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    TR,  and TR.  are  two sets of trajectories  of cp and  cq  for performing  the
    ma&  task  !fw.
    Type is one of the elements of the  following  set,  (sing&+g-imrtion,
    multipeg_insertion,  stack. screw, pcsh_and_huist), which  is a  set of
    most common types  of mating  tasks  in  industry  [SI. This  set can be
    expanded to  accommodate  other types  of  mating  tasks.
    Stability E  [stable, unsrcrble). A  stable task will always maintain the
    Elationship once established regardless of the  orientation  of the  assem-
    bly.
    There  are  three  basic elements in an  assembly  cek  assembly machines,
    fixtures, and  assembly tools. An assembly machine can  be  a robot  or  any
    mechanism which performs assembly functions. A fixture  can  be  modular
    fixture for gend  assembly or  speciallydesigned fixture  for some dedicate
    product The planning system described in this  paper will deal  with logical
    design of  fixtures rather than  physical design  of fixtmes. It is assumed that
    all  logical fixtures can  be  realized.  eitha constructed  from modular fixtures
    or  customized fixtures.  In case  Some of the  fixtures  cannot be  realized.  their
    setup costs can  simply be  assigned to  have an  in!ini@  value such that  the
    assembly planner will never select  them.  A set  of  tools am  also available in
    the  assembly cell. In this paper it is assumed  that the  selection  of tools can
    be done  simply by  direct mapping from  assembly tasks  to the  tool to  be
    used according to manuals or experience. A unified  representation of  the
    knowledge of these  resources will be  presented  in Section  2.4. Figure 1  shows an example assembly of six  components  with its Fcl,
    tional model.  For this example only translations motionS BIC umsidexd,
    thus  the  trajecmies  in TRb  and  TRh  are  3x1  pition  vectors.
    2.2.  Slmullrted  World Model
    The simulated world model is a  dambase which desaibes  and kceps
    track of  the state  of the assembly and various  resixum  (fixlures  and took)
    by simulating the assembling  process.  and  is updated according to the  tasks
    that have been  perfamed.
    An assembled  produd.  is a set  ofumnected  components. Then arc  two
    ways  of  representing  the state  of  an assembly.  The  6rsl rcpreSentatian
    describes the state of an assembly  through the aonditions  of  the msks
    between components.  In this  paper  we assume that  a task  is  either done  or
    not done,  and  thaefore  can  be wfiiciently  described by  a predicatG  DONE,
    where  Ti represent a  task. Ler - mpnt  negation and 5 repm-ent  the
    disassemblying of  Ti,  then  DONE(  Ti )  =  TRUE  if Ti  is not done. The
    expressions shown above are called well-formed  formulas (or  simply for-
    mulas). and the assignment  of truth value  to  a well-formed formula acumi-
    ing  to  the condition of  tasks is called an  inrcrpretation of  the formula
    Hence, Using the  above m,  formulas can  be constructed  to describe
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