Lecture 2 - Degrees of Freedom Flashcards
Define: DOF
variable has N degrees of freedom of we need N numbers to distinguish all its possible values
DOF is a measure of ____
a system’s complexity
another word for DOF =
N-dimensional
define: point particle
object with a location but no magnitude (size is negligible)
Point particle has ___ DOF
3
define: constraint
condition which a system satisfied (DOF may be reduced by satisfying this condition)
What is the highest DOF a rigid system can have?
6
define: rigid body
A set of 3 or more non-collinear points, where the distances between all points are fixed
list 3 biological examples of rigid bodies (under normal circumstances)
- Eyes
- Skull
- Bones
The position of any rigid body can be described with ___ numbers.
- How many used to describe ___?
- How many used to describe ___?
- 6
- 3 used to describe location
- 3 used to describe orientation
Position of a rigid body has ___ components. What are they?
- 2
- location & orientation
Location has ___ DOF. Orientation has ___ DOF.
3, 3
Define: linkage
chain of rigid bodies
Linkages are made up of… (define each)
- Links: rigid bodies making up the linkages
- Joints: connections between the links
Human skeleton has ___ kind of joints. What kind of joints are they?
- 6
- Rotary
What are the 6 types of rotary joints in the body? List their DOF.
- Plane joints (0)
- Hinge joints (1)
- Pivot joints (1)
- Saddle joints (2)
- Ellipsoidal joints (2)
- Ball and socket joints (3 DOF)
Plane joints
- Have very little motion
- Adjacent links meet along flat surfaces
Examples of plane joints
- palm of hand
- palm of foot
Hinge joints
- 1 DOF
- swing about single axis relative to one another
example of hinge joints
- humero-ulnar joint of elbow
- 2 distal joints in each finger
Pivot joints
- 1 DOF
- link rotates about its own long axis
example of pivot joint
- radio-ulnar joint of the elbow
what happens at the radio-ulnar joint?
radius turns in a tissue ring attached to the ulna
where is the radius?
forearm bone on thumb side
Saddle joints
- Allow for extension/flexion & adduction/abduction
- Concave & convex directions match up
- 2 DOF
Example of saddle joint
joint between wrist and thumb (on the bottom, closer to wrist)
Ellipsoidal joints
- 2 DOF
- Articular surfaces are ellipsoids (different from ball and socket because not spherical)
- 1 concaves in, one concaves out
Example of ellipsoidal joints
- wrist
- metacarpophalangeal joints (base of each finger)
Ball and socket joints
- 3 DOF
- up-down movement = pitch
- side-side movement = yaw
- turning movement = roll
Examples of ball and socket joints
- shoulder
- hip
- eyeball in socket
How to calculate DOF of a linkage?
Σ DOF of it’s joints
When is a system redundant?
when it has more DOF than it needs for its job
What is the purpose of redundancy?
lets you do things in more than 1 way
- obstacle avoidance
- keep joints near center of range
- provide alternative ways to do things
How many DOF is is vision (gaze)?
2 (we see things in a sphere kind of way)
How many DOF does VOR need?
3
Why is the macula only 5 degrees across?
- If it covered the entire retina, visual input increases by 100x
- visual imput is 1/5 of all sensory input
- 20x increase in sensory input
- overwhelm brain
- optic nerve needs to be huge to send info to brain
- huge blind spot
How do we compensate for small macula?
move our eyes very quickly
Define: saccades
- rapid eye movement that shift the gaze point
- direct fovea at objects
- quickest eye movements
weakness of saccades
we can’t see well during saccades
macula is ___ degrees across? Fovea is ___ degrees across?
5, 2
define: fovea
small, central, high-acuity region of the retina
Speed of saccades? Duration?
500 degrees/second; 20 ms
What kind of axis can be picked for determining eye movement between two points?
any axis that lines in the plane bisecting the angle between the initial and final gaze directions
Size of eye rotation depends on…
axis picked for eye movement
Which axis gives the smallest eye rotation?
axis that is orthogonal (right angles) to both initial and final gaze directions
Shortest path strategy
- Make smallest saccade that brings gaze to target
- Orthogonal axis
- eyeball rotation small as possible
Rationale for shortest-path strategy
save energy & time
What law do we follow to pick axis for gaze direction?
Donders’ Law
Donders’ Law
for any 1 gaze direction, the brain always chooses the same eye position
Rationale for Donders’ law
for any 1 gaze direction there is presumably 1 eye position that is in some sense best, and we should therefore only use that position
Can we use both Donders’ law and the shortest-path strategy?
No, imcompatible
When does Donders’ law hold? When does it not hold?
Holds when:
- fixations
- saccades with head fixed (not moving)
Not holds when:
- head moves
- sleep
Besides eye movement, what else obeys Donders’ law?
Head & arm control (hand adopts same 3D orientation for any 1 direction when straight arm pointing)
Difference between Donders’ law in gaze control and other areas?
In the arms & head, Donders’ law can be broken voluntarily
Define: synergy
laws that steers a system so that its behaviour is lower dimentsional than the max its plant would permit (e.g. Donders’ law)
Donders’ law ____ redundant dimension. What does it do exactly?
- eliminates
- 3D -> 2D
Why do we want synergies?
it simplifies things, so there is less for the brain to worry about
Donders’ law does not specify…
which positions are used, just that the SAME position is used
Which law specifies which position is used when looking at a particular direction (sub of Donders’ Law)
Listing’s law
Listing’s Law
- there is a special eye position called primary position
- there is head-fixed plane called Listing’s plane
- plane is orthoganol to gaze direction in primary position
- eye will only rotate in positions that can be reacked from primary position by some axis on the Listing’s plane
What DOESN’T Listing’s law say?
It doesn’t say that you ALWAYS move tho these positions by a single fixed-axis rotation from primary, but you COULD
When is Listing’s Law broken?
- head moves
- sleep
When does Listing’s law hold?
- fixation
- saccades with head fixed
What area of the brain causes saccades that obey Listing’s law?
stimulating the superior colliculus (at the midbrain)
What area of the brain causes saccades that doesn’t obey Listing’s law?
stimulating short-lead burst neurons in the brainstem
Where isthe control system for Listing’s law?
between the midbrain (superior colliculus) and brainstem (short-lead burst neurons)
What is 1 example where you obey Donders’ law but not Listing’s?
Fick Pattern:
- pan horizontally by pivoting arbout vertical axis
- pan veritcally about horizontal axis
motions intermediate between Fick and Listings are seen in…
- straight arm pointing
- head motion during spontaneous eye-head gaze shifts
Purpose of Listing’s law
- simplify visual processing (movement vs. still) (orientation of image on retina will always be the same because image will fall on same set of retinal receptors)
- motor efficiency (saccades move along shortest path from primary position)
Proof that Listing’s law is important
animals use it (e.g. chameleons)
