Beyond A*: Speeding up pathfinding through hierarchical abstraction. Daniel Harabor NICTA & The Australian National University

Transcription

1 Beyond A*: Speeding up pathfinding through hierarchical abstraction Daniel Harabor NICTA & The Australian National University

2 Motivation Myth: Pathfinding is a solved problem.

3 Motivation Myth: Pathfinding is a solved problem. Reality: Pathfinding is not solved!

4 Motivation Myth: Pathfinding is a solved problem. Reality: Pathfinding is not solved! A* underlies many pathfinding systems but... Many open problems persist!

5 Problem: Efficient pathfinding in very large environments Image: World of Warcraft (Blizzard Entertainment)

6 Problem: Dynamic terrain Image: Company of Heroes (Relic Entertainment)

7 Problem: Different unit classes Image: Red Alert 3 (Electronic Arts)

8 Problem: Cooperative Pathfinding Image: Settlers: Heritage of Kings (Blue Byte Software)

9 Rest of this talk Some background on A* Then, a story in 3 parts: Part 1: Hierarchical pathfinding Part 2: Dealing with diversity Part 3: Probabilistic roadmaps

10 Background

11 A* In Brief [Hart, Nilsson & Raphael, 1968] Heuristic search f(n) = g(n) + h(n), where: g(n) = distance so far h(n) = estimate remaining distance Complete + Optimal

12 A* example S

13 A* example S

14 A* example S

15 A* example S

16 A* example S

17 The problem with A* Time complexity (nodes expanded) Memory overhead (stuff to remember) Good heuristics not enough.

18 A* Example

19 A* Example Cost = An optimal path (straight-line movement only)

20 A* Example Cost = Cost = Another optimal path

21 A* Example Cost = Cost = Lots of optimal paths (and many more exist!)

22 A* Example A* expands: all green tiles + some white

23 Part 1: Hierarchical pathfinding

24 Hierarchical pathfinding Build a smaller, approximate search space. Area-based vs. Tile-based pathfinding. Factored problems are easy! Faster + lower memory requirements.

25 HPA* Overview [Botea, Müller & Schaeffer, 2004] Pre-processing: Build abstract graph (Clusters and Entrances) Run-time processing: Insert start and goal into abstract graph Find abstract solution Refine if necessary (or use cached info)

26 Map: Temple Prime (Red Alert 3)

27 Split map into square clusters (e.g. 10x10) Image: Adi Botea

28 Identify entrances between adjacent clusters Image: Adi Botea

29 Large entrance (length Small entrance (length >= 6) < 6) 1 central transition point 2 perimeter transition points

30 Transition point = 2 abstract nodes + 1 abstract (inter) edge 1.0 Entrance Abstract graph

31 Select transition points for each entrance Image: Adi Botea

32 Connect abstract nodes inside the same cluster Image: Adi Botea

33 Intra-edges (cost = path length) added to abstract graph

34 Intra-edges (cost = path length) added to abstract graph

35 Intra-edges (cost = path length) added to abstract graph

36 8.0 Intra-edges (cost = path length) added to abstract graph

37 8.0 Intra-edges (cost = path length) added to abstract graph

38 8.0 Intra-edges (cost = path length) added to abstract graph

39 Intra-edges (cost = path length) added to abstract graph

40 Intra-edges (cost = path length) added to abstract graph

41 Intra-edges (cost = path length) added to abstract graph

42 Insert intra-edges (of cost = path length) to abstract graph.

43 Insert intra-edges (of cost = path length) to abstract graph.

44 Insert intra-edges (of cost = path length) to abstract graph.

45 Insert intra-edges (of cost = path length) to abstract graph.

46 Insert intra-edges (of cost = path length) to abstract graph.

47 Final abstract graph Image: Adi Botea

48 Runtime processing Insert start + goal into abstract graph. Find an abstract path connecting them. Refine each abstract step using a small A* search to traverse each cluster (free if we cache intra-edge paths).

49 Add Start + Goal as (temp.) abstract nodes Image: Adi Botea

50 Insert temp. edges to connect Start + Goal Image: Adi Botea

51 Run A* to find an abstract path Image: Adi Botea

52 Refine the abstract path before execution. Each step is independent (and fast!) Images: Adi Botea

53 HPA* Advantages Easy to understand + implement Very fast (Beats A* by 10x) Low memory overhead (vs. A*) Complete

54 HPA* Disadvantages Non-optimal (but very close!). Assumes fixed-size 1x1 units. Only supports single terrain type. Insertion effort.

55 Part 2: Dealing with diversity

56 Assumptions in pathfinding literature

57 Assumptions in pathfinding literature Assumption #1: all units are the same size.

58 Assumptions in pathfinding literature Assumption #1: all units are the same size. Assumption #2: any terrain traversable for one unit is traversable for all.

59 Assumptions in pathfinding literature Assumption #1: all units are the same size. Assumption #2: any terrain traversable for one unit is traversable for all. Pathfinding breaks when either assumption is lifted.

60 HPA* Tank Fail HPA* loses information (like sizes of entrances)

61 Challenging pathfinding Unit sizes: Small, Medium, Large Terrains: Snow, Road, Water, Ground Images: Red Alert 3 (Electronic Arts)

62 HAA* Overview [Harabor, Botea, 2008] HPA*-style abstraction & search. Combined with clearance-based pathfinding. Caters for different terrains and unit sizes. Constructs complete abstractions.

63 Modeling diversity Example Map Agent Sizes Ground (white tiles) Water (blue tiles) Obstacles (black tiles) Small Big Movement rules Terrain traversal Capabilities OK OK Not OK {Ground} {Water} {Ground or Water}

64 Clearance-based pathfinding Intuition: Calculate how much clearance (traversable space) exists at a given tile (t) on the map. A tile (t) is traversable by an agent (a) if: 1. terrain(t) is in capability(a) 2. clearance(t, capability(a)) >= size(a)

65 Example Each tile on the path is traversable for the agent Agent size = 1 Capability = {Ground}

66 Computing clearance 1 Initial Clearance. Capability = {Ground} (white tiles)

67 Computing clearance 1 2 Initial Clearance. Capability = {Ground} (white tiles) First Expansion OK

68 Computing clearance Initial Clearance. Capability = {Ground} (white tiles) First Expansion OK Second Expansion OK

69 Computing clearance Initial Clearance. Capability = {Ground} (white tiles) First Expansion OK Second Expansion OK Third Expansion Fail

70 Maps with multiple terrain types Easy! Each tile has a clearance-value for each possible capability. Clearances for {Ground} Capability

71 Maps with multiple terrain types Easy! Each tile has a clearance-value for each possible capability. Clearances for {Ground} Capability Clearances for {Water} Capability

72 Maps with multiple terrain types Easy! Each tile has a clearance-value for each possible capability. Clearances for {Ground} Capability Clearances for {Water} Capability Clearances for {Ground, Water} Capability

73 Dealing with large agents Each 2x2 agent occupies 4 tiles. Which clearance value do we consider? Agent size = 2, Capability = {Ground}

74 Problem reduction Theorem: Any pathfinding problem can be reduced into a canonical problem (agent size = 1, capability = 1 terrain) G A Initial problem

75 Problem reduction Theorem: Any pathfinding problem can be reduced into a canonical problem (agent size = 1, capability = 1 terrain) G A Initial problem G A Reduced size

76 Problem reduction Theorem: Any pathfinding problem can be reduced into a canonical problem (agent size = 1, capability = 1 terrain) G A Initial problem G A Reduced size G A Reduced size and capability

77 Annotated A* Search process: Similar to A*. Extra parameters: Agent's size and capability. Only expand nodes with clearance > agent size. Works great! For small problem sizes...

78 Abstraction Process Similar to HPA*: Divide map into NxN clusters. Identify entrances for all capabilities. Using AA*, find intra-edges for all agent sizes and capabilities Reduce the graph using dominance

79 Clusters C1 C2 C3 C4

80 Entrances Identifying entrances Transition points Abstract graph C C1 C3 E1 {Ground, 2} E2 {Water, 1} E3 {Ground, Water, 5} w E1E3 x y E2 z C3

81 Intra-edges Use AA* to find all possible ways (sizes/capabilities) to traverse across each cluster. For each path found, add a new abstract edge. w C1 E1 E2 E4 E3 E5 y u E6 Edge Annotations {Terrain, Clearance}: E1 = {Ground, 2} E2 = {Ground, 1} E3 = {Ground, Water,2} E4 = {Ground, Water, 2} E3 = {Ground, Water, 1} E4 = {Ground, Water, 1}

82 Initial abstraction C1 C2 C3 C4

83 Compacting the abstract graph Method produces a representationally complete graph but can get rather large. Solutions: Strong dominance Weak dominance

84 Strong dominance Initial graph High Quality graph E1 E2 E4 u E6 E1 u w E3 E5 y w E2 E3 E4 y Edge Annotations [Capability, Clearance, Length]: E1 = [{Ground}, 2, 7.5] E2 = [{Ground}, 1, 4.5] E3 = [{Ground, Water}, 2, 3.0] E4 = [{Ground, Water}, 2, 3.0] E5 = [{Ground, Water}, 1, 3.0] E6 = [{Ground, Water}, 1, 3.0] Edge Annotations [Capability, Clearance, Length]: E1 = [{Ground}, 2, 7.5] E2 = [{Ground}, 1, 4.5] E3 = [{Ground, Water}, 2, 3.0] E4 = [{Ground, Water}, 2, 3.0] Remove edges with smaller clearance (all else being identical)

85 Weak dominance Retain only edges traversable by largest number of agents (aka. prefer freeways where possible) High Quality graph Low Quality graph Lake v E2 w E3 E6 x E6 x E1 u E4 y E5 z u E4 y E5 z Two places of interest: U and X. Path p = {E1, E2, E3}. Length: 15, Max Size = 1, Capability: {Ground, Water} Path q = {E4, E5, E6}. Length: 17, Max Size = 2, Capability: {Ground}

86 Weak dominance High Quality graph Low Quality graph C1 C2 C1 C2 C3 C4 C3 C4 (c) Weak dominance is applied to transitions between adjacent clusters (inter-edges).

87 HAA* Similar search process to HPA* Insert Start + Goal Find + refine abstract path Differences: Add a node to open list only if reachable according to edge annotations.

88 C1 C2 C1 C2 C3 C4 (c) C3 C4 (c) Initial problem Size = 2, Capability = {Ground} Reducing the problem C1 C2 C3 C4 (c) Inserting Start + Goal Solution

89 HAA* Advantages Works for units with different sizes and capabilities. Complete and near-optimal (within 4-8% from optimal on average) Fast (similar performance to HPA*) Low overhead (storing abstract graph requires little memory in practice)

90 HAA* Disadvantages Only works on grids. Need to model units using square bounding volumes. Insertion effort.

91 Part 3: Probabilistic Roadmaps

92 Limitations of grid methods Coarse coverage of underlying terrain. Restricted movement (fixed angles) Paths don t look realistic Often need to apply smoothing Not very useful in 3D spaces

93 vs.

94 vs.

95 vs.

96 Probabilistic Road Maps PRM: Robotics inspired method from mid 90s. e.g. see [Choset et al, 2004]. Pick a robot configuration. Build a graph by taking lots of sample points.

97 Select non-obstacle sample points from the map

98 X X X X X X X X X X X X X X Retain collision-free points for some configuration of an agent X Configuration

99 Connect nearby points to form a graph

100 Pathfinding: Connect start and goal to rest of roadmap

101 Pathfinding: Connect start and goal to rest of roadmap

102 Pathfinding: Connect start and goal to rest of roadmap

103 Pathfinding: Connect start and goal to rest of roadmap

104 Most PRMs are only useful for agent/configuration used during construction.

105 Most PRMs are only useful for agent/configuration used during construction.

106 Most PRMs are only useful for agent/configuration used during construction.

107 Most PRMs are only useful for agent/configuration used during construction.

108 Most PRMs are only useful for agent/configuration used during construction.

109 2 options to fix this problem: Create a PRM for each unit type Opt for a (generalised) Voronoi-based PRM

110 Voronoi diagram: Each point on the medial axis maximises clearance.

111 Intuition: retract PRM (nodes and edges) to the medial axis.

112 Node retraction z y x Identify the closest obstacle then retract the point to a position equidistant to 2 obstacles.

113 Node retraction z y dist(x) = 2 x Identify the closest obstacle then retract the point to a position equidistant to 2 obstacles.

114 Node retraction z y dist(y) = 2 dist(x) = 2 x Identify the closest obstacle then retract the point to a position equidistant to 2 obstacles.

115 Node retraction z y dist(y) = 2 dist(x) = 2 dist(x) = 2 dist(y) = 2 x Identify the closest obstacle then retract the point to a position equidistant to 2 obstacles.

116 Edge retraction z y x Edges too close to obstacles are retracted by splitting.

117 Edge retraction z y x Edges too close to obstacles are retracted by splitting.

118 Edge retraction z y x Edges too close to obstacles are retracted by splitting.

119 Edge retraction z y x Edges too close to obstacles are retracted by splitting.

120 Other improvements: Remove overlapping edges (similar idea to weak dominance) Smooth edges using circular blends

121 Other improvements Remove overlapping edges (similar to weak dominance) Apply circular blends

122 PRM Advantages Fast to construct. Low overhead (especially Vornoi PRMs) Facilitate pathfinding in n-dimensional spaces Generalisable to different unit sizes (Voronoi PRMs only) Nice alternative to navigation meshes

123 PRM Disadvantages Only probabilistically complete* Might need several PRMs (to support units with different capabilities). Non Voronoi PRMs produce ugly paths.

124 Conclusion There s more to pathfinding than A*! Many open problems; some with nice solutions HPA*, HAA*, Roadmaps Pathfinding with abstraction is very effective.

125 References A Formal Basis for the Heuristic Determination of Minimum Cost Paths P.E. Hart, N.J Nilsson, B. Raphael IEEE Transactions on Systems Science and Cybernetics, 1968, Vol 4, Issue 2, pp Near-Optimal Hierarchical Pathfinding A. Botea, M. Muller, and J. Schaeffer Journal of Game Development, Volume 1, 2004 Hierarchical path planning for multi-size agents in heterogeneous environments Harabor D. and Botea, A. IEEE Symposium on Computational Intelligence & Games, 2008 Principles of Robot Motion H. Choset, K. M. Lynch, S. Hutchinson, G. Kantor, W. Burgard, L. E. Kavraki and S. Thrun MIT Press, 2004 Automatic Construction of Roadmaps for Path Planning in Games D. Nieuwenhuisen, A. Kamphuis, M. Mooijekind and M. H. Overmars, International Conference on Computer Games: Artificial Intelligence, Design and Education, 2004 Roland Geraerts and Mark H. Overmars. Creating Small Roadmaps for Solving Motion Planning Problems. IEEE International Conference on Methods and Models in Automation and Robotics (MMAR'05), 2005, pp

126 Questions?