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so next up Bogdan the robots will take
us dancing in the dark Oh chase so first
of all before you start throwing any
tomatoes this is a Windows machine and
thanks for making it work once again a
few questions anybody here familiar with
inertial navigation ok with general
concepts like transformation matrixes
quaternions ok super awesome
anybody flattered believer maybe okay
depending on the model well basically
first before I start I would like to
take thank my colleague once again for
trying to make badges like sort of it a
takeaway for this workshop that we in
the end didn't manage to push through
because of some technical reasons they
should have supposed to have an ESP 12
with the M view 90 to 60 so quite
precise accelerometer gyroscope
magnetometer 9 dove EMU well you can you
can help still the PCB and try to make
it work on your own but it's it's quite
an it was quite a challenging thing and
we almost made it now so something about
myself my name is Bogdan Duras Lovitz I
currently work in Duke engineering here
in Serbia I'm an advanced embedded
engineer and before that I was doing
some maintenance in the aircraft
industry mainly in the outer chapters 20
to 34 which are out of pilots navigation
systems and before that even
I did some aren't in
military company here in Serbia working
on an advanced light attack system with
with the outer pilot and stop there that
was my first ever job super awesome and
there I fell in love with inertial
navigation now so I'm not too big Bruce
Springsteen fan actually
but somehow the name of the topic came
by itself because how do dancing in the
dark basically that's sort of a
romanticized version of what I see as an
inertial navigation we don't have any
external references we are on the
reliant on our look and feel and we are
trying to navigate our West's best way
there and since the this year's balsa
con was all about finding yourself I
think I took it the most literally here
out of the old bunch and you will see
how to find yourself and how to know
where you're heading without any
additional external reference as well at
least for some while so first of all
what is generally navigation about
navigation is the concept of knowing
where you're going well duh but more
than it it has quite some interesting
properties basically most of our
navigation can be done in two fashions
one being other or any sort of let's
let's call it reference navigation
navigation where you have references
like stars like maybe beacons radio
beacons were used for ships or GPS or
GLONASS or any sort of those Baidu
Galileo GNSS systems anybody familiar
with the concepts of GPS systems ok so
you know that there are two nations that
are mostly dominant for the GPS system
one is well it's a thing owned by the
American government
like GLONASS is basically owned by
Russian government
there is no guarantee of data precision
at any moment or the availability there
is some push these days to get at least
the civilian GPS to be fully available
so l1 l3 satellites second of all any
GPS that's more precise than some range
that I forgot that can measure it more
than 10 Hertz that can travel at more
than some 500 meters per second or above
10 kilometers is considered ammunition
and therefore you cannot easily trade it
you cannot operate with it and you
cannot resell such equipment there are
FPGA implementations which are quite
interesting but they also have
limitation by the sheer design of the
signals and GPS other types who are of
navigation are out of data so maybe
somebody will out of sheer change of
heart pull his sextant out in in the
middle of the sea but you typically
don't expect to see that these days on
the other hand like we also heard in
Jaws talk yesterday inertial navigation
is pretty damn precise when done
correctly you cannot interfere with it
it's based on intrinsic properties of
our universe and how we work and physics
and it's that you cannot it's passive
completely you cannot interfere with it
and within it within any reasonable way
you would have to have Hardware access
to it and if we are talking about fast
moving vehicles like let's say rockets
you have pretty slim of a chance of
getting it just to interfere with its
GPS if you are getting it you have other
means of getting it in the first place
so
reliability is the key aspect also
mentioned in the Zoe's lecture yesterday
it was the aircraft GPS typically let's
say Airbus will have what's called an
audio unit so air data inertial
reference unit or actually three of them
which will correlate and try to find
where you are going where you are
heading and error should be slim up to
0.1 percent and it's considered again
ammunition tactical resource you cannot
get those components of the shelf and
one more thing that's good about
inertial navigation you can go and
extremely high rates for an al analogous
systems you had basically have no speed
limitation you can measure at the rates
which coincide with your sampling rates
or even better if you are competent
enough you can make a kuleana level
system using those properties
well negative effect effects are that
it's quite hefty in the price tag
especially for for the more advanced
version which we'll cover later
and once you're lost you're lost you
have no way of collecting recording your
course you have no way of actually
knowing how it correlates with the real
world and typically it will be a
scenario of analysis in a post-mortem
analysis so once everything failed
you'll see how and why it failed
typically it will be self-evident simple
and stupid
but at the point in time when you are
doing that that's quite quite hard to
expect and one of the key issues in any
such system is calibration calibration
calibration typically has to be done per
units per piece and there is no easy way
around it and no easy way just doing it
right it's a lot of crazy math now how
the navigation actually works let's take
a step back to look at some classics of
physics anybody here familiar with the
second Newton's law okay anybody here
knows this formula well you know anybody
here knows that it's actually wrong form
of it we are taught this in school as a
second Newton's law it actually sounds a
bit different and this is with some
assumptions and please be careful about
those because precisely those
assumptions make rockets fly and if you
look at it we define force as thing
giving an acceleration to the mass
interim acceleration is nothing but
change of the velocity which in term is
change of the position and we have a way
of getting the acceleration
and just by simple integration turns out
not to be so simple but by simple means
of integration we can get back to our to
where are we going now how much of you
forget the constant while integrating
while preparing this talk interesting
thing I made a small octave demo for you
shifted from MATLAB to octave and some
sort of rotation and so on and I spent
two hours troubleshooting the thing that
I missed initial conditions therefore I
missed constant and it took more than
expected so high school physics getting
a revenge and me now
the accelerometer is the device that can
measure specific force why this is
important it can measure something that
we think it's basically force causing
the movement of the object
why specific because somehow gravity is
also there and funny thing I don't know
if you know it but accelerometer
standing still will actually measure the
G alternate not moving and if you throw
it in the air it will happily measure
the zero acceleration while it's in the
free four so freefall so in sort of the
apogee why is that it's because
accelerometers are actually measuring
the total force applied on the object
now what you know of the gravity
is also a bit wrong because turns out
it's not as simple as one might think
well gravity should point straight down
and it's not it's pointing a bit slanted
because there's a thing called
centrifugal centripetal I won't get into
it force because if I see either of the
two I think somebody will burn me any of
anybody in the camp for the centrifugal
force here
centripetal okay glad now I'm glad it so
regarding the flat earthers based on the
distance you want to cover a gravity
model might change a bit so if you are
talking some let's say navigation a few
kilometers away we can happily agree all
that the earth is flat surface
completely flat you may feel free to
draw any tomatoes at me now
please don't and that gravity is a
homogeneous field pointed down whatever
down maybe and the earth is also like
suspended on three elephant trunks and
so on if you want to go a bit further
away you need to use what's called
spherical models so something that we
learned this call just a big blob pretty
round pointing towards the center but it
turns out the earth is not completely
round it is actually more of the Alex
elliptical shape or rotational ellipsoid
even funnier that is also not the case
and if you want to go really really
wrong long-range or really really
precise light it interconnect an
intercontinental ballistic missile
precise you have to factor in everything
and mathematicians have a way around it
they first went about well okay there is
something that is more closely related
to on earth and that's an out period
model which turns out just also not to
be correct and there was like a super
out or referent definition that there is
a gear G right geoid gear it whatever we
want to call it that it's basically the
mathematical shape of the earth so earth
is a shape of the gyrate and G it is
precisely the shape of the earth if the
earth changes by any means the Giri it
also changes to follow the earth and I
think that's one of the best recursive
definitions that I've heard and because
of all that finally in the year eighty
four so three years before my time there
was this bunch of smart people decided
to me decided to make a standard out of
it and basically what it is it's a
referential ellipsoid with the gravity
in all the possible places so like doing
anything with the constants don't be too
smart about it go for this one it's
simple you just punch in the latitude
longitude altitude and you basically get
your great effects the effective gravity
back if you try to calculate to
compensate to do anything you will
surely fail you will be closely failing
but failing nonetheless now
there are various types of
accelerometers around and I listed some
of the devices that I am not even sure
how they work like just yesterday I
found out about heated gas accelerometer
which is quite an exotic technology
basically a bubble of gas that is heated
in the center and then based on the
acceleration the temperature flux will
change and that had been detected one of
the prime versions of the accelerometers
widely used today is MEMS MEMS is
basically micro electrical micro oh my
god so it's micro electrical mechanical
system a tiny little thing basically a
quartz crystal on the substrate that
based on the acceleration will give out
some voltage and widely-used
are the one in phones with dimensions
right ranging in millimeter ranges big
accelerometers tend to be like this big
and why do they need do we need them and
where do we need them that's the top
priority question we can only use
accelerometers in referential planes or
in the inertial planes which are
completely static to the universe
there is one such plane and I will show
a lot of slides I won't read them
through I don't think that's necessarily
but eventually if you want to follow it
and do something based on this maybe it
will serve a purpose
now there's an inertial system this is
ideal construct of an earth
static within the universe everything
else rotating around it
and basically what it does it's there so
we can do all the math nicely in this
system second Newton law actually
applies in system where we are standing
it does not because it's not inertia it
does not in the form that we know it it
applies with applying additional
elements like the Coriolis force now
you'll notice here one interesting
aspect again north there's a concept of
true north
why true north there are actually many
North's to begin with what you see when
you get your compass out it's magnetic
north which is highly imprecise if
you're getting close to the pole which
by the way also not only is imprecise is
shifting over time so a lot of things
like gideros have to compensate with
magnetic variation of or Mackworth
tables and so on and finally it's it's
not easily doable to have a precise
magnet magnetic navigation device
because a lot of soft iron and hard hard
iron problems with it if anybody has
tried doing magnetic navigation you will
know that just having a screwdriver
nearby will mess your readings
tremendously because planet is in some
millimeter Tesla ranges and it's so damn
weak that's basically just any sort of
magnets which we now have in a
especially the strong neodymium few
Tesla types will easily mess with the
second type of north is gia magnetic
north which is the approximation of what
we think north magnetic north should be
and where it should be most oftenly it's
again just nice concept for the
astrophysics but for navigation there's
no purpose and there is only one true
rotational north which is the axis among
the which planet spins you cannot
determine that by magnets that's
completely impossible you either have to
know it or you can use some smart
methods so another reference system that
we use is basically the Earth's center
earthquake system it is the same as a
neural system but with a tiny quirk of
rotating with the earth so the only
difference between inertial and etc
system is that you just have to apply
angular rotation to it among the the
concretely in this case the true north
axis or the jet axis now those were all
global systems and did not help to
determine where we actually are so we
need additional thing we need something
which is local and based on the
literature that you're using you will
find that typically in the West you you
will use the Northeast down reference
plane or in the east especially with the
Russian authors you will use the east
north up coordinate system reasons very
one of the prime definition is based
basically that depends which army type
used it the first so if you're looking
at artillery there will typically use
the e nu if you're looking at aircraft
modeling and so on they will use the
northeast down there completely
analogous and easily to go between and
finally we have some some where we need
somewhere to position our
fluorometers and to start integrating we
have our body coordinate system here
there's quite a nice rockety thingie and
the idea is that basically we are always
in the Cartesian referent planes so we
have three axes of freedom to cover that
we need three accelerometers and now we
have a problem
it turns out that we only know how are
what you call what a heart center of
mass is moving we don't know anything
about the whole body structure and that
is precisely the three degrees of
freedom navigation versus the six
degrees of freedom historically first
ever guided rocket fell to - to use
accelerometers and to use guidance
actually used the suspended JimBob
gyroscope platform for that so what did
it
it had a hollow area inside it and
suspended accelerometer so that they are
free and that you can all the time have
the platform static relatively to the
outer shell of the rocket and basically
integrate your merry way to it through
the flight and hopefully hit the London
at that time but now in modern systems
we don't use such game vault
contraptions because it's highly complex
and highly expensive the typical
approach is to use the strap strap down
navigation system which what it does
it's I had does all the calculations and
basically forms a virtual platform which
is analogous to one being in the middle
and it does it so by constantly rotating
the thing these are some reference
matrixes basically in order to be able
to rotate about about axis you need to
use this anybody familiar with the 3d
graphics will notice that it is pretty
much the same as as you use to rotate
images there is and even a vacation is
actually simpler because skewing is not
possible for rotation matrixes there is
no scaling there is no lot of these
eigenvectors cool features you can
forget all that and just use simple
rotation and translation and typical
typically Euler angles how they are
called are named alpha beta and gamma
but much more widely used and the thing
that I'm citing here are a roll pitch
and yaw which are basically the three
properties of the body navigating so the
roll is being the rotation among its
dominant axis so typically about about
axis of symmetry let's say of a missile
the pitch will be its tendency to go up
and down and the yaw will be its
tendency to go left and right and you'll
notice that in missiles it is decoupled
while in aircraft it's highly coupled
it's typically rolling your you cannot
that easily ship that system now okay so
how do we get our strap down navigation
working at all well there is this Euler
angular formula which is quite analogous
Newton's second law basically what it
does it dictates just how you can find
out what is your current angular
acceleration by the means of knowing the
inertial moment which is the fundamental
inertial criteria for the body or
fundamental property like masses for the
point and basically you just calculate
all the outer forces that are affecting
you you you will notice that here we
just can stop on the on the first
integration but we can discuss it and
now we've got all six equations
everything there that we need and how do
we proceed about it well we just
integrate our Mary Mary way along one
interesting property here is there how
many mechanical engineers are here super
awesome so what what's interesting here
symmetrical systems or highly
symmetrical systems are much more easier
to compute than non symmetrical one so
inertial tensor the matrix big blasted
thing there tends to degrade to a
smaller vector if you are using myself
so there's a lot of of good things going
simple so aircraft would not be as
simple as a missile now
for the programmers out it's not all
hardcore math it turns out that we can
just discretize what does it mean
basically what's an integral integral is
nothing but the cumulative sum and
that's it
if you are fast enough if you're precise
enough you don't have to be super smart
about it you don't have to estimate it
you just tend to be at least ten times
faster than then the response of your
system if you are good there and if you
can provide that speed you're safe to
integrate and that's it also with the
derivatives derivatives are nothing but
the sample of our sample changes the
discrete derivative is just that you
just take the two values and and look at
them how they are now for the angles we
use gyros and gyros are extremely
interesting piece of technology they are
basically fall into two types one being
the absolute gyro or the position gyro
which is what you know as a spinning top
think trying to stay in the place as it
was and the other one being the rate
gyros which are actually telling you
what the rate of the movement is I
didn't want to go fully exotic for this
gyros so I just choose chose some types
here there is again MEMS so mixer I like
to atomic electromechanical systems I
got it right this time there's a
dynamically tuned gyro which is super
cool contraption a mechanically balanced
wheel which has a precise turning to to
precisely turn tuning frequency turning
frequency so it knows all its mechanical
failures it's again expensive to produce
and it has a limited life lifecycle and
there there are more exotic than not
technologies like ring laser gyro
in the fiber-optic gyro what are they
are about the thing called sonic effect
or I hope at least the guys problem name
is pronounced like that saganaki effect
is basically bouncing light beams over
through the mirrors and measuring the
phase shift and basically by difference
in the phase shift you can see how you
are moving with the ring laser gyro and
it's a chef teah contraption
it's basically having prisms precisely
positioned and lasers with a fog or a
fab fiber-optic gyros it's a big
continuous trend of of fiber optic fiber
one want in tightest possible fashion
and it tends to use up space pretty
quickly so there is less problem in
manufacture then in ring laser gyro but
there is a lot more problems calibrating
and adjusting for it now with those
gyros comes a great responsibility and
now there is that it is quite hard to
get them you will typically be bound by
laws by the ether law or international
arm traffic trafficking arms regulation
and by the missile technology control
regime there was a quite an interesting
debate yesterday over the civil licenses
currently in action there are already
licenses prohibiting use of specific
military use of a different technologies
going for a precise gyroscope will
require you to sign an overlap EULA
which is really serious it's not like
just click Next to continue
you could be really really legally
liable for what you sign there and you
will typically have to provide proof of
usage for for per piece for example in
the maintenance industries
and now final thing which we hate the
most
errors there are everywhere everything
we did now was super cool
nice polite spherical frictionless in
vacuum in reality things degrade much
much quicker so there is a lot of noise
so accelerometers tend to be noisy even
more so than gyros there is a drifting
drifting actually physical drifting was
firstly described precisely in the self
sustained self self balancing platforms
people would notice that over time gyro
gimballed platforms tended to drift
physically or to change their position
because of some sort of small
imperfections like for example friction
which is actual thing in the world world
around us maybe not in physics but in
mechanic's stance tends to have quite an
impact
everything is biased not just people
gyros as well and there are a lot of
nonlinear and linearities you never know
how well device will behave
it can even behave differently on how
you are going through it based on it
bandwidth best based on things like
hysteresis so you can go one way around
its the second way when you're returning
and most of our while the error
accumulates extremely quickly
so let me formulate it this way there is
imagine there is some small angular
deviation in your virtual platform and
let's say that that's a function of time
you integrate it so okay you have some F
of timer now imagine that there's even
worse than accelerometer one D so that's
T squared if you've got three
accelerometers that's three T on the
third on the power of three times the T
so you
somewhere in the range 2d on the power
of four it's so if you have anywhere of
orion's in your system you are pretty
much slowly drifting away and further
further away that that you can know of
typical mechanism of compensating this
is through some filtering notable
examples of filtering are for example
the multi platform approach where
basically we know that any movement is
expected to be nice so you know in
physics think things cannot just shift
radically and you are searching for for
the pattern in the noise which best fits
your movement other than that there is
also like Kalman filtering based on
external sensor sensors but you have to
have external sensors which is which you
can really trust which is a problem and
I would like to show you some small
thing if it works
first of all please do work
okay
so this come on
this is an actual data of
three accelerometers
okay sorry my octave is a bit laggy
today
and it will full-screen eventually okay
yes
this is the data that was taken from the
real world
gyros rate gyros to be precise and based
of the of those I want battery because
it's really slow now
I wanted to actually do this real-time
but there was so much data that my
octave was processing it like three to
five minutes and I think that's super
annoying and especially would make a
heck of a demo I would help to whistle
all along
so based on those data we get a flight
profile and this was provided by my
ex-boss and colleague so thanks
if you look at the flight profile it's
pretty characteristic for four it's
basically a mess up nothing to do
strange about it you can notice the the
period of getting to to a certain speed
or to certain altitude in this case
cruising at predefined
parameters and then homing for the
target somewhere in this area
and another thing that I would like to
show and give
is
a demo
let's hope my laptop does not die
yes
more or less
there is quite a disparity between what
they expect when typing and what's
happening now
okay so
okay
here is a small simple system rotating
anybody interested in code can get it
off me and it's been written completely
in octave here there's just my small
check to make sure that it's not that
the reference frame is not in any way
degrading over time which can happen
here you can take a look of basically
what what it is it is the rotation
matrixes I know that there are packages
doing it probably better than me but for
this presentation I wanted to do it from
scratch why not and we can fiddle around
with it we can for example change a
scale and give it some sort of
acceleration don't ask me about the
numbers they're pretty much based on
what I feel should be right
and if we cross our fingers and run it
yes it did not explode
so the thing will start to move
in space and we still didn't lose our
token ality so
one final thought is on quaternions
eventually while while redoing the
lecture for the format I decided to omit
the curtains as it would be too much
math but I just want to give you a food
for thought what actually it is it's
widely using in 3d graphics and it's
super awesome
again it's it has two approaches
navigation and 3d vector graphics if we
go back to to how we position our
platform in in Euler angles what we do
is we have a components of rotation and
each rotation can be separated into
three components your patron what
quaternions do is take the shortcuts and
find the one perfect rotation that gets
your system into place and how they do
it by magic it is basically a 4d system
with three of the axes having -1
intensity art having -1 intensity
everything it's called hamiltonian space
so basically it transforms your physical
coordinates into higher plane of
reference do some mathematical magic
which there is extremely elegant so
everything is multiplication and and
addition unlike Euler angles where
everything is sine cosine and if you get
close to 90 degrees everything explodes
in quaternions you don't have that issue
everything plays nice but you have to
transition into this abstract abstract
mathematical space no space construct
and when you transition to and from it
every single transition rubs you a bit
of precision you can normalize and
renormalize but it will eventually
accumulate and you will feel it as your
system with losing orthogonality for
example
so that's all I had to say Q&A
are there any questions going once going
twice
thanks a lot
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