l6_bw_gppos

l6_bw_gppos - Global Path Planning via Optimal Search and...

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Brian Wil iams, Fal 05 4 Brian Wil iams, Fal 05 1 Global Path Planning via Optimal Search and Shortest Paths Brian C. Williams 16.410 / 13 Se ssion 6 Slides draw upon material from: 6.034 Tomas Lozano Perez and Winston, Brian Wil iams, Fal 05 2 Assignment • Reading: • Problem Sets: Russell and Norvig AIMA Cormen, Leiserson and Rivest ItA – Lecture notes plus… – Informed search and exploration: AIMA Ch. 4.1-2 – Computing Shortest Paths: Cormen, Leiserson & Rivest, (optional) “Introduction to Algorithms” Ch. 25.1-.2 – PS #2 due today, – PS#3 out today. Due Wednesday, October 5th. Brian Wil iams, Fal 05 3 Plan Execute Monitor & Diagnosis Locate in World Plan Routes Map Maneuver and Track Communicate and Interpret Agent Architecture and Building Blocks courtesy NASA Ames courtesy NASA Lewis How Do We Maneuver? Brian Wil iams, Fal 05 6 Questions About Maneuvering • What is the most effective route? Today • How do we quickly compensate for error? Today (time permitting) • How do we anticipate adversaries? Monday • How do we anticipate the effects of error? Monday 1
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Brian Wil iams, Fal 05 8 10 Brian Wil iams, Fal 05 7 L6 and L7: Global Maneuvering using Roadmaps courtesy NASA Ames courtesy NASA Lewis using Linear Programs L8 and L9: Fine-grained Maneuvering Answering Questions Using Roadmaps • What is the most effective route? Find the shortest path from S to G • How do we quickly compensate for error? Precompute action policies (shortest path trees) • How do we anticipate adversaries? Perform adversarial game tree search • How do we anticipate the effects of error? Perform stochastic game tree search Brian Wil iams, Fal 05 9 Brian Wil iams, Fal 05 Outline Today: Encoding roadmaps as weighted graphs Finding a shortest path – Branch and bound – Approximate methods (appendix) Compensating for error using policies – Computing shortest path trees (optional) Next Lecture: Anticipating adversaries Anticipating environmental error Generating road maps Best-first search Courtesy of NASA. Roadmaps are Formalized as Weighted Graphs u v s 1 2 10 5 7 9 2 3 4 6 x y Graph G = <V, E> Weight function w : E ? ´ Brian Wil iams, Fal 05 11 Weighted Graphs Have Weighted Paths u v s 1 2 10 5 7 9 2 3 4 6 x y Path p = < v o , v 1 , … v k > Path weight w (p) = S w(v i-1 ,v i ) Example: p = <s, u, v, y> w(p ) = 10 + 1 + 4 Brian Wil iams, Fal 05 12 2
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Weighted Graphs Have Shortest Paths u v 1 x y 2 10 5 7 9 2 3 4 6 s Shortest path weight d(u,v) = min { w (p) : u ? p v } else 8 Example: d(u,v) = min { w (<u,v>), w(<u, x, v>), w(<u,x,y,v>)} = min {1, 2+9, 2+2+6} = 1 Brian Wil iams, Fal 05 13 Notational Note • x (lower case) An element • S (upper case) A set • x in S “x is an element of set S” {x : p(x)} “The set of all elements x such that p(x) holds. u
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l6_bw_gppos - Global Path Planning via Optimal Search and...

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