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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
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