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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
2
2/12/20
2
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
4
2/12/20
3
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
6
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4
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
8
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5
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
10
2/12/20
6
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
12
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7
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)
14
2/12/20
8
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
16
2/12/20
9
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
18
2/12/20
10
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
20
2/12/20
11
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( )
22
2/12/20
12
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
24
2/12/20
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
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
26
2/12/20
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
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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
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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
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
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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
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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
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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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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
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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
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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
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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
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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
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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
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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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29
Next Time:Spacecraft Control
57
57
Supplemental Material
58
58
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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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2/12/20
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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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