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n (n, - l)n, M 12. Since n + 1 2 n,, the time complexity of this algorithm
is
(9) T(n)=Mn (nc- l)nc/2SMnn(n+ 1)/2
where M depends on the number of components, the geom& shape of
each component. and spatial relationships among components. Its analyti-
cal expression is hard to find, but usually the number of useful operations is
limited and it can be set by the designer to be independent of the above
three factors. Although the algorithm ACQUISITION-FOST is a general
algorithm, in this system we shall limit the trajectories of mating tasks to be
straight-line paths. ?his is not a severe limitation because most mating tasks
in industrial applications have straight-line trajectaies. Appendix A lists the
precedence knowledge for the example shown in Figwe 1.
2.32. Fixture Speclficatkn Mechanism
The function of fixtures in assembly process is to hold workpieces or
subassemblies by applying contact forces on the surfam of these work-
pieces. In practice s~ne of them are dedicazedfuhpes designed for special
shapes and products. others arc modularjm~es consimted from some
standard components. The general procedure in fixm design is to first
decide the locations of fixture elements on the workpiece and then physi-
cally design the fixture. either dedicated or modular. In this mearch only
the first step is considered, while physical design of future is beyond the
scope of this paper.
Figure 2 shows a subassembly Ci being fixed by a set of fixture ele-
ments. Ovz is the world system while O‘iy’z’ is the coordinate
system anached to Ci. Let x,, = (xo ,yo , zo lr be the origin of OYy’z’
expressed with respect to the oxyz coordinate system and e= ( e, 4. vlT
be the three independent angles representing the orientatip of o’iy’z’ with
respect to Oxyz. Then the relationship between the “am in Oxy2 and
O’iy’z’ becomes
x = R(6)x’ + q, (10)
=MO (n3).
where R(0) is a3x3 rotation matrix, x= (x , y , z )‘andx’= (i, y’, s’)~
are coordinates in Oxyz and o‘iy’z’. respectively.
Asada [I] derived the kinematic constraint relations of the
configurations for gend shape subassemblies when the subawxnblies are
in contact with fixtures. Suppose
g(x?=O (1 1)
is the equation of the boundary of Ci and any point x’ outside the boundary
will have g(x‘) 2 0. Suppose xt is the contact point between the kth fixture
element and Ci, and Ci is fixed.
(7'') should be performed after at least one of its precondition states and
before any one of its postandition states appear. The precondition and
postcondition states form a complete set of pncedence constraints. how-
ever, redundancies exist benveen the information contained in precondition