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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.