peds and paths: small group behavior in urban environments
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Peds and Paths: Small Group Behavior in Urban
Environments
Joseph K. KearneyHongling WangTerry Hostetler
Kendall Atkinson
The University of Iowa
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Pedestrian Activity in Urban Environments
• Couples walking down a sidewalk• Families window shopping• Commuters queuing at a bus stop• Friends stopping to chat
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Related Research
• Social psychology (McPhail)• Flocking (Reynolds, Tu & Terzopoulos, Brogan and
Hodgins)• Vehicle and crowd simulation (Musse &
Thalmann, Thomas & Donikian, Sukthankar)
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Public Gatherings
• Mix of singles and small groups of companions
• Majority of people are in clusters of two to five
• Frequency of occurrence of a cluster is inversely proportional to size
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What is a Group?
• Proximity• Coupled Behavior• Common Purpose• Relationship Between
Members
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Moving Formations
• Pairs: Side by side• Triples: Triangular shape
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Stationary Formations
ArcFixed Center of Focus
Conversation Circle
Group Center is Focus
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Modeling Walkways and Roads as Ribbons in Space
walkwayaxis
Object
offset
distance
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Curvilinear Coordinate System
• Defines geometry of navigable surfaces• Give a local orientation to the path• Channels traffic into parallel streams• Frame of reference for spatial relations
– Obstacle avoidance– Navigation
walkwayaxis
Object
offset
distance
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Arc-Length Parameterization
• Parametric spline curves for ribbon axis– Flexible– Differentiable
• Must relate parameter to arc length • Current approaches impractical for
real-time applications
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Traditional approach of arc-length parameterization for parametric curves
• Compute arc length s as a function of parameter t s=A(t)
• Compute the inverse of the arc-length function
• Replace parameter t in Q(t)=(x(t),y(t),z(t)) with )(1 sAt
)))(()),(()),((()( 111 sAzsAysAxsP
)(1 sA
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Problems with traditional approach
• Generally integral for A(t) does not integrate• Function is not elementary function• Solutions by numeric methods impractical for
real-time applications
)(1 sAt
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Related work• Numerical methods for mappings between
parameter and arc length, e.g., [Guenter 90]– Impractical in real-time applications
• Build 2 Bezier curves for mappings between arc length and parameter, one for each direction, e.g., [Walter 96]– Error uncontrolled– Possible inconsistency between the 2 mapping
directions– No guarantee of monotonicity
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Approximately arc-length parameterized cubic spline curve
(1) Compute curve length
(2) Find m+1 equally spaced points on input curve
(3) Interpolate (x,y,z) to arc length s to get a new cubic spline curve
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Compute Curve length
• Compute arc length of each cubic spline piece with Simpson’s rule– Adaptive methods can be used to control the
accuracy of arc length computation• Lengths of all spline pieces are summed• Build a table for mappings between
parameter and arc length on knot points
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Find m+1equally spaced points
• Problem– Mappings from equally spaced arc-length values to parameter values
• Solution:– Table search to map an arc length value to a
parameter interval– Bisection method to map the arc length value to a
parameter value within the parameter interval
mLmmLmL /*...,,/*2,/*1,0
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Compute an approximate arc-length parameterized spline curve
• m+1 points as knot points • Using cubic spline interpolation
– End point derivative conditions• Direction consistent with input curve• Magnitude of 1.0
– Not-a-knot conditions
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Errors
• Match error– Misfit of the derived curve from an input curve
• Arc-length parameterization error – Deviation of the derived curve from arc-length
parameterization
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Errors analysis• Match error
– Match error is difference between the two curves at corresponding points, |Q(t)-P(s)|
• Arc-length parameterization error– For an arc-length parameterized curve,
– Arc-length parameterization error measured by
0.1)()()( 222 dsdz
dsdy
dsdx
0.1)()()( 222 dsdz
dsdy
dsdx
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Experimental results
(1) Experimental curve (2) Curvature of the curve
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Experimental results (cont.)
(1) m=5 (2) m=10
Experimental curve(blue) and the derived curve (red)
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Experimental results (cont.)
(1) m = 5 (2) m = 10
Match error in the derived curve
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Experimental results (cont.)
(1) m=5 (2) m=10
Arc-length parameterization error in the derived curve
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Error factors in experimental results
• Both errors increase with curvature• Both errors decrease with m
– Maximal match error decreases 10 times when m doubled
– Maximal arc-length parameterization error decreases 5 times when m doubled
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Strengths of this technique
• Run-time efficiency is high– No mapping between parameter and arc-length needed– No table search needed for mapping from curvilinear
coordinates to Cartesian coordinates– Mapping form Cartesian coordinates to curvilinear
coordinates is efficient (introduced in another paper)• Time-consuming computations can be put either in
initialization period or off-line
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Strengths of this technique (cont.)
• Higher accuracy can be achieved– By computing length of the input curve more
accurately– By locating equal-spaced points more accurately– By increasing m
• Burden of higher accuracy is only more memory– Doubling m requires doubling the memory for spline
curve coefficients
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Walking Behavior
• Influenced by constraints on movement• Control Parameters
– Speed• Accelerate, Coast, or Decelerate
– Orientation• Turn Left, No Turn, or Turn Right
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Action Space
Accelerate Accelerate Accelerate Turn Left No Turn Turn Right
Coast Coast Coast Turn Left No Turn Turn Right
Decelerate Decelerate Decelerate Turn Left No Turn Turn Right
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Distributed Preference Voting
• Delegation of voters: Constraint Proxies• A proxy votes on all cells of the action space• Votes are tallied• Winning cell represents best compromise
among competing interests
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Vote Tabulation
1.0
Pursuit Point
Tracking
Maintain Formation
Inertia
Centering
Maintain Target
Velocity
Avoid Peds
Winning Cell
Electioneer
1.01.0
2.0
2.0
4.0
5.0
Avoid Obstacles
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Pursuit Point
• Located a small distance ahead of pedestrians on their target path
• Shared by all members of a group
walkwayaxis
pursuit point
ped
target path
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Pursuit Point Tracking
• Pursuit Direction – vector from group’s center to the Pursuit Point
• This proxy votes to align a walker’s orientation with the group’s Pursuit Direction
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One Pedestrian Following a Path
walkwayaxis
pursuit point
ped 1
pursuit direction
offset
distance
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Two Pedestrians Following a Path
walkwayaxis
pursuit point
ped 1
pursuit direction
ped 2
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Vote to Turn Right
Turn No Turn Left Turn Right
Accelerate
Coast
Decelerate
-1.0 -1.0 1.0
-1.0 -1.0 1.0
-1.0 -1.0 1.0
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Maintain Formation
• Group Slip – maximum distance a pedestrian is allowed to move
in front of or behind the rest of the group• If group slip is violated, this proxy votes to
accelerate or decelerate to catch up with the group
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Group Slip
groupslip
walkway axis
pursuit point
Two pedestrians in formation
groupslip
pursuit point
Three pedestrians in formation
groupslip
pursuit point
Two pedestrians not in formation
walkway axis
walkway axis
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A Group of Two Following a Path
ped 1
walkway axis
pursuit point
Winning vote = Accelerate/Turn Right
Election for ped 1
ped 2
-1.0 -1.0 +1.0-1.0 -1.0 +1.0-1.0 -1.0 +1.0
Pursuit Point Tracking
+1.0 +1.0 +1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0
Maintain Formation
+1.0 +1.0 +3.0 -3.0 -3.0 -1.0 -3.0 -3.0 -3.0
2.01.0
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Avoiding Pedestrians
• Activated when a companion intrudes
• Repulsion can lead to undesirable equilibria of forces
• By adding a small orthogonal force we rotate out of local minima
walkwayaxis
ped 2ped 1ped 3
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Vote to Avoid a Companion
-.67 -.67 -.67
0 0 0
.67 .67 .67
-.33 0 .33
-.33 0 .33
-.33 0 .33
-1 -.67 -.33
-.33 0 .33
.33 .67 1
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Scenarios
• Following a circular path• Avoiding an obstacle• Passing through a constriction
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Following a Circular Path
• Target path is formed by the series of pursuit points
• Parameters– turn angle increment– look-ahead distance– path curvature
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Following a Circular Path -- Trajectory
target path
ped 1
walkway axis walkway axis
ped 1
target path
Large look-ahead distance Small look-ahead distance
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Avoiding an Obstacle
• Avoid Obstacle proxy steers pedestrian to an obstacle’s nearest side
• Pursuit point’s offset is shifted around large obstacles
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Avoiding an Obstacle -- Trajectory
Small look-ahead distance Large look-ahead distance
ped 1
ped 2
walkway axis walkway axis
ped 1
ped 2
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Passing Through a Constriction
• Groups – compress at the entrance– move nearly single file down the corridor– reform as a group as they emerge
• State change: suspending Maintain Formation proxy produces smoother motion
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Passing Through a Constriction -- Trajectory
Maintain Formation proxy voting
Maintain Formation proxy not voting
walkway axis walkway axis
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Interaction Between Pairs -- 1
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Interaction Between Pairs -- 2
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Interaction Between Pairs -- 3
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Work in Progress
• Interactions among groups • Stationary formations
– New action space for fine movement– State machine manages transition
• Aggregation and disaggregation
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Conclusions
• Small pedestrian groups can be simulated that– maintain formation while walking– negotiate obstacles together – pass through constrictions
• Distributed preference voting is a promising method for finding good compromise solutions
• State changes can help resolve conflicts between behaviors
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Acknowledgements
• This work is supported in part through National Science Foundation grants INT-9724746, EAI-0130864, and IIS-0002535.
• Jim Cremer and Pete Willemsen made significant contributions to the development of the Hank simulator.
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