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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
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