Analysis of Greedy Robot-Navigation Methods

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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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1st 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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2nd 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
• Dont 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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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-guideThrun et al. 98
• Nomad 150 mobile robots Koenig-Likhachev 02
• Super Scouts Romero et al. 01
• Mars Rover Prototype
• Nourbakhsh and Genesereth96

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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 dont know
  whether or not you can get to it, it isnt 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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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 2V/t
v
w
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Lemma bombs with radius t 2V/t
w
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Lemma bombs with radius t 2V/t
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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
  Eulers formula, Alon et als thm 01.

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Why bomb radii dont work for target
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Telescope idea for target
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(10 - 2) (22 6) within V of total
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Time reversal and cycles
Reverse time add the failed edges
22 6 length shortest cycle containing new
edge - 4
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Time reversal and cycles
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 dgt2 has girth logd-1V
• Sorting, etc. gives O(log2 V)
• Planar graphs O(log V) by considering faces
  and Eulers formula

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The cycle game
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The cycle game
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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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