5-axis CNC Whirlwind Milling Model for Complex Freeform Surfaces

Xianzhong Yi Yang Lin School of Mechanical Engineering Production Department, Tianjin Company

Yangtze University China National Offshore Oil Corporation Limited No. 1, Nanhuan Road, Jingzhou, Hubei 434023, China No. 688 Bohai Oil Road, Tanggu,Tianjin 300452, China

yixzhong@sohu.com linyang2@cnooc.com.cn

Shengli Fu Jingkun Zhao and Xiuqing Hu Dagang Oilfield Postdoctoral Workstation Liaohe Petroleum Exploration Bureau

China National Petroleum Corporation China National Petroleum Corporation No. 3 Yard, Dagang Distrct, Tianjin 300280, China Xinglongtai Town, Panjin, Liaoning 124010, China

fushli@163.com Zhaojk_lh2003@sina.com

Abstract - Whirlwind milling model is a new milling technique proposed in this paper based on 5-axis CNC horizontal bed-type milling machines. The machining strategy of this model is to make a real-time adjustment of cutter orientation of whirlwind milling tool so as to create a close approaching envelopment to the surface being machined with the milling-tool rotational surface. Then, a full approach between these two surfaces can ensure maximizing the material removal on the machined surface at point of contact and improve the machining quality. A whirlwind milling tool with standard-sized circular edge cutters is introduced to side-mill complex freeform surfaces. The fundamentals, kinematics relationship, cutter location equations and tool path generation about this new milling model are analyzed, and some real application is presented.

Index Terms - Whirlwind milling model, freeform surface, 5-axis CNC milling machine-tools, tool path planning.

I. INTRODUCTION

It is clear that the optimization of machining processes in finishing the surfaces has two main objectives: one to achieve the high accuracy and the other to reduce the overall time of machining. As is well known, surface finish is the most important parameter during the finishing cuts. And a lower feed rate has to be used and exact cutter location data must be supplied as so to avoid tool interference and to give the required roughness. Thus, for reaching the goal of shorting the machining time, it is only natural that the material removal rate at the point of contact should be maximized while the cutting tool moves form one point to the next [1-6].

The whirlwind milling model proposed in the paper is an alternate method which can raise the productivity of freeform surfaces and avoid cutter over-cutting and gouging in machining. This new method can locally increase the material removal at the point of contact and further reduce the machining time. So, a less number of tool passes will be needed to obtain the same surface finish, and total the cutting length will in turn be cut down. Furthermore, it can save time and cost.

II. BACKGROUND

A. Surface Definition Theoretically three-dimensional surfaces of the workpiece to be machined are generally defined parametrically, algebraically, explicitly or implicitly. In the parametric definition of an arbitrary surface S , the x , y and zcoordinates of a Cartesian frame attached to this workpiece are related to the parameters u and v by some functions as shown below [7]:

( )( )( )( )

==vuhvugvuf

zyx

vur

w

w

w

w

,,,

, ( )1

The specific functional relationships rely on the type of surface fitting method used, namely, B-spline, Bezier or NURBS. Note that it is assumed that the isoparametric lines are regular and are not tangential to one another.

B. Surface properties From the above definition all characteristics of the surface

can be derived. The unit normal n and the tangent plane SΤ at

any point ( )vuP , on the surface S are given by the following formulas [7]:

wvwu

wvwu

rrrr

n××

= ( )2

( ) 0=−⋅ PTS rrn ( )3

where vector TSr locations on the plane SΤ ; Pr represents the

position of point P in the Cartesian coordinate system ( )wwwww zyxoS ,,, . In addition, wur and wvr are the partial

derivatives of the surface with respect to u and v , and are

evaluated at the point P .The curvature characteristics of the surface can be studied

by investigating the surface curve passing through a given point P . This curve is generated from the intersection of the surface with a cutting plane which includes the surface normal n at the given point. A point on a surface will have different

1-4244-0466-5/06/$20.00 ©2006 IEEE1953

Proceedings of the 2006 IEEEInternational Conference on Mechatronics and Automation

June 25 - 28, 2006, Luoyang, China

radius of curvatures depending on the direction of the cutting plane. The orientation of this plane is defined by an angle σwhere ( ) ( )dvdu=σtan . And this curvature distribution has a maximum and a minimum which are called the principal curvatures ( 1K and 2K ). These appear at the angles 1σ and 2σwhich are the angles of principal curvature, respectively. So, these angles can be used to describe the tangent vectors 1e and

2e in the direction of principal curvatures [7]:

wvwu rre ⋅+⋅= 111 sincos σσ ( )4

wvwu rre ⋅+⋅= 222 sincos σσ ( )5

C. Surface machining In planning the tool paths of freeform surfaces, there is a great choice of tool motion modes to be taken. Usually, a rectangular grid on the freeform surface is utilized and the tool-center placement relative to the surface is positioned along this grid. As the tool moves along the grid line, it approximates the grid curve with linear sections. This linear interpolation introduces some error in the machined surface. Moreover, as parallel cuts are made some un-machined materiel called scallop or cusp is left between two consecutive grid lines. The highest point of the scallop characterizes the difference between the machined surface and the theoretical surface, and this scallop height is a value by which to measure surface finish. In tool path generation, this rectangular grid is quite right the cutter contact (CC) data on the machined surface. The step-forwards (or step length) of CC-data are calculated in accordance of the allowable dimensional tolerance and the cutting error, and its step-overs (or path interval) are determined by the scallop height [8-10]. NC machining of freeform surfaces is typically carried out using ball-nose end-mill cutters, but small cutting velocity in the region of the axis of rotation of the ball-nose causes some problems when these cutters are milling flat regions of the surface. Then flat end-milling cutters are developed to machining these complex 3D surfaces in a 4 or 5-axis machine because of no vanishing cutting speed at any tip of the cutter tool. In addition, as the tilt angle between the axial line of a flat end-milling cutter and the normal of the machined surface are chosen accurately, this will supply a larger radius of curvature in the profile view of the milling cutter to fit the surface curvature being machined closely. The method has the advantages of producing a uniform surface finish. However, the flat end-milling cutter suffers from a corner between the flat bottom plane and the cylinder, it will leave some material in the wake of the tool as it moves from one point on the grid to another. Therefore, a toroidal milling cutter with a rounded radius around its bottom edge is proposed [11,12]. This cutter, sometimes by name filleted end-mill cutter, has the advantages of having no corner and having no point where the cutting velocity is zero. The larger the diameter of a cutter tool, the smaller the resulting scallop height for the same tool path. Moreover, larger tools will reduce the cutting time taken. All the above-mentioned cutters are functional milling schemes for some

typical surfaces. Nevertheless, their rotational radii are restricted within the minimal radius ( 11 K ) of curvature of the surface to be machined, and there still is a lot of room for improvement.

III. WHIRLWIND MILLING TOOL

In this paper, a whirlwind milling tool (shown in Fig. 1) with standard circular cutting-edge cutters (or standard-sized circular cutting inserts) is proposed to machine complex freeform surfaces. It has larger rotational diameter and more narrow cutting width. In addition, it is very simple to rectify its radial dimensional error and little grinding operation is to be need over damn-type side-milling cutters [13].

Generally the whirlwind milling tool is used in 5-axis horizontal CNC machines to side-mill the surface to be machined, its cutter arbor is initially destined to slightly the direction of minimal radius of curvature of the workpiece. Thus, the rotary radius of whirlwind milling tool is subjected to the limitation of maximal radius ( 21 K ) of curvature on the desired workpiece instead of its minimal radius ( 11 K ) of curvature. As a result of this, a relatively larger cutting rotational radius of milling tool is available than other kinds of milling cutters. In feed-in motion, the larger cutting diameter of whirlwind milling tool can provide a higher circular cutting speed and enhance the productivity. And, its narrow cutting width is beneficial to gouging avoidance into the desired

surface [15]. The rotational surface Ω of the cutting edge of whirlwind

milling tool can be described by the following:

( )( )( )

⋅⋅⋅+−⋅⋅+−

==ν

ηνην

νηε

εε

εε

sinsincoscoscos

,

2

222

222

D

DDD

DDD

c

c

c

cc

c

zyx

r ( )6

where the parameters η and v are angle variables, πη 20 ≤≤ , πνπ ≤≤− . The milling tool has a definite shape

at the point of contact with the surface being machined. And neither zero cutting velocity nor a sharp corner happens at any point of its teeth (or edges).

Fig.1. Whirlwind milling cutting tool with standard round-edge cutters

cx

cycz

co

0Cν

ω

εD

cD

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IV. WHIRLWIND MILLING TECHNIQUE

In the whirlwind milling technique, the symmetric surface Ω of milling tool engages with the surface S being machined at the point P , and their tangent planes are represented by ΩTand PT (Fig.2), respectively [14,15]. The straight line L is the intersection of these two planes, and angle ϕ is the included-angle. Suppose that iso-parametric curves lying on the surface S in Fig.2 are grid lines of tool paths obtained by numerical computation. Then, the milling tool will move along these curves and they are real cutter-contact (CC) paths with interference-free. Next, assuming the vector τ is the unit tangential vector of any tool path Γ at P , and angle φ is the included angle between τ and 1e . Here, the vectors 1e is the direction of maximum principal curvature ( 1K ), and the frame

nee 21 comprises a right-hand rectangular coordinate system.

ζ is the included angle between the normal n and the wz -axis in the frame ( )wwwww zyxoS ,,, . Based on the tool path geometry, the cutter location (CL) data (alternatively, the moving locus of the fixed point 0C on the axial center of the milling tool shown in Fig.1) in the frame nee 21 can be defined as follows:

( ) ( ) ( )⋅⋅−+⋅= φεε

nDDD

C Anneer c222210

( ) 122 eAe ⋅−−⋅ ϕπ ( )7

( ) ( ) ( ) 1221 eAAneez enc ⋅−⋅= ϕφ ( )8

where the transformation matrix ( )ϑμA denotes to rotate an angle variable ϑ along the coordinate axis μ . Through some mathematical manipulation, the CL offsetting algorithm is found:

( ) ( )⋅−+⋅= 222210εε DDD

Ccnneer

( )nee ϕϕφϕφ cossinsinsincos 21 +−−⋅ ( )9

( ) +⋅⋅= 121 coscos eneezc ϕφne ⋅+⋅⋅+ ϕϕφ sincossin 2 ( )10

Based on the above algorithm, the cutter location and orientation trajectories are determined and are represented by linear or spline motion commands [12,14]. At the CAM stage, the tool path is defined in a coordinate frame ( )wwwww zyxoS ,,, , which is attached to the workpiece or the working table on the machine tools (Fig.2); ( )ccccc zyxoS ,,,denotes a coordinate frame fixed to the given point on the whirlwind milling tool (Fig.1). It is worth noting that the three components of orientation vector cz can be reduced to two Euler angles. At the CNC machining process, the table is driven by three sliding axes so as to produce the desired CC path with respect to coordinate system wS ; two other rotating axes control the axial line cz of the milling tool. This means that three degrees of freedom are applied on the working table and two degrees of freedom on the cutter tool. In this paper, the Euler angles, α and γ , are so defined that the axial center cz of milling tool, originally in the wxdirection, first rotates along the wz -axis and then along wy -axis in the frame wS . To conduct the 5-axis machining, a coordinate conversion must be implemented in the CNC system. Obviously, this coordinate conversion depends on the machine tool structure. With the above definition, the kinematics equations that relate the machine-tool motion to the cutter path can be derived:

( ) ( ) ( ) ( )( ) ( )

++=+=

⋅+=⋅⋅⋅⋅+=

φσψγϕζα

ϕγαππ

1

21010

20

neerArrrAAAArr

CewPwC

cyxzywCwc

( )11

where the vector wPr represents the position of the point of contact P in the coordinate system wS ; ψ is the included angle between the partial derivative vector wur and wx -axis.

V. PRINCIPAL CURVATURE APPROXIMATION MODEL

The important advantage of the whirlwind milling technique proposed in this paper lies in its machining strategy which is able to create a close fit between the surface being machined S and the rotational surface Ω of whirlwind milling tool. This strategy is achieved by aligning the directions of principal curvature of the tool surface Ω so as to generate a full approach with the surface S but no interference happens with each other. Therefore, this machining method is named for the principal curvature approximation method in the paper.

As stated earlier, the surface characteristics of the whirlwind milling tool in Fig.1 are easily to be found. The direction of its maximum principal curvature ce1 is along the lines of constant values of η and the direction of its minimum principal curvature ce2 along the lines of constant ν . The

Fig.2 Coordinate frames for whirlwind milling technique

wxψ

wur

n

ST

ϕ

ζ

ΩTτ1e

1σ

wz

wy

2eΓ

φL

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principal curvatures at any point on the surface Ω are given in the following [7]:

( )( ) ( ) ( )[ ]−⋅−=

=νν ε

ε

cos1cos2

222

1

DDc

c

cKDK

( )12

It is clear that the maximum curvature of cutting surface corresponds to the constant ( )εD2 and is always in the plane containing the normal at the point of contact and the tool axial line. As the point of contact varies from 0=ν to ( )2π , the minimum curvature changes from ( )cD2 into zero and the plane of the osculating circle goes from being vertical to being horizontal.

As the whirlwind milling tool moves along the tool path Γ the real cutting plane is in the direction perpendicular to the tangent vector τ . The curve scΓ is the intersection between the extending plane of cutter teeth and the machined surface S , passing through the point P of contact. According to Euler formula, the normal curvature nK of the surface S along the direction of the curve scΓ can be written as:

( ) ( )φφ ππ +⋅++⋅= 2221 sincos KKKn ( )13

Moreover, the curvature nscK of the curve scΓ is obtained on the basis of Meusnier theorem:

( ) ϕcos,Pr ⋅=′′= nscwscn KrnojK ( )14

In addition, assume that the starting orientation of the milling arbor, which the whirlwind milling tool rotates along in the alignment of curvatures, is designed to the direction of maximum principal curvature 1e of the surface S . And when this machined surface S and the cutting surface Ω comes in contact, their normals are initially coincident. The point P is the common point of contact between the two surfaces. During the practical machining process of whirlwind milling technique, the normal curvature ncK of the milling tool at the point of contact in the cutting plane can also be described as:

( ) ( )φφ ππ +⋅++⋅= 2221 sincos ccnx KKK ( )15

The key of principal curvature approximation presented in this paper is to make an attempt at minimizing the difference between the curvature ncK and the curvature nscK so as to reach the material removal rate as large as possible. This machining strategy can be realized only when the following equation exist:

nscnc KK = ( )16

Finally, manipulating (13), (14), (15), and (16), the main variables φ and ϕ of principal curvature approximation in whirlwind milling technique can be solved for.

VI. APPLICATION

The principal curvature approximation model proposed in the paper is to closely envelope the surface being machined with the cutting surface. It only requires of correctly matching

the orientation components of the whirlwind milling-tool in the coordinate system of the surface to be machined. This machining technique has been used in 5-axis CNC horizontal milling machines to machine the freeform surfaces [12-14].

A comparison is made in order to evaluate the advantages

of the proposed technique over traditional ball-nose end-milling cutters. The freeform surface illustrated in Fig.3(a ) is a partial shape of the die for a specific automobile body. A whirlwind milling tool with the diameter cD =315mm and

εD =32mm is used to machine a series of points on this parametric surface. The dimensional tolerance limit of the freeform surface is 0.02mm and maximal scallop height 0.0032mm. As the milling tool cuts along the direction parallel to its width, the step-forward is 7.27mm and the step-over 1.96mm. And the number of tool-path passes is 411 and total cutting length 212.0m. While a ball end-milling cutter was employed, the shortest cutting travel was obtained by moving the tool along the direction parallel to the length of the surface. Usually the ball milling-cutter is 50mm in diameter. If a tool path is generated with the same surface finish as the above precision, its step-forward is 17.86mm and step-over 0.56mm. And the total cutting length of 737m with 908 passes is needed. Thus, it can be seen that the whirlwind milling

Fig.3(b) The cutter location tool paths for machining the surface shown in Fig.3(a) with principal curvature approximation of whirlwind milling technique ( 10mm)

Fig.3(a) A special parametric surface ( 10mm)

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technique enable larger rotational diameters of milling tool to cut the workpiece. And its principal curvature approximation method can raise the removal rate at the point of contact and lower the total cutting length about 70 percent over ball end-mill cutters in the typical example.

This whirlwind milling technique has found a wider use in manufacturing of helix surfaces such as positive displacement motors and progressive cavity pumps in petrochemical industry and environmental protection engineering. The details of the implementation are presented in [15,16].

VII. CONCLUSION

A new milling technique for machining complex freeform surfaces is described in this paper, and its principal curvature approximation model is established after a careful analysis of the geometrical properties of the surfaces to be machined. The whirlwind milling tool with standard-sized circular cutters is recommended because it has a wide range of curvature distributions available.

The applications in practical manufacturing of freeform surfaces make clear that the machining strategy of aligning the directions of principal curvatures increases the volume of removal. Moreover, such a good matching between the machined surface and the milling tool will improve the machining quality. If a tool path is developed with the similar surface finish, then the number of passes needed to achieve this goal will be reduced and in turn cut down the machining time spent. This technique for aligning the principal curvatures is based on the coordinative movements of five degrees of freedom in 5-axis CNC milling machines. There still remain the open topics about the optimization of whirlwind milling tool parameters and tool path planning in further research, and these will be discussed in the subsequent work.

ACKNOWLEDGMENT

The partial work of this paper is supported by CPSF (China Postdoctoral Science Foundation) at grant number 2005037348 and by CNPC (China Natural Petroleum Corporation) Innovation Foundation at grant number 2002Z0601.

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