Geometric Containment Analysis for Rotational Parts

1
Geometric Containment Analysis for Rotational
Parts

• Mukul V. Karnik
• Iktara and Associates, LLC
• Satyandra K. Gupta, Edward B. Magrab
• University of Maryland

2
Role of containment in part reuse

• We say that Q is contained in P, if Q can be
  machined from P by performing only material
  removal operations on P.

Q is contained in P
P
Q
Cut second slot
Cut first slot
Q
Q?
P
Q can be produced by cutting two slots in P
Q is contained in P
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Mathematical definition of containment
Y
V
Y
U
V
LP
LQ

• Containment
• Solid Q at a location (t, ?) with respect to
  coordinate system of P is contained in P if
• Q? ? P Q?
• Where Q? TQ

X
U, X
Query Solid - Q
Database Solid - P
Z
W
W
Z
4
Problem statement

• If Q is a newly designed part, then one should
  attempt to locate a set of parts P P1,
  P2,,Pm in a database of n parts such that Q is
  contained in P, P ? P. Determine the rank order
  of parts in P based on their volume difference
  with Q.
• Assumptions
• Parts being considered are limited types of
  rotational solids

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Single axis and rotational solids

• Single axis solid is a solid whose boundary
  consists of only axisymmetric faces such that any
  cross-section that includes the axis of rotation
  will generate the same 2D region
• Rotational solid is a single axis solid with some
  off-axis features

Axis of rotation
Cross-section
Off-axis feature
Axis of rotation
Off-axis feature
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Restrictions on rotational solids being
considered

• Internal and external surfaces belonging to the
  boundary of the underlying single axis solid that
  are of the following type
• Type 1 Cylindrical
• Type 2 Conical
• Type 3 Spherical
• Type 4 Toroidal
• Type 5 Planar
• Off-axis features that are of the following type
• Axis-parallel cylindrical holes
• Axis-parallel internal and external slots

Axis-parallel cylindrical holes
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System working modes
Working mode 2 Find solids in database that
contain the query solid
Working mode 1 Register solids with database
Query solid
Generation of signature
Generation and registration of signatures
Determine containment
Output
CAD models
Database of parts
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Approach for generating and registering signature

1. Determine axis of rotation
2. Find bounding cylinder
3. Create single axis solid
4. Extract off-axis features
5. Compute Volume
6. Enter signatures from steps 2-5 along with solid
   model in the database

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Basic idea behind determining containment

• Goal For every solid P in database
• Examine signatures of P to determine if there
  exists a location (t,?) for Q such that P
  contains Q.
• We refer to the set of all possible locations
  where P contains Q as the feasible transformation
  space of Q.
• Our approach is based on
• Showing that the feasible transformation space is
  empty, or
• Explicitly constructing the feasible
  transformation space

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The extent of transformation space that needs to
be examined

• Locating Q outside this transformation space will
  indicate either of the following
• Q is not contained in P
• Location is a repeat of a location in the maximum
  feasible finite transformation space

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Main steps in approach
Determining containment
Compare zones (prune solids if transformation
space for single axis solid is empty)
Rank order remaining solids based on their
volume difference
Compare bounding cylinder (prune solids with
empty transformation space)
Compare off-axis features (prune solids
if transformation space for off- axis features
is empty)
Signature of query solid
Step 4
Step 1
Step 2
Step 3
Output solids remaining
Database
Pruned solids
Pruned solid
Pruned solid
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Step 1Compare bounding cylinder

• Objective
• Quickly determine if Q has an empty
  transformation space
• If LQ gt LP or RQ gt RP, then Q does not have a
  feasible transformation space ? solid P cannot
  contain Q and hence it is pruned

Database solid
Not pruned
LP
Pruned
Query solid Q
Database solid
RP
RQ gt RP
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Step 2 Compare zones of single axis solid

• Objective
• To determine the feasible transformation space in
  which single axis solid Qs is contained in single
  axis solid Ps
• If no feasible transformation space exists then
  prune P
• Due to rotational symmetry, the transformation
  space for a single axis solid is independent of ?

Single axis database solid Ps represented as zones
Single axis query solid Qs represented as zones
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Step 2 Details of approach

• Determine the containment status of Qs at t 0
• Determine the next location t? of Qs farthest
  from t such that containment status of Qs with
  respect to Ps is still the same as at t
• If Qs is contained in P at t, then Qs is
  contained in Ps at all locations between t and t?
  
• If Qs is not contained in Ps at t, then Qs is not
  contained in Ps at all locations between t and t?
  
• If t? gt LP LQ then stop
• Else, set t t? and go to Step b

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Step 2 Feasible transformation space for single
axis solid
Y
V
X
U
W
Z
Single axis query solid - Qs
Single axis database solid - Ps

• Transformation space for single axis solid
  extends to 2? due to rotational symmetry

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Step 3Pruning based on off-axis features

• Objective
• To determine the feasible transformation space
  for which Q is contained in P
• If no feasible transformation space exists then
  prune P
• Presence of features changes feasible
  transformation space

Change in transformation space due to presence
of features
Transformation space for solid without features
(single axis solid)
Transformation space for solid with features
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Step 3Basic idea

• The feasible transformation space of Q should be
  such that no feature of P intersects Q
• If Q is located at location (t, ?), then for
  containment of Q in P, feature ?Q? ?,
  where Q? TQ
• Thus for containment of Q in P, each feature of P
  is either
• Not intersecting single axis solid Qs
• Contained in feature of Q, or
• Partially contained in feature of Q and remaining
  portion not intersecting single axis solid Qs.

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Step 3Detailed approach

• For each feature of P, determine the
  transformation space of Q for the cases when
• does not intersect single axis solid Qs
• is a subset of another feature in Q
• Combine feasible transformation spaces of all the
  features

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Step 3a (i) Feature does not intersect single
axis solid Qs

• Determine the transformation space of Q such that
  feature does not intersect single axis
  solid Qs

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Step 3a (ii)Feature subset of another feature
Feasible transformation space
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Step 3a Example -Feasible transformation space
for in P
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Step 3b Combining feasible transformation
spaces of all features

• Combine feasible transformation spaces for each
  of the features in P such that only intersecting
  feasible transformation spaces remain

Feature
Feature
Combining transformation spaces
Feasible transformation space for solid Q
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Reverse solid Q

• Solid Q is rotated by 180 about the Z-axis and
  the parts are aligned again
• Steps 2 and 3 are repeated for the reversed solid
  QR and solid P and the feasible transformation
  space is determined

Solid P
Solid QR
V
Y
U
X
Z
W
Y
V
U, X
W
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Step 4Rank order by volume difference

• Objective
• To rank order the set of database solids P that
  contain the query solid Q
• Sort the solids in P based on the volume
  difference
• Where, VP Volume of database solid stored as a
  signature
• VQ Volume of query solid stored as
  a signature

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Implementation

• Implemented on Windows 2000 platform using Visual
  C 6.0
• Libraries used
• Microsoft Foundation Classes (MFC)
• ACIS
• OpenGL
• More than 10,000 lines of code

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Results
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Computational Experiment
Q1
Q5
Database 1
Q4
Database 2
Q2
Q3
Q2
Database 3
Q1
Increasing Complexity
Q3
Q4
Q5
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Summary and Future Work

• The system can test if the newly designed part is
  contained in an existing part.
• some material removal operations can be performed
  on the existing part to obtain the newly designed
  part.
• reduces part inventory
• The transformation space of features can provide
  suggestions for part redesign.
• If the new part is not contained in an existing
  part, then it can be redesigned
• The type of features supported need to be
  extended.