Geometric reasoning about mechanical assembly

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Geometric reasoning about mechanical assembly

• By Randall H. Wilson and Jean-Claude Latombe

Henrik Tidefelt
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Topics

• Automatic generation of assembly algorithms
• Characterization of the complexity of assembly
  designs

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Assembly tree

• An assembly algorithm (plan) is constructed by
  splitting the target into smaller and smaller
  subsystems.
• This yields a partial order in time of assembly
  instructions.

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Assembly planning using the generate-and-test
strategy

• Relax some constraints to come up with candidate
  algorithms (assembly trees).
• At least, the constraints imposed by the
  manipulating system are ignored.
• Search the candidates for globally feasible ones.
  (Motion planning including the manipulating
  system.)

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NDBG

• The non-directional blocking graph represents how
  the parts in an assembly are constraining each
  other.
• It is useful for efficient generation of
  candidate algorithms during assembly planning,
  and also for complexity evaluation of mechanical
  assemblies.
• The NDBG - and hence its interpretation - is a
  function of the family of motions that is
  considered.

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DBG

• The directional blocking graph only takes into
  consideration motions in a particular direction
  d.
• There is one node per part, and an arrow from
  part p1 to part p2 if p2 is blocking p1 in the
  direction d.

DBGs for infinitesimal translation along d
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DGB

• A strongly connected component can not be
  (dis)assembled along the direction d.

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DBG

• A subset with no outgoing arcs is locally free to
  translate in the direction d,
• But there is no guarantee that this cut
  corresponds to a globally feasible assembly plan.

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NDBG

• S, the set of all directions, can be divided into
  intervals over which the DBG is constant. These
  intervals are called regular regions.
• The NDBG is a structure associating each regular
  region with a corresponding DBG.

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Computing the NDBG

• Given an assembly of parts, we can find the
  regular regions by cutting S in every direction
  that is parallel with an edge in contact with
  another part.
• Every cutting direction is a regular region, as
  are the open intervals separated by the cuts.
• Compute DBG for each regular region.

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Computing the DBG

• The DBG is represented as an n by n adjacency
  matrix, where n is the number of parts.
• First, clear the whole matrix.
• Then, pick an arbitrary direction d in the region
  and evaluate each edge contact.

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Time complexity

• Let c be the number of edge contacts.
• There are O(c) regular regions.
• Each DBG is computed by considering c edge
  contacs.
• Computing a NDBG in this way takes O(c2) time.

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Improvements

• Instead of considering all c directions of edge
  contact, we can look at each pair of parts in
  contact and find the direction along which they
  can slide along each other.
• These are the only directions that need to be
  considered.
• Let there be r pairs of parts in contact.
  Finding the r directions takes O( c r log r )
  time, and after addition of the r2 time it takes
  to compute all adjacency matrices, the total time
  becomesO( c r2 ).
• It is guaranteed that r c, and in many cases r
  ltlt c.

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More improvements

• It is possible to exploit the fact that adjacent
  DBGs are similar, resulting in an O( r2 )
  algorithm.
• If the application only makes use of one DBG at a
  time, and can do that in an order so that
  subsequent regular regions are also adjacent,
  only one DBG needs to be stored at any time.

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Other kinds of NDBGs

• So far, we have only seen NDBGs representing
  local freedom of translation in the plane, i e
  limitations on infinitesimal translations in the
  plane, but the NDBG is suitable for other kinds
  of motions too
• Infinitesimal generalized motions (local freedom
  of translation and rotation)
• Infinite translations
• Extension to 3D

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Complexity evaluation

• Aiming at supporting the designer of mechanical
  assemblies to create products that are easy to
  mass-produce and maintain.
• Compare with the importance of knowing the time
  and space complexity of a computer algorithm.
• To automate the complexity evaluation we need
  algebraic complexity measures.

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Complexity measures
There exist an algorithm where each instruction
involves at most p 1 moving subsets, and p - 1
hands are not sufficient.

• p-handed
• Monotonic?
• m-prismatic
• Stack?
• Length
• Linearizable?
• Degree of form closure

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Complexity measures

• p-handed
• Monotonic?
• m-prismatic
• Stack?
• Length
• Linearizable?
• Degree of form closure

Every instruction moves a subassembly to its
final position relative some other subassembly.
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Complexity measures

• p-handed
• Monotonic?
• m-prismatic
• Stack?
• Length
• Linearizable?
• Degree of form closure

There exist an algorithm where the instructions
move each subset in a way that can be described
by a sequence of at most m extended translations.
m - 1 is not enough.
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Complexity measures

• p-handed
• Monotonic?
• m-prismatic
• Stack?
• Length
• Linearizable?
• Degree of form closure

Length of longest sequence of instructions.
1-handed, moving only one part per instruction.
Fingers needed to grasp subassemblies with form
closure.
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Example

• The assembly has three parts, so if it is
  admissible, it will be at least 2-handed
  monotonic.
• Given that we may only do translations, is it
  1-handed monotonic?
• Given translation and rotation?

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2-handed monotonic
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1-handed translations not monotonic
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1-handed translation and rotationmonotonic
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NDBGs and complexity evaluation

• All 1-handed assembly algorithms that are correct
  for infinitesimal translations can be extracted
  from the assemblys NDBG.
• This leaves a polynomial set of algorithms to try
  to see if the assembly is 1-handed monotonic
  prismatic, or linearizable for translations.

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NDBGs and complexity evaluation

• The NDBG can be used to compute an upper bound on
  the number of fingers required to give a
  subassembly form closure
• By identifying loose parts of the assembly in the
  NDBG, we can find appropriate placed to place
  fingers on.
• We add a finger at a time until the assembly has
  the form closure property, updating the NDBG
  after each modification.
• (Force closure might be more practical.)

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NDBGs and complexity evaluation

• All 1-handed monotonic 1-prismatic assembly
  algorithms can be extracted from the NDBG of
  infinite translations.
• The product is a stack iff such an algorithm can
  be extracted from a single DBG.

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Conclusions

• The NDBG can be used both for generation of
  assembly algorithms and for complexity evaluation
  of mechanical assemblies.
• Assembly planning using NDBGs is done in
  generate-and-test fashion.
• Complexity evaluation can help designers design
  products that are suitable for mass-production
  and easy to maintain.