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Minkowski Sum

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Polygonal moving object translating in 2-D workspace. Polygonal obstacles ... Anvil (144 tris) Spoon (336 tris) 23. Knife (516 tris) Scissors (636 tris) Union of ... – PowerPoint PPT presentation

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Title: Minkowski Sum


1
Minkowski Sum
  • Gokul Varadhan

2
Last Lecture
  • Configuration space

workspace
configuration space
3
Problem Configuration Space of a Translating
Robot
  • Input
  • Polygonal moving object translating in 2-D
    workspace
  • Polygonal obstacles
  • Output configuration space obstacles represented
    as polygons

4
Configuration Space of a Translating Robot
Workspace
Configuration Space
Robot
Obstacle
C-obstacle
Robot
y
x
  • C-obstacle is a polygon.

5
Minkowski Sum
B
A
6
Minkowski Sum
7
Minkowski Sum
8
Minkowski Sum
9
Configuration Space Obstacle
C-obstacle is
Classic result by Lozano-Perez and Wesley 1979
C-obstacle
Robot R
Obstacle O
10
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
11
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.


12
Gauss Map
  • Gauss map of a convex polygon
  • Edge ? point on the circle defined by the normal
  • Vertex ? arc defined by its adjacent edges

13
Gauss Map Property of Minkowski Sum
  • pq belongs to the boundary of Minkowski sum
  • only if the Gauss map of p and q overlap.

14
Computational efficiency
  • Running time O(nm)
  • Space O(nm)

15
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)

16
Worst case example
  • O(n2m2) complexity

2D example Agarwal et al. 02
17
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

18
3D Minkowski Sum
  • Non-convex case
  • O(n3m3) complexity
  • Computationally challenging
  • Common approach resorts to convex decomposition

19
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

20
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

21
Rod (24 tris)
Brake Hub (4,736 tris)
Union of 1,777 primitives
22
Spoon (336 tris)
Anvil (144 tris)
Union of 4,446 primitives
23
Scissors (636 tris)
Knife (516 tris)
Union of 63,790 primitives
24
444 tris
1,134 tris
25
Union of 66,667 primitives
26
Offsetting
Cup Offset
Cup (1,000 tris)
Gear Offset
Gear 2,382 tris)
27
Configuration Space Approximation - 3D Translation
Obstacle O
Robot R
28
Assembly
Robot
Obstacle
29
Assembly
Obstacle
Start
Goal
Roadmap 16 secs
Path Search 0.22 secs
30
Assembly
31
Path in Configuration Space
32
Other Applications
  • Minkowski sums and configuration spaces have also
    been used for
  • Interference Detection
  • Penetration Depth
  • Packing
  • Morphing
  • Tolerance Analysis
  • Knee/Joint Modeling

33
Applications - Dynamic Simulation
  • Interference Detection
  • Penetration Depth
  • Computation

Kim et al. 2002
34
Morphing
A
B
Morph
35
Applications - Packing
36
Next lecture
  • Configuration space of a polygonal robot capable
    of translation and rotation
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