Fast Marching and Fast Driving

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Fast Marching and Fast Driving

• Combining off-line search and reactive A.I.

Robert McDowell, Real Time WorldsDaniel
Livingstone, University of Paisley
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Fast Marching

• Introduction to the FMM
• FMM for path planning
• Travel times and optimal paths
• Strengths and Weaknesses
• Application Reactive racer AI
• Extensions
• Conclusions

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The Fast March Method

• Mathematical approach to solving evolving front
  problems (Sethian)
• Stationary approach to non-stationary problem
• Complex derivation
• Simple to apply
• Huge range of applications
• From seismology and combustion models to computer
  graphics
• Some interesting features

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Travel-Times

• Whatever the application the FMM works using the
  principle of travel time
• How long an expanding wavefront takes to reach a
  point
• Always start from a single known point with a
  known travel time, work outwards, calculating
  travel times of neighbouring points
• All points are either known, near or far
• Near unknown, but adjacent to a known point
• Far unknown and not adjacent to a known point

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Application to path-planning

• Straightforward
• Terrain cost slowness at a point
• Known point goal
• Work outwards from goal
• Expanding wavefront comparable to BFS
• Dijkstras
• Some important differences

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FMM algorithm

• Start at Goal, travel-time 0
• Add neighbouring points to near list (and remove
  from far list), estimating t-time
• Repeat
• Select node with smallest travel-time value.
  Remove from near list, add to known.
• Compute travel-time values for each neighbour of
  the selected point (recalculates values for any
  neighbours already in near list)
• Add neighbouring points to near list (and remove
  from far list), estimating t-time
• Until near list is empty

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Calculating the Travel Time

• FMM solves underlying continuous problem
• not constrained by node connectivity!
• Every node has known slowness, s
• Need to find travel time u at point i,j uij
• Will have already found travel times for at least
  one neighbour use neighbouring known values in
  calculations

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Calculating the Travel Time

• Find neighbouring nodes in x and y with minimum
  travel times
• ux min (ui-1,j , ui1,j )
• uy min (ui,j-1 , ui1,j1 )
• Then, it can be shown that
• ( ui,j - ux )2 ( ui,j - uy )2 s2
• Solve for uij

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Solving for uij

• Gives a quadratic equation
• aui,j2 bux ui,j c 0
• For partial difference in x
• a 1 , b -2ux , c ux2
• (If no neighbouring node in x has a known travel
  time, a b c 0)
• Repeat for p.d. in y, summing a, b and c
• Subtract s2 from c
• Finally, solve using quadratic formula

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Assumptions and simplifications
From Travel-Times to Paths

• Travel time gradient gives direction to the goal

• Far points have infinite travel time
• Points off edge of map have infinite slowness,
  travel time
• Blocking terrain has infinite slowness

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Terrain with uniform slowness
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Terrain with Obstacles
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Strengths and Weaknesses

• Guaranteed to find optimal path
• Generates smooth paths
• Finds path to goal from ALL points
• Slow (In comparison to A using good heuristic)

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Sample application Off-road racing

• Off-road racing should allow racers to deviate
  from track
• Always know best route to goal from current
  position
• Common goal for all vehicles, known at compile
  time
• Reactive A.I.
• Often absent in racers
• Should try to avoid collisions with other racers

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The Reactive Racer

• Same controls as player
• Tries to get to goal (finish line)
• Look ahead for corners
• If projected position requires turn, start turn
• Avoid obstacles and other vehicles
• Overrides other behaviour
• Braking and swerving

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Reactive AI and FMM

• Swerve to avoid collision could force racer round
  other side of object / onto alternative path
• FMM pre-computed
• In game search O(1)
• Additional data overhead
• store direction at each point
• Interpolate nodes to reduce number?

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Multiple routes
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Practical Issues

• Can find unexpected shortcuts
• Omit large parts of track
• Set slowness to higher values to block routes
• Can still allow racers sent off track to discover
  the shortcut
• Level Design
• Levels should be designed with AI in mind
• Features may not be exploited to their full

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Extensions

• Additional dimensions
• 3D
• Facing / rotation
• Triangulated domains
• As found in typical 3D models

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Other game applications?

• Good for off-line path planning
• Better than A?
• A is not always optimal depends on heuristic
• FMM should give optimal paths
• FMM not constrained by node connectivity
• FMM isnt really that hard
• On-line (real-time)
• Too slow equivalent to BFS
• Multiple units to single target?

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Conclusions

• FMM
• Yet another search
• Optimal but slow
• Extensible
• Has applications
• Horses for courses