Graphs, Goals and NPCs

1. Graphs, Goals and NPCs Advanced Programming for 3D Applications CE00383-3

2. Path Planning and Search Solution Path Spatial Environment discretisation Grid of cells mapped on the environment Search Tree search Path

3. Overview • Alternative Search Space Representations • Grids • Graphs • Meshes… • When we know what the world looks like, how can we move through it? • A* • Precomputed Pathfinding with Navigation Sets • Potential Fields

4. Introduction • In many video games, artificial characters have to find their way through spatial layouts: • rooms (DOOM-like games) • terrain models, maps, etc. (strategy games)

5. Search & Path Planning • A significant number of games involve moving in buildings, battlefields, etc. • Choosing the best path within a determined space is a classic AI problem • Algorithms have been developed that “construct” the best path by exploring possible directions on the basis of individual cells’ properties

6. Discretisation grid cells hexagonal cells G 3D discretisation S

7. Regular Grids

8. Regular Grids • How do we generate? • Advantages • Random access lookup (O(1)) to determine what tile lies at any coordinate • Complete • Negatives • Usually requires large number of nodes to accurately represent world • Path Quality • Agent can only walk in four cardinal directions? That’s no fun. • Let them walk diagonals! Still not much fun

9. Regular Grids NWSE Diagonals String-Pulling Catmull-Rom Spline

10. Grids as Graphs • Everything else we look at will be a graph • Grids are graphs too • Each cell is a node and edges are adjoining cells • Maybe we should just handle grid as a graph • Undirected graph could tell us about topography, etc, whereas array can’t

11. Grids as Graphs

12. Corner Graphs • Waypoints around obstacles

13. Corner Graphs • How do we generate? • Identify convex corners, can character walk in straight line between? Add edge if possible • Advantages • Less memory • Faster generation • Negatives • Character will “walk on a rail” hugging edges of obstacles instead of walking through open space • And what about different sized characters? • Lookup is O(n2), have to check every node in graph against every other. Imagine a world with a boat load or corners!

14. Corner Graphs

15. Corner Graphs

16. Waypoint Graphs • Place nodes in middle of rooms instead of at convex corners

17. Waypoint Graphs • How do we generate? • Place nodes wherever we want (suits 3-D worlds  But requires hand tuning to be effective ) • Advantages • Reduce memory footprint from regular grids, reduce wall hugging from corner graphs • Work well in “human” architectures • Negatives • Still O(n2) • Path Quality vs. Simplicity • Works poorly in open areas

18. Waypoint Graphs

19. Circle-Based Waypoint Graphs • Add radius parameter to indicate open space near waypoint

20. Circle-Based Waypoint • Advantages • Only look at overlapping circles, alleviating O(n2) problem from before • Easier to obtain optimal paths • Works well in open terrain • Negatives • Doesn’t work as well in human architectures that aren’t circle friendly

21. Space-Filling Volumes • Use rectangles or 3-D Boxes instead of circles

22. Space-Filling Volumes • How do we generate? • Seed and Grow • Make Grid and Merge • Very similar to circle-based, but handles angles better

23. Navigation Meshes • Let’s try and cover walk able surfaces with convex polygons • Character can travel between adjoining polygons

24. Navigation Meshes

25. Navigation Meshes • How do we generate? • By hand (time consuming) • Automated tools to analyze and optimize geometry of world • Too complex and not represented as a single polygon mesh, instead may be overlapping, etc

26. Navigation Meshes • Advantages • Quickly find optimal paths independent of character shapes and capabilities • Handle indoor and outdoor terrains well • Negatives • Can become complex and expensive memory wise • Difficult to generate

27. Problem with N-Sided Meshes

28. Interacting with Pathfinding • What about dynamic objects in world? • All the representations discussed have been static and obviously can’t handle a dynamic world directly • These representations need to be able to provide information to system determining path • Waypoint Graphs and Corner Graphs don’t illustrate walk able surfaces • Meshes and Grids do map every walk able surface • Space-filling volumes and Circle Waypoints provide some representation of walk able areas

29. Path finding • Now that we have world represented, how do we plan movement? • A*, Depth-First, Dijkstra • Dynamic path finding is expensive • Precompiled solutions can eliminate runtime cost, but memory expensive • Article suggests technique of Navigation Set Hierarchy

30. Path finding • Determine best paths leading from source node to boundary of source set • Determine best path from source set boundary to goal set boundary • Determine best path from goal set boundary to goal node • Compile list of complete paths and choose one with least cost

31. Transition Table • Main element in pre computed solutions is a lookup table • Each entry represents next step to take from one node to some goal node

32. Transition Table

33. Transition Table • Do not need full search of nodes at run time, just series of lookups • Fast, but becomes memory expensive as size of world grows • How expensive? n2 • …solution to shrink transition tables? Hierarchy!

34. Navigation Set • Self-contained collection of nodes that requires no links to external nodes to complete a path • Nodes can be members of more than one set • Goal: Find someway to partition large Navigation Sets into smaller ones

35. Complete Hierarchy

36. Interface Nodes and Sets • Need to account for paths that cross navigation sets • Any node that connects to a node in another navigation set is an interface node • Have a second layer of nodes in addition to navigation sets, called interface set • Interface set itself is a navigation set • Therefore, can make transition table for it too

37. Complete Hierarchy • 21 nodes • 1 Navigation Set = 441 Table Entries (21*21) • 4 Navigation Sets = 183 Table Entries (7*7 + 7*7 + 7*7 + 6*6)

38. Constructing the Hierarchy Two goals to process • How many tables to create? • Amount of data needed is 1/n of original size + interface set • As size of navigation sets increase, cost of interface set becomes less a factor • Where to place boundaries? • Keep interface nodes as low as possible

39. Constructing the Hierarchy

40. Potential Fields • Reactive approach to path finding • Setup virtual potential or force field around objects • Must define field and agents reaction to field • Interactions in general are explicitly stated and movement “emerges” • Paths are not planned explicitly

41. Potential Fields • Think of world as matrix • Each point tells has value to represent strength of field under it • Possibly assign potential based on distance from goal • Might get stuck behind obstacle • Result/Goal: “Glide down gradient/path of least resistance”

42. Potential Fields • Advantages? • Works in continuous space! No need to make grid, place waypoints, etc • Disadvantages? • Can be trapped in local minima • It’s emergent, not sure what it will do

43. Path planning on a grid (1/2) • Going from A to B avoiding obstacles • It is only possible to move from one cell to an adjacent cell • How to find: • the shortest path? • In the shortest time? • At the lesser cost? B A

44. B A Path planning on a grid (2/2) • From each node of the grid, it is possible to go only to neighbouring nodes • Path planning is the process by which a path is assembled from A to B through authorised moves

45. Search Trees • From a given node, it is possible to generate a set of neighbours. For instance a given node on the grid has for neighbours corresponding to the N, S, E, W, NW, NE, SE, SW moves • The process of generating all the neighbours of a given node is called node expansion.

46. Grid Moves and Search Trees B path A

47. Search Algorithms and Path Planning • Because the path on the grid is formally equivalent to a path on the tree, tree-searching algorithms can be used for path planning • The path found on the tree does correspond to a path on the grid, hence in the game environment

48. Why Search? • Search is a key technique for AI, in particular real-time AI • Applications in computer games include: • Path planning • Moving Target Search • Two-player games (othello, chess, etc.) • Search-based planning for animation or intelligent actor behaviour

49. Search • Search consists of a set of technique that explore graphs and extract solutions, i.e. sub-graphs satisfying some desirable properties • Solutions can be: • Nodes (e.g. n-queens) • Paths (e.g. n-puzzle)

50. State-space Search • In the state space representation of a problem, the nodes correspond to partial problem solution states and the arcs correspond to steps in the problem solving process (or operators) • This makes possible to apply the same graph-search algorithms for finding a solution to these problems (various optimisation problems, games, etc.)