The Hierarchical Generalized Voronoi Graph (HGVG)

1
The Hierarchical Generalized Voronoi Graph (HGVG)

• Presentation by Bernice Ma

2
Overview
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Introduction
• Distance Function
• GVG and Pre-image Theorem
• Numerical Curve Tracing
• Definition of HGVG

3
Introduction Roadmaps
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• one dimensional curves
• important environmental features
• properties
• accessibility
• departability
• connectivity.

4
HGVG as Roadmap
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• HGVG is a roadmap, constructed incrementally
  using sensor data.

5
Assumptions
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Robot is a point
• Workspace contains only convex obstacles
• Non-convex obstacles are modeled as the union of
  convex obstacles
• Bounded space

6
Basic Definitions
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Single-object distance function

7
Basic Definitions (cont.)
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Multi-object distance function

8
X for X-ray
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Point not within line of sight

9
Visible Distance Functions
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Single-object
• Multi-object

10
Two-Equidistant
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Two-equidistant surface
• Convex obstacles?
• Two-equidistant surjective surface

11
More Rigorous Definition
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Going through obstacles
• Two-equidistant face

12
General Voronoi Diagram
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
13
General Voronoi Graph
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• In 3-Dimensions
• In m-Dimensions

14
GVD vs. GVG
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
Equidistant (obs) Dim Codim
GVD 2 m-1 1
GVG m 1 m-1

• Proofs by Pre-Image Theorem to come

15
Pre-Image Theorem
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
16
Proof for GVD
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
17
Proof for GVG
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• For 3-Dimensions

18
Proof for GVG (cont.)
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• For m-Dimensions

19
More on GVG
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Is the GVG a 1-Dimension manifold?

20
More on GVG (cont.)
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Is the GVG a 1-Dimension manifold?

No, but its the union of 1D manifolds
21
Traceability
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• x is a point on the GVG
• normal slice plane
• sweep coordinate
• Define

22
Traceability (cont.)
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Predictor-corrector scheme
• Take small step, in direction.
• Correct using iterative Newtons Method

23
Accessibility
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Gradient Ascent Cascading Sequence of Gradient
  Ascent Operations
• Move until
• Maintain 2-way equidistant while

24
Demo
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
25
GVD Connected?
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Proof
• Im is continuous (Prof. Yap, NYU)
• Im of a connected set, under a continuous map, is
  a connected set

26
GVG Connected?
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
27
GVG Connected?
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• is not connected

28
GVG2
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Second-order two-equidistant surface

29
Linking to GVG Cycle
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Detect GVG Cycle
• Gradient Descent
• increases distance to
• decreases distance to
• projection
• tangent space of
• projection onto the tangent space

30
From GVG2 to GVG Cycle
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
31
Two Problems
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Gradient goes to 0?
• Going on top of the box
• Define occluding edges

32
Finding Occluding Edges
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
33
Visible Distance Revisited
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Single-object
• Multi-object

34
Occluding Edges (cont.)
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Change in second closest object
• GVG two-equidistant edges (continuous)
• Occluding edges (not continuous)
• Questions?
• When to link?
• Do we have all possible edges?

35
More Linking
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• GVG2
• Occluding edges
• GVG Edge
• Boundary Edge
• Floating boundary edge

36
Basic Links
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
37
Room with Box
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
38
Floating Box
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
39
Box with Opening
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG
40
HGVG is Connected
Introduction Distance Function GVG and Pre-Image
Theorem Numerical Curve Tracing Definition of HGVG

• Connect boundaries of GVG2 to form a connected
  roadmap.

41
References

• H. Choset and J. Burdick. Sensor Based Motion
  Planning The Hierarchical Generalized Voronoi
  Graph. The International Journal of Robotics
  Research, 19, no. 2 (2000) 96-125