Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and...
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Transcript of Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and...
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