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Geometric Motion Planning : Finding Intersections

Geometric Motion Planning : Finding Intersections. MichaelEClarke @ flickr. Motivation – Finding Intersections One -Dimensional Agents No simultaneous movement Simultaneous movement Two -Dimensional Agents Outlook. Motivation. Motivation. planning motions for mobile agents :

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Geometric Motion Planning : Finding Intersections

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  1. Geometric Motion Planning:FindingIntersections

  2. MichaelEClarke @flickr Motivation – Finding Intersections One-Dimensional Agents • Nosimultaneousmovement • Simultaneousmovement Two-Dimensional Agents Outlook

  3. Motivation

  4. Motivation • planningmotionsfor mobile agents: • motion primitives • sensors • communication • here: agentsperformgeometric primitives • movetoanother agent • move on raybetweentwootheragents • move on a circle • whatcanweachievewiththis model? • intersectionpointoftrajectoriesoftwoagents

  5. FindingIntersections • twocurves C1 and C2 • twoagents A1and A2 • agent‘sminimumtraveldistanceisitsdiameter  discretesearchspace: integer grid A1 C1 C2 A2

  6. Bilder – fälle grid und einbettung FindingIntersections – Search Space One open, oneclosedcurve: Twoclosedcurves: Intersectionpoint=suche

  7. Findingintersections • searching on an infinite integer grid was consideredbyBaeza-Yates et al. (1993): • any online strategyforfinding a pointwithindistanceatmostk (in L1-metric) needsat least 2k²+O(k) steps • strategy NSESWSNWN: • visitspoints on diamondaroundorigin in distancek • requires 2k²+5k+2 steps Markiere diamonds, und 4k+3 für schicht k only 4k+3

  8. SearchCompetitivity • searching in the plane is not constantcompetitive • searchcompetitivityasqualitymeasure (Fleischer et al. 2008) • Wecomparethepathofthe online searchstrategy • NOT totheshortestpath • but tothebestpossible online searchpath • searchratiosr: • goal: sr(ALG) ≤ c∙sr(OPT)+a • ≤ constant  ALG searchcompetitive Folie zu serachcomp.? online strategy‘spathto p |Π(p)| ALG shortestpathto p suppG |sp(p)| OPT environment

  9. MichaelEClarke@flickr One-Dimensional Agents Two-Dimensional Agents

  10. MichaelEClarke @flickr One-Dimensional Agents

  11. MichaelEClarke @flickr One-Dimensional Agents • nosimultaneousmovement

  12. One-Dimensional Agents • closedcurvesofequallengthl • anyalgorithmthatfinds an intersection in distanceatmostkneedsat least • 2k² + 2k - 4 steps (k<n) • 2n² + 4zn + 2n - 2z² - 2z - 4 steps (n<k, k=n+z) • strategyusesatmost • 2k² + 5k + 2steps (k<n) • 2n² + 4zn + 7n - 2z² - 3z + 2 steps (n<k, k=n+z) • strategyis 13/4 searchcompetitive 4k MichaelEClarke @flickr k

  13. One-Dimensional Agents • closedcurvesof different length • strategyusesatmost • 2k² + 5k + 2steps (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) • anyalgorithmthatfinds an intersection in distanceatmostkneedsat 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) • thestrategyis 11/2 searchcompetitive MichaelEClarke @flickr

  14. MichaelEClarke @flickr One-Dimensional Agents • simultaneousmovement

  15. One-Dimensional Agents • agentsmovealternatingly  all pointsofequaldistancetothestart on a diamond • agentsmovesimultaneously  all pointsofequaldistance on a square MichaelEClarke @flickr

  16. One-Dimensional Agents • twocurvesofequallength • an optimal strategymoves on a rectangular spiral-likesearchpattern: • targetatsomeunknown finite distancek • if agent knowsupperboundk‘ • does not visitpoints in distancek‘ + 1 • ifagentsdoes not know an upperbound: agent hasto cover eachlayerofpointsofthe same distance, beforevisiting a pointofthenextlayer • connectionoftwolayers: 1 step  squared spiral optimal MichaelEClarke @flickr

  17. One-Dimensional Agents Theorem: Even if the agents are allowed to move simultaneously, there is an optimal strategy in which the agents move alternatingly. MichaelEClarke @flickr

  18. Two-Dimensional Agents

  19. Two-Dimensional Agents • agent = diskofradius R • curves – circlesofradius r • searchspace: torus • but: infinite numberofrendezvouspoints. • setofrendezvouspoints: nomorethan 2 connectedcomponents (CCs) • goal: find a convexregionofcertainsize (in CCs) • inspect finite pointset on grid or  move on Archimedean spiral r R

  20. Two-Dimensional Agents Case 1: |paqb| ≤ 2R Case 2: |paqb| > 2R

  21. Two-Dimensional Agents 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.

  22. Outlook

  23. Outlook • Relatedgeometricproblems Baeza-Yates et al. variantsofstrategiespresentedtoday today

  24. Thankyou.

  25. MichaelEClarke @flickr Motivation – Finding Intersections One-Dimensional Agents • Nosimultaneousmovement • Simultaneousmovement Two-Dimensional Agents Outlook

  26. Motivation

  27. Motivation • planningmotionsfor mobile agents: • motion primitives • sensors • communication • here: agentsperformgeometric primitives • movetoanother agent • move on raybetweentwootheragents • move on a circle • whatcanweachievewiththis model? • intersectionpointoftrajectoriesoftwoagents

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