Lecture 3 - Coordinate Systems Flashcards
define: coordinate system
a scheme for representing things as sets of numbers, or coordinates
How many coordinates do you need to represent something?
as many as that thing’s DOF
location is usually represented using ____ coordinates
Cartesian
Cartesian coordinates
vectors i, j, k
Using the Cartesian system, any location realtive to the origin can be expressed uniquely as a…
weighted sum of the vectors
v = v1i + v2j + v3k
What are the coordinates of vector v?
v = (v1, v2, v3)
Right-handed Cartesian coordinates
index finger = i
middle finger = j
thumb = k
How to calculate the sum of 2 vectors, w = v + u?
w1 = u1 + v1
w2 = u2 + v2
w3 = u3 + v3
how to calculate vector length?
|v| = (v12 + v22 + v32)1/2
Pythagoras’s formula
What does the dot product measure?
how close to paralle 2 vectors are
Formula for dot product
u • v = u1v1 + u2v2 + u3v3
u • v = |u| |v| cosø (angle between u and v)
Dot product: 2 vectors →
number
Formula for cross product
u x v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)
Cross product: 2 vectors →
vector
the cross product is non- ______
commutative (order matters)
u x v!= v x u
Length of cross product u x v
u x v | = |u| |v| |sinø| (angle between u and v)
What vector does the cross product give?
a vector that is orthogonal to both u and v
A coordinate system is often linked to…
aka
the coordinate system expreses locations ______
a reference frame (some object)
…
relative to its frame of reference
What reference frame is used by the brain to process visual memory?
eye-fixed frame
Cartesian coordinates
how far up?
how far sideways?
Polar coordinates
How far from center & what angle
saccades happen every _____ per second
2-3 times
What are some arguments against an eye-fixed frame for visual memory?
- head-fixed or torso-fixed frame would mean less processing time (they move less often than the eyes)
- Visual data can’t be combined with auditory data easily (b/c auditory is in head-fixed frame)
What are some reasons why the eye-fixed frame is used for visual memory?
Easier combination with incoming visual data
What reference frame does auditory data use?
head-fixed frame
what is the simple arm model called?
planar 2-link manipulator
according to the planar 2-link manipulator, how many DOF does the arm have? Details.
2 DOF
Both the sholder and elbow are hinge joints with 1 DOF each
according to the planar 2-link manipulator, what is the length of each limb?
upper & lower arms are the same length, L
What are the 2 coordinate systems we can use to describe arm position?
- joint angles
- hand location
Joint coordinates
if we know ø1 and ø2 (angle of shoulder and elbow respectively)
Hand coordinates
specifying arm position using x, y coordinates of the hand in a Cartesian frame centered on the shoulder
direct kinematics
Joint angle → hand coodinates
Inverse kinematics
hand position → joint angle
what are the 2 hypothesis for how the brain drives arm movement?
- minimum-jerk hypothesis
- joint interpolation
minimum-jerk hypothesis
arm movements minimize the jerk of the hand
What is bad about the minimum-jerk hypothesis?
Brain has to do inverse kinematic calculations at every hand position along the straight path to find the angle of the shoulder & elbow in order to move it properly (this is a lot of work!)
joint interpolation
brain drives arm along straight lines in joint coordinates
Benefit of joint interpolation
- Brain only uses inverse kinematics twice (initial & final position)
- brain plots straight line in joint coordinates between the intial & final positions (easy computations)
In reality, do we use minimum-jerk hypothesis or joint interpolation?
Both, it’s complicated
How did the joint interpolation theory get revised? What is it called?
Staggerd joint interpolation:
joint-space paths may curve, but no point reverses its motion within a movement (waste of effort)
How correct is the staggered joint inerpolation?
- Only true in reaching
- Other tasks like throwing, we reverse joint motion
- not a universal principle of motor control!
