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    clearly stating the applications. The APT language was
    then the only tool to program five-axis contouring appli-
    cations. The problems in postprocessing were alsoclearly stated by Sim [2] in those earlier days of numeri-
    cal control and most issues are still valid. Boyd in Ref.
    [3] was also one of the early introductions. Beziers’ book
    [4] is also still a very useful introduction. Held [5] gives
    a very brief but enlightening definition of multi-axis
    machining in his book on pocket milling. A recent paper
    applicable to the problem of five-axis machine work-
    space computation is the multiple sweeping using the
    Denawit-Hartenberg representation method developed
    by Abdel-Malek and Othman [6].
    Many types and design concepts of machine tools
    which can be applied to five-axis machines are discussed
    in Ref. [7] but not specifically for the five-axis machine.
    The number of setups and the optimal orientation of
    the part on the machine table is discussed in Ref. [8]. A
    review about the state of the art and new requirements
    for tool path generation is given by B.K. Choi et al. [9].
    Graphic simulation of the interaction of the tool and
    workpiece is also a very active area of research and a
    good introduction can be found in Ref. [10].
    4. Classification of five-axis machines’ kinematic
    structure
    Starting from Rotary (R) and Translatory (T) axes four
    main groups can be distinguished: (i) three T axes and
    two R axes; (ii) two T axes and three R axes; (iii) one
    T axis and four R axes and (iv) five R axes. Nearly all
    existing five-axis machine tools are in group (i). Also a
    number of welding robots, filament winding machines
    and laser machining centers fall in this group. Only lim-
    ited instances of five-axis machine tools in group (ii)
    exist for the machining of ship propellers. Groups (iii)
    and (iv) are used in the design of robots usually with
    more degrees of freedom added.
    The five axes can be distributed between the work-
    piece or tool in several combinations. A first classi-
    fication can be made based on the number of workpiece
    and tool carrying axes and the sequence of each axis in
    the kinematic chain. Another classification can be based
    on where the rotary axes are located, on the workpiece
    side or tool side. The five degrees of freedom in a Car-
    tesian coordinates based machine are: three translatory
    movements X,Y,Z (in general represented as TTT) and
    two rotational movements AB, AC or BC (in general rep-
    resented as RR).Combinations of three rotary axes (RRR)
    and two linear axes (TT) are rare. If an axis is bearing
    the workpiece it is the habit of noting it with an
    additional accent. The five-axis machine in Fig. 1 can
    be characterized by XYABZ. The XYAB axes carry the
    workpiece and the Z-axis carries the tool. Fig. 3 shows
    a machine of the type XYZAB, the three linear axes
    carry the tool and the two rotary axes carry the work-
    piece.4.1. Classification based on the sequence of workpiece
    and tool carrying axes
    Theoretically the number of possible configurations is
    quite large if the order of the axes in the two kinematic
    chains of the tool and workpiece carrying axes is counted
    as a different configuration. Also the combinations with
    only two linear axes and three rotary axes are included.
    One tool carrying axis and four workpiece carrying
    axes can be combined in a five-axis machine as follows:
    for each possible tool carrying axis X,Y,Z,A,B,C the other
    four workpiece carrying axes can be selected from the
    five remaining axes. So the number of combinations of
    four axes out of five with considering different permu-
    tation as another configuration is 5×4!=120 for each
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