Building Surveillance Graphs for GRAPH-CLEAR

FRC Nav. Reading Group - Nov. 2, 2009

Building Surveillance Graphs forGRAPH-CLEAR

Andreas Kolling & Stefano CarpinUC Merced

Presented by Pras Velagapudi


Problem: Pursuit-evasion

• Spatial (typically 2D) world• Assumptions about intruders:

– Know where robots are– Can move unboundedly

• Objective:– Find a sequence of actions that detects all

intruders using the least number of robots


GRAPH-CLEAR

• Formalized subclass of pursuit evasion• Environment is weighted graph• Two actions

– Blocking• Robot sits on edge, intruder can’t pass

– Sweeping• Robot searches vertex, determines if intruder is present


GRAPH-CLEARIn order to sweep, all edges must be blocked


Surveillance Graph

• Blocking edges– Edges are blocked when their corresponding areas

are completely segmented by sensor coverage• Sweeping vertices

– Any local clearing strategy for region corresponding to vertex– Simple bounding box sweeping

used here


Moving to the real world…


Moving to the real world…

Step 1:Surveillance

Graph

GRAPH-CLEARSolver

Step 2:???

Step 3:Profit!


Step 1: Surveillance GraphStart with generalized Voronoi graph


Step 1: Surveillance Graph

Minima requirements:

•1 nearby point is farther from Voronoi edge

•No nearby points are closer to Voronoi edge

Find local clearance minima


Step 1: Surveillance Graphclearanceminima

Find local clearance minima


Step 1: Surveillance GraphPartition along minima to create initial SG


Step 1: Surveillance GraphCompute edge weights

w(e) = 2

e


Step 2: Improving the graph

• Auto-generated graph can be inefficient– e.g. GVG + aliasing = extra vertices

• So, do some optimization!



• Collapse leaf nodes

• Collapse chains




Step 3: GRAPH-CLEAR solving

• Start with weighted surveillance graph

• Edges = narrow corridors between regions• Vertices = wide and open regions• Weights = # of robots to sweep/block area

• Vertices = {Contaminated, Clear}• Edges = {Contaminated, Clear, Blocked}



Shove robots in doors until you have a tree graph

Efficient solution for tree graphs

• NP-complete, however, efficient for trees:



• Tree solution overview:1. Compute bidirectional labels for each edge

(Labels represent the cost of clearing the subtree on the other side of the edge, if the source node is already cleared)

2. Find lowest cost label3. Start clearing tree from that label





• Leaf node:

• Internal node:



• Actually, there is an optimal O(n2) algorithm– Compute cuts on the graph for cleared vertices– Subdivide the problem into that of solving the

clearing problem for each subtree


Results

• UC Merced


Results

• Radish (sdr_site_b)


Results

sens

ing r

ange

# ini

tial v

erts

# fina

l ver

ts# r

obot

(ini

tial)

# rob

ots (

final)

# cyc

lesco

st of

cycle

sm

ap co

vera

gem

ap co

vera

ge w

/ non

-MST

ro

bots


Conclusion

• GRAPH-CLEAR is applicable to real-world problems

• Can construct efficient surveillance graphs using simple methods

• Future work– Probabilistic variants– Local optimizations in search patterns

Documents

Building Surveillance Graphs for GRAPH-CLEAR