Finding an Unpredictable Finding an Unpredictable Target in a Workspace with Target in a Workspace with

ObstaclesObstaclesLaValle, Lin, Guibas, Latombe, and Motwani, 1997

CS326 Presentation by David Black-Schaffer

OverviewOverview

• Searching a complicated environment in such a way that an “evader” can’t “sneak” by.

• Applies to: adversarial situations, locating items which may move during the search

The StrategyThe Strategy

Courtesy of Professor Latombe

Related ProblemsRelated Problems

• Homicidal Chauffeur (no Geometry)– Fast car vs. slow maneuverable human

• Art Gallery (no Motion)– How many observers needed to cover the whole space?

M. Falcone

Homicidal Chauffeur Art Gallery

TopicsTopics

• Bounds on how many pursuers are needed

• Information space representation

• How to find a path

AssumptionsAssumptions

• Target motion is continuous• 2D, omnidirectional unlimited distance

sensors

Evader

Pursuer

Algorithm GoalsAlgorithm Goals

• A fast, efficient solution strategy

• Bounds on the number of pursuers needed in terms of the geometry

Number of PursuersNumber of Pursuers

• Depends on the geometry and topology of the free space

• Crucial to issues of “completeness” of the algorithm

Upper BoundsUpper Bounds

• Simply-connected: n edges, O(lg n)• With holes: h holes, n edges: O(lg n + sqrt(h))

Simply-connected Hole

Lower BoundsLower Bounds

• Parson’s Problem: depth k, O(k+1)– Connected graph evasion

– Can be converted into corridor with four bends

Parson’s ProblemParson’s Problem

Finding a SolutionFinding a Solution

• Information Space State Representation

• Only keep Critical Information Changes

Information SpaceInformation Space

• Incomplete knowledge of state– Where is the evader?

• Work with what we do know and can compute:– Location of the Pursuer

– Visibility Region

• Define our State based on:– Current Free Space location

– State of the Free Space Edges at that location (contaminated/clean)

Information StateInformation State

• 4 possible Information States at this location:– (0,0), (0,1), (1,0), (1,1)

• By knowing the location in the Free Space and the state of the gap edges we uniquely define the Information State of the system.

1 or 0

1 or 0

(x,y)

Key PointKey Point

• Multiple Information Space Points may map to the same Cartesian Point

Critical Information ChangesCritical Information Changes

• Information State only changes when a gap edge appears or disappears.

• Conservative Cell Partitioning• Keep track of just these transitions to simplify

without losing completeness.

Information State: (x1,y1,0,1)Information State: (x2,y2,0,1)Information State: (x3,y3,0,1)Information State: (x4,y4,0)Information State: (x3,y3,0,0)Information State: (x,y,x, x)

Clean

Contaminated

PartitioningPartitioning

• Shoot rays off edges in both directions if possible and from vertices if no collisions in either direction

Finding a PathFinding a Path

• Move between the Free Space centriods of the partitions

• How to plan a path in Information Space?

Information State GraphInformation State Graph

• Connects all possible Information Space States– All edge gap contaminated/clean combinations at all points– A point with 2 edge gaps will have four nodes (00, 01, 10, 11) in this graph– Can grow exponentially

• Keep track of gap edges splitting or merging– Connections between Information Space States– Number of gaps may change; need to preserve the connectivity– Preserve contamination

• Search the graph for a solution (Dijksta’s Algorithm)– Initial State will have all contaminated edges (11…)– Goal State will have all clean edges (00…)– Each vertex will only be visited once– Cost function based on Euclidian distance between points

SolutionSolution

Clean

Contaminated

Visible

In More DetailIn More Detail

Re-contaminationRe-contamination

Multiple PursuersMultiple Pursuers

• Do one as best you can (greedy algorithm)• Add another to cover the missed spaces• Less complete, but works pretty well

ConclusionsConclusions

• Works well in 2D with simple geometry and perfect vision– Fast (a few seconds on a 200MHz RISC machine)

– Very effective for cases requiring only 1 robot

– Elegant approach

However…However…

• Requires a simple, 2D geometry– Can simplify more complex geometry

– Need to watch out for collisions

• Information State Graph can be very big• Deterministic: not adaptable to partial information• Real-world vision is not perfect

– Can deal with cone vision

2 Robots2 Robots

Courtesy of Professor Latombe

Animated VisibilityAnimated Visibility

Courtesy of Professor Latombe