Content MM2 Repetition Euler angels Quarternions Principal axis and angle (ê, θ ) Quarternions Kinematics expressed in DCM Euler Quaternions MMS I, Lecture 2

DIRECTION COSINE MATRIX (DCM) u3 {A} {A} {A} {U} u2 ^ AX U T AYU T AZU T ^ ^ ^ ^ C UA = [ UXA UYA UZA ] = ^ Three column vectors z z u1 {U} {A} y Three row vectors y X x MMS I, Lecture 2

ELEMENTARY ROTATIONS -90 [0,1,1] RA RB {B} {A} a a a MMS I, Lecture 2 3 [0,1,1] RB RA {B} a {A} 2 a 1 MMS I, Lecture 2

ELEMENTARY ROTATIONS NB MMS I, Lecture 2 {A} {U} θ x θ x z z Black board: U unit vectors in A CAU A U CUA MMS I, Lecture 2

ROTATION ORDER(important) MMS I, Lecture 2

Transportation Theorem Theorem:(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 the U frame as: â1 û3 â2 ωAU û2 û1 {A} {U} P r r = r1â1+r2â2 + r3â3 = r = CUA ( + ωAU x r A) d r t U · A MMS I, Lecture 2

Example · d r t U d r t A CUA = r = ( ( + ωAU x r A) CUA ) {A} {U} (3) A dt dr ωAU x r A = (-1 1 0) (2) = (1 1 0) (1) rA = (2 2 0) Φ CUA = Cos(θ) Sin(θ) 0 - Sin(θ) Cos(θ) 0 0 0 1 For θ = ( 0, Φ, 90 0 ) MMS I, Lecture 2

Second derivety = CUA + ω x r d dr d dr dt dt dt dt U A d2r d2r dr dr · = CUA + ω x + ω x + ω x r ω x ω x r U A d2r d2r dr dt2 dt2 dt · = CUA + 2ω x + ω x r + ω x ω x r U A Please put on bars and double bars on vectors and matrixes MMS I, Lecture 2

EULER ANGLES (BODY FIXED 3-2-1) MMS I, Lecture 2

EXAMPLE, SHUTTLE (Body Fixed (2-3-1) Flaps Ф2 (Pitch), Ф3 (Yaw). Rudder Ф1 (Roll) gives CUA MMS I, Lecture 2

EULER ANGLES (BODY FIXED 3-1-3) ø φ θ Perihelion Inclination Ascending MMS I, Lecture 2

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

Euler Principal Axis and angle 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. θ Trick: Move â1 to ê1 â1 â2 â3 {A} ê1 e1 e2 e3 ê2 = R21 R22 R23 ê3 R31 R32 R33 â1 â2 â3 ê1 θ {U} û3 û2 û1 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+R212+R312= 1; RTR=I ; e1= R22R33-R23R32 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 MMS I, Lecture 2

Euler Principal Axis and angle cont. cθ + e1(1- cθ) e1e2(1- cθ) + e3sθ e1e3(1- cθ) – e2sθ CUA = 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 0 -e3 e2 e3 0 -e1 -e2 e1 0 E = CUA(ê, θ) = cos(θ) I + (1- cos(θ) eeT – sin(θ) E From a known DCM determine ê, θ: cos(θ) = ½( Trace CUA - 1) ê = CUA C23 - C32 C31 - C13 C12 - C21 1 2sin(θ) θ ≠ ± n·180 MMS I, Lecture 2

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) ˆ ˆ q1:3 q4 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 = MMS I, Lecture 2

Quarternions cont. CUA(ê, θ) = cos(θ) I + (1- cos(θ)) êêT – sin(θ) E ; 0 -e3 e2 e3 0 -e1 -e2 e1 0 CUA(q1:3,q4) = ( q42 - q1:3 T q1:3)I + 2q1:3q1:3T - 2q4Q Q = s(/2) 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 = = 1 4q4 q4 = ½ ( Trace C + 1)1/2 0 ≤ θ ≤ 180 MMS I, Lecture 2

Quarternions cont. q(t+Δt) q(Δt) q(t) = 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) q(t+Δt) = q(Δt) q(t) Compare with C.6 and C.7 in appendix C in literature MMS I, Lecture 2

Kinematics rigid bodies is given ωBA {A} ωBA Direct Cosine Matrix {B} b = CBA(t)â â = CTBA(t)b → = CBAT(t)b + CBAT(t)(ωBAxb) → 0 = CBAT(t)b + CBAT(t)(S(ωBA)b) CBA(t) = - S(ωBA) CBA(t) dâ dt ˆ · 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) MMS I, Lecture 2

Kinematics cont. ω is given 3 {V} Euler Angles {A} θ3 {U} CUA = CUVCVWCWA = C1(θ1)·C2(θ2)·C3(θ3) · · 1 2 ωWA = θ3â3A = θ3â3W ωVW = θ2â2W = θ2â2V ωVU = θ1â1V = θ1â1U · · 2 θ2 · · θ1 {W} 1 1 · ω1 θ1 0 0 ω2 = 0 + C1(θ1) θ2 + C1(θ1)·C2(θ2) 0 ω3 0 0 θ3 · · ω1 1 0 -sin(θ2) θ1 ω2 = 0 cos(θ1) sin(θ1) cos(θ2) θ2 ω3 0 -sin(θ1) cos(θ1) cos(θ2) θ3 · Euler rates MMS I, Lecture 2

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) ω1(t) ω2(t) ω3(t) · 1 cos(θ2) · Singularity Known ((C1(θ1)·C2(θ2)·C3(θ3))T = C3(θ3)T C2(θ2)T C1(θ1)T Advantages: Only 3 differential equations Drawback: Unlinear Singularity MMS I, Lecture 2

Kinematics cont. MMS I, Lecture 2

Kinematics cont. Quaternions dq(t) dt q(t+Δt) - q(t) q(Δt) q(t) – q(t) = = limΔt→0 êsin(Δθ/2) cos(Δθ/2) êΔθ/2 1 q(Δt) ~ Not fractions!! = Calculations: See the notes app.C -ωT = (e1, e2, e3)T Δθ/Δt dq(t) dt -S(ω) ω -ωT 0 = 1/2 q(t) Thomas Bak: As seen the same · q13 = ½(q4 ω – ω x q13) q4 = - ½ ωT q13 · MMS I, Lecture 2

Rotation Representations 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 Quaternions 4 Easy orthogonality Not singular No clear physical representation One redundant parameter Widely used in simulation Preferred for global rotation MMS I, Lecture 2