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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