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Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE342.html

Spacecraft DynamicsSpace 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

1

1

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 × vm( )dm= r × vo +ω × r( )( )dm

2

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Integrate moment of linear momentum of differential particles over the body

h = r × vo +ω × r( )( )dmBody∫

= r × vo( )dmBody∫ + r × ω × r( )( )dm

Body∫

= 0 − r × r × ω( )( )dmBody∫ = − r × r( )dm × ω

Body∫

≡ − !r!r( )dmωBody∫

Angular Momentum of an Object

ω =ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

3

h =hxhyhz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

3

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 dmBody∫ ω

= −0 −z yz 0 −x−y x 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

0 −z yz 0 −x−y x 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥zmin

zmax

∫ymin

ymax

∫xmin

xmax

∫ ρ(x, y, z)dxdydz

4

=(y2 + z2 ) −xy −xz

−xy (x2 + z2 ) −yz

−xz −yz (x2 + y2 )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

dmBody∫ ω = I ω

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Inertia Matrix

I =(y2 + z2 ) −xy −xz

−xy (x2 + z2 ) −yz

−xz −yz (x2 + y2 )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

dmBody∫ =

Ixx −Ixy −Ixz−Ixy Iyy −Iyz−Ixz −Iyz Izz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Moments of inertia on the diagonalProducts of inertia off the diagonal

If products of inertia are zero, (x, y, z) are principal axes

IP =Ixx 0 00 Iyy 0

0 0 Izz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

5

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 frameRate of change of angular momentum = applied moment (or torque), m

dhdt

=d Iω( )dt

=M =

Mx

My

Mz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

d Iω( )dt

= dhdt

= d Idt

ω + I !ω

Chain Rule

d Idt

≠ 0

In an inertial frame

6

I !ω =M − d Idt

ω

!ω = I−1 M − d Idt

ω⎛⎝⎜

⎞⎠⎟

Inertial-frame solution for angular rate

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How Do We Get Rid of dI/dt in the Angular Momentum Equation?

Write the dynamic equation in a body-referenced frame

7

§ Inertia matrix is ~unchanging in a body frame

§ Body-axis frame is rotating

§ Dynamic equation must be modified to account for rotation

7

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)

hB = H IBhI

ω B = H IBω I

8

hI = HBI hB

ω I = HBIω B

H IB : Rotation transformation from inertial frame ⇒ body frame

HBI : Rotation transformation from body frame ⇒ inertial frame

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Euler Angles

9

ψ : Yaw angleθ : Pitch angleφ : Roll angle

• Body attitude measured with respect to inertial frame• Three-angle orientation expressed by sequence of

three orthogonal single-angle rotations

Inertial⇒ Intermediate1 ⇒ Intermediate2 ⇒ Body

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

ψ : Yaw angle1

θ : Pitch angleφ : Yaw angle2

9

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

Yaw rotation (ψ) about zI axis

Pitch rotation (θ) about y1 axis

Roll rotation (ϕ) about x2 axis

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥1

=cosψ sinψ 0−sinψ cosψ 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥I

=xI cosψ + yI sinψ−xI sinψ + yI cosψ

zI

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥2

=cosθ 0 −sinθ0 1 0sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥1

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥B

=1 0 00 cosφ sinφ0 −sinφ cosφ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥2 10

r1 = H I1rI

r2 = H12r1 = H1

2H I1⎡⎣ ⎤⎦rI = H I

2rI

rB = H2Br2 = H2

BH12H I

1⎡⎣ ⎤⎦rI = H IBrI

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

Yaw rotation (ψ) about zI axis

Pitch rotation (θ) about x1 axis

Yaw rotation (ϕ) about z2 axis

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥1

=cosψ sinψ 0−sinψ cosψ 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥I

=xI cosψ + yI sinψ−xI sinψ + yI cosψ

zI

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

11

r1 = H I1rI

r2 = H12r1 = H1

2H I1⎡⎣ ⎤⎦rI = H I

2rI

rB = H2Br2 = H2

BH12H I

1⎡⎣ ⎤⎦rI = H IBrI

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥B

=cosφ sinφ 0−sinφ cosφ 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥2

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥2

=1 0 00 cosθ sinθ0 −sinθ cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥1

11

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

H IB(φ,θ ,ψ)=H2

B(φ)H12 (θ )H I

1 (ψ)

=1 0 00 cosφ sinφ0 −sinφ cosφ

#

$

%%%

&

'

(((

cosθ 0 −sinθ0 1 0sinθ 0 cosθ

#

$

%%%

&

'

(((

cosψ sinψ 0−sinψ cosψ 00 0 1

#

$

%%%

&

'

(((

=

cosθ cosψ cosθ sinψ −sinθ−cosφ sinψ + sinφ sinθ cosψ cosφ cosψ + sinφ sinθ sinψ sinφ cosθsinφ sinψ + cosφ sinθ cosψ −sinφ cosψ + cosφ sinθ sinψ cosφ cosθ

#

$

%%%

&

'

(((

12

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Rotation Matrix from I to BSpacecraft Convention (3-1-3)

H IB(φ,θ ,ψ)=H2

B(φ)H12 (θ )H I

1 (ψ)

13

=cosφ sinφ 0−sinφ cosφ 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1 0 00 cosθ sinθ0 −sinθ cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

cosψ sinψ 0−sinψ cosψ 00 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

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

13

Properties of the Rotation Matrix

Orthonormal transformationAngles between vectors are preserved

Lengths are preserved

rI = rB ; s I = sB∠(rI ,s I ) = ∠(rB ,sB ) = xdeg

r s

14

H IB(φ,θ ,ψ ) =

h11 h12 h13h21 h22 h23h31 h32 h33

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥I

B

H IB(φ,θ ,ψ ) = H2

B(φ)H12 (θ )H I

1 (ψ )

H IB(φ,θ ,ψ )⎡⎣ ⎤⎦

−1= H I

B(φ,θ ,ψ )⎡⎣ ⎤⎦T= HB

I (ψ ,θ ,φ)

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Rotation Matrix InverseInverse relationship: interchange sub- and superscripts

Because transformation is orthonormalInverse = transpose

Rotation matrix is always non-singular

rB = H IBrI

rI = H IB( )−1 rB = HB

I rB

HBI = H I

B( )−1 = H IB( )T = H1

IH21HB

2

HBI H I

B = H IBHB

I = I15

15

Vector Derivative Expressed in a Rotating Frame

Chain Rule

!hI t( ) = HB

I t( ) !hB t( ) + !HBI t( )hB t( )

Effect of body-frame rotation

Rate of change expressed in body frame

Alternatively

!hI = HB

I !hB +ω I × hI = HBI !hB + "ω IhI

!ω =

0 −ω z ω y

ω z 0 −ω x

−ω y ω x 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

hI t( ) = HBI t( )hB t( )

16

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Angular Rate Derivative in Body Frame of Reference

Constant body-axis inertia matrix

!hB = H IB !hI + !H I

BhI = H IB !hI − ω B × hB

= H IB !hI − "ω BhB = H I

BM I − "ω BIBω B

Angular momentum change

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

Angular rate change !hB t( ) = IB !ω B t( ) =MB t( )− "ω B t( )IBω B t( )

17

!hB =MB − "ω BIBω B

17

Euler-Angle Rates and Body-Axis Rates

Body-axis angular rate vector (orthogonal) ω B =

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

Euler-angle rate vector

Form a non-orthogonalvector of Euler angles Θ =

φθψ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

3−2−1

or ψθφ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

3−1−3

!Θ =

!φ!θ!ψ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

3−2−1

or !ψ!θ!φ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

3−1−3

≠ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥ I18

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

• measured in Inertial Frame• measured in Intermediate Frame #1• measured in Intermediate Frame #2

Inverse transformation [(.)-1 ≠ (.)T]

€

˙ φ

€

˙ θ

€

˙ ψ

pqr

!

"

###

$

%

&&&= I3

φ00

!

"

###

$

%

&&&+H2

B

0θ0

!

"

###

$

%

&&&+H2

BH1200ψ

!

"

###

$

%

&&&

pqr

!

"

###

$

%

&&&=

1 0 −sinθ0 cosφ sinφ cosθ0 −sinφ cosφ cosθ

!

"

###

$

%

&&&

φθψ

!

"

###

$

%

&&&=LI

B Θ

φθψ

$

%

&&&

'

(

)))=

1 sinφ tanθ cosφ tanθ0 cosφ −sinφ0 sinφ secθ cosφ secθ

$

%

&&&

'

(

)))

pqr

$

%

&&&

'

(

)))=LB

I ωB

19

19

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

• measured in Inertial Frame• measured in Intermediate Frame #1• measured in Intermediate Frame #2

Inverse transformation [(.)-1 ≠ (.)T]

€

˙ φ

€

˙ θ

€

˙ ψ

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

= I300!φ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+H2

B0!θ0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+H2

BH12

00!ψ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

=sinθ sinφ cosφ 0sinθ cosφ −sinφ 0cosθ 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

!ψ!θ!φ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= LI

B !Θ

!ψ!θ!φ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= 1sinθ

sinφ cosφ 0cosφ sinθ −sinφ sinθ 0−sinφ cosθ cosφ cosθ sinθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

= LBI ω B

20

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21

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

21

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)

HBI hB = ω IHB

I hB

!H IB t( ) = − "ω B t( )H I

B t( ) = −

0 −r t( ) q t( )r t( ) 0 − p t( )−q t( ) p t( ) 0 t( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

H IB t( )

Consequently

22H IB 0( ) = H I

B φ0,θ0,ψ 0( )

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Body-Axis Angular Rate Dynamics[(3-1-3) convention]

For principal axes

!ω x t( )!ω y t( )!ω z t( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

Mx t( ) / I xx

My t( ) / I yy

Mz t( ) / I zz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−

I zz − I yy( )ω y t( )ω z t( ) / I xx

I xx − I zz( )ω x t( )ω z t( ) / I yy

I yy − I xx( )ω x t( )ω y t( ) / I zz

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

ω B =

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

!ω B = IB−1 MB − "ω BIBω B[ ]

23

23

Small Perturbations from Nominal Angular Rate

d ω xo+ Δω x( ) dt

d ω yo+ Δω y( ) dt

d ω zo+ Δω z( ) dt

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

Mx / I xx

My / I yy

Mz / I zz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−

I zz − I yy( ) ω yo+ Δω y( ) ω zo

+ Δω z( ) / I xx

I xx − I zz( ) ω xo+ Δω x( ) ω zo

+ Δω z( ) / I yy

I yy − I xx( ) ω xo+ Δω x( ) ω yo

+ Δω y( ) / I zz

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

ω xo+ Δω x

ω yo+ Δω y

ω zo+ Δω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Products of small perturbations are negligible

Δω xΔω y = Δω xΔω z = Δω yΔω z ! 024

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13

Small Perturbation Equations for Spacecraft Spinning about z Axis

25

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

Then

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

Δω x

Δω y

ω zo

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

!ω x

!ω y

!ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

Δ !ω x

Δ !ω y

!ω zo

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=Mx I xx

My I yy

0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−

ω zoI zz − I yy( )Δω y

⎡⎣ ⎤⎦ I xx

ω zoI xx − I zz( )Δω x⎡⎣ ⎤⎦ I yy

0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

25

2nd-Order Model of Pitch and Yaw Perturbations

26

Linear, Time-Invariant (LTI) Ordinary Differential Equation !r t( ) = 0

Δ !ω x t( )Δ !ω y t( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥=

0ω zo

I yy − I zz( )I xx

ω zoI zz − I xx( )I yy

0

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

Δω x t( )Δω y t( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥+

Mx t( )I xx

My t( )I yy

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

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14

Laplace Transforms of Scalar Variables

s: Laplace operator, a complex variable

L x(t)[ ] = x(s) = x(t)e− st dt

0

∞

∫ , s =σ + jω , ( j = i = −1)

Sum of transforms

L x1(t)+ x2 (t)[ ] = x1(s)+ x2 (s)

Multiplicationby a constant

L a x(t)[ ] = a x(s)

27

L x(t)dt∫⎡⎣ ⎤

⎦ = x1(s) s

Transform of a derivative

L !x(t)[ ] = sx(s)− x 0( )Transform of an integral

27

Laplace Transforms of Vectors and Matrices

Laplace transform of a vector variable

L x(t)[ ] = x(s) =x1(s)x2 (s)...

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Laplace transform of a matrix variable

L A(t)[ ] = A(s) =a11(s) a12 (s) ...a21(s) a22 (s) ...... ... ...

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Laplace transform of a time-derivative

L !x(t)[ ] = sx(s)− x(0)28

28

2/12/20

15

Δ!x(t) = FΔx(t)+GΔu(t)Time-Domain LTI Dynamic Equation

Laplace Transform of LTI Dynamic Equation

sΔx(s)− Δx(0) = FΔx(s)+GΔu(s)

Transformation of the Dynamic Equation

29

Δx(t) : Dynamic StateΔu(t) : Control Input

29

Laplace Transform of the State Vector Rearrange Laplace Transform

of Dynamic EquationsΔx(s)− FΔx(s) = Δx(0)+GΔu(s)sI− F[ ]Δx(s) = Δx(0)+GΔu(s)

Δx(s) = sI− F[ ]−1 Δx(0)+GΔu(s)[ ]

sI− F[ ]−1 = Adj sI− F( )sI− F

(n x n)

Inverse of characteristic matrix

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

30

30

2/12/20

16

Eigenvalues of the Dynamic System sI− F[ ]−1 = Adj sI− F( )

sI− F(n x n)

Characteristic polynomial of the system, [Δ(s)]

sI− F = det sI− F( )≡ Δ(s) = sn + an−1s

n−1 + ...+ a1s + a0

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

= s − λ1( ) s − λ2( ) ...( ) s − λn( ) = 0

Characteristic equation of the system, [Δ(s) = 0]

Factors (or roots) of Δ(s) are the eigenvalues31

31

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

32

Δ!x(t) = FΔx(t)+GΔu(t)

Δu(t) =ΔMx t( )ΔMy t( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

Δx(t) =Δω x t( )Δω y t( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

G=

1/ I xx 00 1/ I yy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

F=0

ω zoI yy − I zz( )I xx

ω zoI zz − I xx( )I yy

0

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

Δ !ω x t( )Δ !ω y t( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥=

0ω zo

I yy − I zz( )I xx

ω zoI zz − I xx( )I yy

0

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

Δω x t( )Δω y t( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥+

ΔMx t( )I xx

ΔMy t( )I yy

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

32

2/12/20

17

LTI Model of Spacecraft Spinning about z Axis (Transform)

33

Δω x s( )Δω y s( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥=

s −ω zo

I yy − I zz( )I xx

−ω zo

I zz − I xx( )I yy

s

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

–1

Δω x 0( )Δω y 0( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥+

ΔMx s( )I xx

ΔMy s( )I yy

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

Δu(t) =ΔMx (s)ΔMy(s)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δx(s) =Δω x (s)Δω y(s)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δx(s) = sI− F[ ]−1 Δx(0)+GΔu(s)[ ]

sI− F[ ]−1 =

sI yy − I zz( )

I xx

ω zo

I zz − I xx( )I yy

ω zos

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

s2 −I zz − I xx( ) I yy − I zz( )

I xxI yy

ω zo2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

33

Eigenvalues

λ1,2 = ±ω zo

I zz

I xx

−1⎛⎝⎜

⎞⎠⎟1− I zz

I yy

⎛

⎝⎜⎞

⎠⎟rad / sec

Characteristic equation, with Ixx ≠ Iyy ≠ Izz

34

Characteristic Equations and Eigenvalues

Δ s( ) = s2 −

I zz − I xx( ) I yy − I zz( )I xxI yy

ω zo2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 0

34

2/12/20

18

Eigenvalues of the Spinning Spacecraft

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

• If Izz is between Ixx and Iyy, eigenvalues are real, and one is positive (i.e., unstable)

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

minimum moment of inertia

Δω x (t) = Acos ω nt( ) = A2e− jωnt + e+ jωnt⎡⎣ ⎤⎦

Δω x (t) =A2e−σ t + Be+σ t( )

35

35

Axisymmetric Spacecraft Spinning About z Axis

36

I xx = I yy

Imaginary rootsNeutral, oscillatory stability

λ1,2 = ±ω zo

I zz

I xx

−1⎛⎝⎜

⎞⎠⎟1− I zz

I yy

⎛

⎝⎜⎞

⎠⎟= ±ω zo

− 1− I zz

I xx

⎛⎝⎜

⎞⎠⎟

2

= ± jω zo1− I zz

I xx

⎛⎝⎜

⎞⎠⎟rad / sec

j = −1

36

2/12/20

19

Spin Stability Eigenvalues define natural frequency of

an undamped oscillation

Δω x (t) = Asin ω zo1− I zz

I xx

⎛⎝⎜

⎞⎠⎟t

⎡

⎣⎢

⎤

⎦⎥ + Bcos ω zo

1− I zz

I xx

⎛⎝⎜

⎞⎠⎟t

⎡

⎣⎢

⎤

⎦⎥

Δω y(t) = Acos ω zo1− I zz

I xx

⎛⎝⎜

⎞⎠⎟t

⎡

⎣⎢

⎤

⎦⎥ − Bsin ω zo

1− I zz

I xx

⎛⎝⎜

⎞⎠⎟t

⎡

⎣⎢

⎤

⎦⎥

Motion is oscillatory but neutrally stable

37

λ1,2 = ± jω zo

1− I zz

I xx

⎛⎝⎜

⎞⎠⎟= ± jω nrad / sec

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

a2+ y

2

b2+ z

2

c2=

I xxx2 + I yyy

2 + I zzz2 = 1

38

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2/12/20

20

Angular Momentum Ellipsoid Locus of all angular rate combinations with

constant angular momentum (principal axes, in the body frame or reference)

I xx2ω x

2 + I yy2ω y

2 + I zz2ω z

2 = h2 = hTh

39

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

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

39

Angular Momentum DistributionMagnitude of angular momentum constant and identical in body and inertial reference frames

I xx2ω x

2 + I yy2ω y

2 + I zz2ω z

2( )B = hB2 = hBThB = hI2 = Constant2

40

h B = h I = Constant

… but individual components may varyAxisymmetric spacecraft spinning about z axis

hB = IB

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

= I xx

ω x

ω y

0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+ I zz

00ω z

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥! hxy + hz

40

2/12/20

21

Angular Momentum of Spinning Axisymmetric Spacecraft

41

I xx2 ω x

2 +ω y2( ) + I zz

2ω z2 = hxy

2 + hz2

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

tanθ =hxyhz

=I xx ω x

2 +ω y2

I zzω z

= Constant

γ ! Precession Angle:angular rate orientation w.r.t. body axes

tanγ =ω x

2 +ω y2

ω z

= Constant

41

Body and Space Cones

42

Angular momentum is fixed in inertial frame

Retrograde PrecessionIzz > Ixx (Disc)

Direct PrecessionIzz < Ixx (Rod)

Define ∠hI = ∠zI for diagrams

42

2/12/20

22

Angular Motion in Space

43

Body-axis rates in terms of Euler-angle rates

Θ = ψθφ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

3−1−3

ω B =

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

=sinθ sinφ cosφ 0sinθ cosφ −sinφ 0cosθ 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

!ψ!θ!φ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= LI

B !Θ

With !θ = 0, ω x2 +ω y

2 = constant, ω z = constant

ω B =

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

=!ψ sinθ sinφ!ψ sinθ cosφ!φ + !ψ cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

43

Angular Motion in Space

44

ω BTω B =ω B

2 = constant

!ω x

!ω y

!ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

=

!!ψ sinθ sinφ + !ψ !φ sinθ cosφ!!ψ sinθ cosφ − !ψ !φ sinθ sinφ

!!φ + !ψ cosθ

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

d ω BTω B( )dt

= 2 ω BT !ω B( ) = 0

ω x !ω x +ω y !ω y +ω z !ω z = 0

ω BT !ω B = !ψ !!ψ sin

2θ = 0

Body-axis rate is constant

Body-axis acceleration

Consequently

44

2/12/20

23

Angular Motion in Space

45

!ω x

!ω y

!ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥B

=!ψ !φ sinθ cosφ− !ψ !φ sinθ sinφ

" 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

ω BT !ω B = !ψ !!ψ sin2θ = 0

for θ ≠ 0, !ψ !!ψ = ddt!ψ 2

2⎛⎝⎜

⎞⎠⎟= 0

∴ !ψ = constant = Precession Speed

… and precession speed is constant

∴ !!φ = 0

45

Angular Motion in Space

46

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

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

I xx !ψ !φ sinθ cosφ + I zz − I xx( ) !ψ sinθ cosφ( ) !φ + !ψ cosθ( ) = 0

Which reduces to

!ψ =I zz!φ

I xx − I zz( )cosθ or 0

Retrograde PrecessionIzz > Ixx (Disc)

Direct PrecessionIzz < Ixx (Rod)

sgn !ψ( ) = sgn !φ( ) sgn !ψ( ) = −sgn !φ( )

!φ =

I xx − I zz( )I xx

ω z !ψ =

I zz

I xx cosθω z or 0

46

2/12/20

24

Poinsot (Energy) Ellipsoid

I xxω x2 + I yyω y

2 + I zzω z2 = 2 × Kinetic Energy = 2T = ωT Iω

Locus of all angular rate combinations with constant angular energy

Polhodes: Paths of angular rate oscillations on Inertia

Ellipsoid

StablePoint

UnstablePoint

StablePoint

47

47

The Strange Case of Explorer IExplorer I spun about its axis of minimum moment

of inertia when inserted into orbitWithin a short time, it went into a flat spin, rotating

about its maximum moment of inertiaWhy?

48

48

2/12/20

25

Shifting Rotational Equilibrium of Explorer I

Whipping antennas dissipate angular energyAngular 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

49

49

Dual-Spin Satellite Dynamics

Satellite has spinning and non-spinning components

Angular momentum and rate in non-spinning frame of reference

INTELSAT-IVA

!hB = IB !ω B =MB − "ω B IBω B + hrotor( )!ω B = IB−1 MB − "ω B IBω B + hrotor( )⎡⎣ ⎤⎦

hrotor =00

hrotor

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Angular momentum added by portion

spinning about z axis

!ω x

!ω y

!ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

I xx 0 00 I yy 0

0 0 I zz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−1Mx

My

Mz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−

0 −ω z ω y

ω z 0 −ω x

−ω y ω x 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

I xx 0 00 I yy 0

0 0 I zz

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ω x

ω y

ω z

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟+

00

hrotor

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

Ixx, Iyy, and Izz are moments of inertia of the entire satellite50

50

2/12/20

26

Perturbations in Dual-Spin Satellite Angular Rate

Perturbations (or nutations) in roll and pitch rate

Δ !ω x

Δ !ω y

Δ !ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

I xx 0 00 I yy 0

0 0 I zz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−1ΔMx − hrotorΔω y

ΔMy + hrotorΔω x

ΔMz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

ω x

ω y

ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

ω xo+ Δω x

ω yo+ Δω y

ω zo+ Δω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=Δω x

Δω y

Δω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

With zero nominal rates

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

51

Eigenvalues of UndampedDual-Spin Satellite

Eigenvalues

Δ !ω x

Δ !ω y

Δ !ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=0 −hrotor / I xx 0

hrotor / I yy 0 0

0 0 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Δω x

Δω y

Δω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

+

ΔMx t( ) / I xx

ΔMy t( ) / I yy

ΔMz t( ) / I zz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Δ(s) = sI− F =s hrotor / I xx 0

hrotor / I yy s 0

0 0 s

= s3 + hrotor2 I xxI yy( )s = s s2 + hrotor2 I xxI yy( ) = 0

λ1,2,3 = 0, ± j

hrotorI xxI yy

Natural frequency of small nutation orbits about the equilibrium point

ω n =

hrotorI xxI yy

52

52

2/12/20

27

Dual-Spin Satellite with Nutation Damper

ks is a “soft” centering spring. Neglecting ks, the mass’s reaction torque on the spacecraft introduces damping, d

Δ !ω x

Δ !ω y

Δ !ω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=0 −hrotor / I xx 0

hrotor / I yy −d / I yy 0

0 0 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Δω x

Δω y

Δω z

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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

mΔ!!zm = −kd Δ!z − Δ!zS/C( )− ks Δzm − ΔzS/C( )= −kd Δ!z − bΔqS/C( )− ks Δzm − bΔθS/C( )

53

53

Natural Frequency and Damping Ratio of Dual-Spin

Satellite with Nutation Damper

Damper prevents small nutations from becoming large enough to shift

equilibrium spin axes

Δ(s) = sI− F =

s hrotor / I xx 0

hrotor / I yy s + d / I yy( ) 0

0 0 s

= 0

= s2 s + d / I yy( ) + s hrotor2 I xxI yy( ) = s s2 + sd / I yy + hrotor2 I xxI yy( )

≡ s s2 + 2ζω ns +ω n2( )

ω n =

hrotorI xxI yy

54

ζ =

d / I yy

2hrotor / I xxI yy

54

2/12/20

28

Gravity-Gradient Effect on Spacecraft Attitude

g = − µr2; ∂g

∂ r= 2µr3

• Gravitational field and gravity gradient

55

• 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

55

Gravity-Gradient Stabilization about the Local Vertical

With Ixx = Iyy, restoring torques produce a librational oscillation with natural frequency

Mx =3µ2r3

I zz − I yy( )sin2φ cos2θMy =

3µ2r3

I zz − I xx( )sin2θ cosφ

Mz =3µ2r3

I xx − I yy( )sin2θ sinφ

Gravitational torques on satellite, (1-2-3) Euler angles

ω n =

3µ2r3

1− I zz

I xx

⎛⎝⎜

⎞⎠⎟rad / sec

56

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2/12/20

29

Next Time:Spacecraft Control

57

57

Supplemental Material

58

58

2/12/20

30

r × v =i j kx y zvx vy vz

= yvz − zvy( )i + zvx − xvz( ) j + xvy − yvx( )k

= yvz − zvy( )100

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+ zvx − xvz( )

010

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+ xvy − yvx( )

001

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=

yvz − zvy( )zvx − xvz( )xvy − yvx( )

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=0 −z yz 0 −x−y x 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

vxvyvz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= !rv

• Cross-product-equivalent matrix

Cross-Product-Equivalent Matrix

!r =0 −z yz 0 −x−y x 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

• Cross product ijk

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= unit vectors along (x, y, z)

59

59

Eigenvalues (or Roots) of the Dynamic System

The picture can't be displayed.

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

= s − λ1( ) s − λ2( ) ...( ) s − λn( ) = 0

s Plane

�

δ = cos−1ζ

• Roots may be real or complex• Real and imaginary parts of the eigenvalues

can be plotted in the s plane• Real roots

– 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

shown

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Modes of Motion

• 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

x(s) = Adj sI− F( )Δ(s)

x(0)+Gu(s)[ ]

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