Geometric Motion Planning : Finding Intersections
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Transcript of Geometric Motion Planning : Finding Intersections
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Sándor P. Fekete, Henning Hasemann, Tom Kamphans, Christiane SchmidtAlgorithms GroupBraunschweig Institute of Technology
Geometric Motion Planning:Finding Intersections
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Motivation – Finding Intersections
One-Dimensional Agents
– No simultaneous movement
– Simultaneous movement
Two-Dimensional Agents
Outlook
MichaelEClarke@flickr
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Motivation
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Motivation• planning motions for mobile
agents:– motion primitives– sensors– communication
• here: agents perform geometric primitives– move to another agent– move on ray between two other
agents– move on a circle
• what can we achieve with this model?
• intersection point of trajectories of two agents
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Finding Intersections• two curves C1 and C2
• two agents A1 and A2
• agent‘s minimum travel distance is its diameter
discrete search space:integer grid
C1
C2
A1
A2
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Finding Intersections – Search Space
One open, one closed curve:
Two closed curves:
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Finding intersections• searching on an infinite
integer grid was considered by Baeza-Yates et al. (1993):– any online strategy for finding a
point within distance at most k (in L1-metric) needs at least 2k²+O(k) steps
– strategy NSESWSNWN:• visits points on diamond
around origin in distance k• requires 2k²+5k+2 steps
only 4k+3
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• searching in the plane is not constant competitive• search competitivity as quality measure (Fleischer et al. 2008)
We compare the path of the online search strategy• NOT to the shortest path• but to the best possible online search path
– search ratio sr:
– goal: sr(ALG) ≤ c sr(OPT)+a∙
– ≤ constant ALG search competitive
Search Competitivity
|Π(p)||sp(p)|
suppG
environment
online strategy‘s path to p
shortest path to p
ALGOPT
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MichaelEClarke@flickr
One-Dimensional Agents Two-Dimensional Agents
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One-Dimensional AgentsMichaelEClarke@flickr
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One-Dimensional AgentsMichaelEClarke@flickr
no simultaneous movement
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One-Dimensional Agents1. closed curves of equal
length l • any algorithm that finds an
intersection in distance at most k needs at least– 2k² + 2k - 4 steps (k<n)– 2n² + 4zn + 2n - 2z² - 2z - 4 steps
(n<k, k=n+z)
• strategy uses at most– 2k² + 5k + 2 steps (k<n)– 2n² + 4zn + 7n - 2z² - 3z + 2 steps
(n<k, k=n+z)
• strategy is 13/4 search competitive
k
4k
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One-Dimensional Agents2. closed curves of different
length• strategy uses at most
– 2k² + 5k + 2 steps (k≤n)– 6n² + 7n + 2j(n+3) + 4nz‘ + 2j - 2 steps
(n<k=n+z‘, 2j-1<z‘≤2j)– 5mn + n² + 4zn + 4n + 3m - 2z² - 2z + 2
log(m-n) - 2 steps (k=m+z)
• any algorithm that finds an intersection in distance at most k needs at least– 2k² + 2k - 4 steps (k≤n)– 2n² + 2n + z‘(4n+2) - 4 steps
(n<k=n+z‘≤m)– 4mn - 2n² + 4zn - 2z² - 2z + 2m – 4 steps
(k=m+z)
• the strategy is 11/2 search competitive
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One-Dimensional AgentsMichaelEClarke@flickr
simultaneous movement
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One-Dimensional Agents• agents move alternatingly all points of equal distance to
the start on a diamond• agents move simultaneously all points of equal distance on
a square
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One-Dimensional Agents• two curves of equal length• an optimal strategy moves on a
rectangular spiral-like search pattern:– target at some unknown finite
distance k– if agent knows upper bound k‘ does not visit points in distance
k‘ + 1 if agents does not know an upper
bound:agent has to cover each layer of points of the same distance, before visiting a point of the next layer
– connection of two layers: 1 step squared spiral optimal
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Theorem:Even if the agents are allowed to move simultaneously, there is an optimal strategy in which the agents move alternatingly.
One-Dimensional Agents
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Two-Dimensional Agents
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Two-Dimensional Agents• agent = disk of radius R• curves – circles of radius r• search space: torus• but: infinite number of
rendezvous points.• set of rendezvous points: no
more than 2 connected components (CCs)
• goal: find a convex region of certain size (in CCs) inspect finite point set on gridor move on Archimedean spiral
R r
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Two-Dimensional Agents
Case 1: |paqb| ≤ 2R
Case 2: |paqb| > 2R
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In the search space there is a square of size at least 2R x 2R such that all points inside the square are rendezvous points.
Two-Dimensional Agents
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Outlook
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• Related geometric problems
Outlook
infinite/infinite infinite/finite finite/finite
open/open
open/closed
closed/closed todayvariants of strategies presented today
Baeza-Yates et al.
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Thank you.
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Motivation – Finding Intersections
One-Dimensional Agents
– No simultaneous movement
– Simultaneous movement
Two-Dimensional Agents
Outlook
MichaelEClarke@flickr
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Motivation
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31MichaelEClarke@flickr
Motivation• planning motions for mobile
agents:– motion primitives– sensors– communication
• here: agents perform geometric primitives– move to another agent– move on ray between two other
agents– move on a circle
• what can we achieve with this model?
• intersection point of trajectories of two agents