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

Lecture 6 - Dynamics Flashcards by Airi -

dynamics deal with systems that…

unfold in time (differential equations / rates of change)

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Dynamics is needed to understand…

sensors - muscles - commands

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dynamic operators

differentiator (simplest) 2. integrator (simple) 3. leaky integrator (slightly complex)

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what does a differentiator do?

input: u output: x where x = u’

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defining property of a dynamic operator

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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Derivative of a ramp

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

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Derivative of a sinusoid

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

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

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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Derivative of a higher-frequency sinusoid

larger amplitude sinusoid, 90 degrees out of phase

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Define: integrator

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

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Properties of integrator as input changes

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

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Given zero input, an integrator…

maintains its current output forever

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Explain leaky integrator

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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Equation for leaky integrator / general equation

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

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another name for leaky integrator. Why?

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

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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Analogy for when input drops in a leaky integrator

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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What are lowpass filters good for?

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

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Noise

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

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highpass filters respond…

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

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equation for highpass filter

x’ = bu’ - cx

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when input changes suddenly, highpass filters…

respond but then quickly fade away

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highpass filter is interested in…

changes

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

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

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equation for elastic system

u = kx (no differential equations) k = stiffness

How does an elastic system respond to a step input?

snaps to new equilibrium instantly (position vs time graph)

equation for viscous system

u = rx' r = viscosity

How does a viscous system respond to a step input?

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)

In an inertial system, driving force determines...

acceleration

what dominates in an inertial system?

mass

formula for inertial system

u = mx'' m = mass

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

constant acceleration (exponential graph of position vs time)

what is a viscoelastic system?

has viscous and elastic forces, but negligible mass

what is the formula for viscoelastic system?

u - kx = rx'

biological example of viscoelastic plant

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

what type of dynamic operator is a viscoelastic system?

leaky integrator

how does viscoelastic system react to input step up?

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

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

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

responses of a leaky integrator are characterized by...

its time constant

explain time constant

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

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

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

time constant of a leaky integrator =

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**

equation for viscoelastic system leaky integrator

rx' = u - kx ↓ x' = (1/r)u - (k/r)x

What is time optimal control?

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

Assumption of time-optimal control

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

time-optimal force for elastic plant

step

Time optimal control for a viscous plant

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

Time optimal control for an inertial plant

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

Time optimal control for a viscoelastic system

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

What if muscles used step instead of pulse-step?

sluggish drift towards equalibrium

r/k ratio of eyeball (value) What does this mean?

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

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

20 - 100 ms (0.02 - 0.1 s) vs 0.6 - 0.8 s

step signal is used to...

make vergence (slow) eye movements * cross/uncross eyes to focus on targets of varying distances

Normal motion of vergence? Lab?

Normal: horizontal Lab: 6 degrees vertical possible

Decelerating profile of vergence

0.6 s

why does vergence not use pulse-step?

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)

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

* malfunctions the most * Last to develop in children * first to be affected by fatigue, drugs, alcohol

vestibular rotation sensors are LOW/HIGH pass filters?

high ## Footnote **Semiciruclar canal output is a highpass filtered version of head velocity**

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

6 s ~20 s

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

60 s Dynamic distortion caused by sensors is partially corrected by velocity storage

Velocity storage

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

Why is velocity storage not perfect?

integrator is leaky

why are intergrators leaky?

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

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

yaw most common motion

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

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

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

perception declines quickly to a NON-ZERO level and HOLDS INDEFINITELY. Gravity - head changes position causing vestibular organs in inner ears to sense movement

Lecture 6 - Dynamics Flashcards

(67 cards)