Roland Geraerts Seminar Crowd Simulation 2011

1. Path Planning with Explicit Corridor Maps Related work Constructing Explicit Corridor Maps Corridor Map Method Exploiting Explicit Corridors Roland Geraerts Seminar Crowd Simulation 2011

2. Related work: A* • Method • Construction phase: create a grid, mark free/blocked cells • Query phase: use A* to find the shortest path (in the grid) • Advantage • Simple • Disadvantages • Too slow in large scenes • Ugly paths • Little clearance to obstacles • Unnatural motions (sharp turns) • Fixed paths • Predictable motions

3. Related work: Potential Fields • Method • Goal generates attractive force • Obstacles generate repulsive force • Follow the direction of steepest descent of the potential toward the goal • Advantages • Flexibility to avoid local hazards • Smooth paths • Disadvantages • Expensive for multiple goals • Local minima

4. Related work: Probabilistic Roadmap Method • Method • Construction phase: build the roadmap • Query phase: query the roadmap • Advantages • Reasonably fast • High-dimensional problems • Disadvantages • Ugly paths • Fixed paths • Predictable motions • Lacks flexibility when environment changes

5. Related work: Probabilistic Roadmap Method • Method • Construction phase: build the roadmap • Query phase: query the roadmap • Advantages • Reasonably fast • High-dimensional problems • Disadvantages • Ugly paths • Fixed paths • Predictable motions • Lacks flexibility when environment changes

6. State-of-the-art: Navigation meshes • Method • Create a representation of the "walkable areas" of an environment • Extract the path • Advantages • General approach • Construction is fast due to use of GPU • Examples and source code can be found on http://code.google.com/p/recastnavigation • Disadvantages • Often needs a lot of manual editing • Current techniques are imprecise • Bad support for non-planar surfaces Obstacles Walkable voxels Voxel regions Polygonal regions Convex regions A path

7. State-of-the-art: Navigation meshes • Some open problems • Automatic annotation of the map • Areas: walk, climb, jump, crouch, “avoid” … • Special places: hiding and sniper spots, … • Handle large (dynamic) changes • Efficiently updating the data structure and paths • Improve the efficiency of mesh generation (large scenes) • Wrong coverage/connectivity due to confusing elements • Steep stairs, ramps, hills, curved surfaces, gaps • The mesh is only a data structure storing the walkable areas • How to create visually convincing paths?

8. Towards a new methodology: Requirements • Fast and flexible path planner • Real-time planning for thousands of characters • Dealing with local hazards • Global path • Natural paths • Smooth • Short • Keeps some distance toobstacles • Avoids other characters • … Titan Quest: Immortal throne

9. Capturing the free space • Requirements of the data structure representing the free (walkable) space • Existence of a path • Contains all cycles • Short global paths, alternative paths • Provides high-clearance paths (corridors) • Provides maximum local flexibility • Small size • Fast extraction of paths • A good candidate • Generalized Voronoi Diagram + annotation

10. Voronoi Diagram • Some inspiration from natural objects… maple leaf drying mud bacteria colonies wasps nest giraffe

11. Voronoi Diagram • Definitions • Voronoi region: set of all points closest to a given point • Voronoi diagram: union of all Voronoi regions Voronoi sites: (red) points

12. Voronoi Diagram • Approximation of the Voronoi Diagram • Compute a distance mesh for each point • Render each mesh in a different color by using the GPU • Using the Z-buffer, only pixels with the lowest distance values attribute to a pixel in the Frame buffer • A parallel projection of the meshes gives the diagram Perspective view (Z-buffer) Top view (Frame buffer)

13. Generalized Voronoi Diagram • Generalized Voronoi Diagram supports any type of obstacles • Point, disk, line, polygon, … • Convert concave polygons into convex ones, otherwise edges do not run into all corners

14. Generalized Voronoi Diagram • Generalized Voronoi Diagram supports any type of obstacles • Point, disk, line, polygon, … • Convert concave polygons into convex ones, otherwise edges do not run into all corners • Distance meshes • Point: cone • Disk: lifted cone • Line: tent + 2 cones • Polygon: n (point + line meshes) • Literature • [Hoff et al., 1999] • [Geraerts and Overmars, 2010]

15. From GDV to Medial axis • Generalized Voronoi Diagram (GVD) • Render distance meshes for each obstacle • Boundaries: bisectors between any two closest obstacles • Medial axis • Yields bisectors between any two distinct closest obstacles • Extraction of the medial axis • Edge: trace pixels between Voronoi regions; continue tracingwhen closest points on the obstacles are equal • Vertex: end point of an edge Medial axis GVD

16. Medial axis • The good • Existence of a path: full coverage/connectivity • Contains all cycles: yes • Provides high-clearance paths: yes • Small size: yes (linear) • Fast extraction of paths: yes • The bad • Unclear how to extract short(est) paths • Moving along 1D-curves limits flexibility • The ugly • Deal with robustness

17. Explicit Corridor Map • Basis: Medial Axis • Plus: annotated event points on the edges • Points where the type of bisector on the edge changes • Straight lines versus parabola’s (bisector of point and line) • Changes occur at crossing between site normal and edge • Annotation: its two closest points on the sites • Equals: planar subdivision (or navigation mesh) • Memory footprint The storage is linear in the number of obstacle vertices. • NoteThere is no need for storing pixels.

18. Explicit Corridor Map: closest points • Computation of the closest points • Look up incident colors at the event point’s position • Each color was linked to an unique obstacle • Compute the (left and right) closest points to each obstacle using simple linear algebra

19. Explicit Corridor Map: experiments • Performance • Setup • NVIDIA GeForce 8800 GTX graphics card • Intel Core2 Quad CPU 2.4 GHz, 1 CPU used • Experiments • McKenna: 200x200 meter, 1600x1600 pixels, 23 convex polygons

20. Explicit Corridor Map: experiments • Performance • Setup • NVIDIA GeForce 8800 GTX graphics card • Intel Core2 Quad CPU 2.4 GHz, 1 CPU used • Experiments • McKenna: 200x200 meter, 1600x1600 pixels, 23 convex polygons time: 0.03s

21. Explicit Corridor Map: experiments • Performance • Setup • NVIDIA GeForce 8800 GTX graphics card • Intel Core2 Quad CPU 2.4 GHz, 1 CPU used • Experiments • City: 500x500 meter, 4000x4000 pixels, 548 convex polygons

22. Explicit Corridor Map: experiments • Performance • Setup • NVIDIA GeForce 8800 GTX graphics card • Intel Core2 Quad CPU 2.4 GHz, 1 CPU used • Experiments • City: 500x500 meter, 4000x4000 pixels, 548 convex polygons time: 0.3s

23. Explicit Corridor Map: experiments • Supports large environments • E.g. 1 km2 • Millimeter precision • However, there must be at leasttwo pixels in between two obstacles to discover an edge

24. Explicit Corridor Map: recent work • Extension to 2.5D (multi-layered) environments • Technique • Result (46 ms) Updated medial axes for Li and Lj Connection scene Multi-layered environment Partial medial axes for Li and Lj

25. Explicit Corridor Map: recent work • Handling dynamic changes • Technique for adding a point/line • Result (1 – 2.7 ms per update) A Finding closest site Continue in 1 dir. w1 has been reached Updated VD (point) Updated VD (line)

26. Compare with the old approach • Disadvantages Implicit Corridor Map • More than linear storage (due to discrete sample points) • Non-exact representation • Additional parameters requiredfor local sampling density

27. Explicit Corridor Map: some thoughts • Open questions • Can we make a dynamic version of the ECM? • Dimensionality • Can we generalize the ECM data structure to 2.5D? • How should we add height information? • How can we handle these extensions? • Terrains (elevation) • Different topological spaces • Different terrain types (grass, road, pavement) • How should we handle holes and enable jumping? Examples of different topological spaces (plane, sphere, cylinder, torus, Möbius strip, Klein bottle)

28. The Corridor Map Method • Construction phase (offline) • Build Explicit Corridor Map • Build kd-tree that stores the ECM • Query phase (on-line) • Construct indicative route • CMM: Medial axis

29. The Corridor Map Method • Query phase (on-line) • Construct indicative route • CMM: Medial axis • Compute a corridor • Compute a path • Construction phase (offline) • Build Explicit Corridor Map • Build kd-tree that stores the ECM • Distinguish three scales1. Macro (corridor)2.Meso (indicative route)3. Micro (local behavior)

30. The Corridor Map Method • Query phase (on-line) • Retract the start and goal to the medial axis • Query the kd-tree • Connect the start and goal to the Corridor Map • Compute the shortest backbone path (using A*) Explicit Corridor Map Query Corridor with its backbone path

31. The Corridor Map Method • Query phase (on-line) • Compute the path • While the corridor determines the character’s global path, forces determine its local path • The forceF(x)=Fa(x)+Fo(x) causes the character to accelerate, pulling it toward the goal. The variable x is the character’s position • Using Newton's Second Law, we have F = Ma, where M = mass = 1 and a = acceleration • Hence, the force F can be expressed as: • Combining these expressions gives us: A smooth path

32. goal α(x) x The Corridor Map Method • Query phase (on-line) • Compute the path: forces • The attraction force steers the character toward the goalFa(x) = , where f controls the magnitudea(x) =attraction point: the furthest point on the backbone path whose disk encloses the character. • Note on old approachThis is an discrete corridor instead of continuous explicit corridor.

33. R[t] goal α(x) d x The Corridor Map Method • Query phase (on-line) • Compute the path: forces • The boundary force keeps the character inside the corridorThis force is hidden inside the attraction force: (r=character’s radius) f=0, when the character is positioned at its attraction point (i.e. d=0)f=∞, when the character touches the disk’s boundary • Note on old approachThis is an discrete corridor instead of continuous explicit corridor.

34. The Corridor Map Method • Query phase (on-line) • Compute the path • Solving the equation gives us the character’s positions • Cannot be done analytically • revert to a numerical approximation A smooth path

35. The Corridor Map Method • Choice of forces • Combining these forces and using disks was a bad choice • This is solved by the IRM, which uses Explicit Corridors • Comparison of their vector fields Vector field: CMM force Vector field: IRM force

36. The Corridor Map Method: Examples • Query phase (on-line) • Compute the path: forces • Adding/changing forces leads to other “behavior” Smooth path Short path Obstacle avoidance Obstacle avoidance (Helbing model) + path variation = crowd? Coherent groups Path variation Camera path

37. The Corridor Map Method: Examples • Query phase (on-line) • Compute the path: forces • Adding/changing forces leads to other “behavior” Stealth-based path planning

38. Exploiting Explicit Corridors • The Corridor’s boundaries are given explicitly • Construction • Convenient representation • Small storage: linear in the number of samples (i.e. events) • Computation of closest points in O(1) time (on average) • Allows computing shortest minimum-clearance paths

39. Explicit Corridors: Obtaining clearance • Minimum clearance in explicit corridors • For each closest point cp, move cp toward its center point c • The displacement equals the desired clearance clmin • Insert event point(s) if clmin > distance(c, cp) c cp Explicit Corridor Shrunk corridors Shrinking a corridor

40. Explicit Corridors: Shortest paths • Computing the shortest path • Construct a triangulation • 2ith triangle: (li , ri , li+1) ; 2i+1th triangle: (ri , li+1 , ri+1) • If the start [goal] is not included, add triangle (s, l1 , r1) [(ln , rn ,g)] • Compute the shortest path • Funnel algorithm [Guibas et al. 1987] ri+1 li+1 g ri li s Triangulation Shortest path Explicit Corridor

41. Explicit Corridors: Shortest paths • Sketch of the Funnel algorithm • Funnel • Tail: computed shortest path from start to apex • Fan: 2 outward convex chains plus one diagonal • The fan keeps track of all possible shortest paths • Algorithm • Add diagonals iteratively whileupdating the funnel • Algorithm is linear in the number of diagonals • (or events) start tail apex fan diagonal goal

42. Explicit Corridors: Shortest paths • Computing the shortest minimum clearance path • Shrink the corridor • Construction time: linear in the number of event points • Compute the shortest path • Adjust Funnel algorithm to deal with circular arcs • Construction time: linear in the number of event points Left/right closest points Triangulation Shortest path

43. Improving the CMM: IRM • Compute a smooth path: Indicative Route Method • Compute the shortest minimum-clearance path • Define the attraction force • Pulls the character toward the goal • Define the boundary force • Keeps the character inside the corridor • Time-integrate the forces • Yields a smooth (C1-continous) path

44. The Query Phase: Experiments • Performance • Setup • Intel Core2 Quad CPU 2.4 GHz, 1 CPU • Experiments • City: 500x500 meter, 1.000 random queries • Results (average query time)

45. The Query Phase: Experiments • Performance • Setup • Intel Core2 Quad CPU 2.4 GHz, 1 CPU • Experiments • City: 500x500 meter, 1 query • Results (query time) • 2.8 ms ECM (0.3s) Explicit corridor Shrunk corridor Triangulation Shortest path Smooth path

46. Integration in Second Life • Implementation in Second Life Virtual World Bitmap Interface Camera path Path planning on server:http request

47. Conclusion • Advantages • Fast and flexible planner creates visually convincing paths • Computation of smooth, short minimum clearance paths • The algorithms run in linear time and are fast • The algorithms are simple • Open problems • Automatic annotation of the navigation mesh • Handling 3D spaces • Handling extensions • Handling character behavior • E.g. shopping and beach behavior • Interaction between different entities (human, car, bicycle)