1 / 50

Motion Planning for Point Robots

Motion Planning for Point Robots. CS 659 Kris Hauser. Agenda. Skimming through Principles Ch. 2, 5.1, 6.1 . Fundamental question of motion planning. Are the two given points connected by a path?. Feasible space. collision with obstacle lack of stability lost sight of target ….

kaiya
Download Presentation

Motion Planning for Point Robots

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Motion Planning for Point Robots CS 659 Kris Hauser

  2. Agenda • Skimming through Principles Ch. 2, 5.1, 6.1

  3. Fundamental question of motion planning • Are the two given points connected by a path? Feasible space collision with obstaclelack of stabilitylost sight of target… Forbidden region

  4. Basic Problem • Statement:Compute a collision-free path for a rigid or articulated object among static obstacles • Inputs: • Geometry of moving object and obstacles • Kinematics of moving object (degrees of freedom) • Initial and goal configurations (placements) • Output: Continuous sequence of collision-free robot configurations connecting the initial and goal configurations

  5. Why is this hard?

  6. Tool: Configuration Space • Problems: • Geometric complexity • Space dimensionality

  7. Tool: Configuration Space

  8. s Problem free space obstacle free path obstacle g obstacle

  9. s Problem Semi-free path obstacle obstacle g obstacle

  10. Local constraints: e.g., avoid collision with obstacles Differential constraints: e.g., bounded curvature Global constraints: e.g., minimal length Types of Path Constraints

  11. Motion-Planning Framework Continuous representation Discretization Graph searching (blind, best-first, A*)

  12. Path-Planning Approaches • RoadmapRepresent the connectivity of the free space by a network of 1-D curves • Cell decompositionDecompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells • Potential fieldDefine a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

  13. Roadmap Methods • Visibility graphIntroduced in the Shakey project at SRI in the late 60s. Can produce shortest paths in 2-D configuration spaces

  14. Simple (Naïve) Algorithm • Install all obstacles vertices in VG, plus the start and goal positions • For every pair of nodes u, v in VG • If segment(u,v) is an obstacle edge then • insert (u,v) into VG • else • for every obstacle edge e • if segment(u,v) intersects e • then goto 2 • insert (u,v) into VG • Search VG using A*

  15. Complexity • Simple algorithm: O(n3) time • Rotational sweep: O(n2 log n) • Optimal algorithm: O(n2) • Space: O(n2)

  16. Rotational Sweep

  17. Rotational Sweep

  18. Rotational Sweep

  19. Rotational Sweep

  20. Rotational Sweep

  21. can’t be shortest path Reduced Visibility Graph tangent segments  Eliminate concave obstacle vertices

  22. Generalized (Reduced) Visibility Graph tangency point

  23. Shortest path passes through none of the vertices locally shortest path homotopic to globally shortest path Three-Dimensional Space Computing the shortest collision-free path in a polyhedral space is NP-hard

  24. Voronoi diagramIntroduced by Computational Geometry researchers. Generate paths that maximizes clearance. O(n log n) timeO(n) space Roadmap Methods

  25. Roadmap Methods • Visibility graph • Voronoi diagram • SilhouetteFirst complete general method that applies to spaces of any dimension and is singly exponential in # of dimensions [Canny, 87] • Probabilistic roadmaps

  26. Path-Planning Approaches • RoadmapRepresent the connectivity of the free space by a network of 1-D curves • Cell decompositionDecompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells • Potential fieldDefine a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

  27. Cell-Decomposition Methods Two classes of methods: • Exact cell decompositionThe free space F is represented by a collection of non-overlapping cells whose union is exactly FExample: trapezoidal decomposition

  28. Trapezoidal decomposition

  29. Trapezoidal decomposition

  30. Trapezoidal decomposition

  31. Trapezoidal decomposition

  32. critical events  criticality-based decomposition Trapezoidal decomposition

  33. Trapezoidal decomposition Planar sweep  O(n log n) time, O(n) space

  34. Cell-Decomposition Methods Two classes of methods: • Exact cell decomposition • Approximate cell decompositionF is represented by a collection of non-overlapping cells whose union is contained in FExamples: quadtree, octree, 2n-tree

  35. Octree Decomposition

  36. Sketch of Algorithm • Compute cell decomposition down to some resolution • Identify start and goal cells • Search for sequence of empty/mixed cells between start and goal cells • If no sequence, then exit with no path • If sequence of empty cells, then exit with solution • If resolution threshold achieved, then exit with failure • Decompose further the mixed cells • Return to 2

  37. Path-Planning Approaches • RoadmapRepresent the connectivity of the free space by a network of 1-D curves • Cell decompositionDecompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells • Potential fieldDefine a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

  38. Potential Field Methods • Approach initially proposed for real-time collision avoidance [Khatib, 86]. Hundreds of papers published on it. Goal Robot

  39. Attractive and Repulsive fields

  40. Local-Minimum Issue • Perform best-first search (possibility of combining with approximate cell decomposition) • Alternate descents and random walks • Use local-minimum-free potential (navigation function)

  41. Sketch of Algorithm (with best-first search) • Place regular grid G over space • Search G using best-first search algorithm with potential as heuristic function

  42. 2 1 2 3 1 1 2 2 3 3 4 4 Simple Navigation Function 0 5

  43. Simple Navigation Function 2 1 2 3 1 0 1 2 2 3 3 4 5 4

  44. Simple Navigation Function 2 1 2 3 1 0 1 2 2 3 3 4 5 4

  45. Completeness of Planner • A motion planner is completeif it finds a collision-free path whenever one exists and return failure otherwise. • Visibility graph, Voronoi diagram, exact cell decomposition, navigation function provide complete planners • Weaker notions of completeness, e.g.:- resolution completeness (PF with best-first search)- probabilistic completeness (PF with random walks)

  46. A probabilistically complete planner returns a path with high probability if a path exists. It may not terminate if no path exists. • A resolution complete planner discretizes the space and returns a path whenever one exists in this representation.

  47. Preprocessing / Query Processing • Preprocessing: Compute visibility graph, Voronoi diagram, cell decomposition, navigation function • Query processing:- Connect start/goal configurations to visibility graph, Voronoi diagram- Identify start/goal cell- Search graph

  48. Some Variants • Moving obstacles • Multiple robots • Movable objects • Assembly planning • Goal is to acquire information by sensing • Model building • Object finding/tracking • Inspection • Nonholonomic constraints • Dynamic constraints • Stability constraints • Optimal planning • Uncertainty in model, control and sensing • Exploiting task mechanics (sensorless motions, under-actuated systems) • Physical models and deformable objects • Integration of planning and control • Integration with higher-level planning

  49. Issues for Future Lectures on Planning • Space dimensionality • Geometric complexity of the free space • Constraints other than avoiding collision • The goal is not just a position to reach • Etc …

  50. Readings • Principles Ch. 3.1-3.4 • *Lozano-Perez (1986) • *Lozano-Perez (1983)

More Related