mms i, lecture 21 content mm2 repetition euler angels –principal axis and angle (ê, θ )...
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![Page 1: MMS I, Lecture 21 Content MM2 Repetition Euler angels –Principal axis and angle (ê, θ ) Quarternions Kinematics expressed in –DCM –Euler –Quaternions](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f4e5503460f94c703c1/html5/thumbnails/1.jpg)
MMS I, Lecture 2 1
Content MM2
• Repetition
• Euler angels– Principal axis and angle (ê, θ )
• Quarternions
• Kinematics expressed in – DCM – Euler – Quaternions
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MMS I, Lecture 2 2
DIRECTION COSINE MATRIX (DCM)
a1
a3
a2
b1
b3
b2
{A}
{U}
u3
u2
u1
C UA = [ UXA UYA UZA ] =
AX U T
AYU T
AZU T
Three column vectors
Three row vectors
{A}
{A}{A}
{U}
X
z
y
z
x
y
^ ^ ^
^
^
^
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MMS I, Lecture 2 3
ELEMENTARY ROTATIONS
a1
a3
a2
[0,1,1]-90
{A}{B}
RARB
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MMS I, Lecture 2 4
ELEMENTARY ROTATIONS
{U}{A}
θ
Black board: U unit vectors in A CAU
A U CUA
y
x
z
y
xz
NB
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MMS I, Lecture 2 5
ROTATION ORDER(important)
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MMS I, Lecture 2 6
Transportation TheoremTheorem:(Transport Theorem) Let frame A rotate relativ to frame U with angular velocity ωAU and let r be a vector following the A coordinate system. The time derivative of r in the A frame is then related to the derivative of r in theU frame as:
r1â1+r2â2 + r3â3r =
â1
â1
û3
â2
ωAU
û2û1
{A}
{U}P
r
= r = CUA ( + ωAU x r A) d rd t U
· d rd t A
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Example
MMS I, Lecture 2 7
= r = ( ( + ωAU x r A) d rd t U
· d rd t A
CUA
{U}
{A}
(1)
(2)
ωAU = (0 0 1/2)
Adtdr
rA = (2 2 0)
ωAU x r A = (-1 1 0)
(3) = (1 1 0)
Φ
CUA
CUA =
Cos(θ) Sin(θ) 0- Sin(θ) Cos(θ) 0 0 0 1
For θ = ( 0, Φ, 90 0 )
)
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MMS I, Lecture 2 8
Second derivety
= CUA + ω x r d dr d drdt dt dt dt
AU
d2r d2r dr drdt2 dt2 dt dt
= CUA + ω x + ω x + ω x r ω x ω x r
d2r d2r dr dt2 dt2 dt
= CUA + 2ω x + ω x r + ω x ω x r
·
·
U
U
A
A
Please put on bars and double bars on vectors and matrixes
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MMS I, Lecture 2 9
EULER ANGLES (BODY FIXED 3-2-1)
3
2
1
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MMS I, Lecture 2 10
EXAMPLE, SHUTTLE (Body Fixed (2-3-1)
Flaps Ф2 (Pitch), Ф3 (Yaw). Rudder Ф1 (Roll) gives CUA
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MMS I, Lecture 2 11
EULER ANGLES (BODY FIXED 3-1-3)
3
1
3
ø φ
θ Ascending
Perihelion
Inclination
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MMS I, Lecture 2 12
EULER ANGLE REPRESENTATIONS
Further 12 Euler Angle matrices exsist given in fixed axis
From now on we expect that the rotation matix exist CUA
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MMS I, Lecture 2 13
Euler Principal Axis and angle
CUA = RTC(θ)R
cθ + e1(1- cθ) e1e2(1- cθ) + e3sθ e1e3(1- cθ) – e2sθ = e2e1(1- cθ) - e3sθ cθ + e2(1- cθ) e2e3(1- cθ) + e1sθ e3e1(1- cθ) + e2sθ e3e2(1- cθ) - e1sθ cθ + e3(1- cθ)
2
2
2
Theorem: (Euler’s Eigenaxis Rotation) A rigid body or a coordinate frame fixed in a point P can be brought from any arbitrary initial orientetion to an arbitrary final orientation by a single rotation about a principal axis ê through the point P.
ê1
â1
â2
â3{A}
û3
û2
û1{U}
ê1 e1 e2 e3
ê2 = R21 R22 R23
ê3 R31 R32 R33
â1
â2
â3θ
P e1 R21 R31
e2 R22 R32
e3 R23 R33
e1 e2 e3
R21 R22 R23
R31 R32 R33
1 0 0 0 cos θ sin θ
0 – sin θ cos θ
Support equations of the type: e12+R21
2+R312= 1; RTR=I ; e1= R22R33-R23R32
Trick: Move â1 to ê1
θ
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MMS I, Lecture 2 14
Euler Principal Axis and angle cont. cθ + e1(1- cθ) e1e2(1- cθ) + e3sθ e1e3(1- cθ) – e2sθCUA = e2e1(1- cθ) - e3sθ cθ + e2(1- cθ) e2e3(1- cθ) + e1sθ e3e1(1- cθ) + e2sθ e3e2(1- cθ) - e1sθ cθ + e3(1- cθ)
2
2
2
CUA(ê, θ) = cos(θ) I + (1- cos(θ) eeT – sin(θ) E 0 -e3 e2
e3 0 -e1
-e2 e1 0E =
From a known DCM determine ê, θ:
cos(θ) = ½( Trace CUA - 1)
ê =
C23 - C32
C31 - C13
C12 - C21
12sin(θ) θ ≠ ± n·180
CUA
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MMS I, Lecture 2 15
Quarternions
Known: ê, θ, CUA
Define: q = ( q1,q2,q3,q4 )T = (q4 + î q1+ j q2 + k q3)T = =
e1sin(θ/2)e2sin(θ/2)e3sin(θ/2) cos(θ/2)By using: sin(θ) = 2sin(θ /2)cos(θ /2)
cos(θ) = cos2(θ /2) - sin2(θ /2) = 2cos2(θ /2) -1 = 1 - 2sin2(θ /2)In:
cθ + e12(1- cθ) e1e2(1- cθ) + e3sθ e1e3(1- cθ) – e2sθ
e2e1(1- cθ) - e3sθ cθ + e22(1- cθ) e2e3(1- cθ) + e1sθ
e3e1(1- cθ) + e2sθ e3e2(1- cθ) - e1sθ cθ + e32(1- cθ)
CUA =
q1:3
q4
ˆ ˆ
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MMS I, Lecture 2 16
Quarternions cont.
CUA(ê, θ) = cos(θ) I + (1- cos(θ)) êêT – sin(θ) E ;
CUA(q1:3,q4) = ( q42 - q1:3
T q1:3)I + 2q1:3q1:3T - 2q4Q Q =
0 -e3 e2
e3 0 -e1
-e2 e1 0
From a known DCM CUA determine q1:3 , q4:
e1sin(θ/2)e2sin(θ/2)e3sin(θ/2)
q1
q2
q3
C23 - C32
C31 - C13
C12 - C21
q1:3 = = = 14q4
q4 = ½ ( Trace C + 1)1/2
0 ≤ θ ≤ 180
s(/2)
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MMS I, Lecture 2 17
Quarternions cont.
i2 = j2 = k2 = -1; ij = k = -ji ; q = q”O q’ = (q”1i, q”2j, q”3k, q”4)(q’1i, q’2j, q’3k, q’4)
Compare with C.6 and C.7 in appendix C in literature
q(t+Δt) q(Δt) q(t) =
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MMS I, Lecture 2 18
Kinematics rigid bodies is given
{B}
{A} ωBADirect Cosine Matrix
0 - ω 3 ω 2
ω3 0 - ω1
- ω 2 ω 1 0
S(ωBA) =
Advantages: Linear No singularities
Drawback: 9 differential coupled equations (redundancies)
b = CBA(t)ââ = CT
BA(t)b → = CBAT(t)b + CBA
T(t)(ωBAxb)
→ 0 = CBAT(t)b + CBA
T(t)(S(ωBA)b)
CBA(t) = - S(ωBA) CBA(t)
dâdt
ˆˆ ˆ ˆ
ˆˆ
·
·
·
ωBA
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MMS I, Lecture 2 19
Kinematics cont. ω is given
Euler Angles
CUA = CUVCVWCWA = C1(θ1)·C2(θ2)·C3(θ3)
1
2
3
θ3
1
2θ2
1 θ1
{A}
{W}
{V}
{U}
ωWA = θ3â3A
= θ3â3W
ωVW = θ2â2W
= θ2â2V
ωVU = θ1â1V
= θ1â1U
ω1 θ1 0 0 ω2 = 0 + C1(θ1) θ2 + C1(θ1)·C2(θ2) 0ω3 0 0 θ3
··· ·· ·
··
·
Euler ratesω1 1 0 -sin(θ2) θ1 ω2 = 0 cos(θ1) sin(θ1) cos(θ2) θ2 ω3 0 -sin(θ1) cos(θ1) cos(θ2) θ3
···
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MMS I, Lecture 2 20
Kinematics cont.
Euler Angles cont.
θ1 cos(θ2) sin(θ1)sin(θ2) sin(θ2) cos(θ1) θ2 = 0 cos(θ1)cos(θ2) -sin(θ1) cos(θ2) θ3 0 sin(θ1) cos(θ1) ···
1cos(θ2)
ω1(t)ω2(t)ω3(t)
Singularity
((C1(θ1)·C2(θ2)·C3(θ3))T = C3(θ3)T C2(θ2)T C1(θ1)T
Known
Advantages: Only 3 differential equations
Drawback: Unlinear Singularity
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MMS I, Lecture 2 21
Kinematics cont.
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MMS I, Lecture 2 22
Kinematics cont.
Quaternions
dq(t) dt limΔt→0
q(t+Δt) - q(t) q(Δt) q(t) – q(t) Δt Δt ==
q(Δt) =êsin(Δθ/2) cos(Δθ/2)
êΔθ/2 1
~
Calculations: See the notes app.C
dq(t) dt
=-S(ω) ω -ωT 0 q(t)1/2
Thomas Bak: As seen the same
q13 = ½(q4 ω – ω x q13)
q4 = - ½ ωT q13
·
·
Not fractions!!
-ωT = (e1, e2, e3)T Δθ/Δt
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MMS I, Lecture 2 23
Rotation RepresentationsRepresentation Par. Characteristics Applications
Direction Cosine matrix
9 • Nonsingular• Intuitive• Six redundant parameters
• Analytical studies
Euler Angles 3 • Minimal set• Clear physical
representation• Trigonometric functions in
rotation matrix• Singular
• Analytical studies
Quaternions 4 • Easy orthogonality• Not singular• No clear physical
representation• One redundant parameter
• Widely used in simulation
• Preferred for global rotation