Minimum Link Paths for tactical game scenarios

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Minimum Link Pathsfor tactical game scenarios

• Project Final Report
• Nicolas Galoppo von Borries

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Motivation for Minimum Link Paths

• Minimum link paths minimize of straight-line
  segments
• Length ? shortest path
• Along corners and borders of objects
• Long, straight segments
• MLP look more like tactical paths and promise
  good threat avoidance
• Artificial agents stay closer to objects ? under
  cover
• Paths form long straight fast bridges between
  the objects

Voronoi
MLP
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The theory

• Minimum Link Paths
• Exact computation is difficult
• Best 2D algorithm known O(n4) worst case
  (Mitchell, 1992)
• Doesnt promise good stuff for 3D (games)
• Do we need exact MLP?
• Not really, all we want is
• Collision free path, or almost collision-free
  (if used as a heuristic)
• An approximation of a minimum link path
• Path is composed of straight-line segments
• The number of segments should be minimal

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How can we approximate a MLP?

• Use the boundary of the generalized Voronoi
  diagram (outer medial axis)
• Useful in presence of narrow passages
• Fast (GPU) implementation (HAVOC3D, DiFi)
• Isnt really what we want
• Not close to objects
• Typically shows many curves, could be complex
• We want straight segments, minimizing turns

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Instead

• Preprocess subdivide straight line segments

Store the result in a tree Converges to
the medial axis
Find boundary search in fixed of discrete
directions
We propose discrete method without the need for
explicit analytic representation of the medial
axis.
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Path reconstruction

• Start at root of the tree
• Perform link query
• Free, then were done, otherwise recurse
• Stop as soon as the segment is free

• This process limits the number of links,
  therefore giving preference to long segments at
  the top of the tree

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Collision-free links

• When is a potential link suitable as path
  segment?
• When no collision of link with objects
• Visibility-based approach
• Use hardware (GPU) occlusion queries to determine
  visibility between two potential path points

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MLP approximation strategy

• REFINEPATH (start position, target position)
• Connect start and target position with straight
  line s
• IF (fixed depth not yet reached)
• Take midpoint m on segment s
• Look for closest boundary point p in ms
  neighborhood
• (in fixed of discrete directions)
• child1 ? (start, p) child2 ? (p, end)
• REFINEPATH(child1)
• REFINEPATH(child2)
• ? Hierarchical representation of segments
  representing a variety of possible paths

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Path reconstruction

• REFINE(tree)
• link ? root of tree
• If (link intersects with environment)
• P1 ? REFINE(left_subpath)
• P2 ? REFINE(right_subpath)
• path ? P1 U P2
• Else path ? link
• Return path

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Performance issues (1)

• Computing the entire 3D GVD for each frame is too
  costly
• Compute discrete approximation one 2D slice at a
  time
• We only compute 1 slice per frame
• Update the segment tree each time a full 3D GVD
  is ready

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Performance issues (2)

• We can limit the number of Voronoi sites in very
  complex environments
• only static and slowly moving objects are
  part of the scene
• fast moving objects do not influence the Voronoi
  diagram, only used in the link queries

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Implementation on GPU

• We use the GPU for two purposes
• Fast discrete GVD computation
• HAVOC3D (Hoff et al., 1999)
• DiFi (Sud et al., 2003)
• Link-environment intersection
• Visibility based approach segment is collision
  free if start point sees endpoint (i.e. no object
  is occluding the view path)
• Place virtual camera at start position, look in
  direction of endpoint
• Visibility queries on GPU NV_occlusion_query

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Hardware Depth Test
Fly passes, bunny is too fat
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Results / Conclusions

• Basic infrastructure implemented based on simple
  subdivision scheme
• 3D workspace, translational agent with any size
  and geometry
• Limited sized tree is sufficient (4-6 levels)
• Real-time computation
• GPU based
• We update 1 Voronoi slice at a time

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What could be next steps?

• Add rotational DOF
• Try other path refinement schemes
• Local search heuristic
• Use distance field to compute gradient descent
  towards Voronoi boundary

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Thanks

• References
• The Open Problems Project
• (ORourke, Mitchell, Demaine 2003)
• Fast Computation of Generalized Voronoi Diagrams
  Using Graphics Hardware
• (Hoff et al., SIGGRAPH 1999)
• Randomized Path Planning for a Rigid Body Based
  on Hardware Accelerated Voronoi Sampling
• (Pisula et al., 1999)
• Interactive Motion Planning Using
  Hardware-Accelerated Computation of Generalized
  Voronoi Diagrams
• (Hoff et al, ICRA 2000)
• Fast and Simple Occlusion Culling Based on
  Hardware Depth Queries
• (Karl Hillesland, Brian Salomon)
• DiFi Fast Distance Field Computation Using
  Graphics Hardware
• (Avneesh Sud, Dinesh Manocha)
• Acknowledgements
• Kenneth Hoff (HAVOC3D), Mark Foskey (Pointers to
  SMA, GVG), Avneesh Sud (DiFi, HAVOC3D)

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Questions?