface at different locations during grinding;and at a wheel location, the envelope element is a curve, which iscalled the effective grinding edge in this work. An effective grinding

edge is defined as a 3D curve on the wheel surface such that thesurface normal on each point of the curve is perpendicular to theinstantaneous wheel velocity at a wheel position. A wheel positionrefers to the wheel center location and the wheel axis direction inthe tool coordinate system <T. Therefore, the wheel swept surfacecan be represented by finding the effective grinding edge of thewheel at any moment of the machining. Figure 7 shows agrinding-wheel cut between position 1 and position 2 in a 5-axisgrinding process. In this cut, the effective grinding edges at thetwo locations are plotted, so is the wheel swept surface betweenthe edges. Due to the complicated kinematics of the 5-axis grind-ing, the effective grinding edge is not simply the wheel profile; itvaries according to the wheel moving directions at different posi-tions, and it is often a 3D curve on the wheel surface. Therefore,the wheel swept surface could be quite complex in shape. In this

Fig. 5 Profiles of four standard grinding-wheels

Fig. 6 The effective cutting edge and the envelope of the toolrevolving surface at two adjacent locations

Fig. 7 The effective grinding edge and the envelope of thegrinding-wheel revolving surface at two adjacent locations

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work, we derive closed-form equation of the effective grindingedge in Sec. 4.

4 CNC Programming for the 5-Axis Flute Grinding

of the Cylindrical End-Mill

The objective of the CNC programming for the 5-axis flutegrinding is to determine the grinding-wheel position, includingthe wheel location and orientation, and the wheel path prior toactual machining. Since the side cutting edges are the guides ofthe flutes of a cylindrical end-mill, they are used as the wheelpaths. However, it is quite challenging to determine the wheel ori-entation and location in terms of the tool. Due to the tight pre-scribed tolerances of the core radius and the rake angle, the wheellocation and orientation should be accurately calculated in theCNC programming. In other words, the specified core radius andthe normal rake angle are the criteria for the CNC programming.In this section, the geometric models of grinding are establishedin order to determine the wheel position for the 5-axis flute grind-ing of a cylindrical end-mill.

4.1 Parametric Representation of a Standard Grinding-Wheel. In industry, there are many types of standard grinding-wheels, and they are often used to produce solid carbide end-millssince these wheels are cheaper compared to nonstandard grinding-wheels. Figure 8 shows three standard grinding-wheels, which areoften used to grind the flutes. Since the grinding-wheel inFig. 8(b) is in the generic shape of the three wheels, the parametricequation of this wheel can be used to represent the other twowheels. Thus, the parametric equation is derived here. First, awheel coordinate system <G ¼: ðO;X; Y; ZÞ is established; theaxes are denoted with the capital letters. The origin O of <G islocated on the right face of the grinding-wheel, and the Z-axis isaligned with the wheel axis and pointing to the left. The X- and Y-axes are on the right face and are perpendicular with each other.The parameters, R0;tG; aG; and T, are labeled in the diagram. Inthis coordinate system, the parametric representation of the wheelis provided in the following:

W u; vð Þ ¼R uð Þ � cos v

R uð Þ � sin v

u

264

375 (16)

where 0 � v � 2p

RðuÞ ¼R0 þ

u

tan aG

0 � u � tG � sin aG

xGj j þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2 � u� zGð Þ2

qtG � sin aG � u � T

8><>: (17)

and pG ¼xG

yG

zG

24

35

4.2 Wheel Orientation Determination. Solid carbide end-mills have different normal rake angles, and these end-mills areused to cut different metals. To machine a part of a specific metal,an appropriate rake angle can effectively reduce the cutting forcesand the cutting edge temperature in optimal balance with the cut-ting edge strength and the tool life. To grind a rake face with aspecified normal rake angle along its corresponding side cuttingedge, the selected grinding-wheel should be properly oriented sothat its lateral face always contacts with the rake face at any pointon the side cutting edge. Thus, the geometric principle of grindingthe rake face is that the unit normal of the rake face is in the oppo-site direction of the unit normal of the wheel lateral face. Basedon this principle, the wheel axis direction can be found; this direc-tion is the wheel orientation in the 5-axis grinding.

To establish the mathematical model of grinding the rake face,the geometric relationship between the grinding-wheel and theflute of the tool is illustrated in Fig. 9. First, a point W u0; v0ð Þ onthe wheel surface is chosen and is represented in the wheel coordi-nate system <G. Second, the surface normal N u0; v0ð Þ at pointW u0; v0ð Þ is found. Then, assuming the tool coordinate system <T

is stationary, the grinding-wheel is reorientated so that the wheelsurface normal N u0; v0ð Þ is aligned with the rake face normaln h0ð Þ but in the opposite direction. Since N u0; v0ð Þ is representedin <G, it has to be transformed into the tool coordinate system <T.More specifically, to align N u0; v0ð Þ with n h0ð Þ in the way afore-mentioned, three steps are proceeded in a consecutive way: (1) tocoincide the two coordinate systems, <G O;X;Y; Zð Þ with<T o; x; y; zð Þ; (2) to rotate N u0; v0ð Þ about the x-axis by angle l,and (3) to rotate N u0; v0ð Þ about the z-axis by angle g.

Using Eq. (15), the unit normal n h0ð Þ of the rake face at pointp :¼ c h0ð Þ can be calculated in the tool coordinate system <T.The unit normal vector N u0; v0ð Þ at W u0; v0ð Þ in the wheel coordi-nate system <G can be derived as

N u0; v0ð Þ ¼@W

@u� @W

@v@W

@u� @W

@v

��������

��������u¼u0v¼v0

(18)

Fig. 8 Profiles and geometric parameters of three standardgrinding-wheels

Fig. 9 The geometric principle of determining the wheel orien-tation for 5-axis grinding of the cylindrical end-mill flutes

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First, N u0; v0ð Þ is rotated about the x-axis by angle l, the transfor-mation matrix is

ROTx lð Þ ¼

1 0 0 0

0 cos l � sin l 0

0 sin l cos l 0

0 0 0 1

26664

37775 (19)

Then, N u0; v0ð Þ is rotated about the z-axis by angle g, the transfor-mation matrix is

ROTz gð Þ ¼

cos g � sin g 0 0

sin g cos g 0 0

0 0 1 0

0 0 0 1

26664

37775 (20)

Finally, N u0; v0ð Þ can be represented in the tool coordinate system<T as

NT u0; v0ð Þ ¼ ROTz gð Þ � ROTx lð Þ � N u0; v0ð Þ1

� �(21)

Therefore, the wheel surface normal is in the opposite direction ofthe rake face normal. Since they are represented in the tool coordi-nate system <T, the mathematical equation of grinding the rakeface is

n h0ð Þ þ NT u0; v0ð Þ ¼ 0 (22)

By using above equation, the two rotation angles, l and g, can besolved, and the grinding-wheel can be reorientated so that the nor-mal rake angle of the rake face can be attained in the 5-axisfluting.

By applying the above equations, the angles, l and g, of thewheel orientation can be explicitly expressed as

l ¼sin�1 cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ b2p� �

� /

p� sin�1 cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2p� �

� /

8>><>>: (23)

where

sin/¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffia2þb2p ; cos/¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2þb2p ; and

a¼ cosaG

b¼�sinaG � sinv0

c¼ cosan � sinw

8<:

(24)

For angle g

g ¼sin�1 c1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a21 þ b2

1

p !

� /1

p� sin�1 c1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1 þ b21

p !

� /1

8>>>><>>>>:

(25)

where

sin/1 ¼a1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a21þ b2

1

p ; cos/1 ¼b1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a21 þ b2

1

p ;

and

a1 ¼ sinaG � cos v0

b1 ¼� cosaG � sinl� cosl � sinaG � sin v0

c1 ¼�cosw � cosan � sinh0 þ cosh0 � sinan

8<:

(26)

or the angle g is

g ¼sin�1 c2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a22 þ b2

2

p !

� /2

p� sin�1 c2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

2 þ b22

p !

� /2

8>>>><>>>>:

(27)

where

sin /2 ¼a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a22 þ b2

2

p ; cos /2 ¼b2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a22 þ b2

2

p ;

and

a2 ¼ cos aG � sin lþ cos l � sin aG � sin v0

b2 ¼ sin aG � cos v0

c2 ¼ cos w � cos an � cos h0 þ sin h0 � sin an

8<:

(28)

Thus, the orientation of the wheel axis is expressed as

IG ¼ ROTz gð Þ � ROTx lð Þ �

0

0

1

1

26643775 (29)

By coinciding W u0; v0ð Þ with point p c h0ð Þð Þ at the cutting edge,the grinding-wheel location O in the tool coordinate system canbe derived.

OT ¼

OTx

OTy

OTz

2664

3775 ¼

rT � cos h0 � u0 � sin l � sin gþ R0 þu0

tan aG

� �� cos l � sin g � sin v0 � cos g � cos v0ð Þ

rT � sin h0 þ u0 � sin l � cos g� R0 þu0

tan aG

� �� cos l � cos g � sin v0 � sin g � cos v0ð Þ

h0

tan w� rT � u0 � cos l� sin l � sin v0 � R0 þ

u0

tan aG

� �

2666666664

3777777775

(30)

4.3 Mathematical Model of the Effective GrindingEdge. At this point, the grinding-wheel location and orientationare obtained for any point p on the cutting edge in h domain.Thus, establishing a relationship between h and time t is requiredto represent the wheel location and orientation for real time grind-ing. This relationship can be expressed as

dc

dh

�������� ¼ dc

dt

�������� � dt

dh(31)

wheredc

dt

�������� is the feed f . Thenð

dc

dh

�������� � dh ¼

ðf � dt (32)

For cylindrical flat end-mill, the relationship can be derived andexpressed as

h ¼ f sin wrT

t (33)

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Suppose point W u0; v0ð Þ of the grinding-wheel is machiningthe flute at point p c h0ð Þð Þ on the side cutting edge at time t, thewheel orientation, lðtÞ and gðtÞ, and its location OT tð Þ in the toolcoordinate system can be calculated using the aforementionedequations. Thus, the grinding-wheel surface W u; vð Þ at any time tcan be represented in the tool coordinate system as

WT u;v; tð Þ¼Tran OT tð Þ

�ROTz gðtÞð Þ �ROTx lðtÞð Þ �W u;vð Þ

1

� �(34)

where

Tran OT

¼

1 0 0 OTx

0 1 0 OTy

0 0 1 OTz

0 0 0 1

2666664

3777775 (35)

As mentioned before, the geometric principle of an effectivegrinding edge EGE at a side cutting edge point is that the unit nor-mal at the grinding-wheel points are perpendicular to the feeddirection of the grinding-wheel at a moment (Fig. 10). In general,the EGE at a point c hð Þ can be formulated as

EGE u; v; tð Þ ¼EGEx

EGEy

EGEz

24

35 :

@WT

@u

@WT

@v

dWT

dt

�������� ¼ 0 (36)

By simplifying the above equation, the following equation isobtained as:

a � cos vþ b � sin v ¼ c (37)

where

abc

2435¼

O•

x cosgþO•

y �sing�u �g• �sinlþR �@R

@u�g• �sinl

O•

y �cosl �cosg�l• �uþO

•

z �sinl�O•

x �cosl �sing�R �@R

@u�l•

�O•

z �@R

@u�coslþO

•

y �@R

@u�sinl �cosg�O

•

x �@R

@u�sinl �sing

2666664

3777775

(38)

The dot notation in Eq. (38) represents the time derivatives of theparameter.

If c=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2p�� �� � 1, the following relationship of the EGE will

be obtained:

v u; hðtÞð Þ ¼sin�1 cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ b2p� �

� /

p� sin�1 cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2p� �

� /

8>><>>: (39)

where

sin / ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2p and cos / ¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ b2p (40)

Hence, at a side cutting edge point p: c h0ð Þð Þ, given a value of pa-rameter u, the parameter v can be calculated. The pair of u and vrepresents a point of the EGE on the grinding-wheel.

4.4 Determination of the Wheel Location. Generally, a cy-lindrical end-mill has a cylindrical core with radius of rC, and thecore dimension should be ensured. In the above sections, a grind-wheel point W u0; v0ð Þ is assumed to contact with a side cuttingedge point c hð Þ. If the point W u0; v0ð Þ is changed, the flute shapewill be changed and the core size will be changed accordingly.Thus, to ensure the core size, the point u0; v0ð Þ should be deter-mined so that the minimum distance between the tool axis and theEGE should be equal to the core radius. After the point is opti-mized, the grinding-wheel location can be calculated usingEq. (30). Therefore, the optimization model of finding the pointu0; v0ð Þ is

Minimize d u0; v0ð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEGE2

x þ EGE2y

q� rC

��� ��� (41)

This optimization problem can be solved easily. A conventionallinear search method can be applied, and the details are not dis-cussed here. With the calculated point u0; v0ð Þ and the side cuttingedge point c h0ð Þ, the grinding-wheel location OT in the tool coor-dinate system can be found. Before machining the flute, thegrinding-wheel orientation and location in the tool coordinate sys-tem are converted into the machine coordinate system with a postprocessor. Therefore, the CNC programming for the 5-axis grind-ing of a cylindrical end-mill flute is conducted.

5 Applications

To validate the new approach to automate and accurate CNCprogramming for the 5-axis flute grinding of the cylindrical end-mills, three grinding-wheels are adopted and the CNC program-ming is conducted for machining a flute of a cylindrical end-mill.Then, machining simulation is carried out, and the solid models ofthe machined flutes are rendered for discussion. In Fig. 11, thestandard grinding-wheels are plotted with dimensions, and theyare used to grind the flute in such a way that the tool rake angleand the core radius are ensured during machining.

For the flutes to be machined with the grinding-wheels, themain parameter values are listed in Table 1. In this example, thegrinding-wheels a, b, and c are used to machine the flutes in job 1,2, and 3, respectively. By applying this approach, the CNC pro-gramming is conducted for the 5-axis grinding of the flutes. Withthe grinding-wheel position and path, each job of flute grinding issimulated, and the results are verified and discussed.

In job 1, the grinding-wheel (a) is used to machine the flutewith the normal rake angle of 10 deg and the core radius of12.5 mm. Using this approach, the orientation is calculated toensure the normal rake angle; the angle l is 70.32 deg, and theangle g is 10.63 deg. Due to the cylindrical end-mill, the helicalside cutting edge is the wheel path. To cut the first point of theside cutting edge, the wheel location is (69.72 mm, 14.48 mm, and�9.07 mm) in the tool coordinate system so that the core radius isFig. 10 The ground flute shape

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ensured, and the parameters u0; v0ð Þ of the grinding-wheel pointcontacting the side cutting edge point are (47.75 mm, 168.34 deg).To verify the CNC programming, a machining simulation is car-ried out, and the simulation results are shown in Fig. 12. In thesimulated tool model, the core radius is 12.5 mm, and the flute istangent to the cylindrical core. At the same time, the normal rakeangle is checked in the simulated tool model, and it is verified thesame as the prescribed value.

To discuss the important relationship between the grinding-wheel contact point and the core radius, points within the region

of the grinding-wheel marked as thick line in Fig. 11 and boundedby umin � u � umax and 165 deg � v � 180 deg are taken as possi-ble contact points. For each point, the machined core radius isevaluated, and the deviation between the actual core radius andthe prescribed is the machining error. Figure 13 shows the rela-tionship between the grinding-wheel contact point and the core ra-dius. It is evident that there are a group of grinding-wheel contactpoints where the core radii are the same as the prescribed. In thiswork, the parameter v of the grinding-wheel is set as 180 deg, andthe parameter u is optimized.

In job 2, the grinding-wheel (b) is used to machine the flutewith the normal rake angle of 12 deg and the core radius of12.5 mm. In CNC programming, the orientation is calculated asthe angle l is 78.69 deg and the angle g is 42.67 deg. To cut thefirst point of the side cutting edge, the wheel location is(31.07 mm, 58.93 mm, and �1.06) in the tool coordinate system,and the parameters u0; v0ð Þ of the grinding-wheel contact point are(5.87 mm, 180 deg). To verify the CNC programming, a machin-ing simulation is carried out, and the simulation results are shownin Fig. 14. In the simulated tool model, the core radius is

Fig. 12 A cylindrical flat end-mill having accurate core radius ground using standard grinding-wheel a

Fig. 13 Contour plots of the core radii deviation errors for cy-lindrical end-mill ground in job 1

Fig. 11 Standard grinding-wheels used to grind the end-millflutes

Table 1 The parameter values of the standard grinding-wheels

JobGrinding

wheel

Toolradius(mm)

Coreradius(mm)

Helicalangle

w (deg)Normal rake

angle an (deg)umin

(mm)umax

(mm)

1 a 25 12.5 20 10 45 502 b 25 12.5 10 12 0 8.83 c 25 12.5 30 20 35 44

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12.5 mm, and the flute is tangent to the cylindrical core. At thesame time, the normal rake angle is checked in the simulated toolmodel, and it is verified the same as the prescribed value.

In job 3, the grinding-wheel (c) is used to machine the flutewith the normal rake angle of 20 deg and the core radius of12.5 mm. In CNC programming, the orientation is calculated asthe angle l is 75.08 deg and the angle g is 6.211 deg. To cut thefirst point of the side cutting edge, the wheel location is(62.49 mm, 9.52 mm, and 0 mm) in the tool coordinate system,

and the parameters u0; v0ð Þ of the grinding-wheel contact point are(36.8 mm, 180 deg). To verify the CNC programming, a machin-ing simulation is carried out, and the simulation results are shownin Fig. 15. In the simulated tool model, the core radius is12.5 mm, and the flute is tangent to the cylindrical core. At thesame time, the normal rake angle is checked in the simulated toolmodel, and it is verified the same as the prescribed value. Havingconducted simulation for the 5-axis flute grinding, the true 3Dsolid models of the flutes machined by using the different

Fig. 14 A cylindrical flat end-mill having accurate core radius ground using standard grinding-wheel b

Fig. 15 A cylindrical flat end-mill having accurate core radius ground using standard grinding-wheel c

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grinding-wheels are built. With the 3D models of the flutes, theirperformance in terms of chip breaking and evacuation can beevaluated in a better way, compared to the current methods,according to the engineers’ experience. Therefore, this can greatlyhelp the tool engineers choose the grinding-wheel geometry andthe tool parameters.

6 Conclusions

A new approach to automate and accurate CNC programmingfor 5-axis grinding of the cylindrical end-mill flutes is proposedand verified. This approach is the first CNC programming methodproposed for 5-axis flute grinding. In this work, a virtual curve onthe grinding-wheel in 5-axis grinding—effective grinding edge—is proposed. Geometrically, the effective grinding edges are criti-cal in determining the shapes of the tool features, and its close-form equations are derived. Subject to the prescribed normal rakeangle, the equations of the grinding-wheel orientation is rendered.A mathemtical model is formulated to calculate the wheel locationto ensure the prescribed core radius. This approach has laid afoundation for the CNC progamming of 5-axis tool grinding, andsubstantially improve the design and manufacturing of cylindricalend-mills. The futue development of this approach will focus onfull control of the flutes shapes by controlling the grinding-wheelgeomteric parameters and path during grinding, extend theapproach to other end-mills, and build a gauging criteria able todetermine the critical size of the grinding-wheel for gauging-freeflute grinding.

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