Coverage. Ioannis Rekleitis
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1 Coverage Ioannis Rekleitis
2 Coverage A task performed quite often in everyday life: Cleaning Painting Plowing/Sowing Tile setting etc. CSCE 774: Robotic Systems 2
3 Motivation Humanitarian Demining CSCE 774: Robotic Systems 3
4 Motivation Lawn Mowing CSCE 774: Robotic Systems 4
5 Motivation Vacuum Cleaning CSCE 774: Robotic Systems 5
6 Robotic Coverage More than 5 million Roombas sold! Automated Car Painting CSCE 774: Robotic Systems 6
7 Roomba Costumes CSCE 774: Robotic Systems 7 From:
8 Coverage First Distinction Deterministic Random Second Distinction Complete No Guarantee Third Distinction Known Environment Unknown Environment CSCE 774: Robotic Systems Demining Vacuum Cleaning 8
9 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 CSCE 774: Robotic Systems Ant Robotics: I. Wagner, IBM & Technion 9
10 Deterministic Coverage Complete Algorithm Guarantees Complete Coverage CSCE 774: Robotic Systems 10
11 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 CSCE 774: Robotic Systems 11
12 The way of the Ox! Ioannis Rekleitis
13 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. CSCE 774: Robotic Systems 13
14 Single Robot Coverage Critical Point Cell 0 Direction of Coverage Cellular Decomposition CSCE 774: Robotic Systems Reeb graph Vertices: Critical Points Edges: Cells 14
15 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 CSCE 774: Robotic Systems 15
16 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 CSCE 774: Robotic Systems 16
17 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). CSCE 774: Robotic Systems 17
18 Offline Analysis Algorithm Offline Analysis Online Trajectory Control Direction Alignment (Optional) Cellular Decomp. Chinese Postman Problem Per-Cell Planner Non- Holonomic Controller CSCE 774: Robotic Systems 18
19 Offline Analysis Algorithm Offline Analysis Online Trajectory Control Direction Alignment (Optional) Cellular Decomp. Chinese Postman Problem Per-Cell Planner Non- Holonomic Controller Input: binary map separating obstacle from free space Boustrophedon Cellular Decomposition (BCD) : intersections = vertices : cells = edges CSCE 774: Robotic Systems 19
20 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) CSCE 774: Robotic Systems 20
21 Per-Cell Coverage Planner Offline Analysis Online Trajectory Control Direction Alignment (Optional) Cellular Decomp. Chinese Postman Problem Per-Cell Planner Non- Holonomic Controller Seed Spreader: piecewise linear sweep lines Footprint width CSCE 774: Robotic Systems 21
22 Coverage Direction Alignment Offline Analysis Online Trajectory Control Direction Alignment (Optional) Cellular Decomp. Chinese Postman Problem Per-Cell Planner Non- Holonomic Controller Static alignment methods Default Obstacle Boundaries Free Space Distribution Alignment with average wind heading (pre-flight) CSCE 774: Robotic Systems 22
23 Non-Holonomic Robot Controller Offline Analysis Online Trajectory Control Direction Alignment (Optional) Cellular Decomp. Chinese Postman Problem Per-Cell Planner Non- Holonomic Controller Turning strategies Greedy Waypoint Controller Curlicue Controller CSCE 774: Robotic Systems 23
24 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 CSCE 774: Robotic Systems 24
25 Double Coverage of a Single Cell By dividing the cell diagonally we control the beginning and end of the coverage CSCE 774: Robotic Systems 25
26 Double Coverage of a Single Cell By dividing the cell diagonally we control the beginning and end of the coverage CSCE 774: Robotic Systems 26
27 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 CSCE 774: Robotic Systems 27
28 Traversal order of the Reeb graph CSCE 774: Robotic Systems 28
29 Example CSCE 774: Robotic Systems 29
30 Example: Boustrophedon Decomposition CSCE 774: Robotic Systems 30
31 Example: Reeb Graph CSCE 774: Robotic Systems 31
32 Example: CPP solution CSCE 774: Robotic Systems 32
33 Example CSCE 774: Robotic Systems 33
34 Example CSCE 774: Robotic Systems 34
35 Example CSCE 774: Robotic Systems 35
36 Example CSCE 774: Robotic Systems 36
37 Example CSCE 774: Robotic Systems 37
38 Example CSCE 774: Robotic Systems 38
39 Example CSCE 774: Robotic Systems 39
40 Example CSCE 774: Robotic Systems 40
41 Example CSCE 774: Robotic Systems 41
42 Example CSCE 774: Robotic Systems 42
43 Example CSCE 774: Robotic Systems 43
44 Example 2 CSCE 774: Robotic Systems 44
45 Example 2 Boustrophedon Decomp. CSCE 774: Robotic Systems 45
46 Example 2 CSCE 774: Robotic Systems 46
47 Example 2 CSCE 774: Robotic Systems 47
48 Example 2 CSCE 774: Robotic Systems 48
49 Example 2 CSCE 774: Robotic Systems 49
50 Example 2 CSCE 774: Robotic Systems 50
51 Example 2 CSCE 774: Robotic Systems 51
52 Example 2 CSCE 774: Robotic Systems 52
53 Example 2 CSCE 774: Robotic Systems 53
54 Example 2 CSCE 774: Robotic Systems 54
55 Example 2 CSCE 774: Robotic Systems 55
56 Example 2 CSCE 774: Robotic Systems 56
57 Example 2 CSCE 774: Robotic Systems 57
58 Example 2 CSCE 774: Robotic Systems 58
59 Example 2 CSCE 774: Robotic Systems 59
60 Example 2 CSCE 774: Robotic Systems 60
61 Example 2 CSCE 774: Robotic Systems 61
62 Example 2 CSCE 774: Robotic Systems 62
63 Example 2 CSCE 774: Robotic Systems 63
64 Example 2 CSCE 774: Robotic Systems 64
65 Example 2 CSCE 774: Robotic Systems 65
66 Example 2 CSCE 774: Robotic Systems 66
67 UAV-Optimal Coverage CSCE 774: Robotic Systems 67
68 UAV-Optimal Coverage UAVs non-holonomic constraints require special trajectory planning 120 Km of flight during coverage CSCE 774: Robotic Systems 68
69 Image Mosaic CSCE 774: Robotic Systems 69
70 Multi-UAV CSCE 774: Robotic Systems 70
71 Video at ICRA 2011 CSCE 774: Robotic Systems 71
72 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 CSCE 774: Robotic Systems 72
73 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 CSCE 774: Robotic Systems 73
74 Trapezoidal decomposition CSCE 774: Robotic Systems 74
75 Spatial decompositions Dividing free space into pieces and using those... start end CSCE 774: Robotic Systems 75
76 Spatial decompositions Dividing free space into pieces and using those... start end Exact cell decomposition sweepline algorithm Running time? CSCE 774: Robotic Systems 76
77 Spatial decompositions Dividing free space into pieces and using those... start end Exact cell decomposition sweepline algorithm Running time? Path? O( N log(n) ) CSCE 774: Robotic Systems 77
78 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 CSCE 774: Robotic 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? via centroids + edge midpoints CSCE 774: Robotic Systems 79
80 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? CSCE 774: Robotic Systems 80
81 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. CSCE 774: Robotic Systems there may be more detail in the world than the task needs to worry about... 81
82 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 CSCE 774: Robotic Systems 82
83 further decomposing... Approximate cell decomposition Quadtree: recursively subdivides each mixed obstacle/free (sub)region into four quarters... CSCE 774: Robotic Systems 83
84 further decomposing... Approximate cell decomposition Quadtree: recursively subdivides each mixed obstacle/free (sub)region into four quarters... CSCE 774: Robotic Systems 84
85 further decomposing... Approximate cell decomposition Quadtree: recursively subdivides each mixed obstacle/free (sub)region into four quarters... CSCE 774: Robotic Systems 85
86 further decomposing... Approximate cell decomposition Again, use a graph-search algorithm to find a path from the start to goal Quadtree CSCE 774: Robotic Systems is this a complete path-planning algorithm? 86 i.e., does it find a path when one exists?
87 Octree Decomposition CSCE 774: Robotic Systems 87
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