DEEP : Dualspace Expansion for Estimating Penetration depth between convex polytopes

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

• Jae Hoon Jung
• POSTECH VRPM
• Advanced Computer Haptics Class
• 2006.4.26

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Outline

• Background
• Previous Work
• Preliminaries
• Minkowski Sum
• GJK Algorithm
• Brute-Force PD computation
• Gauss Map
• Width Computation
• DEEP
• Algorithm
• Analysis
• Performance
• Application to 6DOF Haptic Rendering
• Summary and Future work
• Reference

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Background

• Need for a distance measure for the extent of
  interpenetration between objects
• Applications 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
• O(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
  (Configuration Space Obstacle)
• PD minimum distance between OQ-P and ?(P-Q).

Boundary of Minkowski difference(P,Q)
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Preliminaries - Minkowski Sum

• Minkowski Sum (A, B) a b a ? A, b ? B
• Minkowski Diff (A,B) a - b a ? A, b ? B
  Minkowski Sum (A, -B)
• A and B collide iff Minkowski Diff(A,B) contains
  an origin.

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Preliminaries - Minkowski Sum
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Preliminaries - Minkowski Sum

• Distance penetration-depth computation in
  Minkowski sum are of inherently different nature
  even for convex models
• Distance
• one local minimum
• Penetration Depth
• many local minima

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Preliminaries - Minkowski Sum

• Expensive to compute the boundary of Minkowski
  difference
• Structure of Minkowski difference changes if
  objects rotate independently
• For two convex polyhedra with m and n vertices,
  Minkowski Difference may take O(m n)
• O(n6) at Non convex polygon
• However, GJK algorithm uses Minkowski difference
  quite efficiently to find distance between two
  convex polyhera, computing it on demand.

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Preliminaries - GJK Algorithm

• GJK algorithm solves proximity queries for two
  convex polyhedra
• Computing distance d
• Finding closest pair of points PA, PB

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Preliminaries - GJK Algorithm
Terminology
Supporting (or extreme) point
for direction
returned by support mapping function
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Preliminaries - GJK Algorithm
Terminology
Point set C
Convex hull, CH(C)
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Preliminaries - GJK Algorithm
Terminology

• A d-simplex is the convex hull of d1 affinely
  independent points in d-dimensional space.

• A set of k1 points are said to be affinely
  independent if they generate an affine space of k
  dimension.

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Preliminaries - GJK Algorithm

• GJK Algorithm
• Initialize the simplex set Q with up to d1
  points from C (in d dimensions)
• Compute point P of minimum norm in CH(Q)
• If P is the origin, exit return 0
• Reduce Q to the smallest subset Q of Q, such
  that P in CH(Q)
• Let VSC(P) be a supporting point in direction
  P
• If V is no more extreme in direction P than P
  itself, exit return P
• Add V to Q. Go to step 2

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Preliminaries - GJK Algorithm
INPUT Convex polyhedron C given as the convex
hull of a set of points
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Preliminaries - GJK Algorithm

• 1. Initialize the simplex set Q with up to d1
  points from C (in d dimensions)

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Preliminaries - GJK Algorithm
2. Compute point P of minimum norm in CH(Q)
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Preliminaries - GJK Algorithm

• 3. If P is the origin, exit return 0
• 4. Reduce Q to the smallest subset Q of Q, such
  that P in CH(Q)

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Preliminaries - GJK Algorithm

• 5. Let VSC(P) be a supporting point in
  direction P

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Preliminaries - GJK Algorithm

• 6. If V is no more extreme in direction P than P
  itself, exit return P
• 7. Add V to Q. Go to step 2

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Preliminaries - GJK Algorithm
2. Compute point P of minimum norm in CH(Q)
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Preliminaries - GJK Algorithm

• 3. If P is the origin, exit return 0
• 4. Reduce Q to the smallest subset Q of Q, such
  that P in CH(Q)

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Preliminaries - GJK Algorithm

• 5. Let VSC(P) be a supporting point in
  direction P

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Preliminaries - GJK Algorithm

• 6. If V is no more extreme in direction P than P
  itself, exit return P

DONE!
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Preliminaries - GJK Algorithm
Generalization

• A and B intersecting iff AB contains the origin.
• Distance between A and B given by point of
  minimum norm in AB!
• Use the previous procedure on AB.
• Only change needed
• Support mapping is separable, so we can form it
  by computing support mapping for A and B
  separately.

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Preliminaries -Brute-Force PD Computation
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Preliminaries -Brute-Force PD Computation

• Fastest theoretical algorithm to compute PD
• Time Complexity O( ) ,
• e Any positive constant
• Not implemented

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Preliminaries -Gauss Map
V

• 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
V
S2
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Preliminaries -Gauss Map
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Preliminaries - Width computation

• The definition Given a set of points
  Pp1,p2,..,Pn the width of P, W(P),is defined
  as the minimum distance between parallel planes
  supporting P.
• Antipodal Pairs Let F and F be faces of CH(P).
  Then F and F are called antipodal pairs, if
  there exist two parallel supporting hyperplanes h
  ? h such that
• Special pairs
• Vertex-face pairs (VF-pair)
• Edge-edge pairs (EE-pair)
• Vertex-edge pairs (VE-pair)

????
VE Antipodal pairs
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Preliminaries - Width computation
Houle and Toussaints width computation algorithm

• Lemma 1
• The width of a set of points P in 3D is the
  minimum distance between supporting planes
• The planes having the minimum interdistance
  (i.e., width) are realized only either by
  antipodal VF pair or by antipodal EE pair.
• Idea for The algorithm Enumerate all VF and EE
  pairs

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Preliminaries (Houle and Toussaints width
computation algorithm)

• Houle and Toussanints Algorithm in 3D
• Construct Convex hull in O(n log n) time using
  Preparata and Hongs algorithm
• Transform the polyhedron into two planar
  subdivisions in O(n) time as described
• Compute the overlay of the two planar
  subdivisions using Guibas and Seidels algorithm
  in O(n I) time. (I is the number of antipodal
  edge-edge pairs)
• Examine all vertices and all parallel rays of
  unbounded regions of the overlay to generate V-F
  and E-E pairs in O(I) time, and report the
  smallest antipodal pair distance as being the
  width.

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Preliminaries (Houle and Toussaints width
computation algorithm)
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Preliminaries (Houle and Toussaints width
computation algorithm)

• The width of a polyhedron P in three dimensions
  can be computed in O(n I) time where I is the
  number of antipodal edge-edge pairs of CH(P).
• In the worst case, I is O(n2),

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Globally Optimal Solution of PD

• Explicitly compute Minkowski difference of two
  convex polytops
• Find the smallest distance between Minkowski
  difference origin and CSO(Configuration Space
  Obstacle) from all vertices using a simple
  brute-force algorithm
• O(n2) computation time is unaffordable for
  interactive applications.

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DEEP Overview

• Localized computations of PD using Gauss maps and
  their overlay
• Iterative Optimization
• Identify an initial feature for walk
• Measure the current PD
• March toward the local optimum

CSO
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DEEP Initialization

• Find a subset of the overlay.

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DEEP Initialization

• A good initial guess can lead to empirically
  almost constant running time
• Worst case can lead to O(n2) running time
• The goal is estimate optimal pentration direction
  and use it to form vertex hub pair
• Mostly,c2-c1 and c1-c2 is chosen from each object
  and assigned as an initial vetex hub pair

C means Centroid of object
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DEEP Initialization

• Guessing an Optimal PD direction

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DEEP Iteration
OQ-P
P - Q
CSO
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DEEP Iteration

• Construct a local Gauss map each for v1 and v1
  (Step 1 of Algorithm 3.2).
• Project the Gauss maps onto z 1 plane and call
  them G and G respectively. G and G correspond
  to convex polygons in 2D (Step 1 of Algorithm
  3.2).
• Compute the intersection between G and G using a
  linear time algorithm. The result is a convex
  polygon and we label each vertex comprising the
  intersection ui. These ui s correspond to the VF
  or EE antipodal pairs in object space (Step 2 of
  Algorithm 3.2).
• In object space, determine which ui corresponds
  to the best local improvement in PD (Step 3 of
  Algorithm 3.2).
• Set an adjacent vertex pair (adjacent to ui) to
  (v2, v2) (Algorithm 3.1).

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DEEP Iteration
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DEEP Iteration
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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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Analysis of DEEP

• The worst-case time complexity is O(n2) time
  complexity because the algorithm is very similar
  to Houle and Toussaints width computation
• Empirically the algorithm is usually terminated
  after less than five iteration

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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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Computation time - Fixed PD
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Computation time -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
A decomposed torus
Pairwise estimation
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Force Computation

• Non convex surface decomposition -gt small convex
  patches around concavities
• To avoid instability due to large stiffness ,
  the stiffness of different contact is added up to
  formulate a single stiffness
• In order to represent all clusters to one
  contact, contact position and direction are
  computed in terms of a weighted average.

dPD ndirection pcontact position tdistance
tolerance for defining when two disjoint objects
are in contacts
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Force computation
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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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References

• Eric Larsen Collision Detection for Real-time
  Simulation -HCTS 1999
• Christer Ericson The Gilbert-Johnson-Keerthi
  (GJK) Algorithm SIGGRAPH 04 COURSE NOTE
• M.E. Houle and G.T. Toussain Computing the
  Width of a Set IEEE Trans. Pattern Analysis and
  Machine Intelligence,1998
• DEEP Y. J. Kim, M. C. Lin, and D. Manocha,
  "DEEP Dual-Space Expansion for Estimating
  Penetration Depth between Convex Polytopes," in
  Proceedings of the IEEE International Conference
  on Robotics and Automation, 2002,
• DEEP Y. J. Kim, M. A. Otaduy, M. C. Lin, and D.
  Manocha, "Six-Degree-of-Freedom Haptic Display
  Using Localized Contact Computations," in
  Proceeeding of the International Symposium on
  Haptic Interfaces for Virtual Environment and
  Teleoperator Systems, 2002, pp. 209-216.
• DEEP ICRA presntation PPT Y.J.KIM