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can be positions of surface in the Xp-Yp-Zp coor-
dinate. From interference check by the extended
z-map method, angular positions of apath and
acpath are obtained by adjustment of the milling
cutter direction using the following procedure. As
shown in Fig. 3, if the bottom plane of the milling
cutter interferes with surfaces only at the left
region, angle apath is adjusted in a positive direc-
tion. Likewise, if the bottom plane interferes with
surfaces only at the right region, angle apath is
adjusted in a negative direction. However, if the
bottom plane interferes with surfaces at the left
and right region simultaneously, angle acpath is
adjusted in the negative direction. Therefore, the
cutter axis direction vector can be obtained in the
XT -YrZT coordinate.
ZCt =MA(A)MB(B)Zc
ZCt = MB (-B)MA (-A)Zc
Zet =MB(B)Md-C)ZcType I.Type 2.Type 3.
2.2 Cutter axis direction vector
In this study, the end mill cutter was used for
2.3 Cusp height prediction
Generally, cusp heights in machining that util-izes the ball end mill cutter can be estimated
explicitly, but the prediction of cusp heights in
five-axis machining with the end mill cutter is
very complex (Vickers et ai, 1989)_ In this study,
a common plane is defined for quantitative analy-
sis of scallop, and a scallop on the common plane
can be described from mathematical modelling.
Thus, a cusp height can be obtained from the
scallop on the common plane.
As shown in Fig. 4, the common plane is
expressed as the plane that is formed by the vector
connecting two cc-points between present tool
path and next tool path and the summation vector
of normal vectors at two cc-points. In Fig. 4, N is
normal vector, 0, is the vector connecting two
cc-points, Om is the summation vector of two
normal vectors, and On is the vector perpen-
dicular to 0, on the common plane. If i notes a
tool path and j notes a cc-point being in a tool
path, the position vector of (i,j)-th cc-point is
Pccll.i). Also, the position vector Pcoll .
j) of the
common plane formed by (i,j)-th cc-point andwhere r is from 0 to cutter radius R, and B is from
0° to 360°. PXyzc(l,j) can be converted to PXYZt(l,j)
by a coordinate translation. The machined surface
is made by the motion of cutter. Therefore, the
position vector of the end mill cutter changes
along the tool path, the motion of the cutter.
The cusp heights on machined surface can be
obtained from the trace of the bottom plane of the
end mill cutter on the common plane. The trace
can be obtained by following Eq, (7).
(7)
where Slfn(l,j)(Ot(l,j) is minimum value among
On(I,i) components of e lfn(l,Hk)( B) values when the
k changes from 0 to e at the given Ot(l,i). Also, at
(i + I,j) cc-point, the scallop is obtained such as
Eq. (8). As a result, the scallop of the machined
surface is the less value of both scallops at (i,j)-th
and (i+ I,j)-th tool paths. Therefore, the cusp
height on the machined surface can be obtained
from the scallop through the calculation process.
where index k, the infinite element of the tool
path, is the range from start point 0 entering into
the common plane to end point e getting out the
common plane. If the position vector obtained
from Eq. (7) is eXYZt(l,i+k)(B) for a tool path
passing through the common plane, the vector is
transferred to e lfn(l,Hk)( B) in the local coordinate
of the common plane. Thus, the scallop at (i,j)-th
cc-point is minimum value (On(i,j» of e lfn (l,i+k)
(B) vectors when the end mill cutter moves from
its starting point to its ending point along i-th
tool path. The scallop is described by