understanding perception and action using the kalman filter mathematical models of human behavior...
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Understanding Perception and Action Using the Kalman filter
Mathematical Models of Human Behavior
Amy Kalia
April 24, 2007
Learning in the Context of Action
• What do you need to know to accomplish an action?– Reaching for a glass– Walking in a straight line
• How about without vision?
– Finding your way to the nearest restroom?
Possibilities
• Understanding of the motor system (arm, locomotor)
• accuracy of system
• means of correcting the system
• cognitive map, current location and orientation
Overview
• Overview of an algorithm useful for modeling actions (Kalman filter)
• Application to reaching
• Application to the more complex problem of navigation
Kalman Filter Basics
Occurs in discrete time steps.
Kalman Filter Basics
X is the state at step k
A relates x at the previous time step to x at the current step.
B relates control input u to current state
Q is the process noise covariance
Kalman Filter Basics
H relates the state to the measurement z at step k.
R is the measurement noise covariance.
Estimating the State of a Walker
• Define the state?
Estimating the State of a Walker
• Define the state:X = [position; velocity]
Estimating the State of a Walker
• Define the system model:System dynamics
xt = Axt-1 (ignoring control input)
A = [1 Δt 0 1]
System noiseQ = [0 0
0 0.5]
Estimating the State of a Walker
• Define the measurement model:Zk = H’xk + noiseSensory information from visual, proprioceptive and
vestibular cues.H = [1 0 0 0 position measurement 0 1 1 1] velocity measurement
Measurement noiseR = [1 0 0 0
0 0.1 0 0 0 0 0.5 0
0 0 0 1.5] vestibular cue is noisiest
Estimating the State of a Walker
• Run model for 20 steps
Position Velocity
Estimating the State of a Walker
• What happens when measurement noise increases?
Position Velocity
Estimating the State of a Walker
• What happens when measurement noise is small?
Position Velocity
Summary of Kalman Filter Basics
• Model of state dynamics
• Correction of predicted state using measurement
• Weighted by Kalman gain, K
• Weighting depends on the noisiness of the state model vs. measurement
Application to Perception and Action
• Forward models- the motor system has a model of its dynamics
• Uses sensory feedback to correct errors
Forward Model of Reaching
Wolpert, et. al. (1995)
Wolpert, et. al. (1995)
Model Data
Human Data
How do you walk a straight line while blindfolded?
• People can’t, but instead they veer.– No consistent directional bias
• Why?
How do you walk a straight line while blindfolded?
• People can’t, but instead they veer.
• Why?– Proposed Explanations:
• Differences in leg length? (“Why Lost People Walk in Circles”, 1893)
• Biomechanical asymmetries (leg strength, dominance of one side over another)
How do you walk a straight line while blindfolded?
• Ability to walk a straight line depends on…– The ability to execute the motor commands
necessary– Sensory information about walking direction
• Vision, proprioception, vestibular cues
– Sounds familiar?
Accumulation of Motor Noise
Kallie, Schrater & Legge (2007)
Results
Kallie, Schrater & Legge (2007)
Accumulation of Motor Noise in Length Dimension
Also can explain the increase in variability in path length with distance when subjects are asked to look at a target and walk to it blindfolded.
Navigation Using Dead Reckoning
• Dead reckoning (path integration) is one type of navigation that requires knowledge of your actions => direction and distance traveled.
Gallistel (1990)
Dead Reckoning
Muller & Wehner (1988)
Behavior seen in ants, honeybees, golden hamsters, funnel-web spider, and several species of geese.
Ant Odometry: Estimating Distances
The ant’s odometer does not record the uphill-downhill distance, but rather the horizontal projection of the path (ground distance).
Dead Reckoning in Ants
Muller & Wehner (1988)
Dead Reckoning in Humans
Angular error: 26 deg
Distance error: 175 cm
Angular error: 35 deg
Distance error: 250 cm
Possible Solution: Landmarks
• Landmarks, once learned, can provide a “position fix,” thereby reducing positional uncertainty.
What is a Landmark?
What is a Landmark?
Stankiewicz & Kalia (in press)
Error correction with Landmarks
Error correction with Landmarks
Etienne, et. al. (2004)
Error correction with Landmarks
Error Correction with Landmarks in Humans
Philbeck & O’Leary (2005)
Error Correction with Landmarks
Philbeck & O’Leary (2005)
Conclusions
• Dynamic models (Kalman filter) provide a method for approaching problems in perception and action
• It is necessary to specify a model of the system dynamics, sensory information, and the noisiness of these processes.
• The Kalman filter helps explain several behaviors by describing the interaction of internal processes with external information.