Lecture 3 - Coordinate Systems Flashcards

1
Q

define: coordinate system

A

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

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2
Q

How many coordinates do you need to represent something?

A

as many as that thing’s DOF

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3
Q

location is usually represented using ____ coordinates

A

Cartesian

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4
Q

Cartesian coordinates

A

vectors i, j, k

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5
Q

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

A

weighted sum of the vectors

v = v1i + v2j + v3k

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6
Q

What are the coordinates of vector v?

A

v = (v1, v2, v3)

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7
Q

Right-handed Cartesian coordinates

A

index finger = i

middle finger = j

thumb = k

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8
Q

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

A

w1 = u1 + v1

w2 = u2 + v2

w3 = u3 + v3

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9
Q

how to calculate vector length?

A

|v| = (v12 + v22 + v32)1/2

Pythagoras’s formula

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10
Q

What does the dot product measure?

A

how close to paralle 2 vectors are

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11
Q

Formula for dot product

A

uv = u1v1 + u2v2 + u3v3

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

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12
Q

Dot product: 2 vectors →

A

number

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13
Q

Formula for cross product

A

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

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14
Q

Cross product: 2 vectors →

A

vector

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15
Q

the cross product is non- ______

A

commutative (order matters)

u x v!= v x u

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16
Q

Length of cross product u x v

A

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

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17
Q

What vector does the cross product give?

A

a vector that is orthogonal to both u and v

18
Q

A coordinate system is often linked to…

aka

the coordinate system expreses locations ______

A

a reference frame (some object)

relative to its frame of reference

19
Q

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

A

eye-fixed frame

20
Q

Cartesian coordinates

A

how far up?

how far sideways?

21
Q

Polar coordinates

A

How far from center & what angle

22
Q

saccades happen every _____ per second

A

2-3 times

23
Q

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

A
  • 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)
24
Q

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

A

Easier combination with incoming visual data

25
Q

What reference frame does auditory data use?

A

head-fixed frame

26
Q

what is the simple arm model called?

A

planar 2-link manipulator

27
Q

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

A

2 DOF

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

28
Q

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

A

upper & lower arms are the same length, L

29
Q

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

A
  1. joint angles
  2. hand location
30
Q

Joint coordinates

A

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

31
Q

Hand coordinates

A

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

32
Q

direct kinematics

A

Joint angle → hand coodinates

33
Q

Inverse kinematics

A

hand position → joint angle

34
Q

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

A
  1. minimum-jerk hypothesis
  2. joint interpolation
35
Q

minimum-jerk hypothesis

A

arm movements minimize the jerk of the hand

36
Q

What is bad about the minimum-jerk hypothesis?

A

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!)

37
Q

joint interpolation

A

brain drives arm along straight lines in joint coordinates

38
Q

Benefit of joint interpolation

A
  • Brain only uses inverse kinematics twice (initial & final position)
  • brain plots straight line in joint coordinates between the intial & final positions (easy computations)
39
Q

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

A

Both, it’s complicated

40
Q

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

A

Staggerd joint interpolation:

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

41
Q

How correct is the staggered joint inerpolation?

A
  • Only true in reaching
  • Other tasks like throwing, we reverse joint motion
  • not a universal principle of motor control!
42
Q
A