Understanding Perception and Action Using the Kalman filter Mathematical Models of Human Behavior Amy Kalia April 24, 2007

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.


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


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?


• Define the state:X = [position; velocity]


• Define the system model:System dynamics

xt = Axt-1 (ignoring control input)

A = [1 Δt 0 1]

System noiseQ = [0 0

0 0.5]


• 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


• Run model for 20 steps

Position Velocity


• What happens when measurement noise increases?

Position Velocity


• 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?


• 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)


• 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


Etienne, et. al. (2004)


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.

Documents

Understanding Perception and Action Using the Kalman filter Mathematical Models of Human Behavior Amy Kalia April 24, 2007