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

2. 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.
ICRA 2002
UNC in Chapel Hill

3. Previous Work Cameron and Culley ’86 Dobkin et. al. ’93 Cameron ’97
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.
ICRA 2002
UNC in Chapel Hill

4. Preliminaries (Minkowski Sum)
Minkowski Sum and Minkowski Difference
P+Q = { p+q | 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). P OP OQ
P - Q
OQ-P
ICRA 2002
UNC in Chapel Hill

5. 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 a’s) F S2
ICRA 2002
UNC in Chapel Hill

6. Preliminaries (Minkowski Sum)
Inverse Map
Fig: Bekker ‘01
Overlay
Gauss Map
ICRA 2002
UNC in Chapel Hill

7. DEEP : Why Incremental ?
n2 computation time is unaffordable for interactive applications.
Observations
PD is shallow.
Motion Coherence.
Penetration  Separation
ICRA 2002
UNC in Chapel Hill

8. 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
OQ-P
P - Q PD PD PD
ICRA 2002
UNC in Chapel Hill

9. DEEP: Initialization Find a subset of the overlay. n n -n ICRA 2002
UNC in Chapel Hill

10. DEEP : Initialization Guessing an Optimal PD direction Motion
Coherence
Centroid
Difference
Penetration Witness
Feature
ICRA 2002
UNC in Chapel Hill

11. DEEP : Iteration
OQ-P
P - Q
ICRA 2002
UNC in Chapel Hill

12. DEEP : Local Minima The algorithm can be stuck in local minima.
In practice, we can avoid it by using various heuristics.
ICRA 2002
UNC in Chapel Hill

13. 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.
ICRA 2002
UNC in Chapel Hill

14. Timings (Fixed PD)
ICRA 2002
UNC in Chapel Hill

15. Timings (Variable PD)
ICRA 2002
UNC in Chapel Hill

16. PD Direction Tracking (DEEP)
ICRA 2002
UNC in Chapel Hill

17. PD Direction Tracking (EPA)
ICRA 2002
UNC in Chapel Hill

18. Application to 6DOF Haptic Rendering
The PHANTOM 1.5 Sensable Technology
ICRA 2002
UNC in Chapel Hill

19. 6DOF Haptic Rendering Using Localized Contact Computations
ICRA 2002
UNC in Chapel Hill

20. Summary and Future Work
Incremental Penetration Depth Estimation Algorithm (DEEP)
Library:
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.
ICRA 2002
UNC in Chapel Hill

21. Sponsors
ARO
DOE
NSF
ONR
Intel
ICRA 2002
UNC in Chapel Hill

22. Thanks

23. 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
ICRA 2002
UNC in Chapel Hill

24. DEEP : Local Minima The algorithm can be stuck in local minima.
In practice, we can avoid it by using various heuristics.
Centroid Difference
ICRA 2002
UNC in Chapel Hill

25. Degeneracies Coplanar Faces Central Projection
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.
ICRA 2002
UNC in Chapel Hill