MMS I, Lecture 2 1

Content MM2

• Repetition

• Euler angels– Principal axis and angle (ê, θ )

• Quarternions

• Kinematics expressed in – DCM – Euler – Quaternions

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

^ ^ ^

^

^

^

MMS I, Lecture 2 3

ELEMENTARY ROTATIONS

a1

a3

a2

[0,1,1]-90

{A}{B}

RARB

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

MMS I, Lecture 2 5

ROTATION ORDER(important)

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

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 )

)

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

MMS I, Lecture 2 9

EULER ANGLES (BODY FIXED 3-2-1)

3

2

1

MMS I, Lecture 2 10

EXAMPLE, SHUTTLE (Body Fixed (2-3-1)

Flaps Ф2 (Pitch), Ф3 (Yaw). Rudder Ф1 (Roll) gives CUA

MMS I, Lecture 2 11

EULER ANGLES (BODY FIXED 3-1-3)

3

1

3

ø φ

θ Ascending

Perihelion

Inclination

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

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

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

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

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)

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) =

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

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

···

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

MMS I, Lecture 2 21

Kinematics cont.

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

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