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.

target positionx y

z

Geometric Stage Input -- OutputReaching to Grasp with a Multi-jointed Arm

target orientation, ,α β γ

arm posture

Geometric Stage

Arm postural

Path

Represented as changes in

joint angles

What goes on in

here?

r ≥0 r = f q1 ,q2 ,K ,qn( )

q1 qn

r

Gradient Descent

−∇r q( ) =−

∂r∂q1

, ∂r∂q2

,K , ∂r∂qn

⎛

⎝⎜⎞

⎠⎟

−∇r q( )

q

x2

x1

r

r = hand to target distance

x3

x = x1,x2 ,x3( )

x t = x1t,x2

t ,x3t( )

r =

2xi−xi

t( )i=1

i=3

∑

Hand to target distance

x3

x2

q1

q2

q3

q4 q5

q6

q7

x1

Joint Angles

Posture in 7D Joint Angle Space

Hand position3D Space

f q( )

q f

1q( )

f

2q( )

f

3q( )

x1

x2

x3 x

r=2

xit−fi q( )( )

i=1

i=3∑

r =2

xit−xi( )

i=1

i=3∑

Hand to target distanceAs function of joint angles

Gradient Descent for Simulating Hand Translation

Δq=−η∇r xt ,q( )

∇r xt ,q( )

On each time step the change in joint angles is:

qinitial

qfinal

Posture path

x final

x initial

Hand path

Reconfiguring Joint Angle Space

G =2 1+cos q2( )( ) 1+cos q2( )

1+cos q2( ) 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Example:

x1

x1

x2

x2 r

r ′q =G⋅q

∇'r xt ,q( ) =G−1 ⋅∇r xt ,q( )

x1

Orientation Matching

x2

x3

H⎡⎣ ⎤⎦

H⎡⎣ ⎤⎦= f q( )

O -1⎡⎣ ⎤⎦=g vision( )

O -1⎡⎣ ⎤⎦

cos φ( ) =

12

Tr R⎡⎣ ⎤⎦−1

R⎡⎣ ⎤⎦= O−1⎡⎣ ⎤⎦ H⎡⎣ ⎤⎦

R⎡⎣ ⎤⎦

r = xi

t − fi q( )( )2+α kφ( )

2

i=1

i=3

∑

Distance function for translation and rotation

k A constant chosen so that total distance = total rotation

α Co-articulation parameter

Experiments

Fitting G to Experimental Data

Best Worst

G symmetrical and positive-definite

fastnormal

slow

TARGET

Speed Invariance of Movement Path

One subject, one movement at each speed

Six subjects, six movements each

r = xi

t − fi q( )( )2+α kφ( )

2

i=1

i=3

∑

Co-articulation

Error Correction

Correcting error with retinal image feedback

x

1−x1

t( ) , x2 −x2t( ) , x3 −x3

t( )( ) =12∇r x,xt( )

Hand -Target Offset

∇r xt ,q( )=∇r x,xt( ) J f q( )( ) J = J acobian( )

Using chain rule

J f q( )( ) =

∇ f1

q( )( )

∇ f2

q( )( )

∇ f3

q( )( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Retina(s)

Discussion Break

r 2 = xi

t − fi q( )( )2+α kφ q( )( )

2

i=1

i=3

∑

∇r2 xt ,q( ) =∇ xi

t − fi q( )( )2

i=1

i=3

∑⎛

⎝⎜⎞

⎠⎟+∇ α kφ q( )( )

2⎛⎝

⎞⎠

The gradient of a sum = the sum of the gradients

Each of these terms can be computed in different brain areas

Hidden Units

dq

 

posture target position target orientation

x y

z

, ,α β γ

Preferred Directions Rotate across External Space