Computing Movement Geometry
A step in Sensory-Motor
Transformations
Elizabeth Torres & David Zipser
Sensory Input
Kinematics
Motor output
Postural Path
Geometry
Speed ?
Stuff that’s easy in the
Geometric Stage
•Specifying movement paths.
•Dealing with excess degrees of
freedom.
•Some constraint satisfaction.
•Some error correction.
Geometric Stage Input -- Output
Reaching to Grasp with a Multi-jointed Arm
z
x
y
target position
 , , 
target orientation
arm posture
Geometric
Stage
What
goes on in
here?
Arm
postural
Path
Represented
as changes in
joint angles

Gradient Descent

r  f q1 ,q2 ,K ,qn
r 0
 r r
r 
r q   
,
,K ,

qn 
 q1 q2

r
q
qn
q1
r q

r = hand to target distance

x  x ,x ,x
t
x3
r
x1
t
2
t
3

x  x1 , x2 , x3 
r
x2
t
1


i1
i3
t
xi xi

2
Hand to target distance
Joint Angles
q1
q3
x3
q7
q
q4 5
q2
q6
x2
x1
f q 
Posture in 7D Joint
Angle Space
Hand position
3D Space

f q 
f q 
f1 q
q
2
3
r
t x
x


i
i
i1
i3
2
x1
x2
x3
x
r


 xit  f i q 
i3
i1
2
Hand to target distance
As function of joint angles
Gradient Descent for Simulating Hand Translation
On each time step the change in joint angles is:


q  
r x ,q 
r x t ,q
t
q
initial
q
final
Posture path
x final
x
initial
Hand path
Reconfiguring Joint Angle Space
q  G  q
x2
r
 
 
' r x t ,q  G 1 r x t ,q
x1
Example:

  1 cos q2 

1
 

 2 1 cos q
2
G  
 1 cos q2
r
x2
x1
Orientation Matching
O -1 
 
 H 
 R 
x3

  g vision 

 H   f q
O -1

 R   O 1   H 
x2
x1

1
cos   Tr  R   1
2
Distance function for translation and rotation
r
 x
i3
t
i
    k
 fi q
2
i1
k

A constant chosen so that
total distance = total rotation
Co-articulation parameter
2
Experiments
Fitting G to Experimental Data
Best
Worst
G symmetrical and positive-definite
Speed Invariance of Movement Path
TARGET
fast
normal
slow
One subject, one movement at each speed
Six subjects,
six movements each
Co-articulation
r
 x
i3
t
i
i1
    k
 fi q
2
2
Error Correction
Correcting error with retinal image feedback



x1  x1t , x2  x2t , x3  x3t

 
1
 r x, x t
2
Hand -Target Offset
Using chain rule
Retina(s)



    J  Jacobian
r x t ,q  r x, x t J f q
  f

J  f q     f

   f
1
2
3
q 
q 
q 
Discussion Break
The gradient of a sum = the sum of the gradients
i3

    k q 
r   x  fi q
2
t
i
2
2
i1
i3

2
t
t
r x ,q     xi  fi q


 
i1
 
2





k
q


Each of these terms can be computed
in different brain areas
  
2


dq
Hidden Units
z
x
posture
y
target position
 , , 
target orientation
Preferred Directions Rotate across External Space