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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...
![Page 1: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/1.jpg)
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
![Page 2: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/2.jpg)
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
![Page 3: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/3.jpg)
The StrategyThe Strategy
Courtesy of Professor Latombe
![Page 4: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/4.jpg)
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
![Page 5: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/5.jpg)
TopicsTopics
• Bounds on how many pursuers are needed
• Information space representation
• How to find a path
![Page 6: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/6.jpg)
AssumptionsAssumptions
• Target motion is continuous• 2D, omnidirectional unlimited distance
sensors
Evader
Pursuer
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Algorithm GoalsAlgorithm Goals
• A fast, efficient solution strategy
• Bounds on the number of pursuers needed in terms of the geometry
![Page 8: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/8.jpg)
Number of PursuersNumber of Pursuers
• Depends on the geometry and topology of the free space
• Crucial to issues of “completeness” of the algorithm
![Page 9: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/9.jpg)
Upper BoundsUpper Bounds
• Simply-connected: n edges, O(lg n)• With holes: h holes, n edges: O(lg n + sqrt(h))
Simply-connected Hole
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Lower BoundsLower Bounds
• Parson’s Problem: depth k, O(k+1)– Connected graph evasion
– Can be converted into corridor with four bends
![Page 11: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/11.jpg)
Parson’s ProblemParson’s Problem
![Page 12: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/12.jpg)
Finding a SolutionFinding a Solution
• Information Space State Representation
• Only keep Critical Information Changes
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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)
![Page 14: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/14.jpg)
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)
![Page 15: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/15.jpg)
Key PointKey Point
• Multiple Information Space Points may map to the same Cartesian Point
![Page 16: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/16.jpg)
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
![Page 17: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/17.jpg)
PartitioningPartitioning
• Shoot rays off edges in both directions if possible and from vertices if no collisions in either direction
![Page 18: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/18.jpg)
Finding a PathFinding a Path
• Move between the Free Space centriods of the partitions
• How to plan a path in Information Space?
![Page 19: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/19.jpg)
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
![Page 20: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/20.jpg)
SolutionSolution
Clean
Contaminated
Visible
![Page 21: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/21.jpg)
In More DetailIn More Detail
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Re-contaminationRe-contamination
![Page 23: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/23.jpg)
Multiple PursuersMultiple Pursuers
• Do one as best you can (greedy algorithm)• Add another to cover the missed spaces• Less complete, but works pretty well
![Page 24: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/24.jpg)
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
![Page 25: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/25.jpg)
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
![Page 26: Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d6d5503460f94a4d3eb/html5/thumbnails/26.jpg)
2 Robots2 Robots
Courtesy of Professor Latombe
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Animated VisibilityAnimated Visibility
Courtesy of Professor Latombe