DEEP : Dual-space Expansion for Estimating Penetration depth between convex polytopes

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DEEP Dual-space Expansion for Estimating
Penetration depth between convex polytopes

• Young J. Kim
• Ming C. Lin
• Dinesh Manocha

Dept of Computer Science UNC Chapel Hill
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Background

• Need for the distance measure for the extent of
  interpenetration
• Application Robot Motion Planning, Dynamic
  Simulation, Haptic Rendering, etc.
• Penetration Depth (PD)
• Minimum translational distance to make P and Q
  disjoint over all possible directions
• An incremental PD estimation algorithm, DEEP.

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Previous Work

• Cameron and Culley 86
• n2 algorithm based on explicit Minkowski sum
  computation.
• Dobkin et. al. 93
• Directional PD algorithm.
• Cameron 97
• Rough PD estimation based on the GJK algorithm.
• Agarwal et. al. 00
• Randomized algorithm.
• Bergen 01
• Expanding Polytope Algorithm (EPA). IMPLEMENTED.

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Preliminaries (Minkowski Sum)

• Minkowski Sum and Minkowski Difference
• PQ pq p?P, q?Q
• P-Q p-q p?P, q?Q , a.k.a. CSO or TCSO
• PD minimum distance btwn OQ-P and ?(P-Q).

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Preliminaries (Gauss Map)

• Gauss Map (F ? S2) Dual mapping from feature
  space to normal space
• Face f ? Point n (outward normal of f).
• Edge e ? Great Arc a (locus of normals of two
  adjacent faces).
• Vertex v ? Region r (bounded by as)

F
S2
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Preliminaries (Minkowski Sum)
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DEEP Why Incremental ?

• n2 computation time is unaffordable for
  interactive applications.
• Observations
• PD is shallow.
• Motion Coherence.
• Penetration ? Separation

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DEEP Overview

• Localized computations for Gauss map and Overlay
• Iterative Optimization
• Identify an initial feature for walk
• Measure the current PD
• March toward the local optimum

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DEEP Initialization

• Find a subset of the overlay.

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DEEP Initialization

• Guessing an Optimal PD direction

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DEEP Iteration
OQ-P
P - Q
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DEEP Local Minima

• The algorithm can be stuck in local minima.
• In practice, we can avoid it by using various
  heuristics.

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DEEP Performance

• Random Models with different complexities and
  aspect ratios e.g. sphere, ellipsoid, pen.
• One object revolves around the other object while
  rotating on its center of mass.

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Timings (Fixed PD)
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Timings (Variable PD)
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PD Direction Tracking (DEEP)
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PD Direction Tracking (EPA)
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Application to 6DOF Haptic Rendering
The PHANTOM 1.5 Sensable Technology
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6DOF Haptic Rendering Using Localized Contact
Computations
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Summary and Future Work

• Incremental Penetration Depth Estimation
  Algorithm (DEEP)
• Library http//gamma.cs.unc.edu/DEEP
• Better way to avoid the local minima problem.
• Extension to non-convex polyhedron
• Recently a new method combining the object space
  and image space has been proposed.

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Sponsors

• ARO
• DOE
• NSF
• ONR
• Intel

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Thanks
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DEEP Iteration

• 1. Construct G-map G1 and G2 for V1 and V1

• 2. Do the central projection

• 3. Compute the intersection ui s in O(n)

• 4. In object space, determine which ui produces
  the best local improvement

• 5. Repeat this process until there is no
  improvement

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DEEP Local Minima

• The algorithm can be stuck in local minima.
• In practice, we can avoid it by using various
  heuristics.

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Degeneracies

• Coplanar Faces
• Mapped to the same point on Gauss map
• Simply ignore duplicated points.
• Need to expand the search for neighborhood
• Central Projection
• Equator projected to infinity, crossing arc can
  be broken
• Solved by local projection.