Coverage and Search Algorithms. Chapter 10

Transcription

1 Coverage and Search Algorithms Chapter 10

2 Objectives To investigate some simple algorithms for covering the area in an environment To understand how break down an environment in simple convex pieces To understand how consider searching environments with a limited range and limited direction sensor. 10-2

3 Complete Coverage Algorithms Difficulties and Issues Boustrophedon Coverage Other Coverage Ideas Search Algorithms Searching and Vibility Guard Placement Traveling Salesman Problem Vibility Search Paths What s in Here? Searching With Limited Range Vibility 10-3

4 Complete Coverage Algorithms 10-4

5 Coverage Algorithms A complete coverage algorithm produces a path that a robot must travel on in order cover or travel over the entire surface of its environment. Applications include: vacuum and sweeping painting searching security patrolling map verification etc 10-5

6 Coverage Algorithms How can we determine a valid path that the robot can take cover the whole environment? 10-6

7 Coverage Algorithms One approach simply travel in some fixed direction (e.g., North) until an obstacle encountered, then turn around cover in strips: Can Can get get tricky tricky due due obstacle obstacle and and border border angles. angles. 10-7

8 Coverage Algorithms Even in rectilinear environments, many problems may are: Unreachable Unreachable areas. areas. Painted Painted in in a a corner corner :(. :(. Gaps Gaps left left due due motion motion inaccuracy. inaccuracy. Th Th area area was was msed. msed. Crooked Crooked coverage coverage due due wheel wheel dcrepancy. dcrepancy. 10-8

9 Is there any hope? Coverage Algorithms there will always be some error in terms of coverage. - may still ms close edges and in corners allowing overlapping coverage will help dividing environment in smaller chunks will help For most applications (not painting the floor) being close enough the obstacles sufficient. sensors can pick-up /detect from a certain dtance away. sometimes, a rough coverage enough. 10-9

10 Boustrophedon Coverage Recall the Boustrophedon cell decomposition of a polygonal environment: Critical Critical points points 10-10

11 Boustrophedon Coverage Now connect adjacent cells form a graph and consider an arbitrary ordering of the cells: (e.g., from left right)

12 10-12 Finding a Path Finding a Path Perform a depth-first-search (DFS) on the graph determine an exhaustive walk through the cells: dfs(g) { lt L = empty tree T = empty choose a starting vertex x search(x) WHILE (L nonempty) DO remove edge (v,w) from end of L IF (w not yet vited) THEN add (v,w) T search(w) } search(vertex v) { vit(v) FOR (each edge (v,w)) DO add edge (v,w) end of L } a L a b k d c f j g h i e b T a c e f g L a b k d c f j g h i e b T a c e f i L a b k d c f j g h i e b g T j a c e f i L a b k d c f j g h i e b g j T k a c e f i L a b k d c f j g h i e b g j k T a c e f h L a b k d c f j g h i e b g j k i T etc

13 Finding a Path Here what the DFS ordering may have produced in our example: Cells Cells vited vited in in the the following following order order (blue (blue numbers numbers indicate indicate backtracking): backtracking):

14 Coverage Along a Path Once a path found, the robot vits all of these cells in that order: Vitation Vitation order order may may not not be be the the most most efficient. efficient. There There are are other other ways ways traverse traverse besides besides the the DFS. DFS

15 Coverage Along a Path When coming back cells already vited, it not necessary re-cover the cell again: Need Need compute compute path path back back previous previous cells. cells

16 Coverage Along a Path When entering a cell, the robot performs some simple maneuvers cover the cell s entire area: Usually, Usually, vertical vertical motions motions up up and and down down separated separated by by a a robot robot width. width. Such Such motions motions are are joined joined by by travel travel along along the the obstacle obstacle boundary. boundary. It It may may still still be be necessary necessary make make path path adjustments adjustments due due obstacles. obstacles

17 Coverage Along a Path Must take care when crossing one cell another: Follow Follow a a fixed fixed rule: rule: e.g., e.g., always always cross cross at at p p of of cell, cell, even even if if extra extra coverage coverage necessary. necessary. Sometimes, Sometimes, backtracking backtracking along along same same path path necessary necessary ensure ensure that that nothing nothing msed. msed. Robot Robot must must go go down down then then up up along along th th portion portion in in order order cover cover entire entire cell. cell

18 Coverage Along a Path When backtracking, follow along cell boundaries: Follow Follow fixed fixed rule rule when when backtracking backtracking (e.g., (e.g., travel travel along along p p of of cell cell boundaries) boundaries) Cover Cover other other cells, cells, then then come come back back again again (i.e., (i.e., backtracking backtracking in in red). red). Continue Continue coverage coverage maneuvers maneuvers whenever whenever an an uncovered uncovered cell cell encountered encountered during during the the walk walk through through the the cells. cells

19 Other Coverage Ideas There are other ways decompose the environment in cells and compute a coverage path. For example: circular or diamond-shaped spiral cells We spike cells We will will look look very very briefly briefly at at these these two two brushfire decomposition cells (like GVD) Each of these, however, may require different traversal techniques. Their choice should depend on the robot s sensor charactertics

20 Circular Coverage Patterns We can alternatively create circular cells defined by circles extending outwards from the start location: Cell Cell boundaries boundaries formed formed by by circles. circles. Critical Critical point point defined defined when when circle circle tangent tangent obstacle. obstacle

21 Circular Coverage Patterns Once again, interconnect cells and do DFS find path in graph: Critical Critical point point defined defined when when circle circle tangent tangent obstacle. obstacle

22 Circular Coverage Patterns Traverse each cell by making laps around the cell where each lap separated by the robot width: Alternatively Alternatively can can spiral spiral here. here. When When obstacle obstacle encountered encountered follow follow along along edge. edge

23 Brushfire Decomposition Can even break down in regions based on GVD and then traverse cells around obstacles: When When tracing tracing cell cell around around obstacle, obstacle, robot robot maintains maintains fixed fixed dtance dtance away away from from obstacle obstacle at at all all times times (unless (unless when when adjusting adjusting for for cell cell boundary) boundary) Th Th technique technique best best for for robots robots that that have have dead-reckoning dead-reckoning errors. errors. By By maintaining maintaining fixed fixed dtance dtance from from obstacle, obstacle, robot robot can can readjust readjust its its measurements measurements and and re-confirm re-confirm its its position. position. Robot Robot needs needs long long dtance dtance range range sensor sensor though. though

24 Search Algorithms 10-24

25 Searching Consider covering an environment for the purpose of searching for other robots, fire, intruders, any identifiable object etc Robot equipped with one or more search sensors of some kind which have either: unlimited or limited detection range omni-directional (i.e, 360 ) or limited direction detection capabilities 10-25

26 Searching As the robot moves around in the environment, it able search based on its current vibility: Unlimited Unlimited range, range, Limited Limited direction direction Unlimited Unlimited range, range, Omni-directional Omni-directional (Most (Most common common for for theoretical theoretical models) models) Limited Limited range, range, Omni-directional Omni-directional Limited Limited range, range, Limited Limited direction. direction. (Most (Most common common for for real real robots) robots) 10-26

27 Vibility Consider a simple environment with no obstacles and a robot with omni-directional sensing with unlimited range capabilities. Which environments can it search (i.e., see) completely without moving? star-shaped polygons With With limited limited direction direction capabilities, capabilities, robot robot would would have have rotate rotate view view entire entire environment environment 10-27

28 Vibility The kernel of a star-shaped polygon the area of the polygon from which the robot can see the entire boundary of the environment: Extend Extend lines lines from from each each reflex reflex vertex vertex parallel parallel edges edges containing containing that that vertex. vertex. A reflex reflex vertex vertex one one which which forms forms an an inside inside angle angle > > > 180 Kernel Kernel formed formed as as intersection intersection of of the the resulting resulting half half planes. planes

29 Vibility What if environment not star-shaped or has obstacles? kernel empty (i.e., can t see whole environment from one location) need determine a set of locations (i.e., view points) that cover the entire environment 10-29

30 Vibility Placing robot at each reflex vertex will ensure complete vibility coverage. Do you know why? 10-30

31 Guard Placement Can we cover with less locations? Th problem called the Guard Placement problem or Art Gallery problem. For a simple polygon environment with n vertices: n/3 locations are occasionally necessary and always sufficient have every point in the polygon vible from at least one of the locations: e.g., n = 12 and 12 // 3 = 4 locations are necessary and sufficient 10-31

32 Guard Placement When the environment contains h obstacles and has n edges (including obstacle edges), it can be shown that (n+h)/3 locations are sometimes necessary and always sufficient cover the entire environment: e.g., n = 19, h = 1 then (19+1)//3 = 20//3 = 6 locations are necessary and sufficient. There There are are many many possible possible placements, placements, here here are are two two 10-32

33 Guard Placement How do we compute these locations? Can do a 3-coloring of the triangulation: color each vertex of the triangulation with one of 3 colors no two vertices sharing a triangulation edge should have the same color Coloring Coloring done done through through a a DFS, DFS, but but in in some some cases cases the the straight straight forward forward approach approach does does not not always always work it work it can can be be tricky. tricky. Each Each color color here here indicates indicates a a possible possible set set of of robot robot locations. locations

34 Search Paths To perform an exhaustive search, the robot must move around in the environment shortest watchman route the shortest possible path in the environment such that the robot covers (i.e., sees) all areas in the environment. difficult find exact solution, approximations are usually simpler and acceptable Can solve th problem by finding guard placement locations and then connect them with an efficient path (i.e., travel between multiple goal locations)

35 Traveling Salesman Problem 10-35

36 Traveling Salesman Problem Given a number of locations that the robot must travel, what the cheapest round-trip route that vits each location once and then returns the starting location? (e.g., viting stations in a building for security checks)

37 Traveling Salesman Problem Most direct solution: try all permutations (ordered combinations) and see which one cheapest number of permutations n! for n locations impractical!! There are many approaches th problem many use heurtics and approximations If we don t need the optimal path, we can comprome for some simpler algorithms. u w v Assume triangle inequality holds: uw uv + vw 10-37

38 Traveling Salesman Problem Consider the locations that robot must travel. start Approximate ur based on minimum spanning tree from the start location: start Use Use well-known well-known Prim s Prim s algorithm algorithm or or other other favorite. favorite

39 Traveling Salesman Problem Consider a complete graph of the locations i.e., each location connects every other location The minimum spanning tree a subset of the complete graph s edges that forms a tree that includes every location, where the tal length of all the edges in the tree minimized. 1. Create a tree containing a single arbitrarily chosen location 2. Create a set S containing all the edges in the graph 3. WHILE (any edge in S does not connect two locations in the tree) DO 4. Remove the shortest edge from S that connects a location in the tree with a location not in the tree 5. Add that edge the tree Use Use simple simple heap heap data data structure. structure

40 Traveling Salesman Problem From the root of the minimum spanning tree perform a pre-order traversal of the tree. Connect nodes in order vited. Solution Solution can can be be at at most most twice twice the the best best path path but but usually usually not not so so bad. bad

41 Traveling Salesman Problem Running time O(n 2 ) for n locations. There are other variations of th problem we could spend a whole course dcussing these types of problems. Can we use th algorithm practically? the triangle inequality may not hold since obstacles are often in the way. can still do a minimum spanning tree, but must replace straight line paths with weighted shortest path links

42 Traveling Salesman Problem Solution TSP may yield invalid paths. would have replace point--point costs with shortest path costs 10-42

43 Traveling Salesman Problem Can replace invalid segments with shortest path segments: Thick Thick line line replaces replaces dotted dotted one. one. These These can can be be time time consuming consuming compute compute!!!! 10-43

44 Traveling Salesman Problem The solution the traveling salesman problem does not directly apply our problem since paths may be invalid. Often a simpler, more practical, approach a better one. + easier compute + can ensure complete coverage - may end up with longer path Simplest, most practical approach use the dual graph of the triangulation

45 Vibility Search Paths 10-45

46 Dual Paths Consider a robot with an unlimited range, omnidirectional sensor. First, triangulate the polygon with holes: Use Use a a constrained constrained Delauney Delauney Triangulation Triangulation for for best best results. results

47 Dual Paths We can traverse the dual graph (using a Depth First Search) as a rough path around all obstacles: Robot Robot vits vits the the center center location location of of each each triangle. triangle. From From center center location, location, the the robot robot can can see see the the entire entire triangle triangle and and more. more

48 Vibility Path As the robot travels along the dual graph, it can actually see (i.e., search) a much larger area than the triangles it passes through: Many Many triangles triangles can can be be seen seen (hence (hence searched) searched) without without having having enter enter them. them. However, However, it it difficult difficult keep keep track track of of which which parts parts of of the the triangles triangles are are seen seen from from other other triangles. triangles

49 When robot travels between triangles it can: - search along the way - search only when arriving while traveling Vibility Path at the triangle centers Some Some areas areas are are seen seen twice. twice. Most Most areas areas are are Some Some areas areas are are seen seen multiple multiple Some Some additional additional Some never never seen Some areas areas are are seen times times from from multiple multiple areas areas are are never never seen during during travel. seen only only once. once. travel. locations. locations. seen. seen. + better coverage + less computation - more computation - less coverage 10-49

50 Refining Paths Recall that dual graph paths can be refined by computing a shortest sleeve path: results in a slightly modified area coverage Th Th area area no no longer longer vible vible from from the the new new / / shorter shorter sleeve sleeve path. path

51 Refining Paths Combining all such refined paths leads an efficient path that will guarantee vibility of the entire environment: 10-51

52 Refining Paths Can even trim (i.e., remove edges from) the path: walk through the path, keeping track of which triangles are completely covered along the way. Eliminate edges/vertices that do not add the path s coverage. Do Do you you know know why why we we need need keep keep th th vertex vertex?? 10-52

53 Safer Paths For safety, we can first grow obstacles according robot model allow valid paths that do not collide: Points Points now now away away from from obstacles obstacles allow allow for for safe safe robot robot traversal. traversal. New New points points may may be be added added due due grown grown obstacle obstacle restrictions. restrictions. Some Some points points may may be be gone gone due due grown grown obstacle obstacle restrictions. restrictions

54 Safer Paths A safer/simpler approach: place view locations at midpoints of triangulation diagonal edges and connect viewpoints from edges on the same triangle Provides Provides a a safer safer clearance clearance between between obstacles. obstacles

55 Safer Paths Once again, trim edges by removing ones that do not add the coverage: May May also also put put constraint constraint that that edge edge must must have have certain certain clearance clearance from from obstacles. obstacles. Merge Merge triangles triangles that that form form convex convex polygons polygons since since entire entire convex convex polygon polygon vible vible from from any any point inside it. point inside it

56 Convex Pieces Of course, we can always merge triangles form convex areas before we search the graph reduces need trim off many edges later The The dashed dashed edges edges are are not not necessary necessary since since they they cross cross polygons polygons that that are are already already reached reached by by other other path path portions. portions

57 Convex Pieces How do we merge triangles in convex pieces? traverse dual graph using DFS. build up convex polygon by adding new triangles one at a time if a new triangle ruins convexity, start a new polygon Th Th triangle triangle cannot cannot be be added added since since it it would would destroy destroy the the convexity convexity property. property

58 How do we determine if a polygon convex? There are a variety of ways: check that the line segment between each pair of non-adjacent vertices does not intersect any polygon edge. check that each pair of consecutive edges forms an interior angle 180. traverse the polygon CW and make sure that each consecutive edge makes a right turn. Convex Pieces 10-58

59 Limited Direction Vibility What if the robot cannot sense omni-directionally? Recall that robot can turn at each search point: can be time consuming try minimize search locations Alternatively, some robots are equipped with head turrets that can turn

60 Searching With Limited Vibility Range 10-60

61 Limited Range Vibility The problem changes when the robot has limited sensing range: Traveling Traveling inside inside triangle triangle no no longer longer ensures ensures complete complete vibility vibility of of that that triangle s triangle s area. area

62 Recursively Decompose One option ensure that each triangle small enough be covered by the robot s range: For For each each triangle triangle that that not not covered covered from from its its center center (based (based on on robot s robot s viewing viewing range range d), d), split split triangle triangle in in smaller smaller ones, ones, recursively, recursively, adding adding any any necessary necessary edges. edges. d 10-62

63 Again, form path from dual graph: more loops now Limited Range Paths cannot trim vertices now, only edges

64 Refining Limited Range Paths May trim as many edges as possible, provided that the removal of the edge does not dconnect the graph

65 Convex Pieces Again, we can merge in convex polygons first, provided that the convex polygons are fully vible from each edge: Th Th edge edge not not needed needed since since whole whole polygon polygon vible vible from from the the red red vertex. vertex

66 Limited Range Vibility Coverage Result that entire area covered: 10-66

67 Summary You should now understand: How compute paths that cover an environment Different ways of covering an environment How compute a set of robot locations that see the entire environment A simple way search an environment with robots that have sensors with unlimited or limited range as well as omni-directional or limited direction