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Rotational Kinematics
Descr i pt i on of at t i t ude ki nemat i csusi ng r ef er ence f r ames, r ot at i on
mat r i ces, Eul er par amet er s, Eul erangl es, and quat er ni ons
Recall the fundamental dynamics equations
For both equations, we must relate momentum to
kinematics
~f = ddt
~p
~g = ddt
~h
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Translational vs Rotational Dynamics
Linear momentum
=mass velocity
d/dt (linear momentum)=
applied forces
d/dt (position)
=
linear momentum/mass
Angular momentum
=inertia angular velocity
d/dt (angular momentum)=
applied torques
d/dt (attitude)
=
angular momentum/inertia
~f = ddt
~p ~g =ddt
~h
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Translational Kinematics
Newtons second law can be written in
first-order state-vector form as
~r = ~p/m
~p = ~f
Here, ~p is the linear momentum, definedby the kinematics equation; i.e., ~p = m~r
Thus, the kinematics differential equationallows us to integrate the velocity tocompute the positionThe kinetics differential equation allows usto integrate the applied force to computethe linear momentumIn general, ~f
=~f
(~r
, ~p
)
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Translational Kinematics (2)
Consider ~f = m~a expressed in an inertial frame:
mr1 = f1
mr = f
mr2 = f2
mr3 = f3
Equivalently
p1 = f1p = f p2 = f2
p3 = f3
r1 = p1/m
r = p/m r2 = p2/mr3 = p3/m
We need to develop rotational equations equivalent
to the translational kinematics equations
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Reference Frame & Vectors
i1 i2
i3
~v
v1 v2
v3
Reference frame is
dextral triad oforthonormal unitvectors
Vector can be expressedas linear combination of
the unit vectors
Must be clear aboutwhich reference frame is
used
Fi n
i1, i2, i3
o
~v = v1i1 + v2i2 + v3i3
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Orthonormal
Orthonormal means the base vectors are perpendicular(orthogonal) to each other, and have unit length(normalized)
This set of 6 (why not 9?) properties can be written in
shorthand as
We can stack the unit vectors in a column matrix
And then write the orthonormal property as
i1 i1 = 1 i1 i2 = 0 i1 i3 = 0
i2 i1 = 0 i2 i2 = 1 i2 i3 = 0i3 i1 = 0 i3 i2 = 0 i3 i3 = 1
ii ij =
1 if i = j0 if i 6= j
or ii ij = ij
nio = ni1 i2 i3oT
ni
on
i
oT
=
1 0 00 1 0
0 0 1
= 1
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Dextral, or Right-handed
Right-handed means the ordering of the three
unit vectors follows the right-hand rule
Which can be written more succinctly as
Or even more succinctly as
i1 i1 = ~0 i1 i2 = i3 i1 i3 =
i2
i2 i1 = i3 i2 i2 = ~0 i2 i3 = i1i3 i1 = i2 i3 i2 = i1 i3 i3 = ~0
nion
ioT
=
~0 i3 i2i3 ~0 i1i2
i1 ~0
=
nio
ii ij = ijk ik
ijk =
1 for i,j,k an even permutation of 1,2,31 for i,j,k an odd permutation of 1,2,3
0 otherwise (i.e., if any repetitions occur)
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Skew Symmetry
The [] notation defines a skew-symmetric 3 3
matrix whose 3 unique elements are the
components of the 3 1 matrix []
The same notation applies if the components ofthe 3 1 matrix [] are scalars instead of vectors
The skew-symmetry property is satisfied since
a =
a1a2
a3
a =
0 a3 a2
a3 0 a1a2 a1 0
(a)T = a
nio
=
~0 i3 i2i3 ~0 i1i2 i1 ~0
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Vectors
A vector is an abstract mathematical object with
two properties: direction and length
Vectors used in this course include, for example,
position, velocity, acceleration, force,
momentum, torque, angular velocity
Vectors can be expressed in anyreference frame
Keep in mind that the term state vector refers
to a different type of object -- specifically, a state
vector is generally a column matrix collecting all
the system states
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Vectors Expressed in Reference Frames
i1i2
i3
~v
v1 v2
v3
~v = v1i1 + v2i2 + v3i3The scalars, v1, v2, and v3, arethe components of ~v expressedin Fi. These components arethe dot products of the vector ~vwith the three base vectors ofF
i.
Specifically,
v1 = ~vi1, v2 = ~vi2, v3 = ~vi3
Since thei vectors are unit vec-tors, these components may also
be written as
vj = v cosj, j = 1, 2, 3
where v = k~vk is the magnitudeor length of~v, and j is the angle
between ~v and ij for j = 1, 2, 3
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Vectors Expressed in Reference Frames (2)
Frequently we collect the components of the vector into
a matrix
When we can easily identify the associated reference
frame, we use the simple notation above; however,when multiple reference frames are involved, we use a
subscript to make the connection clear. Examples:
v = v1
v2v3
vi denotes components in Fi
vo denotes components in Fo
vb denotes components in Fb
Note the absence of an overarrow
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Further Vector Notation
Matrix multiplication arises frequently in dynamics and
control, and an interesting application involves the 3 1
matrix of a vectors components and the 3
1 matrixof a frames base vectors*
We frequently encounter problems of two types:
* Peter Hughes coined the term vectrix to denote this object
~v = [v1 v2 v3]
i1
i2
i3
= vTi
nio
= vTi
nio
= vTo {o} = vTb
nbo
=
Given vi and vb, determine the attitude ofFb with respect to Fi
Given the attitude of Fb with respect to Fi, and components vi,determine vb
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Rotations from One Frame to Another
Suppose we know the components of vector ~v in Fb, denoted vb, andwe want to determine its components in Fi, denoted vi
In some sense this problem has three unknowns (the components of
vi); hence we expect to form a set of three equations and three unknowns
Specifically, we note that ~v = vTi
nio
= vTb
nbo
and seek a linear
transformation R such that nio = Rnbo With such a transformation, we can make a substition and have ~v =
vTi Rn
bo
= vTb
nbo
Since the base vectorsn
bo
are orthonormal, the coefficients on both
sides of the equation must be equal, so
v
T
i R = vT
b RT
vi = vb
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Rotations (2)
Problem reminder: Knowing vb, determine vi
We have RTvi = vb
If we know R, then we just have to solve the linear system to determinevi
We know that R is a 3 3 matrix (i.e., R R33), and thatn
io
= Rn
bo
The latter can be expanded to
i1 = R11b1 + R12b2 + R13b3
i2 = R21b1 + R22b2 + R23b3
i3 = R31b1 + R32b2 + R33b3
What, for example, is R11?
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Rotations (3)
Consider just one of the equations involve the components of R:
i1 = R11b1 + R12b2 + R13b3
Comparing this expression with the definition of the components of avector in a specific frame, we see that
R11 = i1 b1, R12 = i1 b2,
and in general, Rij = ii bj
Using direction cosines, R11 = cos11, R12 = cos12, and in general,Rij = cosij, where ij is the angle between ii and bj
Thus R is a matrix of direction cosines, and is frequently referred toas the DCM (direction cosine matrix)
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Rotations (4)
Another way to describe R is to observe that its rows contain thecomponents of the base vectors of Fi expressed in Fb, and that itscolumns contain the components of the base vectors ofFb expressed inFi
These observations mean that the rows and columns are mutuallyorthogonal, and since the base vectors are unit vectors, the rows andcolumns are mutually orthonormal
Recalling earlier notation,n
ion
ioT
= 1, we can also write R as the
dot product ofn
io
withn
boT
, i.e.,
R =n
ion
boT
So, if we know the relative orientation of the two frames, we cancompute R and solve the required linear system to compute vi
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Rotations (5)
Assuming we have computed R, we just need to solve the requiredlinear system to compute vi
The linear system, RTvi = vb, is easily solved because of the previ-
ously observed fact that R is an orthonormal matrix
If we were to return to the beginning of this development and beginwith
RTnio = nbo instead of nio = Rnbowe would find that the same matrix R satisfies both equations, thusproving that R1 = RT, which is perhaps the most useful property oforthonormal matrices
Its application here leads to
vi = Rvb vi = Ribvb
We adopt the notation Rib
to denote the orthonormal matrix that trans-forms vectors from Fb to Fi
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Rotations (6)
Summary of Rotation Notation
Rotation matrix, orthonormal matrix, attitude matrix, and directioncosine matrix are synonymous
The inverse of the rotation matrix is simply its transpose
The orthonormal matrix that transforms vectors from Fb to Fi isdenoted Rib:
vi = Rib
vb and vb = Rbi
vi
The rotation matrix can be computed using
Rib = nio nboT
and Rbi = nbo nioT
Rib =
i1bi2b
i3b
= b1i b2i b3i
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Rotations (7)
More Properties of Rotation Matrices
Since R1 = RT, RTR = 1 (the identity matrix)
The determinant of R is unity: det R = 1
One of the eigenvalues of R is +1
Every rotation corresponds to a rotation about a single axis a through
an angle ; this fact implies that Ra = a, and hence a is the eigenvectorcorresponding to the unity eigenvalue
Rotations multiply; i.e., if Rab relates frames Fa and Fb, and Rbc
relates frames Fb and Fc, then
Rac = RabRbc
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Rotational Kinematics Representations
The rotation matrix represents the attitude
A rotation matrix has 9 numbers, but they are
not independent
There are 6 constraints on the 9 elements of a
rotation matrix (what are they?)
Rotation has 3 degrees of freedom There are many different sets of parameters that
can be used to represent or parameterize
rotations Euler angles, Euler parameters (aka quaternions),
Rodrigues parameters (aka Gibbs vectors),
Modified Rodrigues parameters,
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Euler Angles
Leonhard Euler (1707-1783) reasoned that the
rotation from one frame
to another can bevisualized as a sequence
of three simple rotations
about base vectors
Each rotation is through
an angle (Euler angle)
about a specified axis
Consider the rotation from Fi toFb using three Euler angles, 1, 2,and 3
Thefi
rst rotation is about thei3axis, through angle 1
The resulting frame is denoted
Fi0 or ni0o The rotation matrix from Fi toFi0 is
Ri0i = R3(1)
Ri0i = R3(1) =
cos 1 sin 1 0 sin 1 cos 1 0
0 0 1
vi0 = R3(1)vi
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Visualizing That 3 Rotation
Ri0i = R3(1) =
cos 1 sin 1 0 sin 1 cos 1 0
0 0 1
vi0 = R3(1)vi
i11
1
i2i0
2
i3 and
i0
3 are out of the page
i0
1
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Euler Angles (2)
The second rotation is about thei0
3 axis, through angle 2
The resulting frame is denoted
Fi00 or ni00o The rotation matrix from Fi0 toFi00 is
Ri00i0
= R2(2)
The notation for the simple rota-tions is Ri(j), which denotes a ro-
tation about the ith axis. The sub-script on R defines which axis isused, and the subscript on de-fines which of the three angles in
the Euler sequence used
Ri00
i0
= R2(2)
=
cos 2 0 sin 20 1 0
sin 2 0 cos 2
vi00 = R2(2)vi0 = R2(2)R3(1)vi
Ri00i = R2(2)R3(1)
is the rotation matrix transformingvectors from Fi to Fi00
For an Ri rotation, the ith row andcolumn are always two zeros and
a one. The other two rows andcolumns have cos and sin in aneasily memorized pattern
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Euler Angles (3)
The third rotation is about the i00
1
axis, through angle 3
The resulting frame is denoted Fb
orn
bo
The rotation matrix from Fi00 toFb is
Rbi00
= R1(3)
Rbi00
= R1(3)
=
1 0 00 cos 3 sin 30
sin 3 cos 3
Note that the cos terms are on thediagonal of the matrix, whereas thesin terms are on the off-diagonal
vb = R1(3)vi00
= R1(3)R2(2)R3(1)vi
Rbi = R1(3)R2(2)R3(1)
is the rotation matrix transformingvectors from Fi to Fb
3
3
i002
i00
3
b2
b3
i00
1 and b1 are out of the page
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Euler Angles (4)
We have performed a 3-2-1 rotation from Fi to Fi00
Rbi = R1(3)R2(2)R3(1)
=
1 0 0
0 c3 s30 s3 c3
c2 0 s2
0 1 0s2 0 c2
c1 s1 0
s1 c1 00 0 1
=
c1c2 s1c2 s2
c3s1 + c1s2s3 c1c3 + s1s2s3 c2s3c1s2c3 + s1s3 s1s2c3
c1s3 c2c3
We can select arbitrary values of the three angles and compute a rotationmatrix. For example, let 1 = /3, 2 = /7, and 3 = /5:
R = 0.450484 0.780262 0.4338840.573114 0.625371 0.529576
0.684547 0.010099 0.728899
Conversely, given a rotation matrix, we can extract the Euler angles
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Euler Angles (5)
To extract Euler angles from a given rotation matrix, equate elementsof the two matrices:
Rbi =
c1c2 s1c2 s2
c3s1 + c1s2s3 c1c3 + s1s2s3 c2s3c1s2c3 + s1s3 s1s2c3 c1s3 c2c3
=
0.450484 0.780262 0.4338840.573114 0.625371 0.529576
0.684547 0.010099 0.728899
Choose the easy one first:
R13 : s2 = 0.433884 2 = sin1 0.433884 = 0.448799 /7
R11 : c1c2 = 0.450484 1 = 1.047198
/3
Quadrant checks are generally necessary for the second and third calcu-lations.
Exercise: Select another Euler angle sequence and determine the valuesof the three s that give the numerical R above.
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Euler Angles (6)
We just developed a 3-2-1 rotation, but there are other possibilities
There are 3 choices for the first rotation, 2 choices for the secondrotation, and 2 choices for the third rotation, so there are 322 = 12
possible Euler angle sequences
The Euler angle rotation sequences are
(1 2 3) (1 3 2) (1 2 1) (1 3 1)(2 3 1) (2 1 3) (2 3 2) (2 1 2)(3 1 2) (3 2 1) (3 1 3) (3 2 3)
The two left columns are sometimes called the asymmetric rotationsequences, and the two right columns are called the symmetricsequences
The roll-pitch-yaw sequence is an asymmetric sequence (which one?),whereas the -i- sequence (what is this?) is a symmetric sequence
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Euler Angles (7)
The three simple rotation matricesare
R1() =
1 0 00 cos sin
0 sin cos
R2() = cos 0 sin
0 1 0sin 0 cos
R3() = cos sin 0 sin cos 0
0 0 1
(1,0,0) in 1st row and column
Cosines on diagonal, Sines on off-
diagonal, negative Sine on rowabove the 1st row
(0,1,0) in 2nd row and column
Cosines on diagonal, Sines on off-
diagonal, negative Sine on row
above the 2nd row
(0,0,1) in 3rd row and column
Cosines on diagonal, Sines on off-diagonal, negative Sine on row
above the 3rd row
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Roll, Pitch and Yaw
Roll, pitch and yaw are Euler angles and are sometimes defined as a3-2-1 sequence and sometimes defined as a 1-2-3 sequence
Whats the difference?
The 3-2-1 sequence (we did earlier) leads to
Rbi =
c1c2 s1c2 s2c3s1 + c1s2s3 c1c3 + s1s2s3 c2s3
c1s2c3 + s1s3 s1s2c3 c1s3 c2c3
where 1 is the yaw angle, 2 is the pitch angle, and 3 is the roll angle
The 1-2-3 sequence leads to
Rbi =
c2c3 s1s2c3 + c1s3 s1s3 c1s2c3c2s3 c1c3 s122s3 s1c3 + c1s2s3
s2 s1c2 c1c2
where 1 is the roll angle, 2 is the pitch angle, and 3 is the yaw angle
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Roll, Pitch, Yaw (2)
Note that the two matrices are not the same
Rotations do not commute However, if we assume that the angles are small (appropriate for
many vehicle dynamics problems), then the approximations of the
two matrices are equal
3-2-1 Sequence
cos
1 and sin
Rbi
1 1 21 1 3
2 3 1
where 1 is the yaw angle, 2 is thepitch angle, and 3 is the roll angle
1-2-3 Sequence
cos
1 and sin
Rbi
1 3 23 1 1
2 1 1
where 1 is the roll angle, 2 is thepitch angle, and 3 is the yaw angle
Rbi
1
32
1
Rbi
1
Rbi
1
rollpitchyaw
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Perifocal Frame
Earth-centered, orbit-based,
inertial The P-axis is in periapsis
direction
The W-axis is perpendicularto orbital plane (direction of
orbit angular momentum
vector, )
The Q-axis is in the orbitalplane and finishes the triad
of unit vectors
In the perifocal frame, the
position and velocity vectorsboth have a zero component
in the W direction
P
P
Q
Q W
vrGG
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A One-Minute Course on Orbital Mechanics
i
I
J
K
n
Equatorialplane
Orbitalplane
Orbit is defined by 6 orbital elements(oes):
semimajor axis, a; eccentricity, e;
inclination, i; right ascension of ascending node, ;
argument of periapsis, ; and true anomaly,
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Inertial Frame to Perifocal Frame
R3() =
cos sin 0 sin cos 0
0 0 1
R1(i) =
1 0 00 cos i sin i
0 sin i cos i
R3() =
cos sin 0 sin cos 0
0 0 1
Rpi = R3()R1(i)R3() =
cc ci s s ci c s + c s si ss c ci c s ci c c s s si c
si s si c ci
Use a 3-1-3 sequence
Right ascension of the ascendingnode about K: R3(), rotates Ito n (the ascending node vector)
Inclination i about the node vec-tor: R1(i), rotates K to W (theorbit normal direction)
Argument of periapsis aboutthe orbit normal: R3(), rotates n
to P (the periapsis direction)
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Orbital Reference Frame
Same as roll-pitch-yawframe, for spacecraft
The o3 axis is in the nadirdirection
The o2 axis is in the negativeorbit normal direction
The o1 axis completes thetriad, and is in the velocityvector direction for circularorbits
In the orbital frame, positionand velocity both have zeroin the o2 direction, andposition has zero in the o1
direction as well You will find the rotation
from perifocal to orbitaleasier to visualize if youmake yourself two reference
frames
vG
rG
w
1o
2o
3o
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Perifocal Frame to Orbital Frame
Use a 2-3-2 sequence
270 about Q: R2(270), rotates
W to P (the nadir direction)
90 about the nadir vector:R3(90
), rotates o2 to W (thenegative orbit normal direction)
Negative true anomaly abouto2: R2(), rotates o3 to thetransverse direction
Rop = R2()R3(90)R2(270) =
s c 00 0 1c s 0
R2(270) =
0 0 10 1 01 0 0
R3(90) =
0 1 01 0 0
0 0 1
R2() =
cos 0 sin 0 1 0 sin 0 cos
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Inertial to Perifocal to Orbital
Rpi = R3()R1(i)R3() =
cc ci s s ci c s + c s si s
s c
ci c s ci c c
s s si c
si s si c ci
Rop = R2()R3(90)R2(270) =
s c 00 0 1
c
s 0
Roi = RopRpi
=
su c cu ci s su s + cu ci c cu sisi s si c ci
cu c + su ci s
cu s
su ci c
su si
where u = +
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An Illustrative Example
In an inertial reference frame, an Earth-orbiting satellite has positionand velocity vectors:
~r = 6000~I + 10, 000 ~J + 5, 000 ~K [km]~v =
5~I
2 ~J + 1 ~K [km/s]
The orbital elements are (using elementary astrodynamics)
a = 12, 142 km, e = 0.22026, i = 23.986, = 46.469, = 321.80, = 113.98
We can use i, , and to compute Rpi as
Rpi =
0.95089 0.18060 0.25139
0.094620 0.94286 0.31946
0.29472 0.27998 0.91364
Then we can rotate position and velocity into Fp:
rp = [5156.3 11594 0.037610]T
vp = [5.3670 1.0932 2.7899 107
]T
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An Illustrative Example (continued)We can use = 113.98 to compute Rop as
Rop =
0.91369 0.40642 00 0 1
0.40642 0.91369 0
We can then multiply RopRpi to get Roi, or we can use i, , andu = + to compute Roi:
Roi
= 0.83036 0.54821 0.0998580.29472 0.27998 0.91364
0.47291 0.78808 0.39406
Then we can rotate position and velocity into Fo:
ro = [0.66151 0.037610 12, 689]Tvo = [5.3481 2.7899 10
7 1.1824]T
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Differential Equations of Kinematics
Given the velocity of a point and initialconditions for its position, we can compute itsposition as a function of time by integratingthe differential equation
We now need to develop the equivalentdifferential equations for the attitude whenthe angular velocity is known
Preview:
d/dt (attitude) = angular momentum/inertia
~r = ~v
= 0 sin 3/ cos 2 cos 3/ cos 2
0 cos 3 sin 31 sin 3 sin 2/ cos 2 cos 3 sin 2/ cos 2
123
= S
1
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E l A l d A l V l i
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Euler Angles and Angular Velocity
One frame-to-frame at a time, just as we did for developing rotationmatrices
3-2-1 rotation from Fi to Fi0 to Fi00 to Fb
3-rotation from Fi to Fi0 about i3 i03 through 1
The angular velocity ofFi0 with respect to Fi is
~i0i = 1 i3 = 1 i
0
3
We can express ~i0i in any frame, but Fi and Fi0 are especially simple:
i
0
ii = [0 0 1]T
i0ii0 = [0 0 1]
T
Keep the notation in mind: i0ii is the angular velocity of Fi0 with
respect to Fi, expressed in Fi
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E l A l d A l V l i (2)
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Euler Angles and Angular Velocity (2)
2-rotation from Fi0 to Fi00 about i0
2 i002 through 2
The angular velocity ofFi00 with respect to Fi0 is
~i00
i0
= 2 i02 = 2 i002
We can express ~i00i0 in any frame, but Fi0 and Fi00 are especially
simple:
i00i0
i0 = [0 2 0]T
i00i0
i00 = [0 2 0]T
Keep the notation in mind: i00i0
i0 is the angular velocity ofFi00 with
respect to Fi0 , expressed in Fi0
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E l A l d A l V l it (3)
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Euler Angles and Angular Velocity (3)
1-rotation from Fi00 to Fb about i00
1 b1 through 3
The angular velocity ofFb with respect to Fi00 is
~bi00
= 3 i001 = 3b1
We can express ~bi00
in any frame, but Fi00 and Fb are especiallysimple:
bi00
i00 = [3 0 0]T
bi00
b = [3 0 0]T
Keep the notation in mind: bi00
b is the angular velocity of Fb with
respect to Fi00 , expressed in Fb
AOE 5204
Addi th A l V l iti
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Adding the Angular Velocities
Angular velocities are vectors and add like vectors:
~bi = ~bi00
+ ~i00i0 + ~i
0i
We have expressed these three vectors in diff
erent frames; to add themtogether, we need to express all of them in the same frame
Typically, we want bib , so we need to rotate the 31 matrices we justdeveloped into Fb
We have
i0ii =
i0ii0 = [0 0 1]
T need Rbi0
i00i0
i0
=
i00i0
i00
= [02
0]
T
needRbi
00
bi00
i00 = bi00
b = [3 0 0]T need Rbb = 1
We previously developed all these rotation matrices, so we just needto apply them and add the results
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Complete the Operation
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Complete the Operation
What happens when 2 n/2, for odd n ?
What happens when the Euler angles and their rates are small?
Carry out the matrix multiplications and additions
bib =
bi00
b +i00i0
b + i0ib
and obtain
bib =
3 sin 2 1cos 3 2 + cos 2 sin 3 1 sin 3 2 + cos 2 cos 3 1
=
sin 2 0 1
cos 2 sin 3 cos 3 0cos 2 cos 3 sin 3 0
123
= S()
or
= S1() =
0 sin 3/ cos 2 cos 3/ cos 20 cos 3 sin 3
1 sin 3 sin 2/ cos 2 cos 3 sin 2/ cos 2
12
3
AOE 5204 Kinematic Singularity in the Differential
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Kinematic Singularity in the Differential
Equation for Euler Angles
For this Euler angle set (3-1-2), the Euler rates go to
infinity when cos 2 0
The reason is that near 2 = /2 the first and third
rotations are indistinguishable
For the symmetric Euler angle sequences (3-1-3, 2-1-
2, 1-3-1, etc) the singularity occurs when 2 = 0 or
For the asymmetric Euler angle sequences (3-2-1, 2-3-1, 1-3-2, etc) the singularity occurs when 2 = /2 or
3/2
This kinematic singularity is a major disadvantage ofusing Euler angles for large-angle motion
There are attitude representations that do not have a
kinematic singularity, but 4 or more scalars are
required
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Linearizing the Kinematics Equation
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Linearizing the Kinematics Equation
Exercise: Repeat these steps for a 1-2-3 sequence and for a symmetric sequence.
bi
b
= S()
=
3 sin 2 1cos 3 2 + cos 2 sin 3 1 sin 3 2 + cos 2 cos 3 1
=
sin 2 0 1cos 2 sin 3 cos 3 0cos 2 cos 3 sin 3 0
123
If the Euler angles and rates are small, then sin i i and cos i 1:
bib
321
0 0 10 1 0
1 0 0
12
3
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Refresher: How To Invert a Matrix
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Refresher: How To Invert a MatrixSuppose you want to invert the n n matrix A, with elements Aij
The elements of the inverse are
A1ij =Cji|A|
where Cji is the cofactor, computed by multiplying the determinant ofthe (n 1) (n 1) minor matrix obtained by deleting the jth row andith column from A, by (1)i+j
This formulation is absolutely unsuitable for calculating matrix inversesin numerical work, especially with larger matrices, since it is computa-tionally expensive
One normally uses LU decomposition instead
Elementary row reduction is essentially LU decomposition
Note: In most cases, we do not need the inverse anyway; we need thesolution to a linear system
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Matrix Inversion Application
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Matrix Inversion ApplicationLet us invert the matrix
S() =
sin 2 0 1cos 2 sin 3 cos 3 0
cos 2 cos 3 sin 3 0
In row reduction, we augment the matrix with the identity matrix:
sin 2 0 1 1 0 0
cos 2 sin 3 cos 3 0 0 1 0cos 2 cos 3
sin 3 0 0 0 1
and apply simply row-reduction operations to transform the left 3 3block to the identity, leaving the inverse in the right 3 3 block
In this case, we must swap the first row with one of the other two rows,say the 3rd row, which amounts to a permutation by :
P =
0 0 10 1 0
1 0 0
Note that P
1 = P
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Matrix Inversion Application (2)
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Matrix Inversion Application (2)
cos 2 cos 3 sin 3 0 0 0 1cos 2 sin 3 cos 3 0 0 1 0 sin 2 0 1 1 0 0
Multiply row 2 by sin 3/ cos 3 and add to row 1.
Multiply row 1 by sin 3/ cos 3 and add to row 2.
c2 c3 + c2s3 tan 3 0 0 0 tan 3 1
0 c3 + s3 tan 3 0 0 1 tan 3s2 0 1 1 0 0
Divide row 1 by c2 c3 + c2s3 tan 3 and simplify
Multiply resulting row 1 by s2, add to row 3, and simplify
Divide row 2 by c3 + s3 tan 3 and simplify
1 0 0 0 s3/c2 c3/c20 1 0 0 c3
s3
0 0 1 1 s3 tan 2 c3 tan 2
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Eulers Theorem
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Euler s Theorem
00.5-0.5 00.5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
The most general motion of a rigid body with a fixed point is a
rotation about a fixed axis.
The axis, denoted a, is called the eigenaxis or Euler axis
The angle of rotation, , is called the Euler angle or the principal Eulerangle
Here the black axes arethe base vectors of the in-ertial frame, and the red,blue, and yellow axes arethe base vectors of a bodyframe, rotated about
a = [2/2 2/2 2/2]T
through angle = /4
Why no subscript on a?
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Eulers Theorem (2)
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Euler s Theorem (2)Any rotation matrix can be expressed in terms of a and :
R = cos1 + (1 cos)aaT sina
Since a is an eigenvector of R with eigenvalue 1,
Ra = a
Check this result:
Ra = cos1 + (1 cos)aaT
sina a
= cos1a + (1 cos)aaTa sinaa= cosa + aaTa cosaaTa sinaa= cosa + a cosa sinaa (aTa = 1)
= cosa + a cosa (a
a = 0)= a
Also note that the trace of R is simply
traceR = 1 + 2 cos
AOE 5204
Extracting a and from R,
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g ,
and Integrating to Obtain a and
Given any rotation matrix we can compute the Euler axis a and angle using:
= cos1 1
2 (trace R 1)a =
1
2sin
RT R
What does one do about the sin
= 0 case?
One can show that the kinematics differential equations for a and are:
= aT
a = 12
a cot
2aa
So, this system of equations also has kinematic singularities, at = 0and = 2
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Quaternions (aka Euler Parameters)
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Quaternions (aka Euler Parameters)
Define another 4-parameter set of variables to represent the attitude:
q = a sin
2
q4 = cos
2
The 31 matrix q forms the Euler axis component of the quaternion, also
called the vector component. The scalar q4 is called the scalar component.Collectively, these four variables are known as a quaternion, or as theEuler parameters.
The notation q denotes the 4 1 matrix containing all four variables:
q =
qT q4
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Quaternions (2)
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Quaternions (2)
R(q) and q(R) :
R =
q24 qTq
1 + 2qqT 2q4q
q4 = 12
1 + trace R
q =1
4q4
R23 R32R31 R13R12 R21
Differential equations q :
q = 12
q
+ q41qT = Q(q)
Note that there is no kinematic singularity with these differential equa-tions
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Typical Problem Involving
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Angular Velocity and Attitude
Given initial conditions for the attitude
(in any form), and a time history ofangular velocity, compute R or any
other attitude representation as a
function of time Requires integration of one of the sets of
differential equations involving angular
velocity
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Spherical Pendulum Problem
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p
Illustration from Wolfram Research Mathworld
mathworld.wolfram.com/SphericalCoordinates.html
Use a rotation from Fi to Fr us-
ing two Euler angles, , and
The first (3) rotation is about the
i3 axis, through angle
The second (2) rotation is about
the e axis through
The three unit vectors havederivatives:
er = sine + e,
e = siner cose,e = er + cose
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Linearization for Small
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As with Euler angles, we are frequently interested in small attitude mo-tions.
If is small, then q = a sin(/2) a2
, and q4 1
Hence, for small ,
R =
q24 qTq
1 + 2qqT 2q4q
(1
0) 1 + 2(0)
2q
1 2q
Compare this expression with the previously developed R 1 forEuler angles. What is the equivalent expression for the (a,) representa-
tion?
Small rotations are commutative:
RcbRba
1 2q
2 1 2q
1 = 1 2q
2
2q1
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Other Attitude Representations
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p
We have seen direction cosines, Euler angles, Eulerangle/axis, and quaternions
Two other common representations Euler-Rodriguez parameters
Modified Rodriguez parameters
p = a tan
2
R = 1 +2
1 + pT
p
(pp
p)
p =1
2(ppT + 1 + p)
= a tan
4R =
1
1 + T
(1 (T)2)1 + 2T 2(1 (T)2)
=1
2
1 + T 1 +
T
21
AOE 5204
Typical Problem Involving
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Angular Velocity and Attitude
Given initial conditions for the attitude
(in any form), and a time history ofangular velocity, compute R or any
other attitude representation as a
function of time Requires integration of one of the sets of
differential equations involving angular
velocity
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