Coverage. Ioannis Rekleitis

Transcription

1 Coverage Ioannis Rekleitis

2 Motivation Humanitarian Demining CS-417 Introduction to Robotics and Intelligent Systems 2

3 Motivation Lawn Mowing CS-417 Introduction to Robotics and Intelligent Systems 3

4 Motivation Vacuum Cleaning CS-417 Introduction to Robotics and Intelligent Systems 4

5 Robotic Coverage More than 5 million Roombas sold! Automated Car Painting CS-417 Introduction to Robotics and Intelligent Systems 5

6 Roomba Costumes CS-417 Introduction to Robotics and Intelligent Systems 6 From:

7 Coverage First Distinction Deterministic Random Second Distinction Complete No Guarantee Third Distinction Known Environment Unknown Environment CS-417 Introduction to Robotics and Intelligent Systems Demining Vacuum Cleaning 7

8 Non-Deterministic Coverage S. Koenig Ant Robotics, terrain coverage Complete Random Walk Ant Robotics Leave trail Bias the behavior towards or away from the trails Andrew Russell, Monash University, Australia CS-417 Introduction to Robotics and Intelligent Systems Ant Robotics: I. Wagner, IBM & Technion 8

9 Deterministic Coverage Complete Algorithm Guarantees Complete Coverage CS-417 Introduction to Robotics and Intelligent Systems 9

10 Cell-Decomposition Methods Two families of methods: Exact cell decomposition The free space F is represented by a collection of non-overlapping cells whose union is exactly F Examples: trapezoidal and cylindrical decompositions CS-417 Introduction to Robotics and Intelligent Systems 10

11 The way of the Ox! Ioannis Rekleitis

12 Single Robot Coverage Deterministic algorithm Guarantee of completeness Sensor based Unknown Environment Seed spreader algorithm: Lumelsky et al, Dynamic path planning in sensor-based terrain acquisition, IEEE Transactions on Robotics and Automation, August Boustrophedon algorithm: Choset and Pignon, Coverage path planning: The boustrophedon cellular decomposition, International Conference on Field and Service Robotics,1997. CS-417 Introduction to Robotics and Intelligent Systems 12

13 Single Robot Coverage Critical Point Cell 0 Direction of Coverage Cellular Decomposition CS-417 Introduction to Robotics and Intelligent Systems Reeb graph Vertices: Critical Points Edges: Cells 13

14 Critical Points There are four types of critical points: Forward Concave critical point Reverse Concave critical point Reverse Convex critical point Forward Convex critical point Direction of Coverage CS-417 Introduction to Robotics and Intelligent Systems 14

15 Optimal Coverage Find an order for traversing the Reeb graph such that the robot would not go through a cell more times than necessary Solution Use the Chinese Postman Problem CS-417 Introduction to Robotics and Intelligent Systems 15

16 Chinese Postman Problem The Chinese postman problem (CPP), is to find a shortest closed path that visits every edge of a (connected) undirected graph. When the graph has an Eulerian circuit (a closed walk that covers every edge once), that circuit is an optimal solution. See: J. Edmonds and E.L. Johnson, Matching Euler tours and the Chinese postman problem, Math. Program. (1973). CS-417 Introduction to Robotics and Intelligent Systems 16

17 Direction Alignment (Optional) Offline Analysis Algorithm Offline Analysis Cellular Decomp. Chinese Postman Problem Online Trajectory Control Per-Cell Planner Non- Holonomic Controller

18 Direction Alignment (Optional) Offline Analysis Algorithm Offline Analysis Cellular Decomp. Chinese Postman Problem Online Trajectory Control Per-Cell Planner Non- Holonomic Controller Input: binary map separating obstacle from free space Boustrophedon Cellular Decomposition (BCD) : intersections = vertices : cells = edges

19 Offline Analysis Algorithm (cont.) Offline Analysis Online Trajectory Control Direction Alignment (Optional) Cellular Decomp. Chinese Postman Problem Per-Cell Planner Non- Holonomic Controller Chinese Postman Problem Eulerian circuit, i.e. single traversal through all cells (edges)

20 Direction Alignment (Optional) Per-Cell Coverage Planner Offline Analysis Cellular Decomp. Chinese Postman Problem Online Trajectory Control Per-Cell Planner Non- Holonomic Controller Seed Spreader: piecewise linear sweep lines Footprint width

21 Direction Alignment (Optional) Coverage Direction Alignment Offline Analysis Cellular Decomp. Static alignment methods Chinese Postman Problem Online Trajectory Control Per-Cell Planner Non- Holonomic Controller Default Obstacle Boundaries Free Space Distribution Alignment with average wind heading (pre-flight)

22 Direction Alignment (Optional) Non-Holonomic Robot Controller Offline Analysis Cellular Decomp. Turning strategies Chinese Postman Problem Online Trajectory Control Per-Cell Planner Non- Holonomic Controller Greedy Waypoint Controller Curlicue Controller

23 Chinese Postman Problem The solution of the CPP guarantees that no edge is doubled more than once That means some cells have to be traversed twice Cells that have to be traversed/covered are divided in half CS-417 Introduction to Robotics and Intelligent Systems 23

24 Double Coverage of a Single Cell By dividing the cell diagonally we control the beginning and end of the coverage CS-417 Introduction to Robotics and Intelligent Systems 24

25 Double Coverage of a Single Cell By dividing the cell diagonally we control the beginning and end of the coverage CS-417 Introduction to Robotics and Intelligent Systems 25

26 Optimal Coverage Algorithm Given a known environment: Calculate the Boustrophedon decomposition Construct the Reeb graph Use the Reeb graph as input to the Chinese Postman Problem (CPP) Use the solution of the CPP to find a minimum cost cycle traversing every edge of the Reeb graph For every doubled edge divide the corresponding cell in half Traverse the Reeb graph by covering each cell in order CS-417 Introduction to Robotics and Intelligent Systems 26

27 Traversal order of the Reeb graph CS-417 Introduction to Robotics and Intelligent Systems 27

28 Example CS-417 Introduction to Robotics and Intelligent Systems 28

29 Example: Boustrophedon Decomposition CS-417 Introduction to Robotics and Intelligent Systems 29

30 Example: Reeb Graph CS-417 Introduction to Robotics and Intelligent Systems 30

31 Example: CPP solution CS-417 Introduction to Robotics and Intelligent Systems 31

32 Example CS-417 Introduction to Robotics and Intelligent Systems 32

33 Example CS-417 Introduction to Robotics and Intelligent Systems 33

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41 Example CS-417 Introduction to Robotics and Intelligent Systems 41

42 Example CS-417 Introduction to Robotics and Intelligent Systems 42

43 Example 2 CS-417 Introduction to Robotics and Intelligent Systems 43

44 Example 2 Boustrophedon Decomp. CS-417 Introduction to Robotics and Intelligent Systems 44

45 Example 2 CS-417 Introduction to Robotics and Intelligent Systems 45

46 Example 2 CS-417 Introduction to Robotics and Intelligent Systems 46

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64 Example 2 CS-417 Introduction to Robotics and Intelligent Systems 64

65 Example 2 CS-417 Introduction to Robotics and Intelligent Systems 65

66 UAV-Optimal Coverage CS-417 Introduction to Robotics and Intelligent Systems 66

67 UAV-Optimal Coverage UAVs non-holonomic constraints require special trajectory planning 120 Km of flight during coverage CS-417 Introduction to Robotics and Intelligent Systems 67

68 Image Mosaic CS-417 Introduction to Robotics and Intelligent Systems 68

69 Multi-UAV CS-417 Introduction to Robotics and Intelligent Systems 69

70 Video at ICRA 2011 CS-417 Introduction to Robotics and Intelligent Systems 70

71 Coverage of Known Worlds From: X. Zheng and S. Koenig. Robot Coverage of Terrain with Non-Uniform Traversability. In Proc. of the IEEE Int. Conf. on Intelligent Robots and Systems (IROS), pg , 2007 CS-417 Introduction to Robotics and Intelligent Systems 71

72 Cell decomposition for Path Planning Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells CS-417 Introduction to Robotics and Intelligent Systems 72

73 Trapezoidal decomposition CS-417 Introduction to Robotics and Intelligent Systems 73

74 Spatial decompositions Dividing free space into pieces and using those... start end CS-417 Introduction to Robotics and Intelligent Systems 74

75 Spatial decompositions Dividing free space into pieces and using those... start end Exact cell decomposition sweepline algorithm Running time? CS-417 Introduction to Robotics and Intelligent Systems 75

76 Spatial decompositions Dividing free space into pieces and using those... start end Exact cell decomposition sweepline algorithm Running time? Path? O( N log(n) ) CS-417 Introduction to Robotics and Intelligent Systems 76

77 Spatial decompositions Dividing free space into pieces and using those... start centroids end Exact cell decomposition sweepline algorithm Running time? O( N log(n) ) Path? via centroids CS-417 Introduction to Robotics and Intelligent Systems 77

78 Spatial decompositions Dividing free space into pieces and using those... start centroids + midpoints end Exact cell decomposition sweepline algorithm Running time? O( N log(n) ) Path? via centroids + edge midpoints CS-417 Introduction to Robotics and Intelligent Systems 78

79 Spatial decompositions Dividing free space into pieces and using those... start centroids + midpoints end Exact cell decomposition sweepline algorithm Running time? O( N log(n) ) Path? Why? via centroids + edge midpoints + graph search why else? CS-417 Introduction to Robotics and Intelligent Systems 79

80 Optimality Obtaining the minimum number of convex cells is NP-complete. 15 cells 9 cells Trapezoidal decomposition is exact and complete, but not optimal -- even among convex subdivisions. CS-417 Introduction to Robotics and Intelligent Systems there may be more detail in the world than the task needs to worry about... 80

81 Cell-Decomposition Methods Two families of methods: Exact cell decomposition Approximate cell decomposition F is represented by a collection of nonoverlapping cells whose union is contained in F Examples: quadtree, octree, 2 n -tree CS-417 Introduction to Robotics and Intelligent Systems 81

82 further decomposing... Approximate cell decomposition Quadtree: recursively subdivides each mixed obstacle/free (sub)region into four quarters... CS-417 Introduction to Robotics and Intelligent Systems 82

83 further decomposing... Approximate cell decomposition Quadtree: recursively subdivides each mixed obstacle/free (sub)region into four quarters... CS-417 Introduction to Robotics and Intelligent Systems 83

84 further decomposing... Approximate cell decomposition Quadtree: recursively subdivides each mixed obstacle/free (sub)region into four quarters... CS-417 Introduction to Robotics and Intelligent Systems 84

85 further decomposing... Approximate cell decomposition Again, use a graph-search algorithm to find a path from the start to goal Quadtree CS-417 Introduction to Robotics and Intelligent Systems is this a complete path-planning algorithm? 85 i.e., does it find a path when one exists?

86 Octree Decomposition CS-417 Introduction to Robotics and Intelligent Systems 86