Behavioral Dynamics of Locomotor Path...

William H. WarrenDept. of Cognitive & Linguistic SciencesBrown University

Behavioral Dynamics ofLocomotor Path Formation

Perception Action

Laboratory

With: Brett Fajen, Justin Owens, Jon Cohen, Hugo Bruggeman,Philip Fink, Mike Cinelli, Martin Gérin-Lajoie

Thanks to: NIH, NSF

Motivation: The organization of behavior

“Control lies not in the brain, but in the animal-environmentsystem. Behavior is regular without being regulated. Thequestion is how this can be.”

-- J. Gibson (1979)

• Organized behavior is not prescribed by the brain, butemerges as a stable solution of the system’s dynamics» Exploit physical and informational constraints

Environment Agent

action

information

˙ a =! a, i( )˙ e = ! e, F( )

i = ! e( )

F = ! a( )

Dynamics of perception & action

Perception & Action

x.

x

˙ x = f (x)

Behavioral Dynamics

• Behavior corresponds to solutions of the behavioral dynamics» Goal states = attractors» Avoided states = repellors» Transitions = bifurcations

The problem

How guide locomotion in acomplex, dynamic environment?• World model+path planning• Emergent behavior

goalx

obstacles

• Basic locomotor behaviors1. Steer to goal2. Avoid stationary obstacle3. Intercept moving target4. Avoid moving obstacle5. Follow neighbor?…

The VENLab (12 x 12 m)

SGI Onyx 2 IR

microphonesinertia cubeInterSense 900 Tracker:

sonic beacons

Kaiser HMD(60˚ x 40˚, 60 Hz,50 ms latency)

Optic flow (Gibson, 1950)

• Perception of heading (~1˚)• Is optic flow actually used to control locomotion?

FOE

Steering control (Warren, Kay, Zosh, Duchon, & Sahuc, 2001)

• Optic Flow strategy (Gibson, 1950)

• Egocentric Direction strategy (Rushton, et al, 1998)

• Test: Displace optic flow from the actual direction of walking (10˚)»Optic Flow strategy predicts straight path» Egocentric Direction strategy predicts curved path

Virtual headingWalking direction

Vary Amount of Flow +++ Direction strategy_ _ _ Flow strategy Data

z (c

m)

x (cm)

headingerror = 9˚

z (c

m)

x (cm)

headingerror = 6˚

x (cm)

z (c

m)

headingerror = 4˚

p < .001x (cm)z

(cm

)

headingerror = 2˚

Visual-locomotor adaptation(Bruggeman, Zosh, & Warren, 2007)

• Given a lone target, how know what direction to walk?»Mapping from visual direction to locomotor direction

• Hypothesis: Optic flow serves as a “teaching signal” torecalibrate the mapping (Held & Freedman, 1963)

» Adapt: optic flow displaced 10˚ to right (38 trials)» Test: normal optic flow (10 trials)

• Adaptation is twice as great and 6 times faster with optic flow• Negative a:ereffect is twice as great

Test

relativea:ereffect =65%

relativea:ereffect =33%

p < .0001

relativeadaptation =52%

relativeadaptation =28%

Adaptation

Conclusion

• Both strategies contribute to on-line steering control»Low flow-->Egocentric direction strategy dominates»Rich flow-->Flow strategy dominates

• Optic flow drives visual-locomotor adaptation»Recalibrates the mapping from visual direction to walking direction»Basis for the egocentric direction strategy

• Robust control under varying conditions

Behavioral dynamics of steering (w/ Brett Fajen)

• Steer to goal: φ−ψg=0

goalψg

• Avoid obstacle: φ−ψo>0

heading

refframe

φ v

φ−ψg

headingerror

Dynamics(Schöner, Dose & Engels, 1995)

!

˙ " = #kg " #$ g( )

φ

φ.

ψg

goal =attractor

of heading

φψo

φ.

obstacle =repellor of

heading

1 Stationary goal (Fajen & Warren, JEP:HPP, 2003)

• Walk 1 m, goal appears• Vary initial direction and distance of goal

0 1 20

1

2

3

4 5˚ 25˚

4 m

z (m

)

x (m)

!

˙ ̇ " = #b ˙ " # kg (" #$g )(e#c1dg + c

2)

Goal Model

Distance termstiffness decreases

with distance (TTC)

“Stiffness”angular accelerationincreases with angle

heading

goal

kb

d

v

“Damping”resistance to

turning

Least squares fits:b=3.25 dampingkg=7.5 stiffnessc1=0.4 decay in dc2=0.4 asymptote(constant mean speed)

• Null target-heading error (φ−ψg)• Goal direction as attractor

Mean Pathsz

(m)

0 1 2 30

2

4

6

8

x (m)

2 m

4 m

8 m

Distance

20˚

0 1 2 30

2

4

6

2 m

4 m

8 m20˚

8

x (m)

z (m

)

Distance

0 1 20

1

2

3

4

Direction

5˚ 25˚

4 m

z (m

)

Human data Model

z (m

)

0 1 20

1

2

3

45˚ 25˚

4 m Direction

• Goal behaves as an attractor of heading

0 1 2 3 4-20

0

20

0 1 2 3 4-20

0

20

0 1 2 3 4-20

0

20

2 m

4 m

8 m

Head

ing

erro

r (φ−ψ

g)

t(s) t(s)

0 1 2 3 4-20

0

20

0 1 2 3 4-20

0

20

0 1 2 3 4-20

0

20

2 m

4 m

8 mHead

ing

erro

r (φ−ψ

g) R2=.99

R2=.99

R2=.96

Overall R2=.98

Mean Heading Error

Human data Model

Fly steering control (Reichardt & Poggio, 1976)

• Steer toward a target• “Stiffness” term linear over ±30˚• Similar potential function

Heading error (deg)

Torq

uePo

tent

ial

!

˙ ̇ " = #b ˙ " + ko(" #$

o)e

#c3 |"#$o|(e

#c4do )

2 Stationary Obstacle (F&W, 2003)

Distance (TTC)+ “Stiffness”

heading

obstacle

kb

• Increase heading error φ−ψo• Obstacle direction as repeller

c3=6.5 decay in φko=198.0 stiffness

c4=0.8 decay in d

Head

ing

erro

r (φ−ψ

o)

0

50

10

20

30

40

0 1 2 3 4

t(s)

2˚ 3 m 4 m 5 m

0

50

10

20

30

40

0 1 2 3 4

t(s)

2˚ 3 m 4 m 5 m

Overall R2=.98• Obstacle behaves as a repellor of heading

-0.4 -0.2 0 0.2 0.4 0.60

2

4

6

8

10

5 m 4 m 3 m

x (m)

z (m

)

4°

x (m)

z (m

)4°

2

4

6

8

5 m 4 m 3 m

0

10

0 0.2 0.4 0.6

Human data Model

Exp: Route selection with 1 obstacle

• Switch from outside to inside path?• Walk 1 m, goal and obstacle appear

» Vary goal-obstacle angle, goal distance

00

2

4

6

8

z (m

)x (m)

1 2

Model out

out

in

ψg-ψo

dg

goal(15˚)

obstacle

outin

• At goal-obstacle angle between 2˚- 4˚ (p<.001)• Model switches between 1˚- 4˚ with c4=1.6

» Obstacle repulsion decays faster with distance» Previously tested only outside paths

Human Paths

x (m)

56.1%

71.1%

89.8%

4˚

0 1 20

2

4

6

8

x (m)

z (m

)

0 1 2

2˚

0

2

4

6

8 77.8%65.2%

54.8%

• Switch as goal gets closer (p<.001)

ModelDynamics

• Attractors evolveas agent moves

(a) (b)

(c) (d)

d

b c

Goal

Obstacle

abistable

out

in

in

φ φ

in

φ.

φ.

• Bistability• Tangent

bifurcation» 1 --> 2 attractors

• Route “choice”

Vector fieldsfor angularacceleration

Exp: Route selection with 2 obstacles

Model

0

2

4

6

8

-2 -1 0 1 2

smallopening

Right

-2 -1 0 1 2

mediumopening

Left

-2 -1 0 1 2

largeopening Center

goal

ψg-ψo

obstacles

Human Paths

• Switch Right → Le: → Center as opening increases.

0°

0

2

4

6

8

37% 0% 63%

R

4°2% 29%69%

L

-2 -1 0 1 2

10°

34% 65% 1%

Center

-2 -1 0 1 2

8°

46% 6%48%

switch

2°50% 0% 50%

switch

-2 -1 0 1 2

6°

0

2

4

6

8

19% 16%65%

L

1. Pursuit strategy • Null β • Yields curved “chasing” path

2. Constant target-heading angle • Null β-dot • 2 solutions: lead and lag

3. Constant bearing strategy • Null ψ-dot • Yields straight interception path

3 Moving Target (Fajen & Warren, 2004, 2007)

Interception

β

β

The open-field tackle

Lowell Red Arrows vs. Hastings High, 1998

• People don’t head directly toward the target (heading error≠0)• Pursuit model is inconsistent with data

Pursuit model (null β)

Meanpaths

Meanheadingerror

Constant Target-Heading Angle model (null β).

• People never exhibit lag solution• Constant Target-Heading angle model is inconsistent with data

Meanpaths

Meanheadingerror

!

˙ ̇ " = #b ˙ " # kt(# ˙ $ )(d

t+ c5)

ψ-dot

b=7.75 dampingkt=0.06 stiffnessc5=1 distance

• Latency to detect target motion=0.5 s

• Null ψ-dot• Interception path as attractor

Constant Bearing model (null ψ).

• People turn onto straight interception paths• Constant Bearing model reproduces human paths

Meanpaths

• People lead the target (heading error plateaus >> 0)• Constant Bearing model reproduces human time series

Overall R2=.87RMSE=2.15˚

Meantime seriesof headingerror

Dragonfly interception (Olberg, Worthington, & Venator, 2000)

• Constant bearing strategy» Elevation angle (ψ) constant» Target-heading angle (β) changes

horizonψ

ψ

ψ

ψ

β

• People don’t anticipate target trajectory• Paths consistent with Constant Bearing model

Test: Circular target trajectory (w/ Justin Owens)

Radius2 m

1.5 m

1 m

R2=.96on heading

!

˙ ̇ " = #b ˙ " + km(# ˙ $ )e

# c6 | ˙ $ |(e

# c7dm )

• Avoid constant bearing» Avoid nulling ψ-dot

• Interception path as repeller

km= 176 stiffnessc6= 6.5 decay in φc7= 0.008 decay in d• input mean initial conditions

and mean speed profile

+ “Stiffness”

4 Moving Obstacle (w/ Jon Cohen)

• Model captures dominant path• Route switching at same obstacle speed

Exp: Vary obstacle speed (w/ Jon Cohen)

R2=.98RMSE=3.6˚on obstacle error

Obs

tacle

Dire

ctio

n

Obstacle Speed

Model Data

5 Linear combinations

• Model scales linearly withthe complexity of the scene» Resultant of all spring forces

• Can we predict morecomplex behavior withlinear combinations ofnonlinear components?» No free parameters • Simulated football

• Model reproduces dominant path• Route switching at ~same target-obstacle angle

Exp: Moving target + Stationary obstacle(w/ Hugo Bruggeman)

.6 m/s

R2=.88 (.18)

• Model captures dominant path• Route switching at ~same obstacle and target speeds

Exp: Moving target + Moving obstacle(w/ Jon Cohen & Hugo Bruggeman)

R2=.85 (.79)

*****

*****

But: 2 Moving obstacles (w/ Hugo Bruggeman & Jon Cohen)

• People behave inconsistently with 2 moving obstacles• Not completely accounted for by initial conditions and speed• Attentional effects?

Data

S4

S10

Stationary Target

S4

S3

Moving Target

Claims

1. Steering dynamics» Agent tracks locally specified attractor as dynamics evolve» Path emerges on-line» World model or explicit path planning unnecessary

• Repulsion function assymptotes to zero ~3 m

2. Organization of behavior» Behavior is not centrally controlled, but emerges as a stable

solution of the system’s dynamics» In this sense, behavior is regular without being regulated

Ultimately…

• Extended barriers• Pursuit-evasion games• Interact with model-driven agents in VR• Simulate crowd behavior

» Human “flocking”» Grand Central Station, burning nightclub

Reynolds(1987)

Noise simulations

• Add 10% Gaussian noise to initial parameters & perceptual variables• Model is stable• Reproduces distribution of human paths

Noise simulations

• Add 10% Gaussian noise to parameters & perceptual variables (initial)• Model is stable• Reproduces distribution of human paths

Where do parameter values come from?

• Why walk on particular paths?» Physical constraints of an inertial body» Requires centripetal force to change direction

• Variational principle?» Total impulse and metabolic cost increase with:» path curvature» path length

• Hyp: Calibrate parameters to reduce total cost of path

at

an

!

an

= v2/r

Fn

= man

In

= Fnt