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Analysis of Greedy Robot- Navigation Methods Sven Koenig (USC) Apurva Mugdal (Ga. Tech) Craig Tovey (Ga. Tech)

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Move the robot to solve: Localization Goal-based Planning Given map of 2D or 3D gridworld, determine location Given map minus roadblocks, reach goal

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Graph Model of Robot Motion Occupy one vertex of G=(V,E) at a time. G usually is a gridgraph: V= set of cells. Adjacency is {N,S,E,W} or chess king. Robot has compass Tactile sensors detect neighbors of current vertex v; other sensors may detect more.

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1 st problem: localization Know graph G (input). Robot is at some vertex of G. Problem: determine location by moving about, making observations at each vertex visited. Might conclude that robot cannot uniquely determine its location.

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2 nd problem: goal directed search (target) Know graph G=(V,E) Know initial location s and goal t Robot has compass, or on general graphs, can distinguish among vertices Don’tblockedDon’t know B V, set of blocked vertices If w is blocked and (v,w) E, the robot detects that w is blocked when it scans from v There may be no unblocked s to t path

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S t S t S t

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S t S t S t

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S t S t S t

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S t S t S t

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slow Robots are slow Planning time usually small compared with travel time We can replan as we gain information Our plans can be algorithms

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Greed goes by many names… D* [Stentz 95]; D-Lite [Koenig-Likhachev 02] Greedy mapping [Thrun et al. 98] A* Planning with Freespace Assumption

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Greed is found in many places… Nomad class museum tour-guide[Thrun et al. 98] Nomad 150 mobile robots [Koenig-Likhachev 02] Super Scouts [Romero et al. 01] Mars Rover Prototype Nourbakhsh and Genesereth[96]

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but it is always the same idea Choose the most economical move that improves the situation

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Target: move along a shortest presumed unblocked path to t. Localization: move to the nearest vertex which if scanned eliminates at least one location from set of remaining possibilities. If you don’t know whether or not you can get to it, it isn’t the nearest vertex that reduces uncertainty.

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Mars Rover Prototype

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Main results on greedy algorithms upper bound: goal search on general graphs

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Greed: upper bounds on travel Localization -- O(n log n) bound by covering region with bomb blasts. Applies to greedy mapping too. Target: localization analysis does not apply. Use bounds on girth of graphs instead.

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Analyzing greedy localization for any sensor type Algorithm travels to a nearest informative vertex. That vertex is scanned and becomes uninformative. Other vertices may become uninformative too. Uninformative vertices never become informative.

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v

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v

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v

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If You Take a Large Step, All the Vertices in a Large Area Are Not Informative Define a bombing sequence as a sequence of (vertex, radius, unbombed set) triplets. Drop a bomb on an unbombed vertex with the given blast radius. The adversary maximizes the sum of the blast radii (+1)

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Lemma: # bombs with radius ≥ t ≤ 2|V|/t vw

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w

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Benefits of Greedy Localization Analysis Most current implementations travel to nearest vertex about which there is uncertainty, rather than to nearest informative vertex, for non-tactile sensors Same upper bound holds but we suspect performance is slightly worse Applies to greedy mapping too

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Upper Bounds for Goal Search Why bombing sequence does not apply Telescoping Time reversal: add edges to blocked vertices Adding edges makes cycles. Big steps mean big (long) cycles. Relate to bounds on girth (shortest cycle) from Euler’s formula, Alon et al’s thm [01].

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S t S t S t Why bomb radii don’t work for target

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S t S t S t Telescope idea for target 543 2 5 4 6 1 8 9 10 6 7 4 5 6 8 20 21 22 2324 (10 - 2) + (22 – 6) + … within |V| of total

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S t S t Time reversal and cycles 8 9 6 7 4 5 8 20 21 22 2324 10 22 – 6 · length shortest cycle containing new edge - 4 Reverse time: add the failed edges

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S t S t Time reversal and cycles 8 9 6 7 4 5 8 9 8 7 9 1011 10 7 – 6 · length shortest cycle containing new edge - 4

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Bounding sum of cycle lengths IDEA: as we go backwards in time, graph has more edges. Shortest cycle containing new edge should not often be big Girth: length of shortest cycle Thm [Alon, Hoory, Linial]: Any graph with average degree d>2 has girth · log d-1 |V| Sorting, etc. gives O(log^2 |V|) Planar graphs: O(log |V|) by considering faces and Euler’s formula

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The cycle game

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|V|

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The cycle game |V|/2

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The cycle game |V|/2

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The cycle game |V|/4

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Conclusions Robot motion provides a nice blend of theory and practice Some theoretical justification for greed Idea of visiting informative vertices may slightly improve current implementations of greedy localization (and mapping) Informative vertices might be useful for goal search if |B| is not small.

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S t S t S t

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S t S t S t

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S t S t S t

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lower bounds Essentially one proof for both problems (Target grid graph construction differs significantly)

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Localization Make an extra copy for each branch and block the leaf at its tip X The robot has to check each tip to know which copy it is in

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Lower bound: goal search

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Telescoping

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