Journal of
ELSEVIER
Journal of Materials Processing Technology 48 (1995) 739745
Materials Processing Technology
Gouge detection and tool position modification for fiveaxis NC machining of sculptured surfaces F. Li a, X . C . Wang% S . K . Ghosh b, D . Z . Kong% T . Q . l_ai~and X . T . W u a aDepartment of Mechanical Engineering, Xilan Jiaotong University, Xitan, China q n t e r n a f i o n a l College of Engineering, Lohmar, Germany
GKN Automotive A G ,
Posffach 1 1 5 2 ,
53784
Curvature catering is a highefficiency method for machining sculptured surfaces using 5axis machine tools. By means of this method, the condition of interference in the vicinity of the contact point is critical, cannot be detected using the methods based on curvat!ares at the point. This paper presents the algorithms for detecting the interference due to the r i m , front cutting edge a n d opposite side of a disctype milling cutter, and the methods for eliminating these kinds of gouge. Since toolsize interference can be detected in the process of calculating tool position a n d axis, is not involved in this paper. The interference due to the outer cone and tool axis will rarely occur during machining open surfaces, nor is discussed at this stage. 1. INTRODUCTION Although the gouge problem of m a n u f a cturing sculptured surface on 5  a x i s machines have been discussed in m a n y papers, most of t h e m are limited to using ballended tools, few about disctype cutters. The curvature catering method uses a disc cutter with a concave e n d , b y adjusting the cutter axis to swing relatively to the component being machined according to some special rules during the cutter passes through the surface, the envelope formed b y the trace of the toolnose in each tool pass and the required surface h a v e the same derivatives up to the third order in the plane normal to the feeding direction. Therefore, the steps needed in machining process can be greatly reduced without the decrease of accuracy. Because the cutter and the machined surface are always in the state of critical interference, the method based on osculating quadric can no Elsevier Science S.A. SSDI 09240136(94)01716E
longer be used to detect interference. Several kinds of gouge detection a n d modification methods are presented in this paper. 2. CLASSIFICATION OF G O U G E S CURVATURE CATERING METHOD
IN
2 . 1 . The intersecting line of the tool endplane and the surface crossing over the toolnosetrace circle If the contact point on the surface is a convex elliptic one, the angle ~ between the tool endplane S and the tangent plane of the surface 22 at the contact point M is less t h a n zero ( ~ ~ 0) in the machining process the intersecting line f of S and 27 should be always inside C, as shown in Fig. 1. For other types of point, f should be outside C, as shown in Fig. 2. If f crosses over C , there will be overcut, as shown in Fig3. The intersecting points of C and f is defined as :
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F. Li et at / Journal o f Materials Processing Technology 48 (1995) 739745
3 ~ E ~l ( r   RO . a ~ = 0, I ~  L I (~,;  ~¢,ao) :/: 0
=R, O)
ter.
where ~ denotes the position vector of point M , ~ denotes the position vector of the tool center, ~ denotes the unit vector of the tool axis, n denotes the unit normal at point M. The first equation indicates r E f , the second shows ~ E C , the third states f crosses over C at the point. Because n X ac is the tangent direction of f , if it is not perpendicular to r/L , they are intersected. Fig. 2 A n illustration of machining a concave surface
Fig. 1 A n illustration of machining a convex surface 2. 2. The gouge of front edge and the surface W h e n a convex elliptic point is machined, ~ 0. Even if f is totally enclosed in C, the gouge of the f r o n t edge of the cutter and the surface are still probably exist, as shown in Fig. 4. This kind of gouge is defined as: <
90
(2)
where 90 denotes front relief angle of the cut
Fig. 3 The gouge of tool nose trace. 2 . 3 . The gouge caused b y the opposite point of the contact one on the cutter
F. Li et aL / Journal of Materials Processing Technology 48 (1995) 739745
741
3. DETECTION OF INTERFERENCE. CUTTER
3. 1. The interference of f crossing over C Because the identifying of this kind of gouge is difficult, it is redescribed as follows in practice:
Et ~ E f,{l~~,l > n I~~1 < R \ Fig. 4
The gouge of front edge
Although there is no intersection of f and C or the gouge of f r o n t edge, if the diameter of the cutter is rather large or the shape of the surface varies greatly, the opposite point of the contact one on the cutter will probably gouge the surface, as shown in Fig. 5. This kind of gouge is defined as:
(~<0) (4) (~o)
In order to detect gouge, a high efficiency method for calculating the point of f should be introduced firstly. Let ~ denote a certain point on the intersecting line, the unit tangent at this point is
i=
~~, I/~ x BI
and the unit normal is
= K × ~ =t ? E L ' I ( 2 ~   ~ .  
~.~)
(5)
(6)
~) X ~ = o , ( 2 / ~  
.~<0
(3)
where 2~¢r~ denotes the opposite point of M on the C. The first equation indicates the normal position vector of 7 passing through opposite point of M . The second one means the opposite point beneath the surface.
Although the vector fl is uncertainly towards the concave side of f , there is no influence to the calculation. The normal curvature along t direction is K , = K l ( e l • ~)2 + Kz(e2 ° ~)2
(7)
According to Mesunierls t h e o r e m , the relative curvature of f is CuTTER
K = ~/~ • ~
(8)
Let s denotes the directed arc length f r o m this point to next one of f , the approximate radial vector of this point is GouG~ POINT Fig. 5
The gonge of opposite point
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F. L i et al. / Journal o f Materials Processing Technology 48 (1995) 739745
F
OF 3u
F
3F Ow
k8 2 _
where At" = d + a  B Zd'lt  
Now the approximate point ~'~+xon the surface can be calculated. In order to do so, Au and Aw should be determined firstly. Commonly the vector r'~+a is not on S. Let
dw
(14)
(10) Let u = u + du, w = w + dw and
where r. and ~ denote the partial derivatives. From multiplying b y vector ~ and ~. respectively, the foUowing equations can be obtained AuE + AwF = AZr~ • 7r~ AuF + AwG = Ar" r~
(11)
By using the two linear equations together, the vector r'~+l can be determined. r'~+l = r ( u + A u , w + Aw)
Let F =
(r'~+l   ~ )
(12)
" a~,
the condition for r'~+l = r~+t is F = O . Let u = u + Au and w = w + d w , and search the point along the gradient direction of F on u w  p l a n e . The gradient direction is
dw'du="
where Let
3F Ow
3F
~
3F 3u

(13) 3F
= r. • ac,
aF du + ~ d w
=
3w   r~ • ac
F
the following equations is obtained
T l i~ 1 =
r(u,w) . Repeat the above process until IF[ ~ e , where e is a predetermined positive value. The next step is to check if ~r,+l is inside C, and according to the sing of y to determine if there exists interference. Start from point M , if y ~ 0, f is closed and the interference value is not very big, search the whole f line. Otherwise only search to the edge of surface or to the point where interference value is big enough, then search the opposite direction. 3 . 2 . The interference of front edge and the surface The method to check this kind of interference is very simple. Equation ( 2 ) can be directly used to do so. Because it is easy to check, it had better be done immediately after the cutter location data is obtained. If this kind of gouge exists, the mmodification should be done before checking other kinds of interference. In this w a y , a high efficiency can be expected. 3 . 3 . The gouge of the opposite point of the contact point on C In order to check this kind of gouge, a point which satisfies the following equation should be determined firsty. r E 2:l(2Rcr.,~)
Xn=o
(15)
17. L i et aL / Journal o f Materials Processing Technology 48 (1995) 739745
T h i s e q u a t i o n c a n be r e  d e s c r i b e d as (2~
 r.  r) ///n
Sincen.r~=u.r~= let (2~
(16) 0
F , = r~ • (2Re   r~   r ) , F z = r~ •  ~.  r) (18)
t h e e q u a t i o n c a n be r e  d e s c r i b e d as F, =
(17)
=
F2
743
m o s t feasible w i d t h of m a c h i n e d strips w h i l e the interferenceis eliminated. Because the m o d i f i c a t i o n c a n n o t be d e t e r m i n e d b y trials , s o m e criterions f o r t h e choice of m o d i f i c a tion m e t h o d s according to t h e c h a r a c t e r s
should be established. Gouges caused by front edges and opposite point of the contact point on C are easy to be e l i m i n a t e d will be discussed f i r s t l y . F o r t h e f o r m e r o n e , if ~ is limited as
O.
~~+a~ U s i n g N e w t o n ' s iteration m e t h o d , du a n d clw can be d e t e r m i n e d :
this kind o f gouge c a n be avoid. F o r t h e late r , let
"3 Fldu ~F1 u +  ~ d w =   F1 3 F2du
3F2
(2~~o~).~×
(19)
Let u = u b du and w = w 7t dw , repeat t h e a b o v e process u n t i l IF, I < e a n d IF21 < e . T h e n a c c o r d i n g to t h e v a l u e of ( 2 ~   r~,  r ) • n , w h e t h e r t h e r e exists i n t e r f e r e n c e c a n be detected a n d t h e a m o u n t of i n t e r f e r e n c e c a n be d e t e r m i n e d .
4. T H E M O D I F I C A T I O N M E T H O D S T h e i n t e r f e r e n c e c a n be e l i m i n a t e d b y a d j u s t i n g 0 , ~ or m o d i f y 0 a n d ~ together. A f t e r the modification, the envelope formed by t h e trace of t h e t o o l  n o s e in e a c h step a n d t h e r e q u i r e d s u r f a c e will n o longe h a v e t h e s a m e d e r i v a t i v e s u p to t h e t h i r d o r d e r , t h e w i d t h of m a c h i n e d strip will be reduced. H o w e v e r , if t h e m o d i f i c a t i o n m e t h o d a n d a m o u n t is s u i t a b l e , t h e cutter a n d t h e s u r f a c e a r e still in a s t a t e o f critical i n t e r f e r e n c e , a n d a w i d t h o f t h e m a c h i n e d strip m u c h greater t h a n t h e o t h e r m e t h o d s c a n still be obtained. T h e m o d i f i c a t i o n process s h o u l d keep t h e
(20)
= ~ 
1.2
2R(~ • ~)
(21)
t h e g o u g e c a n be relieved. T h e c o n s t a n t coe f f i c i e n t 1 . 2 is to a v o i d r e p e a t e d l y m o d i f i c a tion a n d e n s u r e t h e rear p a r t o f t h e cutter n o t to s c r a t c h a n d w e a r a w a y o n t h e s u r f a c e . N o w t h e m o d i f i c a t i o n m e t h o d of t h e i n t e r section o f f a n d C is discussed. F o r ~ ~ O, t h e r e c e r t a i n l y exists a p o i n t r o n f , it g u a r a n t e e s t h e distance f r o m r to R~ is t h e m i n i m u m . If t h e r e a r e i n t e r f e r e n c e o n b o t h side o f M , t h e n ~ s h o u l d be m o d i f i e d as f o l l o w s : Let r0 be t h e riearest p o i n t to ~ on f , t h e a n gle b e t w e e n t h e line f r o m t h e p o i n t to ~ a n d t h e line f r o m M to ~ is d e t e r m i n e d b y : 0c = t a n  '
ff0c~0,
] ( h , , F .   R~,~0   ~ ) ]
t h e n 0 c ~ r2
(22)
0c. Let n0 d e n o t e t h e
u n i t n o r m a l vector at ~0 , t h e m o d i f i e d ~ is ,
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F. Li et aL / Journal o f Materials Processing Technology 48 (1995) 739745
~=~+
(n
IRo~01)~ 1.2 ~ / 1 
(no .aoY
R(1   cos~,)fi0 • (23)
F r o m above equation it is known that, for the same value of interference, the smaller the angle between no and a~ , the smaUer the modification is required. Howerer, a small 0¢ will lead to large modification. For the situation of one side interference, the modification of 0 can be used. The amount of modification can be calculated as follows :
AO
=
(R
IL~0l)
G., ~0


"
~., no)
(~~0) •
IR~

no
•
the efficiency of modifying 0 is reduced, and the rear part of the cutter will gouge the surface. The problem can be solved b y modifying ~ in any case. The front and the side interference should by modified according to the angle between no and/~. W h e n ~ < 0 , if the nearest distance from f to ~ is not far f r o m the R , the modification is similar to the above, the only difference is the sing "   " should be add to the Eq. (2 3) and FXl. ( 2 4 ). If ~?< 0 and the point on f i s far from C, two crossing points of f and C should be found. Using P to denote the midpoint of the arc characterized by the two points, a method similar to solving the gouge of the opposite point of the contact point on C can be used. i . e . a point P satisfying the following equation should be f o r n d ,
~ErIGP) xn=6
~0l
(25)
(24) If the absolute value of 0 is reduced after modification, recalculate new y according to the second order catering condition. Otherwise ~ is not modified (Because kl~k2, if the absolute value of 0 gets smaller, y will increase, which is determined by the second order catering condition). The interference detection should be done once again after the modification of 0 . If there is still a little interference, y should be modified. F r o m Eq. ( 2 2 ) it is known t h a t , if the angle between no and ~ is small, high efficiency can be obtained by modify r/. If M is near a shoulder on the surface, the angle between no and a~ is big, high efficiency can be ensured by modifying 0 .
But if @ ~
_676 q
,
then the modified ~ can be calculated as fellows :
1.2(P  7) • n = ~+
(1   cos0c0)(n • ac)
(26)
where 0c0is the angle between the line f r o m P to R and thd the line from rm to ~ .
5. CONCLUSION Interference problems arising in the manufacture of sculptured surface using a 5axis CNC machine and disctype cutter are very complicated. The algorithms for detecting interference due to the r i m , front edge and the opposite side of a disctype milling cut
F. Li et al. / Journal of Materials Processing Technology 48 (1995) 739745
ter, and the methods for eliminating these kinds of gouge have been presented in this paper. These algorithms and the methods have been successfully used in the curvature catering method on 5axis machining. REFERENCES 1. Wang, X. C, Li, Y. B, Ghosh, S. K. and Wu , X . T . "Curvature CateringA New Approach in the Manufacture of Sculptured Surfaces (Part 1. Theorem)" J. Mats. Proc. Tech. , 1993, pp. 159
745
176. 2. Wang~ X. C, Li, Y. B, Ghosh, S . K . and W u , X . T . "Curvature CateringA New approach in the Manufacture of Sculptured Surfaces ( Part 2. Methodology)", J. Mats. Proc. Tech. , 1993, pp. 177194 3. Zhang, D. H, Yang, P. J and Yang, H. C. " A n AntiInterference Method for MultiAxis NC Machining of Sculptured S u r f a c e ' , J. Northwetstern Polytechnical Univ. , Vol. 11, No. 2, 1993, pp. 157162, (in chinese).