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
