1. Computing Movement Geometry A step in Sensory-Motor Transformations Elizabeth Torres & David Zipser

2. Sensory Input Kinematics Motor output Postural Path Geometry Speed ?

3. Stuff that’s easy in the Geometric Stage Specifying movement paths. Dealing with excess degrees of freedom. Some constraint satisfaction. Some error correction.

4. target position Geometric Stage Input -- Output Reaching to Grasp with a Multi-jointed Arm target orientation arm posture Geometric Stage Arm postural Path Represented as changes in joint angles What goes on in here?

5. Gradient Descent

6. r = hand to target distance Hand to target distance

7. Joint Angles

8. Posture in 7D Joint Angle Space Hand position 3D Space Hand to target distance As function of joint angles

9. Gradient Descent for Simulating Hand Translation On each time step the change in joint angles is: Posture path Hand path

10. Reconfiguring Joint Angle Space Example:

11. Orientation Matching

12. Distance function for translation and rotation A constant chosen so that total distance = total rotation Co-articulation parameter

13. Experiments

14. Fitting G to Experimental Data Best Worst G symmetrical and positive-definite

15. fast normal slow TARGET Speed Invariance of Movement Path One subject, one movement at each speed

16. Six subjects, six movements each

17. Co-articulation

18. Error Correction

19. Correcting error with retinal image feedback Hand -Target Offset Using chain rule Retina(s)

20. Discussion Break

21. The gradient of a sum = the sum of the gradients Each of these terms can be computed in different brain areas

22. Hidden Units dq posturetarget positiontarget orientation

23. Preferred Directions Rotate across External Space