Lecture 3 - Coordinate Systems

Question 1

define: coordinate system

Answer 1

a scheme for representing things as sets of numbers, or coordinates

Question 2

How many coordinates do you need to represent something?

Answer 2

as many as that thing’s DOF

Question 3

location is usually represented using ____ coordinates

Answer 3

Cartesian

Question 4

Cartesian coordinates

Answer 4

vectors i, j, k

Question 5

Using the Cartesian system, any location realtive to the origin can be expressed uniquely as a…

Answer 5

weighted sum of the vectors

v = v₁i + v₂j + v₃k

Question 6

What are the coordinates of vector v?

Answer 6

v = (v₁, v₂, v₃)

Question 7

Right-handed Cartesian coordinates

Answer 7

index finger = i

middle finger = j

thumb = k

Question 8

How to calculate the sum of 2 vectors, w = v + u?

Answer 8

w₁ = u₁ + v₁

w₂ = u₂ + v₂

w₃ = u₃ + v₃

Question 9

how to calculate vector length?

Answer 9

|v| = (v₁² + v₂² + v₃²)^1/2

Pythagoras’s formula

Question 10

What does the dot product measure?

Answer 10

how close to paralle 2 vectors are

Question 11

Formula for dot product

Answer 11

u • v = u₁v₁ + u₂v₂ + u₃v₃

u • v = |u| |v| cosø (angle between u and v)

Question 12

Dot product: 2 vectors →

Answer 12

number

Question 13

Formula for cross product

Answer 13

u x v = (u₂v₃ - u₃v₂, u₃v₁ - u₁v₃, u₁v₂ - u₂v₁)

Question 14

Cross product: 2 vectors →

Answer 14

vector

Question 15

the cross product is non- ______

Answer 15

commutative (order matters)

u x v!= v x u

Question 16

Length of cross product u x v

Answer 16

u x v | = |u| |v| |sinø| (angle between u and v)

Question 17

What vector does the cross product give?

Answer 17

a vector that is orthogonal to both u and v

Question 18

A coordinate system is often linked to…

aka

the coordinate system expreses locations ______

Answer 18

a reference frame (some object)

…

relative to its frame of reference

Question 19

What reference frame is used by the brain to process visual memory?

Answer 19

eye-fixed frame

Question 20

Cartesian coordinates

Answer 20

how far up?

how far sideways?

Question 21

Polar coordinates

Answer 21

How far from center & what angle

Question 22

saccades happen every _____ per second

Answer 22

2-3 times

Question 23

What are some arguments against an eye-fixed frame for visual memory?

Answer 23

• 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)

Question 24

What are some reasons why the eye-fixed frame is used for visual memory?

Answer 24

Easier combination with incoming visual data

Question 25

What reference frame does auditory data use?

Answer 25

head-fixed frame

Question 26

what is the simple arm model called?

Answer 26

planar 2-link manipulator

Question 27

according to the planar 2-link manipulator, how many DOF does the arm have? Details.

Answer 27

2 DOF

Both the sholder and elbow are hinge joints with 1 DOF each

Question 28

according to the planar 2-link manipulator, what is the length of each limb?

Answer 28

upper & lower arms are the same length, L

Question 29

What are the 2 coordinate systems we can use to describe arm position?

Answer 29

1. joint angles
2. hand location

Question 30

Joint coordinates

Answer 30

if we know ø₁ and ø₂ (angle of shoulder and elbow respectively)

Question 31

Hand coordinates

Answer 31

specifying arm position using x, y coordinates of the hand in a Cartesian frame centered on the shoulder

Question 32

direct kinematics

Answer 32

Joint angle → hand coodinates

Question 33

Inverse kinematics

Answer 33

hand position → joint angle

Question 34

what are the 2 hypothesis for how the brain drives arm movement?

Answer 34

1. minimum-jerk hypothesis
2. joint interpolation

Question 35

minimum-jerk hypothesis

Answer 35

arm movements minimize the jerk of the hand

Question 36

What is bad about the minimum-jerk hypothesis?

Answer 36

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!)

Question 37

joint interpolation

Answer 37

brain drives arm along straight lines in joint coordinates

Question 38

Benefit of joint interpolation

Answer 38

• Brain only uses inverse kinematics twice (initial & final position)
• brain plots straight line in joint coordinates between the intial & final positions (easy computations)

Question 39

In reality, do we use minimum-jerk hypothesis or joint interpolation?

Answer 39

Both, it’s complicated

Question 40

How did the joint interpolation theory get revised? What is it called?

Answer 40

Staggerd joint interpolation:

joint-space paths may curve, but no point reverses its motion within a movement (waste of effort)

Question 41

How correct is the staggered joint inerpolation?

Answer 41

• Only true in reaching
• Other tasks like throwing, we reverse joint motion
• not a universal principle of motor control!