Understanding Perception and Action Using the Kalman filter Mathematical Models of Human Behavior...
-
date post
21-Dec-2015 -
Category
Documents
-
view
219 -
download
0
Transcript of Understanding Perception and Action Using the Kalman filter Mathematical Models of Human Behavior...
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.