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Peds and Paths: Small Group Behavior in Urban

Environments

Joseph K. Kearney

Hongling Wang

Terry 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 ds

dz

ds

dy

ds

dx

0.1)()()( 222 ds

dz

ds

dy

ds

dx

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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.