Euler Angle Rates & Angular Velocity- Kinematic Differential Equation for Rigid Body Dynamics
Euler Angle Rates & Angular Velocity- Kinematic Differential Equation for Rigid Body Dynamics
Space Vehicle Dynamics ☀️ Lecture 14: Euler angle rates are not equal to the angular velocity. We derive the relationship between them carefully. We start with the fundamental kinematic differential equation for the direction cosine matrix [C]. By going through the successive pure rotations that define the Euler angles, we arrive at a matrix relationship between the Euler angles, yaw (ψ), pitch (θ), roll (Φ) in the 3-2-1 convention, and the angular velocity vector (ω) as seen in the body fixed frame. This is via a matrix we call [B], which depends on the Euler angles. We also discuss the singularity problem of the Euler angles. This suggests we search for other alternative ways to represent rotations based on the principal rotation vector.
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► Dr. Shane Ross 🛩aerospace engineering professor, Virginia Tech
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► References
Schaub \u0026 Junkins, Analytical Mechanics of Space Systems, 4th edition, 2018
https://arc.aiaa.org/doi/book/10.2514…
► Topics covered in course https://is.gd/SpaceVehicleDynamics
- Typical reference frames in spacecraft dynamics
- Mission analysis basics: satellite geometry
- Kinematics of a single particle: rotating reference frames, transport theorem
- Dynamics of a single particle
- Multiparticle systems: kinematics and dynamics, definition of center of mass (c.o.m.)
- Multiparticle systems: motion decomposed into translational motion of c.o.m. and motion relative to the c.o.m.
- Multiparticle systems: imposing rigidity implies only motion relative to c.o.m. is rotation
- Rigid body: continuous mass systems and mass moments (total mass, c.o.m., moment of inertia tensor/matrix)
- Rigid body kinematics in 3D (rotation matrix and Euler angles)
- Rigid body dynamics; Newton’s law for the translational motion and Euler’s rigid-body equations for the rotational motion
- Solving the Euler rotational differential equations of motion analytically in special cases
- Constants of motion: quantities conserved during motion, e.g., energy, momentum
- Visualization of a system’s motion
- Solving for motion computationally
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📚Attitude Dynamics and Control
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📚Nonlinear Dynamics and Chaos
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📚Hamiltonian Dynamics
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📚Three-Body Problem Orbital Mechanics
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📚Lagrangian and 3D Rigid Body Dynamics
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Content
0.08 -> we're going to continue talking about euler
angles and get into how you write the rigid body
7.28 -> kinematic differential equation written in terms
of the euler angles you could write a set of odes
13.44 -> for the euler angles and how they change as
a function of the angular velocity vector but
19.84 -> before we get into that these are the notes from
last time just to familiarize you first we talked
25.2 -> about this c matrix right the rotation matrix
that relates your your body fixed frame blue i
34.32 -> usually use blue compared to some other reference
frame usually an inertial frame and you want to
42 -> write what's the orientation of this triad of
unit vectors with respect to this other triad
47.92 -> so the the kind of straightforward
way to do that would be a a c matrix
55.12 -> but we learned another way to write
that in terms of the euler angles
61.6 -> i guess before that how does the c matrix change
so in general right this matrix the the blue unit
69.52 -> vectors are going to change with
respect to the red unit vectors
73.92 -> and that change happens according
to the angular velocity vector
79.12 -> so we eventually got down to this ode here it's
a matrix ode and it's shorthand for nine scalar
89.12 -> odes it says how the matrix c changes with
time if you have some angular velocity vector
99.68 -> that's also changing with time so you write you
have that angular velocity vector you write in
104.72 -> terms the b frame components that's what gave us
omega and then you use it to populate this omega
110.24 -> tilde matrix which was a it's a skew-symmetric
matrix up here right there in the upper right
120.56 -> so you could write nine scalar odes for how this
changes but that's not what's typically done
128.72 -> what's typically done is we first want
to represent the c matrix in terms of
135.68 -> a smaller set of numbers so instead of nine
numbers right the nine entries of the matrix
139.84 -> we use these euler angles which describe any
orientation as a sequence of three pure rotations
149.92 -> about axes and one of the main ones is the yaw
pitch and roll which we eventually said had a
156.24 -> specific name it's the three two one euler angles
because the first rotation is about the current
162 -> number three axis about a yaw angle and then
the second rotation is about the number two axis
168.64 -> through a pitch angle and the third rotation is
about the number one axis through a roll angle so
176.08 -> we went through that each of these pure rotations
can be written in terms of these m matrices on
182.64 -> the right m sub i means a rotation through the
number i axis through whatever angle you're given
193.28 -> and at the end of that calculation the product of
those three matrices is another way to write the
200 -> c matrix the rotation matrix and we said that
there are 12 different euler angle conventions
208.72 -> written in terms of what the first
axis is the second and the third and
215.44 -> all of the conventions the the c matrix
is written in appendix b of the book
221.84 -> so we mentioned the three two one also there's
the three one three which is used as part of
226.8 -> orbital elements longitude of the ascending
node inclination of an argument of periapsis
233.68 -> actually described the orientation of the
238 -> orbital plane compared to some inertial plane in
terms of three angles and first a rotation about
244.72 -> the number three axis then the
number one and then number three
248.4 -> but there's a problem with euler angles so
there's other parameters that eventually
253.36 -> get used and this problem is that they
have geometric singularities it's always
260.32 -> with the number two angle you know
there's there's two there's three
267.2 -> rotations that we do the singularity is always
some particular angle that gives a problem
274.8 -> in the number two angle so theta two for the
three two one convention it's theta two the pitch
282.32 -> is plus or not minus 90 degrees
or plus or minus pi over two
287.68 -> in which case there's so there's ambiguity in
what the roll and the yaw are so there are some
295.28 -> orientations where you can't uniquely describe
those orientations in terms of euler angles so
300.8 -> that could be a problem if you have something that
could be orienting any which way so eventually
305.44 -> we'll get to other things like quaternions or
the parameters which don't have this problem
312.48 -> and we'll see where that problem shows up in
what's called the kinematic differential equation
318.16 -> so that's what we'll talk about today all right
so this is the we're continuing with rigid body
323.36 -> kinematics but specifically the kinematic
differential equation for euler angles
330.24 -> when we describe rigid bodies you typically
describe their the translational motion and the
337.2 -> rotational motion for the translational motion you
pick some reference point on the body or in the
343.28 -> body often it's the center of mass it doesn't have
to be and you write what's the location of that
350.96 -> center of mass so some position vector then how
does that position vector change we would often
356.64 -> write this vector c i mean rc location of the
center of mass in terms of end frame components
366.56 -> so this would be x y z and let's say all of these
are changing in time so this is changing in time
375.28 -> then the way that we would write how are x y
and z changing we've got the inertial derivative
383.12 -> of the vector r c and that'll give us x dot y dot
z dot time rate of change of all of those and this
393.92 -> equals whatever the velocity is maybe i'll write
the velocity this way here's the inertial velocity
401.2 -> of the center of mass at a given moment so this
is a vector and it's got three components so we
408 -> could either write it as okay here's the vector
and write it with respect to the end frame or
414 -> in terms of components the velocity in the x
direction velocity and the y velocity in the
419.76 -> z and these are all changing in time possibly now
you don't often think about it but what does this
427.84 -> look like if we write it out there's an implied
identity matrix it's in the front the x v y v z
440.08 -> let me always remind myself what frame i'm in so
this would be the root the translational kinematic
447.28 -> differential equation and we i mean we
often don't even write this identity
451.92 -> matrix because it's a waste of time it's the
identity matrix so this is the translational
458.24 -> kinematic enametic differential equation right and
where do the v's come from the way the v changes
467.92 -> in time comes from translational dynamics so from
f equals m a newton's second law let's leave that
477.12 -> for now let's just say okay if i wanted to if
i know how v is changing with time and i wanted
482.64 -> to reconstruct what the position of my rigid
body is so we have the path that it's taking
489.36 -> then i would integrate this set of kinematic
differential equations and the reason i'm
494.24 -> emphasizing there's this three by three identity
matrix is because when we go over to euler angles
503.12 -> instead of there being an identity
matrix here there's something else
509.04 -> so if we've got
513.12 -> for rotational kinematics
519.04 -> so the rotational kinematic differential
equation if we write the orientation of this body
527.04 -> so remember what the orientation
is you've got some body fixed frame
531.44 -> like that blue frame attached to the body we write
how that's oriented with respect to the inertial
537.84 -> frame so it's like you line up the frames exactly
right on top of each other and then calculate this
543.6 -> cosine matrix or the euler angles in terms
of euler angles let's say the three two one
555.2 -> so yaw pitch and roll you'll get a kinematic
differential equation that looks like this so
562.56 -> time rate of change of yaw time rate of
change of pitch time rate of change of roll
570.24 -> those aren't really written with respect to any
particular frame so i don't even have to put a
574.48 -> little vector thing on it but there
will be a three by three matrix here
580.24 -> that is going to be a function
of the um euler angles
590.96 -> so for now i'll just put in stars to represent
that those are components this matrix
601.36 -> in the terminology of the book it calls this
matrix b so this matrix b each of the entries
606.56 -> is going to be a function of the euler angles
so we could say this matrix b is a function of
611.92 -> yaw pitch and roll times instead of translational
velocity that's what v is this is the rotational
621.6 -> or angular velocity so we'll have omega 1 as a
function of time omega 2 as a function of time
630.8 -> omega-3 everything is changing the time so if
i were to emphasize that over here right this
637.36 -> y'all pitch and roll everything's changing with
time and just like before where you get the how is
647.2 -> the velocity changing it comes from translational
dynamics how is the angular velocity changing
652.56 -> that comes from rigid body dynamics which will be
the subject of the next chapter so this right for
661.92 -> right now i'll say this b matrix is unknown and we
want to find out what it is and we always write um
671.04 -> this is these are the components
of the angular velocity
675.6 -> vector written in the b frame so if i were to
go back up here and write the angular velocity
684.64 -> that's the angular velocity of the b frame with
respect to the end frame and we always write it
694.72 -> in terms of b frame components and in general
those will change in time due to whatever the
701.28 -> moments are and other things okay so let's find
out what's this unknown b matrix and it's going
708.56 -> to be different for each of the 12 euler angle
conventions we'll focus on yaw pitch and roll
717.84 -> the b matrices are listed in appendix b
of the book for all of the different 12.
724.4 -> but it's it's probably useful to just
start with y'all pitch and roll so what is
733.28 -> what is the b matrix the way we get at this is we
look at the the euler angle sequence of rotations
744.24 -> and for each individual rotation we can write the
angular rate it'll be the time rate of change of
752.64 -> one of those euler angles and a direction so an
axis of rotation all right for yaw pitch and roll
759.84 -> uh this is the three two one euler
angle sequence or euler angle convention
770.88 -> so the first one you think of that initially the
two frames are aligned right so we've got the
781.84 -> inertial frame and the body fixed frame
you think of those as initially aligned
791.36 -> so these are as aligned as i'm going to be
able to do it b1 b2 b3 n1 and two and three
804.24 -> and then the first rotation is about the
number three axis the number three axis
813.68 -> through an angle phi which means we're
going from here to to what we rotate
824.8 -> through some angle not v psi so that
means we're trying to sketch this
835.84 -> those n directions will just always be
staying the same but this new b directions
843.36 -> i'll call this b1 prime b2 prime and the number
three direction did not change this angle is psi
859.04 -> psi right so we did a rotation through a yaw angle
if we were to write this as a angular velocity
871.6 -> we did the angular velocity from the um i guess
we could call this the b prime frame from the
882.56 -> end frame and so this is just one simple
rotation so we write the time rate of change
891.2 -> of that angle psi dot and then the direction
is we could either write it as b3 or n3
900 -> i mean b3 prime i'll write it as b3 prime and we
may also want to remind ourselves from last time
907.76 -> how are these directions b1 b2 b3 how are
they related to the n directions it's a
918.08 -> clockwise uh no counterclockwise rotation
it's positive rotation in the sense of the
925.12 -> right hand rule about the number three axis
so we had cosine psi sine psi negative sine
934.08 -> psi cosine psi zero zero one times i'll use
the right colors here and one and two and three
946.64 -> we're gonna need this because we're gonna have
to keep track of what these angles are eventually
952.72 -> okay so that's the first rotation what's the
next rotation the next rotation is we rotate
960.64 -> about the new number two direction what's the new
number two direction it's this one here b2 prime
966.4 -> we're going to rotate about that direction
through an angle theta the pitch angle
974.96 -> for this second rotation i'll try
to write b1 prime here b2 prime
985.76 -> b3 prime and we're going to a new set of
directions so we're rotating about this axis
998.72 -> the positive sense is given by the right hand
rule so my thumb points in the direction of
1002.16 -> that b2 prime axis my fingers are curling
in the direction of that theta is increasing
1012.24 -> so what does that mean um we'll call this b2
sorry b3 prime prime so we've rotated through an
1023.12 -> angle theta there and they'll be this will also
rotate through an angle theta this will be b1
1032.8 -> prime prime that's our new b1 direction
so this is the second rotation
1041.04 -> about the new number two axis which is b2
prime it's also equal to since we didn't change
1052.96 -> b2 double prime b2 double prime and that's through
an angle theta i'll keep track of these what's the
1064.16 -> angular velocity the angular velocity going from
the b prime frame to the b double prime frame this
1073.92 -> is theta dot it's the angular rate of rotation and
then the axis we could either write it as b2 prime
1083.2 -> or b2 double prime we'll write b2 double prime
and then again to remind ourselves how are the b
1092.16 -> double prime directions related to the b prime
directions i need to do a rotation matrix so this
1098.72 -> is b1 double prime b2 double prime b3 double
prime equals it's going to be some 3x3 matrix
1107.76 -> times the b1 prime b2 prime b3 prime and this is
a pure rotation about the number two direction
1120.16 -> and this one you i usually just have
to look it up cosine negative sign
1127.28 -> down here will be sine and then cosine
1131.44 -> so good we've got that the last one the third
and last rotation is about the new number one
1138.56 -> axis which is this b one double prime direction
so we'll do a rotation again in the positive
1147.2 -> sense will be my thumb pointing in b1 double prime
fingers curling in the direction of the roll angle
1153.76 -> fee so let me see if i can draw
this down here reliably this is b1
1161.92 -> double prime and i've got b three double prime and
over here b two double prime these don't look very
1172.48 -> 3d anymore we're rotating about that direction
through an angle fee so what does that mean
1181.68 -> that means uh let me write it this way here's
my angle v and we've rotated now i won't put
1189.36 -> any primes on these because these represent the
actual body directions b1 b2 b3 so we've got that
1199.28 -> and then this one has also rotated v now this
is looking complicated now b2 and then this is
1208.4 -> direction b1 so we'll just leave that as it
is so third and final rotation about the new
1218.32 -> number one axis which is b1 double
prime but that's also equal to just b1
1227.2 -> and this is through an angle v so if we were
to write what the angular velocity is this goes
1234 -> from the b double prime to the b frame the
angular velocity for just that rotation would be v
1243.52 -> dot and then i could either write b1 double
prime or b1 well i'm just going to write b1
1251.6 -> for completeness we could write
what the relationship is between the
1255.76 -> b unit vectors b1 b2 b3 and the b double prime
unit vectors b1 double prime b2 double prime
1269.84 -> b3 double prime this is a pure rotation about
the number one direction and this is cosine phi
1279.68 -> sine phi negative sine v cosine phi
okay cool now the total angular velocity
1289.2 -> would be what the total angular velocity now we
use the angular velocity addition formula so the
1296.24 -> total angular velocity is the angular velocity
in going from the the end frame to the b frame
1304.96 -> or b frame with respect to the n frame and we just
add all of these individual angular velocities up
1313.52 -> so we would have angular velocity of
b with respect to b double prime plus
1320.64 -> angular velocity of b double prime with respect
to b prime what color i do the first one just
1328.16 -> black b prime with respect to n you add all those
up and what do you get well this is v dot b1
1339.76 -> what's this one this one was uh theta dot b2
prime prime plus psi dot b3 prime i guess i did
1355.2 -> this one in blue so just for completeness let's
write it in blue there we go okay now we've got
1362.16 -> the the total angular velocity vector before
we've just called this omega omega is omega the
1368.48 -> b frame with respect to the end frame and we want
to write that we want this in terms of just the b
1377.28 -> frame components so instead of having these double
primes and everything these sort of intermediate
1384.32 -> rotation directions we want to write b2
double prime and b3 prime in terms of
1391.28 -> the b1 b2 and b3 directions so this
is what we want we want something
1399.12 -> in the b1 direction plus something in the
b2 direction plus something so these little
1406.64 -> parentheses with stars stand for okay we want
to put something in in that form which means
1410.88 -> we're going to have to rewrite b3 prime and b2
double prime how do we do that let's start with b
1420.64 -> equals so we're talking about this one here
this is actually a pure rotation about the
1426.72 -> number one direction through an angle
phi times the b double prime directions
1437.2 -> now to get the b double prime directions in terms
of the b directions we can take the transpose
1444.48 -> because the transpose is the inverse of the matrix
so if we take uh we can get b double prime equals
1455.92 -> m one phi t t here means transpose times the
b directions and what is it's easy to take
1466 -> the transpose of a matrix so then we'll get
you know b1 double prime b2 double prime b3
1475.6 -> double prime i take the transpose of this
matrix up here and it looks like i just remove
1481.76 -> a a negative sign right there from that one to
the one on the opposite side of the diagonal
1488.4 -> okay i can do that cosine v negative sine phi
1497.12 -> sine phi cosine v b1 b2 b3 and all i
want is the b2 double prime component
1509.52 -> so if i solve this out i get b2 double
prime equals cosine phi times b2
1524.08 -> minus sine phi b3 right so then i would
substitute what i have into here and now
1534.08 -> i'm on my way because now i've got things in terms
of the b directions not b prime not b double prime
1540.56 -> okay what about this one this b three double
prime we could do that um what was this the
1552 -> where was it yeah i'll rewrite this right
what was this pure rotation about the
1558.88 -> number two direction through an
angle theta so i'm going to rewrite
1565.36 -> this but in the shorthand form so that would
be b double prime the vectrix equals m2
1572.72 -> times the b prime vectrix all right because that
comes from that second rotation this is m2 theta
1583.6 -> i don't know if i wrote that one in a
particular color no okay b prime and
1592.72 -> then what do i get i could write this is
b equals um pure rotation through v times
1607.92 -> pure rotation about the number two
axis through theta times um b prime
1621.6 -> and if i want to get the b prime directions then
i just i i'm taking the transpose of this equation
1629.04 -> so the transpose of that will give me
1632.4 -> m 2 theta right when you take the
transpose you reverse the order
1638 -> m1 v transpose times b and then i can work
out what that matrix multiplication gives me
1649.44 -> uh at least for the the only component i
care about which is b3 so this gives b3 prime
1657.76 -> is equal to negative sine theta b1 plus sine
phi cosine theta b2 plus cosine v cosine theta
1677.04 -> b3 so that's let me put a box around these
two because then i'm just substituting back
1685.92 -> into this equation up here so omega which was
omega b frame with respect to the end frame
1696.48 -> is going to equal and now i'm going to group
them as everything that's in the b1 direction
1701.2 -> everything in the b2 and everything in
the b3 i get negative sine theta times
1708.4 -> v dot sorry psi dot plus v dot that's in
the b1 direction plus sine phi cosine theta
1720 -> psi dot plus cosine v theta dot that's
all in the b2 direction and then cosine v
1731.52 -> cosine theta psi dot minus sine theta sine
phi theta dot b3 all right and then just
1741.04 -> identifying what these components are this is the
component of omega in the b1 direction so this is
1747.52 -> just omega 1. this is the component of omega
in the b2 direction so it's omega 2. this is
1755.84 -> the component of omega in the b3 direction
so it's omega-3 so i could summarize this
1762.4 -> in matrix form and it's also tiring writing all
those sines and cosines so i'll just use shorthand
1772.56 -> yeah this shorthand over here this is just you
know s it's going to be sine c is going to be
1781.76 -> cosine so if we were to summarize this we have
omega 1 omega 2 omega 3. and we remind ourselves
1790.88 -> these are the b frame components and this equals
we're going to write this as some 3 by 3 matrix
1800.8 -> times the time rates of change of the euler
angles psi dot or time rate of change of yaw
1808.8 -> theta dot time rate of change of pitch v
dot time rate of change of rule and in here
1815.36 -> this is negative s theta and then zero and one i'm
just sort of rewriting that component up there now
1823.44 -> we write component omega 2 sine phi cosine theta
which will be times psi dot and then cosine v
1837.44 -> times theta dot and then zero and then this last
one this is negative what sine v and zero we're
1847.92 -> almost there we've got this relationship where
we've said okay here's the omega vector it's
1855.84 -> this weird matrix it's a function of the euler
angles time the euler angle rates of change
1862.32 -> and this was just for one of the 12 euler angle
conventions so remember so this isn't what you
1870.32 -> would get for all of the euler angle conventions
this is just for the yaw pitch and roll
1876 -> three two one sometimes people write it not in
terms of the number but in terms of uh the letter
1882.4 -> so they would call this the z y x so you might see
that it means the same thing also there's no uh
1892.24 -> binding rule that you have to call the first
angle yaw that's just another common convention
1900.4 -> or that you even have to label it psi so to if
i were to summarize this in shorthand i would
1905.76 -> call this vector it's usually written as a theta
but as a vector and it's the time rate of change
1915.92 -> so we define this theta vector and it's just made
up of the three angles the first angle that you
1923.68 -> rotate through theta one the second and then
the third which for three two one this is psi
1932.08 -> theta v okay and then this matrix the book
calls this the b matrix uh actually this
1941.84 -> is the inverse of the b matrix inverse of the
matrix and since it depends on the euler angles
1951.44 -> we say it's a function of this theta
vector and then what is this well this
1956.72 -> is the probably the easiest one to write this is
omega vector but written in terms of the b frame
1963.84 -> that's sort of the compact way that we
could write this omega equals b inverse
1975.28 -> times time rate of change of the euler angles
people who are brand new to rigid body dynamics
1981.28 -> sometimes think this vector the time rate of
change of the euler angles equals the angular
1986.88 -> velocity and this exercise is meant to show
you that no it's not there's this weird matrix
1994.64 -> and that's just life if you went from the euler
angle rates of change to the angular velocities
2001.28 -> but often it goes the other way we want to
write what are the euler angle rates of change
2007.36 -> well we just multiply both of these equations by
the b matrix and b times b inverse disappears so
2016.72 -> here's the b matrix times omega so these are this
would be the one that i would call the kinematic
2025.44 -> differential equation i guess we could say it's
the rotational kinematic differential equation
2035.68 -> written in terms of euler's vector
euler angles all right and uh what's
2043.6 -> what i'm not showing here is there's always
going to be something out in front so b theta
2051.36 -> for example will have something like
1 over sine theta 2 times some stuff
2058.88 -> this is just an example so if we had this
then we'd say oh this matrix numerically
2064.32 -> blows up at theta two goes to zero meaning it
has a singularity just because of this term
2073.28 -> blows up blows up just means it goes
to infinity as theta 2 goes to zero
2080.24 -> and that's not good for any kind of numerical
implementation if you're trying to come up with
2086.56 -> some automated algorithm and so this is one of
the the well-known problems with the euler angles
2093.76 -> when things go to infinity that's also called a
singularity so that is um it has a singularity
2105.6 -> so it's not well behaved at all angles if you
for whatever reason no oh i'm always going to
2110.48 -> avoid theta two equals zero or anything close to
zero then okay fine but you don't know that let
2117.04 -> me show you these euler angle conventions so the
appendix b has the direction cosine matrices c
2128.08 -> and either in the same appendix or a nearby one
2135.28 -> it has the b matrices and also the inverse for
each of the conventions so we've been looking at
2140.32 -> three two one right the yaw pitch and roll
so notice what this has this one has a
2147.2 -> the b matrix look like that it actually has a one
over cosine theta two which means this thing has a
2156.16 -> singularity when theta 2 is plus or minus pi over
2 or plus or minus 90 degrees which is what we
2165.92 -> expected from before in that little
illustration so this is the b matrix
2170.8 -> and then this is b inverse over here b inverse
is the one that's easier to calculate and then
2176.96 -> just get the the inverse of it from uh procedures
for getting the inverse of a matrix so you have
2184.56 -> this for all of the conventions look there's
there's three one three right up above it
2189.44 -> appendix of uh schaub and jenkins has
b and b inverse for all the conventions
2198.72 -> this here over here is just for an
example for all 12 euler angle sets
2206.32 -> and i don't know um the details of why you
would pick one set versus another because
2211.92 -> you might wonder why are there 12 am i only going
to use yaw pitch and roll there may be some cases
2218.08 -> where one of them makes sense but i don't have
an answer for that right now we can look at these
2223.84 -> further i don't know what this is these are
three non-linear odes right we're writing
2233.52 -> a like a matrix ode or a vector ode but this
represents 3 non-linear ordinary differential
2244.48 -> equations one for each of the euler angles and
how they change with time and like i said they
2251.36 -> have they all have problems with singularities
this sort of shows you this is not the same as
2257.68 -> this b matrix is not the identity so if we were to
compare i'm going to sneak up to the top up here
2266.72 -> where i think it's useful to compare translational
kinematic equations which are really simple it's
2273.84 -> the translational kinematic equation if we write
it in vector form it's just like rc dot equals vc
2281.84 -> that's super easy it's not so easy for
2285.84 -> euler angles um unfortunately you have this
so now we've we've called this what theta dot
2293.12 -> we've got this as the omega vector there's
always this b matrix that isn't constant
2300.4 -> it's not the identity the entries are all
changing in time and functions of order angles
2307.92 -> oh well and they're non-linear and they have
the problem of singularities euler angles are
2313.92 -> a good conceptual tool to first think about how to
represent rigid body rotations but they aren't the
2320.32 -> the main one that gets used so there are this
is kind of an aside but we will get to it
2326.96 -> there are alternatives to euler angles
that don't have the singularity problem
2335.68 -> and they're based on the principal
2339.28 -> rotation vector i'm just gonna throw out some
names here it's not like you gotta remember
2345.04 -> euler parameters quaternions the way these often
work is instead of having three three parameters
2351.76 -> any set of three parameters is going to have a
singularity so you have to add a fourth parameter
2358.4 -> and so with the sets of four parameters do
not quaternions as you might see from the name
2365.36 -> quaternions they've got four parameters so these
are used and they don't have the singularity
2372.48 -> problem if we were to summarize what we've got for
the rotational kinematic differential equations so
2379.6 -> far purpose is to go from the something you get
from dynamics which is how the angular velocity
2389.6 -> go from the angular velocity which in general
will be changing in time to the current attitude
2397.52 -> orientation which we've written as the matrix c so
there's so far we've talked about two ways to do
2403.52 -> that you could write the matrix differential
equation for c and that'll be something that
2415.04 -> looks like that these are nine linear odes
which means they're not as hard to deal with
2423.04 -> or we've got now this other way i
guess uh written it a few times now
2432.48 -> maybe this seems more straightforward three
non-linear odes and from this right you would
2440.8 -> find out how do the thetas change in time so
this gives how theta i'll write it this way
2448.48 -> for sort of a general set once you know how those
thetas change in time then you could reconstruct
2453.84 -> the matrix the rotation matrix c because c is a
function can be written as a function of those
2462.48 -> then you've got the current orientation
there are other approaches too
2468.4 -> so that what i talked about here was from
section if you're looking to read section 3.3
2476.4 -> we're now going to talk about section 3.4 which
introduces the principal rotation vector and
2483.28 -> this thing called euler's principle uh rotation
theorem the b matrix isn't orthogonal no yeah
2491.68 -> that's why yeah it's a good good question that's
why uh b inverse is not the same as b transpose
2499.68 -> the c matrix is orthogonal because it's a rotation
but this thing b is not a rotation i don't even
2506.08 -> know what it is it might not be any kind of
special matrix at all but yeah good question
2513.04 -> all right in the last 15 minutes i'll talk about
the principal axis and angle these alternatives
2520.4 -> i talked about up here alternatives to euler
angles that don't have the singularity problem
2525.6 -> they're all based on this thing called the
principal rotation vector or the axis um i
2532.4 -> sometimes call it the axis angle let's talk about
that so this is the principal rotation vector
2544.32 -> maybe let's think of it this way we're going
to start with this way of writing the angular
2551.04 -> or the the kinematic differential equation for
the rotation matrix c in some sense this is the
2558.32 -> most fundamental one this one will not steer you
wrong because it's uh it doesn't have any singular
2565.12 -> singularity problems so this is the most
fundamental kinematic differential equation
2569.84 -> of rotation maybe i'll just uh that's hard to
write so right as kde kinematic differential of
2578.8 -> equation of rotation we can come we can combine
well first let me mention euler's theorem
2590.24 -> it's sometimes called euler's rotation theorem
or euler's principle rotation theorem the idea
2597.6 -> is that you could go from any frame to any other
frame you can go from you could write a rotation
2607.6 -> i'll use the typical colors i've
been using you can write the rotation
2618.32 -> going from everything's aligned so the blue
and the red the end frame and the b frame being
2623.76 -> aligned to the b's current orientation instead
of writing that as a sequence of three rotations
2631.6 -> you could just write it as one rotation about
some special direction so there's some special
2637.76 -> direction we write it as a unit vector e so
that special direction there's just one rotation
2647.76 -> uh it's through that special direction
e or about that special direction e
2652.96 -> through an angle capital phi i've said the
essence of the theorem any rotation matrix
2659.44 -> from the then i'm writing the end frame but it's
any frame to any other frame right it's just a
2666.32 -> triad of unit vectors can be written as a single
rotation about a unit vector e through an angle
2681.68 -> capital phi and these are sometimes called
the principal axis and principal angle
2688.88 -> it's somewhat hard to illustrate i i
couldn't find anything online actually uh
2696.64 -> but if i have if i want to write i've got these
two frames oriented in some just kind of random
2703.36 -> way there is some special direction i'll use the
orange vector here and to go from red to blue all
2712.72 -> i needed to do was rotate about that one special
direction so one rotation through some about some
2718.32 -> special angle through some special uh about
some special axis through some special angle
2725.04 -> and that's the theorem that euler derived where
there did a lot there's a lot of low-hanging fruit
2731.44 -> so we can combine these into a special
vector maybe we'll wait on that the
2737.76 -> interesting thing about this e direction is
that it has the same components in both the b
2744.24 -> and the end frame this principal axis we could
write it as e b one in the b one direction
2751.52 -> plus e b two in the b two direction plus e b
three the b three direction we could also write it
2761.76 -> in the n direction en2 in the n2 direction plus
e n 3 and the n3 direction and it turns out
2773.28 -> the components are equal so that means e b sub i
equals e n sub i so we don't even have to put the
2782.64 -> b we'll just write these as e sub i so it's the so
this hopefully makes sense it's the one direction
2791.52 -> because you're rotating about it anything along
there uh won't change the e directions equals
2800 -> this c matrix times the e directions or those
e components it doesn't change if you recall
2807.76 -> eigenvalues and eigenvectors then this looks
like we took the matrix c multiplied by some
2816.16 -> direction and then we have one times e so e is
the eigenvector i mean we should emphasize we
2826.24 -> have to normalize this it's a unit vector it's the
normalized eigenvector corresponding to eigenvalue
2834.24 -> plus one of c so you can find the you can
find this e direction by finding calculating
2843.04 -> the eigenvalues of c there's always going to be
an eigenvalue 1 and there'll be a corresponding
2849.12 -> eigenvector for any matrix any rotation matrix
so that's kind of cool how does this get used
2857.12 -> we can use this by combining the e direction and
this angle into a common vector so combine the
2866.16 -> axis it's just the unit vector and the angle
into one vector which i'll write as gamma
2875.52 -> so if we combine these this is the principal
rotation vector write it this way and we can show
2883.6 -> so this is the rotation vector when omega is
fixed in a direction in space it turns out
2893.12 -> the derivative of this principal rotation vector
equals omega so now this is almost looking like
2901.92 -> the simple translational kinematic ode so so
that's good there's something there's something
2909.04 -> good here so it means there is some special vector
whose time rate of change equals the angular
2915.76 -> velocity and there's a relationship between the
gamma and the c matrix if we were to put this into
2924.88 -> i said was this fundamental kinematic differential
equation up here right this was negative omega
2931.36 -> tilde times c well if we were to write this in
terms of you know using this expression instead
2939.92 -> of omega tilde we could do the derivative of gamma
tilde times c again assuming that only the angle
2950.32 -> is changing not the axis and this matrix ode is
solved by and this is pretty cool c equals e to
2959.36 -> the negative gamma tilde and maybe i guess i'll
write it that way so this is a matrix exponential
2967.6 -> if you haven't seen it if i have a square matrix
a so let's say e to the a e to the a will be
2974.8 -> and let's say that this is a n by n matrix
then you write the taylor series expansion
2982.88 -> as if it were a scalar but then you put in
the matrix so this is going to be 1 plus
2990.8 -> a plus 1 over 2 factorial a squared plus 1 over
3 factorial theta cubed and so on so that's what
2999.76 -> a matrix exponential is you could quickly write
down what's the rotation matrix in terms of the
3005.92 -> principal rotation vector this is just a intro so
for next time read section 3.4 and we'll say more
3015.44 -> about this principle rotation vector and maybe get
into the other alternatives to the euler angles
3022.08 -> that come from this so that principle rotation
vector is important so i'll stop there i see
3027.68 -> some questions uh the expansion would stop at n no
no it keeps on going so this is an infinite series
3038.08 -> just like with any taylor series it turns
out though if a has a special form then
3044.8 -> it will truncate at some point uh what
does the till day mean till day oh well
3052.64 -> if you don't remember if we have a three
vector i'll call it a then the a tilde
3061.6 -> you make a three by three matrix with entries of
a so if a is has entries a1 a2 a3 it's a 3 vector
3071.6 -> then this is negative a 3 a2 negative a1 a1
negative a2 a3 if you remember i think it was like
3083.28 -> either lecture two or lecture three this came from
if you write a cross b so if you want to write the
3090.4 -> cross product you could actually write the cross
product as a matrix multiplication so it'd be this
3096.16 -> a tilde cross b so that's where it comes from
so yeah it's kind of weird why does this this
3103.44 -> matrix that has to do with cross products uh show
up and what's it doing in a matrix exponential
Source: https://www.youtube.com/watch?v=Z8nwjouP58o