Toward More Realistic Pathfinding

1
Toward More Realistic Pathfinding

• Authored by Marco Pinter

2
Path finding

• How do people find paths?
• Local knowledge
• Go downhill
• Follow the person in front of you
• Follow the scent of food
• Global knowledge
• Consider all possible paths from start to goal
• Assign a cost to each leg of the path
• Find path to goal with smallest cost

Continuous
Discrete
3
Algorithms for path finding

• Uniformed
• You dont have any insight other than whats
  immediately before you
• Informed
• You have some insight, or heuristics that help
  you choose where to explore

4
Path finding data structures

• Commonly used data structures
• Start where the search must begin
• Goal where the search must terminate
• Fringe locations in the world that define the
  boundary of knowledge about the paths from Start
  to Goal

5
Uniformed search Uniform-Cost Search

• Always expand the lowest-path-cost node
• Dont evaluate the node until it is expanded
  not when it is the result of expansion

6
Uniform-Cost Search

• Fringe S0
• Expand(S) ? A1, B5, C15

7
Uniform-Cost Search

• Fringe A1, B5, C15
• Expand(A) G11

8
Uniform-Cost Search (review)

• Fringe B5, G11, C15
• Expand(B) ? G10

9
Uniform-Cost Search (review)

• Fringe G10, C15
• Expand(G) ? Goal

10
Best-first Search

• Uniform-cost search explored the node that was
  the closest
• With disregard for how far it was from the goal
• because we didnt know that information
• What if we knew something about the remaining
  distance to the goal?
• How could we take this into account?
• The best node on fringe to explore is one that
  balances cost to reach and cost to goal

11
Evaluation function f(n)

• Combine two costs
• f(n) g(n) h(n)
• g(n) cost to get from start to n
• h(n) cost to get to from n to goal

12
Heuristics

• A function, h(n), that estimates cost of cheapest
  path from node n to the goal
• h(n) 0 if n goal node

13
A (A-star) Search

• Dont simply minimize the cost to intermediate
  state on fringe minimize the cost from start to
  goal
• f(n) g(n) h(n)
• g(n) cost to get to n from start
• h(n) cost to get from n to goal
• Select node from fringe that minimizes f(n)

14
A is Optimal?

• A can be optimal if h(n) satisfies conditions
• h(n) never overestimates cost to reach the goal
• it is eternally optimisic
• f(n) never overestimates cost of a solution
  through n

15
A is Optimal

• We must prove that A will not return a
  suboptimal goal or a suboptimal path to a goal
• Let G be a suboptimal goal node
• f(G) g(G) h(G)
• h(G) 0 because G is a goal node
• f(G) g(G) gt C (because G is suboptimal)

16
A is Optimal

• We must prove that A will not return a
  suboptimal goal or a suboptimal path to a goal
• Let G be a suboptimal goal node
• f(G) g(G) h(G)
• h(G) 0 because G is a goal node
• f(G) g(G) gt C (because G is suboptimal)

• Let n be a node on the optimal path
• because h(n) does not overestimate
• f(n) g(n) h(n) lt C
• Therefore f(n) lt C lt f(G)
• node n will be selected before node G

17
Contours

• Because f(n) is nondecreasing
• we can draw contours
• If we know C
• We only need to explore contours less than C

18
Properties of A

• A expands all nodes with f(n) lt C
• A expands some (at least one) of the nodes on
  the C contour before finding the goal
• A expands no nodes with f(n) gt C
• these unexpanded nodes can be pruned

19
A is Optimally Efficient

• No other optimal algorithm is guaranteed to
  expand fewer nodes than A
• (except perhaps eliminating consideration of ties
  at f(n) C)
• Compared to other algorithms using same heuristic

20
Pros and Cons of A

• A is optimal and optimally efficient
• A is still slow and bulky (space kills first)
• Number of nodes grows exponentially with the
  length to goal
• This is actually a function of heuristic, but
  they all have errors
• A must search all nodes within this goal contour
• Finding suboptimal goals is sometimes only
  feasible solution

21
A for video games

• Is very common
• It will find a path from start to goal
• The world is spatially discretized
• Produces zigzag effect between cells
• Does not produce smooth turns
• Does not check for illegal moves or for abrupt
  turning when a unit has a particular turning
  radius

22
Removing the Zigzag Effect
Can just associate a cost with each turn, but
that does not solve the problem entirely. A
better smoothing process is to remove as many
waypoints as possible.
23
Removing the Zigzag Effect
checkPoint starting point of pathcurrentPoint
next point in pathwhile (currentPoint-gtnext !
NULL)   if Walkable(checkPoint,
currentPoint-gtnext)      // Make a straight path
between those points      temp
currentPoint      currentPoint
currentPoint-gtnext      delete temp from the
path   else      checkPoint
currentPoint      currentPoint
currentPoint-gtnext
24
Removing the Zigzag Effect

• Checks path from waypoint to waypoint. If the
  path traveled intersects a blocked location, the
  waypoint is not removed. Otherwise, remove the
  waypoint.
• Leaves impossible sections as is.

25
Adding Realistic Turns

• Want to be aware of a units turning radius and
  choose shortest route to destination, in this
  case the right turn will lead to the shortest
  route.

26
Adding Realistic Turns

• Calculate the length of the arc plus the line
  segment for both left and right turns. Shortest
  path is the one to use.

27
Adding Legal Turns

• Several Postprocess Solutions (cheats)
• Ignore blocked tiles obviously bad
• Path recalculations associate a byte with each
  tile, each bit denoting the surrounding tiles as
  legal or illegal may invalidate legal paths from
  other directions bad
• Decrease turning radius until it is legal bad
  for units with constant turning radius (e.g.
  truck), good for units such as people
• Backing up make a three point turn good for
  units such as trucks, weird for people

28
Adding Legal Turns

• Problems
• Methods shown previously are basically cheats,
  except for the second one (path recalculations),
  which can fail to find valid paths
• How can we make sure that paths taken are both
  legal and do not violate the turning radius?

29
Directional A
Sometimes the only legal path that does not
violate the turning radius constraints is
completely different from the path that the
standard A algorithm produces. To get around
this, Pinter proposes a modification known as
Directional A.
30
Directional A

• First things first
• Add an orientation associated with each node,
  where the orientation can be any one of the eight
  compass directions
• Each node now represented by three dimensions
  x, y, orientation

31
Directional A

• Check the path from a parent to a child node, not
  just whether or not the child is a blocked tile
• Consider the orientation at parent and child
  nodes as well as the turning radius and size of
  the unit to see if path hits any blocked tiles
• Result a valid path given the size and turning
  radius of the unit

32
Directional A
Directional A does not go directly from a to c
because it sees that an abrupt left turn would
cause the unit to hit blocked tiles. Thus,
waypoint b is found as the shortest legal path.
33
Directional A

• Problem adding an end orientation
• Four common paths

34
Directional A

• Can easily compute the four common paths and
  choose the shortest one
• Can add smoothing algorithm presented earlier,
  with the modification of sampling points along a
  valid curve between two points rather than just a
  straight line between two points

35
Expanding the Search

• So far, have been using a Directional-8 search
• Some paths will fail under this method
• Try searching more than one tile away
• Directional-24 search two tiles away, rarely
  fails to find a valid path
• Directional-48 search three tiles away, almost
  never fails to find a valid path

36
Modifying the Heuristic

• Standard A measures straight line distance from
  start to goal
• Modify so that it measures the shortest curve and
  line segment as calculated earlier, considering
  turn radius
• If units can go in reverse, consider adding a
  cost penalty to encourage units to move in
  reverse only when necessary

37
Directional A Limitations

• Very slow, constantly uses tables as a means to
  optimize it (directional table, heuristic table,
  hit-check table, etc)
• Directional-48 may still possibly (though rarely)
  fail
• The four common paths from origin to destination
  are not the only ones there is an infinite
  number of paths

38
More Optimizations

• Timeslice the search
• Only use Directional-24 and -48 if a
  Directional-8 fails
• Divide map into regions and pre-compute whether a
  tile in one region can reach a tile in another
• Use two matrices to calculate intermediate search
  results, one for slow, timesliced searches and
  the other for faster searches

39
Summary

• Directional A finds legal paths and realistic
  turns
• Directional A is slow
• Examples only take into account a 2D space
  partitioned into a grid, but can be extended to
  support various partitioning methods in 2D or 3D
• Suggest standard A for most units and
  Directional A for various large units

40
Realistic Human Walking Paths

• David C. Brogan
• Computer Science DepartmentUniversity of Virginia

Nicholas L. Johnson School of InformationUniversi
ty of Michigan
41
Motivation for walking characters

• Scientific and Entertainment Applications

Helbing et al. Escape Panic
Helbing et al. Trails
Phantom Menace
42
Realistic walking paths

• Dynamics matters!
• Walking dynamics influence what path is generated
• Can I cut the corner to dart down the hallway?
• Following a path influences walking dynamics
• I must slow down to cut the corner
• What a character can do influences what it does

43
Background

• Robotics, Animation, Practitioners
• Reactive Control
• Reynolds steering behaviors
• Planning
• Bandi and Thalmann A method
• Example-based
• Choi, Lee, and Shin motion capture library

44
What do realistic paths look like?

• An empirical question
• 36 people performed walking experiments
• convenience sample of passersby (unpaid)
• Video taped and rotoscoped for path acquisition

45
Five walking experiments

• Carry piece of paper from A to B
• Return to A andrepeat 3 times
• 7 paths generated

46
Five walking experiments

• No Turn
• 7 participants

47
Five walking experiments

• Wide Turn
• 7 participants

48
Five walking experiments

• Sharp Turn
• 9 participants

49
Five walking experiments

• U Turn
• 6 participants

50
Five walking experiments

• S Turn
• 7 participants

51
Empirical Observations

• Similar speed and acceleration profiles
• Presence of turning constraints

Limit on turning radius
Inertia
52
Pedestrian model outline

• Discrete event simulation of Sim
• Calculate a desired speed
• Proximity to goal and obstacles
• Compute desired heading
• Obey turning constraints and avoid obstacles
• Integrate dynamics
• Obey inertial constraints
• Heavily influenced by experiments

53
Calculate desired speed

• Nominal desired speed 1.36 m/s
• Linear acceleration from start during first 1.82
  m
• Linear deceleration to goal during final 1.63 m
• Desired speed near obstacle 1.10 m/s
• Linear speed modulation within 0.81 m

54
Compute desired heading
55
Compute desired heading

• Heading for each grid cell
• A function of turning constraints
• A function of walking dynamics
• A function of speed
• Shortest path to goal
• If speed were maintained

r
Compute headings for multiple turning constraints
56
Heading for each grid cell

• Flood fill from goal
• Goal heading and position is known

57
Heading for each grid cell

• Flood fill from goal
• Goal heading and position is known

58
Heading for each grid cell

• Recursively flood fill fromeach subgoal
• Heading is overwritten if betterpath is found

59
Heading for each grid cell

• Recursively flood fill fromeach subgoal

60
Heading for each grid cell

• Recursively flood fill fromeach subgoal

61
Heading for each grid cell

• Recursively flood fill fromeach subgoal

62
Heading for each grid cell

• Recursively flood fill fromeach subgoal

63
Heading for each grid cell

• After 2p / q iterations
• Subgoals trace a circle of radius, r

64
Heading for each grid cell

• Recursively flood fill fromeach subgoal

65
Heading for each grid cell

• Recursively flood fill fromeach subgoal

66
Heading for each grid cell

• Obstacles cause shadows

67
Heading for each grid cell

• Obstacles cause shadows
• Subgoal if
• Adjacent to obstacle
• and
• Adjacent to unlabeled grid cell (or grid cell
  pointing to inferior subgoal)

68
Heading for each grid cell

• Obstacles cause shadows

69
Heading for each grid cell

• Obstacles cause shadows

70
Heading for each grid cell

• Obstacles cause shadows

71
Heading for each grid cell

• Obstacles cause shadows

72
Heading for each grid cell

• Obstacles cause shadows

73
Heading for each grid cell

• Obstacles cause shadows
• Some grid cells unlabeled
• r parameter is too high
• Create heading charts with multiple r values

74
Walking dynamics

• Compute Sim turning radius, r
• According to speed and mass
• Look up heading from appropriate heading chart
• Reduce speed if no specified heading for grid
  cell
• Change speed according to schedules
• Influence of inertia
• Integrate Sim dynamics

75
Turning Radius

• To maintain constant centripetal force
• Faster Sim requires larger r
• Assume constant ratio between mass and force
• Larger people exert larger forces

76
Inertia

• Cannot instantly accomplish desired heading
• Weighted average of previous heading and desired
• cinertia and cturn tuned using experiments

77
Validation

• Two experimental conditions

78
Evaluation Metrics

• No perfect metric
• Distance error
• Momentary velocity error leads to sequence of
  distance errors
• Area error
• Ignores speed errors
• Speed error
• Ignores dissimilarity of paths

79
Results

• Compared to A path planning model
• Cell size 0.04 m
• 20 heading charts, turning radii 0 to 1.5 m

Model Distance error Area error Speed error
Pedestrian model .3133 .0204 .2185
A .4683 .0389 .2536
80
Results

• Compared to A path planning model

81
Conclusion

• A method for generating realistic walking paths
• Parameterized from observations
• Shortest path subject to
• Turning limitations
• Inertia

82
Future work

• Better understand evaluation metrics
• Distance is highly correlated to both area and
  speed errors
• Human path planning is complex
• Humans adjust velocity opportunistically during
  path
• Humans do not strive for shortest paths
• Computational techniques complement research from
  biomechanists, perceptual psychologists, and
  cognitive scientists

83
Our goal

• Given
• Initial position, speed, facing direction
• Final position, speed, facing direction
• Accurate model of world
• Find
• Path
• satisfying boundary constraints
• providing realistic walking paths

84
Building a walking path generator

• What can pedestrians do?
• Experimental evaluationof human performance

85
Building a walking path generator

• What paths suit capabilities?
• Search for paths to goal subject to realistic
  walking constraints

86
Building a walking path generator

• Evaluation
• Validate pedestrian modelwith additional
  experimentsand multiple metrics

87
Participant characteristics

• Questionnaire results
• Age 23 (SD 6)
• Height 1.8 m (SD 0.078 m)
• Weight 75 kg (SD 15 kg)
• Shoulder Width 0.25 m (SD 0.31 m)

88
Empirical Observations

• Similar speed profiles
• Mean maximum speed across all tests
• 1.36 m/s (N 285, SD 0.162 m/s)
• Mean speed at closest point to each obstacle
• 1.10 m/s (N 308, SD 0.143 m/s)

89
Empirical observations

• Similar acceleration profiles
• Deceleration begins 1.63 m from goal
• Acceleration stops at 1.82 m from start
• Measured using Hi-Ball tracker in straight-line
  walking

90
Empirical observations

• Turning constraints
• Limit on radius
  Inertia

91
Pedestrian model goal

• Given
• Final position, speed, heading
• Accurate model of world
• Find
• Path
• From any initial position
• Satisfying boundary constraints
• Providing realistic walking paths

92
Validation

• Two experimental conditions
• Hidden camera
• 30 participants
• Digitized paths