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Visibility based Probabilistic Motion Planning

This article discusses visibility-based probabilistic motion planning, focusing on the capture of topology in Configuration Space and the small time controllability of mechanical systems. It explores various methodologies and algorithms for generating guards and connectors, as well as their robustness and computational challenges. The article also covers the use of random sampling and diffusion techniques in answering queries and estimating coverage percentages.

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Visibility based Probabilistic Motion Planning

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  1. Visibilitybased Probabilistic Motion Planning T. Siméon, J.P. Laumond, C. Nissoux, Visibilitybasedprobabilisticroadmaps for motion planning Advanced Robotics Journal, Vol. 14, N°6, 2000 J.P. Laumond, T Siméon, Notes on visibilityroadmaps and path planning in Algorithmic and ComputationalRobotics: New Directions B. Donald, K. Lynch, D. Rus Eds, A.K. Peters, 2001

  2. Configuration Space • Any admissible motion for the 3D mechanical system appears a collision-freepath for a point in the CSpace • Translating the continuousprobleminto a combinatorial one • Capturing the topologyof CSfreewithgraphs

  3. Configuration SpaceTopology and Small Time Controllability • Anysteeringmethodaccounting for the small time controllability of the system induces the sametopology

  4. Configuration SpaceTopology and Small Time Controllability • Exemples: • linear interpolation, • Manhattan paths, • Reeds and Shepp paths, • flatnessbasedmethods Euclidean Manhattan Reeds-Shepp

  5. Manhattan like Euclidean Sametopology, differentvisibility sets Reeds-Shepp

  6. Combinatorialtopology (1) • Existence and robustness of finite coverage Euclidean Manhattan

  7. No Combinatorialtopology (1) • Existence and robustness of finite coverage Euclidean Manhattan

  8. No Combinatorialtopology (1) • Existence and robustness of finite coverage Euclidean Manhattan Euclidean Manhattan

  9. Combinatorialtopology (1) • Existence and robustness of finite coverage Euclidean Euclidean Manhattan Manhattan

  10. Combinatorialtopology (2) • Optimal coverage (related to art gallery problem) Euclidean Manhattan

  11. Combinatorialtopology (2) • Optimal coverage: finite? bounded? Euclidean and 2D polygonal obstacles: finite and bounded

  12. Combinatorialtopology (3) • Optimal coverage: finite? bounded? Euclidean: finite and unbounded

  13. Combinatorialtopology (4) • From (optimal) coverage to (visibility) roadmaps Guards + Connectors

  14. Computational challenge • No explicit knowledge of CS-obstacles • No explicit knowledge of visible (reachable) sets

  15. Computational challenge • Probabilistic method ingredients: • A collision checker • A steering method • Two type methods: • Learning CS topology by sampling • Answering single query by diffusion

  16. Computational challenge Randomsampling

  17. Computational challenge Random diffusion

  18. Visibilybasedsampling • Algorithm: • Generate guards and connectors randomly • Stop after #try failures

  19. Visibilybasedsampling

  20. Visibilybasedsampling

  21. Visibilybasedsampling

  22. Visibilybasedsampling

  23. Visibilybasedsampling • Algorithm: • Generate guards and connectors randomly • Stop after #try failures (#try=10)

  24. Visibilybasedsampling • Algorithm: • Generate guards and connectors randomly • Stop after #try failures (#try=100)

  25. Visibilybasedsampling • Algorithm: • Generate guards and connectors randomly • Stop after #try failures (#try=200)

  26. Visibilybasedsampling • Algorithm: • Generate guards and connectors randomly • Stop after #try failures (#try=500)

  27. Visibilybasedsampling • Algorithm: • Generate guards and connectors randomly • Stop after #try failures (#try=1000)

  28. Visibilybasedsampling • Algorithm: • Generate guards and connectors randomly • Stop after #try failures (#try=10000)

  29. Visibilybasedsampling • Algorithm: • Generate guards and connectors randomly • Stop after #try failures (#try=1000000000!)

  30. Visibilybasedsampling • Algorithm: • Generate guards and connectors randomly • Stop after #try failure • Theorems: • The estimated percentage of non-covered free-space is #try-1 • Probability to find an existing path increases as an exponential function of time

  31. Visibilybasedsampling • Possible online estimation of #try coverage % 100 # guards

  32. Visibilybasedsampling • Real time demonstrations

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