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PSL304 > Lecture 6 - Dynamics > Flashcards

Lecture 6 - Dynamics Flashcards

1
Q

dynamics deal with systems that…

A

unfold in time (differential equations / rates of change)

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2
Q

Dynamics is needed to understand…

A
  • sensors - muscles - commands
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3
Q

dynamic operators

A
  1. differentiator (simplest) 2. integrator (simple) 3. leaky integrator (slightly complex)
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4
Q

what does a differentiator do?

A

input: u output: x where x = u’

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5
Q

defining property of a dynamic operator

A

output at any moment depends not on the instantaneous value of its input, but on how the input unfolds through time (things happening before/after the instance)

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6
Q

Derivative of a ramp

A

constant Positive value if ramp slopes up Neg if ramp slopes down

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7
Q

Derivative of a sinusoid

A

another sinusoid, 90 degrees out of phase (1/4 cycle)

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8
Q

When is the derivative of a sinusoid at max value (peak)? At min value? At 0?

A

max: f(x) slopes up, crossing the axis (biggest pos change) min: f(x) slopes down, crossing axis (biggest neg change) zero: f(x) peaks (no change)

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9
Q

Derivative of a higher-frequency sinusoid

A

larger amplitude sinusoid, 90 degrees out of phase

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10
Q

Define: integrator

A

input: u output x where x = ∫u dt (or x’ = u)

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11
Q

Properties of integrator as input changes

A

HOSE BUCKET ANALOGY Positive u = increasing x Negative u = decreasing x 0 u = constant x

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12
Q

Given zero input, an integrator…

A

maintains its current output forever

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13
Q

Explain leaky integrator

A

Analogy: bucket with hole leak rate depends on size of hole (c) and water pressure/volume of water in the bucket (x) leak rate = cx Rate of change of x = inflow (u) - leak (cx) x’ = u - cx

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14
Q

Equation for leaky integrator / general equation

A

x’ = u - cx x’ = bu - cx (b and c are positive constants)

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15
Q

another name for leaky integrator. Why?

A

lowpass filter Respond strongly to low-frequency inputs, but weakly to high-freq inputs (not enough time due to sluggish reaction time)

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16
Q

how do leaky integrators respond to inputs? Why? (think of water pouring into empty bucket)

A

dragged-out, sluggish way As x increases, bu-cx decreases ∴ x’ decreases x’ decreases as x increases When x is big enough that bu = cx, x’ = 0 x reaches a constant level, which is constant as long as u does not change

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17
Q

Analogy for when input drops in a leaky integrator

A
  1. Turn off hose 2. Water level drops quickly 3. Water level decreases slowly because less volume in bucket = less water pressure = slower x’ = bu - cx = 0 - cx = -cx
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18
Q

What are lowpass filters good for?

A

removing noise from signals (block high frequency signals without distorting lower frequency signals)

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19
Q

Noise

A

When signals are contaminated by noise, noise is usually higher-frequency than the desired signal

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20
Q

highpass filters respond…

A

weakly to low-freq inputs but strongly to high-freq inputs

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21
Q

equation for highpass filter

A

x’ = bu’ - cx

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22
Q

when input changes suddenly, highpass filters…

A

respond but then quickly fade away

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23
Q

highpass filter is interested in…

A

changes

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24
Q

What happens when u increases to a new constant level in a highpass filter?

A

x steps to a new value higher than u, but drops down back to 0 soon after

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25
Q

equation for elastic system

A

u = kx (no differential equations) k = stiffness

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26
Q

How does an elastic system respond to a step input?

A

snaps to new equilibrium instantly (position vs time graph)

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27
Q

equation for viscous system

A

u = rx’ r = viscosity

28
Q

How does a viscous system respond to a step input?

A

Ignoring r, u = x’ (integrator) ∴ viscous system is just an integrator x a constant ∴ when input steps to a new constant, x increases at a constant rate (positive straight slope of position vs time)

29
Q

In an inertial system, driving force determines…

A

acceleration

30
Q

what dominates in an inertial system?

A

mass

31
Q

formula for inertial system

A

u = mx’’ m = mass

32
Q

what happens when a input is stepped to a new constant in an inertial system?

A

constant acceleration (exponential graph of position vs time)

33
Q

what is a viscoelastic system?

A

has viscous and elastic forces, but negligible mass

34
Q

what is the formula for viscoelastic system?

A

u - kx = rx’

35
Q

biological example of viscoelastic plant

A

oculomotor plant - eye in orbit - 6 extraocular muscles per eye - orbital fatty tissue

36
Q

what type of dynamic operator is a viscoelastic system?

A

leaky integrator

37
Q

how does viscoelastic system react to input step up?

A

log scale (fast at first, then slow down to constant)

38
Q

trajectory of viscoelastic system in response to step up input depends on…

A

ratio of viscosity to stiffness (k/r) - k > r = resembles spring system where increase is super fast - r > k = resembles viscous system where increase is more gradual

39
Q

responses of a leaky integrator are characterized by…

A

its time constant

40
Q

explain time constant

A

Time constant = T As long as the input in a leaky integrator stays the same, output will move 63% of the way to its final value every T seconds Equilibrium is reached in 3-4 time constants

41
Q

do highpass filters have time constants? If so, what are they?

A

yes, step responses fade away in 3-4 T s

42
Q

time constant of a leaky integrator =

A

1/c (c comes from x’ = bu - cx)

in a leaky integrator, x’ = (1/r)u - (k/r)x, so c = k/r ∴ time constant = r/k

43
Q

equation for viscoelastic system leaky integrator

A

rx’ = u - kx

↓

x’ = (1/r)u - (k/r)x

44
Q

What is time optimal control?

A

You don’t wait for the system to slowly make its way to the new signal; you change the input to make change happen faster

45
Q

Assumption of time-optimal control

A

there is an upper limit to the amount of driving force that can be exerted

46
Q

time-optimal force for elastic plant

A

step

47
Q

Time optimal control for a viscous plant

A

pulse (max force until B is reachec, then shut off driving force completely)

48
Q

Time optimal control for an inertial plant

A

biphasic (max pos until halfway → max neg until B → stop)

49
Q

Time optimal control for a viscoelastic system

A

pulse-step

non-zero (because elastic forces) → max pos → drop to non-zero, but higher than intial (need to account for additional elastic force)

50
Q

What if muscles used step instead of pulse-step?

A

sluggish drift towards equalibrium

51
Q

r/k ratio of eyeball (value)

What does this mean?

A

0.2

Means it’s time constant is 0.2s, and drift will last 0.6~0.8s

52
Q

duration of a normal saccade vs duration of saccade if step signal was used

A

20 - 100 ms (0.02 - 0.1 s)

vs

0.6 - 0.8 s

53
Q

step signal is used to…

A

make vergence (slow) eye movements

  • cross/uncross eyes to focus on targets of varying distances
54
Q

Normal motion of vergence? Lab?

A

Normal: horizontal

Lab: 6 degrees vertical possible

55
Q

Decelerating profile of vergence

A

0.6 s

56
Q

why does vergence not use pulse-step?

A

Maybe there is another limiting factor we don’t know (even if we used pulse-step, this factor may be slowing it down, so there is no point)

57
Q

Convergence is an evolutionarly new eye mechansim. This is proven because…

A
  • malfunctions the most
  • Last to develop in children
  • first to be affected by fatigue, drugs, alcohol
58
Q

vestibular rotation sensors are LOW/HIGH pass filters?

A

high

Semiciruclar canal output is a highpass filtered version of head velocity

59
Q

Time constant for vestibular rotation sensors? How long before canals don’t report any motion (theoretically)?

A

6 s

~20 s

60
Q

In reality, how long does your vestibular rotation sensors last before you stop feeling motion? Why?

A

60 s

Dynamic distortion caused by sensors is partially corrected by velocity storage

61
Q

Velocity storage

A

perception != signals coming out of canals (or else you would loose sensation after 20s)

There is a sideloop with integrator (1/T)

perception = canal signal + integrator signal

62
Q

Why is velocity storage not perfect?

A

integrator is leaky

63
Q

why are intergrators leaky?

A

non-leaky integrators are hard to build and dangerous toh ave around

64
Q

velocity storage only operates in which motion (yaw, pitch, row)? Why?

A

yaw

most common motion

65
Q

What happens when you roll or pitch about an Earth-vertical axis in the dark?

A

sensation of motion fades in 20s (6 s time constant)

66
Q

What happens when you roll / pitch about an Earth-horizontal axis in darkness? Why?

A

perception declines quickly to a NON-ZERO level and HOLDS INDEFINITELY.

Gravity - head changes position causing vestibular organs in inner ears to sense movement

67
Q
A

PSL304 (10 decks)

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