Lecture 6 - Dynamics Flashcards
dynamics deal with systems that…
unfold in time (differential equations / rates of change)
Dynamics is needed to understand…
- sensors - muscles - commands
dynamic operators
- differentiator (simplest) 2. integrator (simple) 3. leaky integrator (slightly complex)
what does a differentiator do?
input: u output: x where x = u’
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)
Derivative of a ramp
constant Positive value if ramp slopes up Neg if ramp slopes down
Derivative of a sinusoid
another sinusoid, 90 degrees out of phase (1/4 cycle)
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)
Derivative of a higher-frequency sinusoid
larger amplitude sinusoid, 90 degrees out of phase
Define: integrator
input: u output x where x = ∫u dt (or x’ = u)
Properties of integrator as input changes
HOSE BUCKET ANALOGY Positive u = increasing x Negative u = decreasing x 0 u = constant x
Given zero input, an integrator…
maintains its current output forever
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
Equation for leaky integrator / general equation
x’ = u - cx x’ = bu - cx (b and c are positive constants)
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)
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
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
What are lowpass filters good for?
removing noise from signals (block high frequency signals without distorting lower frequency signals)
Noise
When signals are contaminated by noise, noise is usually higher-frequency than the desired signal
highpass filters respond…
weakly to low-freq inputs but strongly to high-freq inputs
equation for highpass filter
x’ = bu’ - cx
when input changes suddenly, highpass filters…
respond but then quickly fade away
highpass filter is interested in…
changes
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
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
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
