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

2. 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?

3. Possibilities Understanding of the motor system (arm, locomotor) accuracy of system means of correcting the system cognitive map, current location and orientation

4. Overview Overview of an algorithm useful for modeling actions (Kalman filter) Application to reaching Application to the more complex problem of navigation

5. Kalman Filter Basics Occurs in discrete time steps.

6. 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

7. Kalman Filter Basics H relates the state to the measurement z at step k. R is the measurement noise covariance.

8. Estimating the State of a Walker Define the state?

9. Estimating the State of a Walker Define the state: X = [position; velocity]

10. Estimating the State of a Walker Define the system model: System dynamics x t = Ax t-1 (ignoring control input) A = [1Δt 01] System noise Q = [00 00.5]

11. Estimating the State of a Walker Define the measurement model: Z k = H’x k + noise Sensory information from visual, proprioceptive and vestibular cues. H = [1000position measurement 0111]velocity measurement Measurement noise R = [1000 00.100 000.50 0001.5]vestibular cue is noisiest

12. Estimating the State of a Walker Run model for 20 steps PositionVelocity

13. Estimating the State of a Walker What happens when measurement noise increases? PositionVelocity

14. Estimating the State of a Walker What happens when measurement noise is small? PositionVelocity

15. 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

16. Application to Perception and Action Forward models- the motor system has a model of its dynamics Uses sensory feedback to correct errors

17. Forward Model of Reaching Wolpert, et. al. (1995)

18. Model Data Human Data

19. How do you walk a straight line while blindfolded? People can’t, but instead they veer. –No consistent directional bias Why?

20. 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)

21. 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?

22. Accumulation of Motor Noise Kallie, Schrater & Legge (2007)

23. Results Kallie, Schrater & Legge (2007)

24. 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.

25. 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)

26. Dead Reckoning Muller & Wehner (1988) Behavior seen in ants, honeybees, golden hamsters, funnel-web spider, and several species of geese.

27. 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).

28. Dead Reckoning in Ants Muller & Wehner (1988)

29. Dead Reckoning in Humans Angular error: 26 deg Distance error: 175 cm Angular error: 35 deg Distance error: 250 cm

30. Possible Solution: Landmarks Landmarks, once learned, can provide a “position fix,” thereby reducing positional uncertainty.

31. What is a Landmark?

32. Stankiewicz & Kalia (in press)

33. Error correction with Landmarks

34. Etienne, et. al. (2004)

35. Error correction with Landmarks

36. Error Correction with Landmarks in Humans Philbeck & O’Leary (2005)

37. Error Correction with Landmarks Philbeck & O’Leary (2005)

38. 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.