Minkowski Sums

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Minkowski Sums

• Dinesh Manocha
• Gokul Varadhan
• UNC Chapel Hill

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Last Lecture

• Configuration space

workspace
configuration space
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Problem Configuration Space of aTranslating
Robot

• Input
• Polygonal moving object translating in 2-D
  workspace
• Polygonal obstacles
• Output configuration space obstacles represented
  as polygons

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Configuration Space of aTranslating Robot
Workspace
Configuration Space
Robot
Obstacle
C-obstacle
Robot
y
x

• C-obstacle is a polygon.

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Minkowski Sum
B
A
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Minkowski Sum
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Minkowski Sum
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Minkowski Sum
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Configuration Space Obstacle
C-obstacle is
Classic result by Lozano-Perez and Wesley 1979
C-obstacle
Robot R
Obstacle O
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Properties of Minkowski Sum

• Minkowski sum of boundary of P and boundary of Q
  is a subset of boundary of
• Minkowski of two convex sets is convex

P?Q
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Minkowski sum of convex polygons

• The Minkowski sum of two convex polygons P and Q
  of m and n vertices respectively
• is a convex polygon P Q of m n vertices.
• The vertices of P Q are the sums of vertices
  of P and Q.

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Gauss Map

• Gauss map of a convex polygon
• Edge ? point on the circle defined by the normal
• Vertex ? arc defined by its adjacent edges

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Gauss Map Property of Minkowski Sum

• pq belongs to the boundary of Minkowski sum
• only if the Gauss map of p and q overlap.

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Computational efficiency

• Running time O(nm)
• Space O(nm)

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Minkowski Sum of Non-convex Polygons

• Decompose into convex polygons (e.g., triangles
  or trapezoids),
• Compute the Minkowski sums, and
• Take the union
• Complexity of Minkowski sum O(n2m2)

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Worst case example

• O(n2m2) complexity

2D example Agarwal et al. 02
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3D Minkowski Sum

• Convex case
• O(nm) complexity
• Many methods known for computing Minkowski sum in
  this case
• Convex hull method
• Compute sums of all pairs of vertices of P and Q
• Compute their convex hull
• O(mn log(mn)) complexity
• More efficient methods are known Guibas and
  Seidel 1987

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3D Minkowski Sum

• Non-convex case
• O(n3m3) complexity
• Computationally challenging
• Common approach resorts to convex decomposition

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3D Minkowski Sum Computation

• Two objects P and Q with m and n convex pieces
  respectively
• Compute mn pairwise Minkowski sums between all
  pairs of convex pieces
• Compute the union of the pairwise Minkowski sums
• Main bottleneck
• Union computation
• mn is typically large (tens of thousands)
• Union of mn pairwise Minkowski sums has a
  complexity close to O(m3n3)
• No practical algorithms known for exact Minkowski
  sum computation

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Minkowski Sum Approximation

• We developed an accurate and efficient
  approximate algorithm Varadhan and Manocha 2004
  
• Provides certain geometric and topological
  guarantees on the approximation
• Approximation is close to the boundary of the
  Minkowski sum
• It has the same number of connected components
  and genus as the exact Minkowski sum

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Rod (24 tris)
Brake Hub (4,736 tris)
Union of 1,777 primitives
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Spoon (336 tris)
Anvil (144 tris)
Union of 4,446 primitives
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Scissors (636 tris)
Knife (516 tris)
Union of 63,790 primitives
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444 tris
1,134 tris
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Union of 66,667 primitives
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Offsetting
Cup Offset
Cup (1,000 tris)
Gear Offset
Gear 2,382 tris)
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Configuration Space Approximation- 3D Translation
Obstacle O
Robot R
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Assembly
Robot
Obstacle
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Assembly
Obstacle
Start
Goal
Roadmap 16 secs
Path Search 0.22 secs
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Assembly
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Path in Configuration Space
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Other Applications

• Minkowski sums and configuration spaces have also
  been used for

• Interference Detection
• Penetration Depth
• Packing

• Morphing
• Tolerance Analysis
• Knee/Joint Modeling

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Applications - Dynamic Simulation

• Interference Detection
• Penetration Depth
• Computation

Kim et al. 2002
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Morphing
A
B
Morph
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Applications - Packing
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Configuration Space of 2T1R Robot

• Dinesh Manocha
• Gokul Varadhan
• UNC Chapel Hill

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Polygonal robot translating rotating in 2-D
workspace
configuration space
workspace
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Polygonal robot translating rotating in 2-D
workspace
q
y
x
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Contact Surfaces (C-surfaces)

• A C-surface arises from a contact between
  features of the robot and the obstacle

R
O
R
O
Type A contact
Type B contact
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Type A Contact Surface
APPLAi,j
vi R(q) . (bj-1 bj ) ? 0 ?
Contact is feasible when
vi R(q) . (bj1 bj ) ? 0
bj-1
ai1(q)
O
R
bj
bj1
vi R(q)
ai(q)
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2D Translation and Rotation
Obstacles
Robot
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Contact Surfaces
3,929 contact surfaces
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Representation of C-obstacle

• How can we represent C-obstacle in terms of
  C-surfaces?
• Non-convex case
• Resort to convex decomposition

For the case of a convex robot and a convex
obstacle,
CONSTAi,j (q) ? CONSTBi,j(q) is true for all
contacts (edge-vertex pairs)
q ? CO
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Free Space and Contact Surfaces

• F is bounded by the C-surfaces

C-surfaces
F
C-obstacle
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Free Space Computation

• To obtain the free space requires computing
  arrangement of the C-surfaces

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Arrangement

• Arrangement A(S) of a set S of geometric objects
  Halperin 1997 Agarwal Sharir 2000

Decomposition of space into relatively open
connected cells of dimensions 0,...,d
Arrangement of lines (clipped within a window)
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Free Space Computation

• Compute an arrangement of the C-surfaces
• Compute intersections between the C-surfaces
• Retain the appropriate portions of the arrangement

F
C-obstacle
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Free Space Computation

• Arrangement computation is difficult
• Computing surface-surface intersection is prone
  to robustness problems
• Typically O(n2) number of contact surfaces
• Contact surfaces are non-linear

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Free Space Approximation

• We have developed an accurate and efficient
  approximate algorithm Varadhan and Manocha 2004
  
• Provides certain geometric and topological
  guarantees on the approximation
• Approximation is close to the boundary of the
  free space
• It has the same number of connected components
  and genus as the exact Minkowski sum

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Free Space Approximation
3,929 contact surfaces
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2T1R Gears
Goal
Start
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2T1R Gears
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2T1R GearsPath in Configuration Space
Path
Goal
Start