11. Spacecraft Dynamics MAE 342 2016

2/12/20

Spacecraft Dynamics Space System Design, MAE 342, Princeton University Robert Stengel • Angular rate dynamics • Spinning and non-spinning spacecraft • Gravity gradient satellites • Euler Angles and spacecraft attitude • Rotation matrix • Precession of spinning axisymmetric spacecraft

Angular Momentum of a Particle

• Moment of linear momentum of differential particles that make up the body – Differential mass of a particle times component of its velocity that is perpendicular to the moment arm from the center of rotation to the particle

dh = (r × dmv) = (r × v m )dm

= (r × (vo + ω × r))dm

1 2/12/20

Angular Momentum of an Object Integrate moment of linear momentum of differential particles over the body

h = r × v + ω × r dm ∫ ( ( o )) Body = r × v dm + r × ω × r dm ∫ ( o ) ∫ ( ( )) Body Body = 0 − ∫ (r × (r × ω))dm = − ∫ (r × r)dm × ω Body Body

⎡ ⎤ ≡ − (r!r!)dmω ⎡ ⎤ ω x ∫ hx ⎢ ⎥ Body ⎢ ⎥ ω = ⎢ ω y ⎥ h = ⎢ hy ⎥ ⎢ ω ⎥ ⎢ h ⎥ ⎣⎢ z ⎦⎥ ⎣⎢ z ⎦⎥

Angular Momentum with Respect to the Center of Mass Choose center of mass as origin about which angular momentum is calculated (= center of rotation) Use cross-product-equivalent matrix to define the inertia matrix, I h = − ∫ r! r! dm ω Body

x y z ⎡ 0 −z y ⎤⎡ 0 −z y ⎤ max max max ⎢ ⎥⎢ ⎥ = − ∫ ∫ ∫ ⎢ z 0 −x ⎥⎢ z 0 −x ⎥ρ(x, y,z)dx dydz xmin ymin zmin ⎢ −y x 0 ⎥⎢ −y x 0 ⎥ ⎣ ⎦⎣ ⎦

⎡ (y2 + z2 ) −xy −xz ⎤ ⎢ ⎥ 2 2 = ∫ ⎢ −xy (x + z ) −yz ⎥dm ω = I ω Body ⎢ 2 2 ⎥ ⎢ −xz −yz (x + y ) ⎥ 4 ⎣ ⎦ 4

2 2/12/20

Inertia Matrix

2 2 ⎡ ⎤ ⎡ (y + z ) −xy −xz ⎤ I xx −I xy −I xz ⎢ ⎥ ⎢ ⎥ 2 2 −I I −I I = ∫ ⎢ −xy (x + z ) −yz ⎥dm = ⎢ xy yy yz ⎥ Body ⎢ 2 2 ⎥ ⎢ ⎥ −xz −yz (x + y ) −I xz −I yz Izz ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥

Moments of inertia on the diagonal Products of inertia off the diagonal If products of inertia are zero, (x, y, z) are principal axes

⎡ I 0 0 ⎤ ⎢ xx ⎥ IP = ⎢ 0 I yy 0 ⎥ ⎢ 0 0 I ⎥ ⎢ zz ⎥ ⎣ ⎦ For fixed mass distribution, inertia matrix is constant in body frame of reference 5

Inertial-Frame Inertia Matrix is Not Constant if Body is Rotating

Newton’s 2nd Law applies to rotational motion in an inertial frame Rate of change of angular momentum = applied moment (or torque), m

⎡ M ⎤ Chain Rule ⎢ x ⎥ dh d(Iω) d = = M = ⎢ M y ⎥ (Iω) dh d I dt dt = = ω + Iω! ⎢ M ⎥ dt dt dt ⎣⎢ z ⎦⎥ Inertial-frame solution for In an inertial frame angular rate d d I I Iω! = M − ω ≠ 0 dt dt −1 ⎛ d I ⎞ ω! = I ⎜ M − ω⎟ ⎝ dt ⎠ 6

3 2/12/20

How Do We Get Rid of dI/dt in the Angular Momentum Equation? Write the dynamic equation in a body-referenced frame

§ Inertia matrix is ~unchanging in a body frame § Body-axis frame is rotating § Dynamic equation must be modified to account for rotation

Expressing Vectors in Different Reference Frames • Angular momentum and rate are vectors – They can be expressed in either the inertial or body frame – The 2 frames are related by the rotation matrix (also called the direction cosine matrix)

B HI : Rotation transformation from inertial frame ⇒ body frame I HB : Rotation transformation from body frame ⇒ inertial frame

B I hB = HI hI hI = HBhB B I ω B = HI ω I ω I = HBω B 8

4 2/12/20

Euler Angles • Body attitude measured with respect to inertial frame • Three-angle orientation expressed by sequence of three orthogonal single-angle rotations

Inertial ⇒ Intermediate1 ⇒ Intermediate2 ⇒ Body

ψ : Yaw angle ψ : Yaw angle1 θ : Pitch angle θ : Pitch angle

φ : Roll angle φ : Yaw angle2

• 24 (±12) possible sequences of single-axis rotations • Aircraft convention: 3-2-1, z positive down • Spacecraft convention: 3-1-3, z positive up 9

Reference Frame Rotation from Inertial to Body: Aircraft Convention (1-2-3)

Yaw rotation (ψ) about zI axis

⎡ ⎤ ⎡ x ⎤ ⎡ cosψ sinψ 0 ⎤⎡ x ⎤ xI cosψ + yI sinψ ⎢ ⎥ ⎢ ⎥ 1 ⎢ y ⎥ ⎢ y ⎥ x sin y cos r = H r ⎢ ⎥ = ⎢ −sinψ cosψ 0 ⎥⎢ ⎥ = ⎢ − I ψ + I ψ ⎥ 1 I I ⎢ z ⎥ ⎢ 0 0 1 ⎥⎢ z ⎥ ⎢ z ⎥ ⎣ ⎦1 ⎣ ⎦⎣ ⎦I ⎣ I ⎦

Pitch rotation (θ) about y1 axis

⎡ x ⎤ ⎡ cosθ 0 −sinθ ⎤⎡ x ⎤ ⎢ ⎥ ⎢ ⎥ 2 2 1 2 y = ⎢ ⎥ y r = H r = ⎡H H ⎤r = H r ⎢ ⎥ ⎢ 0 1 0 ⎥⎢ ⎥ 2 1 1 ⎣ 1 I ⎦ I I I ⎢ z ⎥ ⎢ sinθ 0 cosθ ⎥⎢ z ⎥ ⎣ ⎦2 ⎣ ⎦⎣ ⎦1

Roll rotation (ϕ) about x2 axis

⎡ x ⎤ ⎡ 1 0 0 ⎤⎡ x ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ B B 2 1 B y 0 cosφ sinφ y rB = H2 r2 = ⎣⎡H2 H1 HI ⎦⎤rI = HI rI ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ z ⎥ ⎢ 0 −sinφ cosφ ⎥⎢ z ⎥ ⎣ ⎦B ⎣ ⎦2 ⎣ ⎦ 10

5 2/12/20

Reference Frame Rotation from Inertial to Body: Spacecraft Convention (3-1-3)

Yaw rotation (ψ) about zI axis

Pitch rotation (θ) about x1 axis

⎡ x ⎤ ⎡ 1 0 0 ⎤⎡ x ⎤ ⎢ ⎥ ⎢ ⎥ 2 2 1 2 y = ⎢ 0 cos sin ⎥ y r = H r = ⎡H H ⎤r = H r ⎢ ⎥ ⎢ θ θ ⎥⎢ ⎥ 2 1 1 ⎣ 1 I ⎦ I I I ⎢ z ⎥ ⎢ 0 −sinθ cosθ ⎥⎢ z ⎥ ⎣ ⎦2 ⎣ ⎦⎣ ⎦1

Yaw rotation (ϕ) about z2 axis

⎡ x ⎤ ⎡ cosφ sinφ 0 ⎤⎡ x ⎤ ⎢ ⎥ B B 2 1 B ⎢ y ⎥ ⎢ y ⎥ r = H r = ⎡H H H ⎤r = H r ⎢ ⎥ = ⎢ −sinφ cosφ 0 ⎥⎢ ⎥ B 2 2 ⎣ 2 1 I ⎦ I I I ⎢ z ⎥ ⎢ 0 0 1 ⎥⎢ z ⎥ ⎣ ⎦B ⎣ ⎦⎣ ⎦2 11

Rotation Matrix from I to B Aircraft Convention (1-2-3)

B B 2 1 HI (φ,θ,ψ) = H2 (φ)H1 (θ)HI (ψ)

# 1 0 0 &# &# cosψ sinψ 0 & % ( cosθ 0 −sinθ % ( 0 cos sin % ( = % φ φ (% 0 1 0 (% −sinψ cosψ 0 ( % ( % ( $ 0 −sinφ cosφ '$% sinθ 0 cosθ '($ 0 0 1 ' # cosθ cosψ cosθ sinψ −sinθ & % ( = % −cosφ sinψ +sinφ sinθ cosψ cosφ cosψ +sinφ sinθ sinψ sinφ cosθ ( % ( $ sinφ sinψ + cosφ sinθ cosψ −sinφ cosψ + cosφ sinθ sinψ cosφ cosθ '

6 2/12/20

Rotation Matrix from I to B Spacecraft Convention (3-1-3)

B B 2 1 HI (φ,θ,ψ) = H2 (φ)H1 (θ)HI (ψ)

⎡ cosφ sinφ 0 ⎤⎡ 1 0 0 ⎤⎡ cosψ sinψ 0 ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ 0 cos sin ⎥ ⎢ −sinφ cosφ 0 ⎥⎢ θ θ ⎥⎢ −sinψ cosψ 0 ⎥ ⎢ ⎥ 0 −sinθ cosθ ⎢ ⎥ ⎣ 0 0 1 ⎦⎣⎢ ⎦⎥⎣ 0 0 1 ⎦ ⎡ cosφ cosψ − sinφ cosθ sinψ cosφ sinψ + sinφ cosθ cosψ sinφ sinθ ⎤ ⎢ ⎥ = ⎢ −sinφ cosψ − cosφ cosθ sinψ −sinφ sinψ + cosφ cosθ cosψ cosφ sinθ ⎥ ⎢ sinθ sinψ −sinθ cosψ cosθ ⎥ ⎣ ⎦

Properties of the Rotation Matrix

B ⎡ ⎤ h11 h12 h13 B ⎢ ⎥ HI (φ,θ,ψ ) = ⎢ h21 h22 h23 ⎥ ⎢ ⎥ h31 h32 h33 ⎣ ⎦I B B 2 1 HI (φ,θ,ψ ) = H2 (φ)H1 (θ)HI (ψ )

B −1 B T I ⎣⎡HI (φ,θ,ψ )⎦⎤ = ⎣⎡HI (φ,θ,ψ )⎦⎤ = HB (ψ ,θ,φ) Orthonormal transformation Angles between vectors are preserved r s Lengths are preserved

rI = rB ; sI = sB

∠(rI ,sI ) = ∠(rB ,sB ) = x deg 14

7 2/12/20

Rotation Matrix Inverse Inverse relationship: interchange sub- and superscripts

B rB = HI rI

B −1 I rI = (HI ) rB = HBrB Because transformation is orthonormal Inverse = transpose Rotation matrix is always non-singular

I B −1 B T I 1 2 HB = (HI ) = (HI ) = H1H2HB

I B B I HB HI = HI HB = I 15

Vector Derivative Expressed in a Rotating Frame

I hI (t) = HB (t)hB (t) Effect of Chain Rule body-frame rotation h! t = HI t h! t + H! I t h t I ( ) B ( ) B ( ) B ( ) B ( ) Rate of change expressed in body frame Alternatively h! HI h! h HI h! " h I = B B + ω I × I = B B + ω I I ⎡ 0 −ω ω ⎤ ⎢ z y ⎥ ω! = ⎢ ω z 0 −ω x ⎥ ⎢ ⎥ −ω y ω x 0 16 ⎣⎢ ⎦⎥ 16

8 2/12/20

Angular Rate Derivative in Body Frame of Reference Angular momentum change ! B ! ! B B ! hB = HI hI + HI hI = HI hI − ω B × hB = HBh! − ω" h = HBM − ω" I ω I I B B I I B B B h! M " I B = B − ω B Bω B Constant body-axis inertia matrix ! hB (t) = Bω! B (t) = MB (t) − ω" B (t) Bω B (t) I I Angular rate change

−1 ω! B (t) = IB ⎡MB (t) − ω" B (t)IBω B (t)⎤ ⎣ ⎦ 17

Euler-Angle Rates and Body-Axis Rates

⎡ ⎤ ω x Body-axis angular rate ⎢ ⎥ ω = ω vector (orthogonal) B ⎢ y ⎥ ⎢ ⎥ ω z ⎣⎢ ⎦⎥B

⎡ φ ⎤ ⎡ ψ ⎤ Form a non-orthogonal ⎢ ⎥ ⎢ ⎥ or vector of Euler angles Θ = ⎢ θ ⎥ ⎢ θ ⎥ ⎢ ψ ⎥ ⎢ φ ⎥ ⎣ ⎦3−2−1 ⎣ ⎦3−1−3

⎡ φ! ⎤ ⎡ ψ! ⎤ ⎡ ω ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ x ⎥ Euler-angle ! ! Θ = ⎢ θ ⎥ or ⎢ θ! ⎥ ≠ ⎢ ω y ⎥ rate vector ⎢ ⎥ ψ! ⎢ φ! ⎥ ⎢ ω ⎥ ⎣ ⎦3−2−1 ⎣ ⎦3−1−3 ⎢ z ⎥ ⎣ ⎦I18

9 2/12/20

Relationship Between (1-2-3) Euler-Angle and Body-Axis Rates

• ψ˙ measured in Inertial Frame ! $ ! $ p φ ! 0 $ ! 0 $ ˙ # & # & # & • θ measured in Intermediate Frame #1 B  B 2 # & # q & = I3 # 0 &+ H2 # θ &+ H2 H1 0 • φ˙ measured in Intermediate Frame #2 # & # r & "# 0 %& "# 0 %& # ψ & " % " % €€ ! p $ ! 1 0 −sinθ $! φ $ € # & # &# & B  # q & = # 0 cosφ sinφ cosθ &# θ & = LI Θ # & # &# & " r % " 0 −sinφ cosφ cosθ % ψ " % Inverse transformation [(.)-1 ≠ (.)T] $ ' φ $ 1 sinφ tanθ cosφ tanθ '$ p ' & ) & )& )  I & θ ) = & 0 cosφ −sinφ )& q ) = LBωB & ψ ) & )& ) % ( % 0 sinφ secθ cosφ secθ (% r ( 19

Relationship Between (3-1-3) Euler-Angle and Body-Axis Rates

˙ ⎡ ⎤ • ψ measured in Inertial Frame ω x ⎡ 0 ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ 0 0 • θ˙ measured in Intermediate Frame #1 ⎢ ⎥ B ⎢ ⎥ B 2 ⎢ ⎥ ⎢ ω y ⎥ = I3 0 + H2 θ! + H2 H1 0 ˙ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ • φ measured in Intermediate Frame #2 ⎢ ⎥ ⎢ φ! ⎥ ⎢ 0 ⎥ ⎢ ψ! ⎥ ⎢ ω z ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦B €€ ⎡ ⎤ ω x ⎡ sinθ sinφ cosφ 0 ⎤⎡ ψ! ⎤ € ⎢ ⎥ ⎢ ⎥⎢ ⎥ ω ! B ! ⎢ y ⎥ = ⎢ sinθ cosφ −sinφ 0 ⎥⎢ θ ⎥ = LI Θ ⎢ ⎥ ⎢ cos 0 1 ⎥⎢ φ! ⎥ ⎢ ω z ⎥ ⎣ θ ⎦⎣ ⎦ ⎣ ⎦B Inverse transformation [(.)-1 ≠ (.)T]

⎡ ⎤ ⎡ ψ! ⎤ ⎡ sinφ cosφ 0 ⎤ ω x ⎢ ⎥ 1 ⎢ ⎥⎢ ⎥ I ⎢ θ! ⎥ = cosφ sinθ −sinφ sinθ 0 ⎢ ω y ⎥ = LBω B sinθ ⎢ ⎥ ⎢ φ! ⎥ ⎢ −sinφ cosθ cosφ cosθ sinθ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎢ ω z ⎥ ⎣ ⎦B 20

10 2/12/20

(1-2-3) Euler-Angle Rates and Body-Axis Rates

Options for Avoiding the Singularity at θ = ±90° § Don’t use Euler angles as primary definition of angular attitude § Alternatives to Euler angles - Direction cosine (rotation) matrix - Quaternions (next lecture) Propagation of rotation matrix (1-2-3) (9 parameters) H I h  HI h B B = ωI B B Consequently ⎡ 0 −r(t) q(t) ⎤ ⎢ ⎥ ! B B B HI (t) = −ω" B (t)HI (t) = − ⎢ r(t) 0 − p(t) ⎥ HI (t) ⎢ ⎥ ⎢ −q(t) p(t) 0(t) ⎥ ⎣ ⎦B B B HI (0) = HI (φ0 ,θ0 ,ψ 0 ) 22

11 2/12/20

Body-Axis Angular Rate Dynamics [(3-1-3) convention] ω! = −1 M − ω" ω B IB [ B BIB B ] ⎡ ω ⎤ ⎢ x ⎥ ω B = ⎢ ω y ⎥ ⎢ ⎥ ω z ⎣⎢ ⎦⎥B For principal axes

⎡ ⎤ ⎡ ⎤ ⎡ − ω t ω t / ⎤ ω! x (t) M x (t) / I xx (I zz I yy ) y ( ) z ( ) I xx ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ω! y (t) ⎥ = ⎢ M y (t) / I yy ⎥ − (I xx − I zz )ω x (t)ω z (t) / I yy ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ω! t M t / ⎢ ⎥ ⎢ z ( ) ⎥ ⎢ z ( ) I zz ⎥ I yy − I xx ω x (t)ω y (t) / I zz ⎣ ⎦ ⎣ ⎦ ⎣⎢ ( ) ⎦⎥ 23

Small Perturbations from Nominal Angular Rate ⎡ ⎤ ⎡ ω ⎤ ω x + Δω x ⎢ x ⎥ ⎢ o ⎥ ω ⎢ y ⎥ = ⎢ ω yo + Δω y ⎥ ⎢ ⎥ ⎢ ⎥ ω z ω + Δω ⎣⎢ ⎦⎥ ⎣⎢ zo z ⎦⎥

⎡ d dt ⎤ ⎡ / ⎤ ω xo + Δω x ⎡ ⎤ I zz − I yy ω yo + Δω y ω zo + Δω z I xx ⎢ ( ) ⎥ M x / I xx ⎢ ( )( )( ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ d ω + Δω dt = M y / I yy − − ω + Δω ω + Δω / ⎢ ( yo y ) ⎥ ⎢ ⎥ ⎢ (I xx I zz )( xo x )( zo z ) I yy ⎥ ⎢ ⎥ ⎢ M / ⎥ ⎢ ⎥ d ω + Δω dt ⎢ z I zz ⎥ − ω + Δω ω + Δω / ⎢ ( zo z ) ⎥ ⎣ ⎦ ⎢ (I yy I xx )( xo x )( yo y ) I zz ⎥ ⎣ ⎦ ⎣ ⎦ Products of small perturbations are negligible

Δω xΔω y = Δω xΔω z = Δω yΔω z ! 0 24

12 2/12/20

Small Perturbation Equations for Spacecraft Spinning about z Axis

Assume yaw rate is constant, while pitch and yaw motions are small

⎡ ⎤ ⎡ ω ⎤ Δω x ⎢ x ⎥ ⎢ ⎥ ⎢ ω y ⎥ = ⎢ Δω y ⎥ ⎢ ⎥ ⎢ ω ⎥ ω ⎢ z ⎥ ⎢ zo ⎥ Then ⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ω! ⎤ Δω! x ⎡ ⎤ ⎡ω − Δω ⎤ x M x I xx ⎣ zo (I zz I yy ) y ⎦ I xx ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ω! Δω! ⎢ ⎥ ⎢ y ⎥ = ⎢ y ⎥ = ⎢ M y I yy ⎥ − ⎡ω − Δω ⎤ zo (I xx I zz ) x I yy ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣ ⎦ ⎥ ω! ω! z 0 ⎣⎢ z ⎦⎥ ⎣⎢ o ⎦⎥ ⎣ ⎦ ⎢ 0 ⎥ ⎣ ⎦

2nd-Order Model of Pitch and Yaw Perturbations r! t = 0 ( ) Linear, Time-Invariant (LTI) Ordinary Differential Equation

⎡ ω − ⎤ ⎡ M t ⎤ ⎢ zo (I yy I zz ) ⎥ x ( ) ⎡ ⎤ 0 ⎡ ⎤ ⎢ ⎥ Δω! x (t) ⎢ ⎥ Δω x (t) I xx ⎢ ⎥ I xx ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ + ⎢ ⎥ ⎢ Δω! y t ⎥ ω − ⎢ Δω y t ⎥ M y (t) ( ) ⎢ zo (I zz I xx ) ⎥ ( ) ⎣ ⎦ 0 ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ I yy ⎥ ⎢ I yy ⎥ ⎣ ⎦ ⎣ ⎦

13 2/12/20

Laplace Transforms of Scalar Variables s: Laplace operator, a complex variable

∞ L [x(t)] = x(s) = ∫ x(t)e−st dt, s = σ + jω, ( j = i = −1) 0

Multiplication Transform of by a constant a derivative a x(t) a x(s) L x!(t) = sx(s) − x 0 L [ ] = [ ] ( ) Sum of transforms Transform of an integral

L [x1(t) + x2 (t)] = x1(s) + x2 (s) L ⎡ x(t)dt ⎤ = x (s) s ⎣∫ ⎦ 1 27

Laplace Transforms of Vectors and Matrices Laplace transform of a vector variable

⎡ x (s) ⎤ ⎢ 1 ⎥ x(t) x(s) L [ ] = = ⎢ x2 (s) ⎥ ⎢ ... ⎥ ⎣ ⎦ Laplace transform of a matrix variable

⎡ a (s) a (s) ... ⎤ ⎢ 11 12 ⎥ A(t) A(s) L [ ] = = ⎢ a21(s) a22 (s) ... ⎥ ⎢ ...... ⎥ ⎣ ⎦ Laplace transform of a time-derivative L [x!(t)] = sx(s) − x(0) 28

14 2/12/20

Transformation of the Dynamic Equation Δx(t) : Dynamic State Δu(t) : Control Input Time-Domain LTI Dynamic Equation

Δx!(t) = F Δx(t) + G Δu(t) Laplace Transform of LTI Dynamic Equation sΔx(s) − Δx(0) = F Δx(s) + GΔ u(s)

Laplace Transform of the State Vector Rearrange Laplace Transform of Dynamic Equation sΔx(s) − F Δx(s) = Δx(0) + G Δu(s) [sI − F]Δx(s) = Δx(0) + G Δu(s) 1 Δx(s) = [sI − F]− [Δx(0) + G Δu(s)] Inverse of characteristic matrix

−1 Adj(sI − F) [sI − F] = (n x n) sI − F

Adj(sI − F) : Adjoint matrix (n × n) sI − F = det(sI − F) : Determinant (1×1) 30

15 2/12/20

Eigenvalues of the Dynamic System

−1 Adj(sI − F) [sI − F] = (n x n) sI − F Characteristic polynomial of the system, [Δ(s)]

sI − F = det(sI − F) n n−1 ≡ Δ(s) = s + an−1s + ...+ a1s + a0 Characteristic equation of the system, [Δ(s) = 0]

n n−1 Δ(s) = s + an−1s + ...+ a1s + a0 = 0

= (s − λ1 )(s − λ2 )(...)(s − λn ) = 0 Factors (or roots) of Δ(s) are the eigenvalues

LTI Dynamic Model of Spacecraft Spinning about z Axis (ODE)

⎡ ω − ⎤ ⎡ ΔM t ⎤ ⎢ zo (I yy I zz ) ⎥ x ( ) ⎡ ⎤ 0 ⎡ ⎤ ⎢ ⎥ Δω! x (t) ⎢ ⎥ Δω x (t) I xx ⎢ ⎥ I xx ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ + ⎢ ⎥ ⎢ Δω! y t ⎥ ω − ⎢ Δω y t ⎥ ΔM y (t) ( ) ⎢ zo (I zz I xx ) ⎥ ( ) ⎣ ⎦ 0 ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ I yy ⎥ ⎢ I yy ⎥ ⎣ ⎦ ⎣ ⎦ Δx!(t) = F Δx(t) + G Δu(t) ⎡ Δω t ⎤ x ( ) ⎡ ⎤ Δx(t) = ⎢ ⎥ ΔM x (t) ⎢ Δω y (t) ⎥ Δu(t) = ⎢ ⎥ ⎣ ⎦ ⎢ ΔM y (t) ⎥ ⎣ ⎦ ⎡ ⎤ ω z I yy − I zz ⎢ 0 o ( ) ⎥ ⎡ ⎤ ⎢ I xx ⎥ 1/ I xx 0 F= ⎢ ⎥ G= ⎢ ⎥ ω − ⎢ 0 1/ I yy ⎥ ⎢ zo (I zz I xx ) ⎥ ⎣ ⎦ ⎢ 0 ⎥ ⎢ I yy ⎥ 32 ⎣ ⎦

16 2/12/20

LTI Model of Spacecraft Spinning about z Axis (Transform)

1 Δx(s) = [sI − F]− [Δx(0) + G Δu(s)]

–1 ⎡ ⎤ ⎧ ⎡ ⎤⎫ ω z I yy − I zz ΔM x (s) ⎢ o ( ) ⎥ ⎪ ⎪ ⎡ ⎤ s − ⎡ ⎤ ⎢ ⎥ Δω x (s) ⎢ ⎥ ⎪ Δω x (0) I xx ⎪ ⎢ ⎥ I xx ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎨ + ⎢ ⎥⎬ ⎢ Δω y s ⎥ ω − ⎢ Δω y 0 ⎥ ΔM y (s) ( ) ⎢ zo (I zz I xx ) ⎥ ⎪ ( ) ⎪ ⎣ ⎦ − s ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ I yy ⎥⎪ ⎢ I yy ⎥ ⎩ ⎣ ⎦⎭ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ Δω x (s) I yy − I zz ⎢ s ( ) ⎥ Δx(s) = ⎢ ⎥ ω zo ⎢ Δω y (s) ⎥ ⎢ I xx ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ (I zz − I xx ) ⎥ ω s ⎢ zo ⎥ I yy ⎡ ⎤ −1 ⎣ ⎦ ΔM x (s) [sI − F] = Δu(t) = ⎢ ⎥ ⎡ ⎤ 2 (I zz − I xx )(I yy − I zz ) 2 ⎢ ΔM y (s) ⎥ ⎢s − ω ⎥ ⎣ ⎦ zo ⎣⎢ I xxI yy ⎦⎥ 33

Characteristic Equations and Eigenvalues

Characteristic equation, with Ixx ≠ Iyy ≠ Izz

⎡ ⎤ (I zz − I xx ) I yy − I zz s s2 ( ) 2 0 Δ( ) = ⎢ − ω zo ⎥ = ⎢ I xxI yy ⎥ ⎣ ⎦ Eigenvalues

⎛ ⎞ ⎛ ⎞ λ = ±ω I zz −1 1− I zz rad / sec 1,2 zo ⎜ ⎟ ⎜ ⎟ ⎝ I xx ⎠ ⎝ I yy ⎠

17 2/12/20

Eigenvalues of the Spinning Spacecraft

• If Izz < Ixx & Iyy or Izz > Ixx & Iyy, eigenvalues are imaginary, and neutrally stable oscillation occurs

A − jωnt + jωnt Δω x (t) = Acos(ω nt) = ⎡e + e ⎤ 2 ⎣ ⎦

• If Izz is between Ixx and Iyy, eigenvalues are real, and one is positive (i.e., unstable) A Δω (t) = e−σt + Be+σt x 2 ( )

Therefore, satellite attitude is stable only if it spins about the axis of maximum or minimum moment of inertia

Axisymmetric Spacecraft Spinning About z Axis

I xx = I yy

2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ λ = ±ω I zz −1 1− I zz = ±ω − 1− I zz 1,2 zo ⎜ ⎟ ⎜ ⎟ zo ⎜ ⎟ ⎝ I xx ⎠ ⎝ I yy ⎠ ⎝ I xx ⎠ ⎛ ⎞ = ± jω 1− I zz rad / sec zo ⎜ ⎟ ⎝ I xx ⎠

j = −1 Imaginary roots Neutral, oscillatory stability 36

18 2/12/20

Spin Stability Eigenvalues define natural frequency of an undamped oscillation

⎛ ⎞ λ = ± jω 1− I zz = ± jω rad / sec 1,2 zo ⎝⎜ ⎠⎟ n I xx Motion is oscillatory but neutrally stable

⎡ ⎛ ⎞ ⎤ ⎡ ⎛ ⎞ ⎤ Δω (t) = Asin ω 1− I zz t + Bcos ω 1− I zz t x ⎢ zo ⎜ ⎟ ⎥ ⎢ zo ⎜ ⎟ ⎥ ⎣ ⎝ I xx ⎠ ⎦ ⎣ ⎝ I xx ⎠ ⎦ ⎡ ⎛ ⎞ ⎤ ⎡ ⎛ ⎞ ⎤ Δω (t) = Acos ω 1− I zz t − Bsin ω 1− I zz t y ⎢ zo ⎜ ⎟ ⎥ ⎢ zo ⎜ ⎟ ⎥ ⎝ I xx ⎠ ⎝ I xx ⎠ ⎣ ⎦ ⎣ ⎦ Unfortunate notation overlap:

(ω x ,ω y ,ω z ) are components of rotation rate, rad/s

ω and ω n are oscillatory input frequency and system natural frequency, rad/s 37

Ellipsoid of Inertia Properties of the spacecraft mass distribution For principal axes in the body frame of reference, x2 y2 z2 + + = a2 b2 c2 2 2 2 I xx x + I yy y + I zzz = 1

19 2/12/20

Angular Momentum Ellipsoid Locus of all angular rate combinations with constant angular momentum (principal axes, in the body frame or reference)

2 2 2 2 2 2 2 T I xx ω x + I yy ω y + I zz ω z = h = h h

… but angular momentum vector is fixed in an inertial reference frame

therefore, body reference frame may rotate even with MB = 0 39

Angular Momentum Distribution

Magnitude of angular momentum constant and identical in body and inertial reference frames

2 2 2 2 2 2 2 T 2 2 (I xx ω x + I yy ω y + I zz ω z ) = hB = hBhB = hI = Constant B

h B = h I = Constant … but individual components may vary Axisymmetric spacecraft spinning about z axis

⎡ ⎤ ⎡ ⎤ ω x ω x ⎡ 0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ h ω 0 h h B = IB ⎢ y ⎥ = I xx ⎢ ω y ⎥ + I zz ⎢ ⎥ ! xy + z ⎢ ⎥ ⎢ ⎥ ⎢ ω z ⎥ ω z 0 ⎣ ⎦ ⎣⎢ ⎦⎥B ⎣ ⎦

20 2/12/20

Angular Momentum of Spinning Axisymmetric Spacecraft

2 2 2 2 2 2 2 I xx (ω x +ω y ) + I zz ω z = hxy + hz

θ ! Nutation Angle: body-axis orientation w.r.t. inertial axes

2 2 h I xx ω x +ω y tanθ = xy = = Constant h ω z I zz z γ ! Precession Angle: angular rate orientation w.r.t. body axes

2 2 ω x +ω y tanγ = = Constant ω z 41

Body and Space Cones Angular momentum is fixed in inertial frame

Define ∠hI = ∠zI for diagrams Direct Precession Retrograde Precession I < I (Rod) zz xx Izz > Ixx (Disc)

21 2/12/20

⎡ ψ ⎤ ⎢ ⎥ Θ = ⎢ θ ⎥ Angular Motion in Space ⎢ φ ⎥ ⎣ ⎦3−1−3 Body-axis rates in terms of Euler-angle rates

⎡ ⎤ ω x ⎡ sinθ sinφ cosφ 0 ⎤⎡ ψ! ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ω ! B ! ω B = ⎢ y ⎥ = ⎢ sinθ cosφ −sinφ 0 ⎥⎢ θ ⎥ = LI Θ ⎢ ⎥ ⎢ cos 0 1 ⎥⎢ φ! ⎥ ⎢ ω z ⎥ ⎣ θ ⎦⎣ ⎦ ⎣ ⎦B

! 2 2 With θ = 0, ω x +ω y = constant, ω z = constant ⎡ ⎤ ω x ⎡ ψ! sinθ sinφ ⎤ ⎢ ⎥ ⎢ ⎥ ω B = ⎢ ω y ⎥ = ⎢ ψ! sinθ cosφ ⎥ ⎢ ⎥ ⎢ ! ⎥ ⎢ ω z ⎥ φ +ψ! cosθ ⎣ ⎦B ⎣ ⎦

Angular Motion in Space Body-axis rate is constant T 2 ω B ω B = ω B = constant

d T (ω B ω B ) T = 2 ω ω! = 0 dt ( B B ) ω ω! +ω ω! +ω ω! = 0 x x y y z z Body-axis acceleration

⎡ ω! ⎤ ⎡ ψ!!sinθ sinφ +ψ!φ!sinθ cosφ ⎤ ⎢ x ⎥ ⎢ ⎥ ⎢ ω! y ⎥ = ⎢ ψ!!sinθ cosφ −ψ!φ!sinθ sinφ ⎥ ⎢ ! ⎥ ⎢ !! ⎥ ⎢ ω z ⎥ ⎢ φ +ψ! cosθ ⎥ ⎣ ⎦B ⎣ ⎦ Consequently T 2 ! ! !!sin 0 44 ω B ω B =ψψ θ =

22 2/12/20

Angular Motion in Space … and precession speed is constant

T 2 ω B ω! B =ψ!ψ!!sin θ = 0 d ⎛ψ! 2 ⎞ for θ ≠ 0, ψ!ψ!! = = 0 dt ⎝⎜ 2 ⎠⎟ ! constant = Precession Speed ∴ψ =

⎡ ⎤ ω! x ⎡ ψ!φ!sinθ cosφ ⎤ ⎢ ⎥ ⎢ ⎥ ∴φ!!= 0 ⎢ ω! y ⎥ = ⎢ −ψ!φ!sinθ sinφ ⎥ ⎢ ! ⎥ ⎢ ⎥ ⎢ ω z ⎥ " 0 ⎣ ⎦B ⎣ ⎦ 45

Angular Motion in Space

Recall: I xxω! x + (I zz − I yy )ω yω z = 0

I xx = I yy : I xxω! x + (I zz − I xx )ω yω z = 0

! ! I xxψ!φ sinθ cosφ + (I zz − I xx )(ψ! sinθ cosφ) φ +ψ! cosθ = 0 ( ) Which reduces to ! (I xx − I zz ) I zz I zzφ ! ψ! = ω or 0 ψ! = or 0 φ = ω z z I xx I xx cosθ (I xx − I zz )cosθ

Direct Precession Retrograde Precession I < I (Rod) zz xx Izz > Ixx (Disc) ! sgn(ψ! ) = sgn(φ) sgn ψ! = −sgn φ! ( ) ( )

23 2/12/20

Poinsot (Energy) Ellipsoid Locus of all angular rate combinations with constant angular energy

2 2 2 T I xxω x + I yyω y + I zzω z = 2 × Kinetic Energy = 2T = ω Iω

Polhodes: Paths of angular rate oscillations on Inertia Ellipsoid

Stable Point

Unstable Point Stable Point

47 47

The Strange Case of Explorer I Explorer I spun about its axis of minimum moment of inertia when inserted into orbit Within a short time, it went into a flat spin, rotating about its maximum moment of inertia Why?

24 2/12/20

Shifting Rotational Equilibrium of Explorer I Whipping antennas dissipate angular energy Angular momentum remains constant Equilibrium rotational axis shifts from minimum to maximum moment of inertia (i.e., a flat spin)

Polhode of Explorer I angular rate perturbations

Kaplan

INTELSAT-IVA Dual-Spin Satellite Dynamics Satellite has spinning and non-spinning components Angular momentum Angular momentum and rate in non- added by portion spinning frame of reference spinning about z axis h! = ω! = M − ω" ω + h ⎡ 0 ⎤ B IB B B B (IB B rotor ) ⎢ ⎥ h = 0 ! −1 M " h rotor ⎢ ⎥ ω B = IB ⎣⎡ B − ω B (IBω B + rotor )⎦⎤ h ⎣⎢ rotor ⎦⎥

−1 ⎡ ⎤ ⎡ ⎤ ⎧⎡ ⎤ ⎛ ⎞ ⎡⎛ ⎞ ⎛ ⎞ ⎤⎫ ω! x I xx 0 0 M x 0 −ω z ω y I xx 0 0 ω x ⎛ ⎞ ⎪ ⎢ 0 ⎥⎪ ⎢ ⎥ ⎢ ⎥ ⎪⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ω! y = 0 I yy 0 M y − ω z 0 −ω x ⎢ 0 I yy 0 ω y + 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎨⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ ⎢ ⎥ ⎢ ⎥ ⎪⎢ ⎥ ⎢ ⎜ ⎟ ⎥⎪ ω! 0 0 M ⎜ −ω ω 0 ⎟ ⎜ 0 0 ⎟ ⎜ ω ⎟ ⎝ hrotor ⎠ ⎣⎢ z ⎦⎥ ⎣⎢ I zz ⎦⎥ ⎪⎣⎢ z ⎦⎥ ⎝ y x ⎠ ⎣⎢⎝ I zz ⎠ ⎝ z ⎠ ⎦⎥⎪ ⎩ ⎭

Ixx, Iyy, and Izz are moments of inertia of the entire satellite 50

25 2/12/20

Perturbations in Dual-Spin Satellite Angular Rate With zero nominal rates ⎡ ⎤ ⎡ ω ⎤ ω x + Δω x ⎡ Δω ⎤ ⎢ x ⎥ ⎢ o ⎥ ⎢ x ⎥ ω = ω + Δω = Δω ⎢ y ⎥ ⎢ yo y ⎥ ⎢ y ⎥ ⎢ ⎥ ⎢ ω ⎥ ω + Δω ⎢ Δω ⎥ ⎣⎢ z ⎦⎥ ⎣⎢ zo z ⎦⎥ ⎣⎢ z ⎦⎥ Perturbations (or nutations) in roll and pitch rate

−1 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Δω! x I xx 0 0 ΔM x − hrotor Δω y ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Δω! y ⎥ = ⎢ 0 I yy 0 ⎥ ⎢ ΔM y + hrotor Δω x ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Δω! 0 0 ΔM ⎣⎢ z ⎦⎥ ⎣⎢ I zz ⎦⎥ ⎢ z ⎥ ⎣ ⎦ If z is the axis of minimum inertia (i.e., a “prolate” configuration), nutation damping is required to prevent satellite from entering a flat spin 51

Eigenvalues of Undamped Dual-Spin Satellite

⎡ ! ⎤ ⎡ ⎤⎡ ⎤ ⎡ ΔM t / ⎤ Δω x 0 −hrotor / I xx 0 Δω x x ( ) I xx ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ Δω! Δω ⎢ y ⎥ = ⎢ hrotor / I yy 0 0 ⎥⎢ y ⎥ + ⎢ ΔM y (t) / I yy ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Δω! ⎢ 0 0 0 ⎥ Δω M t / ⎣⎢ z ⎦⎥ ⎣ ⎦⎣⎢ z ⎦⎥ ⎢ Δ z ( ) I zz ⎥ ⎣ ⎦ Eigenvalues Natural frequency of small nutation orbits about the equilibrium point s h / 0 rotor I xx h ω = rotor Δ(s) = sI − F = h / s 0 n rotor I yy I xxI yy

0 0 s

3 2 2 2 = s + hrotor I xxI yy s = s s + hrotor I xxI yy = 0 ( ) ( )

hrotor λ1,2,3 = 0, ± j I xxI yy 52

26 2/12/20

Dual-Spin Satellite with Nutation Damper

Spring-mass-damper mounted on fixed platform Angular motion about the x or y axis disturbs the system

mΔ!z!m = −kd (Δz! − Δz!S/C ) − ks (Δzm − ΔzS/C ) = −k Δz! − bΔq − k Δz − bΔθ d ( S/C ) s ( m S/C ) ks is a “soft” centering spring. Neglecting ks, the mass’s reaction torque on the spacecraft introduces damping, d

⎡ ! ⎤ ⎡ ⎤⎡ ⎤ Δω x 0 −hrotor / I xx 0 Δω x ⎢ ⎥ ⎢ ⎥⎢ ⎥ Δω! Δω ⎢ y ⎥ = ⎢ hrotor / I yy −d / I yy 0 ⎥⎢ y ⎥ ⎢ Δω! ⎥ ⎢ ⎥⎢ Δω ⎥ ⎣⎢ z ⎦⎥ ⎣ 0 0 0 ⎦⎣⎢ z ⎦⎥ 53

Natural Frequency and Damping Ratio of Dual-Spin Satellite with Nutation Damper

s hrotor / I xx 0 Δ(s) = sI − F = = 0 hrotor / I yy (s + d / I yy ) 0 0 0 s

2 2 2 2 = s (s + d / I yy ) + s(hrotor I xxI yy ) = s(s + sd / I yy + hrotor I xxI yy ) 2 2 ≡ s(s + 2ζω ns +ω n )

hrotor d / I yy ω n = ζ = I xxI yy 2hrotor / I xxI yy

Damper prevents small nutations from becoming large enough to shift equilibrium spin axes 54

27 2/12/20

Gravity-Gradient Effect on Spacecraft Attitude

• Gravitational field and gravity gradient µ ∂ g 2µ g = − ; = r2 ∂r r 3

• Dumbbell satellite (equal masses at end of bar) –At horizontal attitude, gravitational effects are equal, and torque is zero –At small angle, forces are unequal, and torque rotates satellite away from horizontal –Near vertical attitude, unequal forces tend to align satellite with the vertical

Gravity-Gradient Stabilization about the Local Vertical Gravitational torques on satellite, (1-2-3) Euler angles

3µ 2 M x = 3 I zz − I yy sin2φ cos θ 2r ( ) 3µ M y = 3 I zz − I xx sin2θ cosφ 2r ( ) 3µ M sin2 sin z = 3 (I xx − I yy ) θ φ 2r With Ixx = Iyy, restoring torques produce a librational oscillation with natural frequency

3µ ⎛ ⎞ 1 I zz rad / sec ω n = 3 ⎜ − ⎟ 2r ⎝ I xx ⎠ 56

28 2/12/20

Next Time: Spacecraft Control

Supplemental Material

29 2/12/20

Cross-Product-Equivalent Matrix

• Cross product ⎡ i ⎤ ⎢ j ⎥ unit vectors along (x, y,z) ⎢ ⎥ = ⎢ k ⎥ i j k ⎣ ⎦ x y z r × v = = (yvz − zvy )i + (zvx − xvz ) j + (xvy − yvx )k

vx vy vz

⎡ 1 ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ = yv − zv ⎢ ⎥ + zv − xv ⎢ ⎥ + xv − yv ⎢ ⎥ ( z y )⎢ 0 ⎥ ( x z )⎢ 1 ⎥ ( y x )⎢ 0 ⎥ ⎣⎢ 0 ⎦⎥ ⎣⎢ 0 ⎦⎥ ⎣⎢ 1 ⎦⎥

⎡ yv − zv ⎤ ⎡ ⎤ ⎢ ( z y ) ⎥ ⎡ 0 −z y ⎤ vx ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = zv − xv = z 0 x vy = r!v ⎢ ( x z ) ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −y x 0 ⎥ ⎢ ⎥ xv − yv ⎣ ⎦ vz ⎢ ( y x ) ⎥ ⎣⎢ ⎦⎥ ⎣ ⎦ ⎡ 0 −z y ⎤ ⎢ ⎥ • Cross-product-equivalent matrix r! = ⎢ z 0 −x ⎥ ⎢ −y x 0 ⎥ ⎣ 59 ⎦

Eigenvalues (or Roots) of the Dynamic System

n n−1 Δ(s) = s + an−1s + ...+ a1s + a0 = 0

= (s − λ1 )(s − λ2 )(...)(s − λn ) = 0 The picture can't be • Roots may be real or complex displayed. • Real and imaginary parts of the eigenvalues can be plotted in the s plane • Real roots δ = cos−1 ζ – are confined to the real axis – represent convergent or divergent modes • Complex roots – occur only in complex-conjugate pairs – represent oscillatory modes – natural frequency and damping ratio as s Plane shown

30 2/12/20

Modes of Motion

Adj(sI − F) x(s) = [x(0) + Gu(s)] Δ(s)

• Eigenvalues characterize the modes of the system

– Mode is stable if Real (λi) < 0

– Mode is unstable if Real (λi) > 0

Response of first-order mode Response of second-order mode