This document discusses spacecraft attitude dynamics and control. It begins by introducing typical modes of spacecraft operation like attitude acquisition and nominal earth pointing. It then covers key topics like reference frames, attitude representation using Euler angles, quaternions and direction cosine matrices, orbital elements, external disturbances, and spacecraft attitude dynamics equations. Quaternion algebra is described for representing attitude and performing successive rotations between frames. Overall, the document provides an overview of fundamental concepts for analyzing and controlling a spacecraft's orientation in space.

1. SPACECRAFT DYNAMIC AND
CONTROLS
By
Zuliana Ismail

2. INTRODUCTION
Modes of Operation
Reference Frame
Satellite Attitude Representation
Orbital Elements
External Disturbances
Dynamics
Kinematics
Satellite Attitude Control

3. INTRODUCTION
Purpose of Attitude Control Systems:
• To stabilize the spacecraft and orients it in desired directions
during the mission despite the external disturbance torques
acting on it.

4. TYPICAL MODES
Attitude Acquisition:
Acquire definite attitude (e.g. sun pointing, earth pointing) from arbitrary
initial dynamic condition (Attitude, angular rate)
Safe attitude:
Sun pointing with S/C z-axis, slow rotation around z-axis to assure safe power
and thermal conditions
Nominal Attitude:
Steady state earth pointing (roll/pitch bias capability) supplying the mission
objectives (e.g. telecommunication)

5. REFERENCE FRAME
Require to describe the motion of a satellite
• Inertial Earth (IE) coordinate system,
• Satellite’s Body (B) coordinate system
• Local-Vertical-Local-Horizontal (LVLH) coordinate system (assigned for nadir
pointing)
XLVLH
YLVLH
ZB
YB
ZLVLH
ZIE
XIE
YIE
earth



XB

6. ATTITUDE REPRESENTATION
• Attitude
•  Orientation with respect to (w.r.t.) a given reference frame
• Satellite’s attitudes (roll, pitch, yaw) are defined with respect to
the LVLH coordinate system.
Attitude representation
•  Orientation of a body fixed axis w.r.t. reference frame
• Three techniques to represent the satellite’s attitude: Euler
Angles, Direct Cosine Matrix and Quaternion.

7. QUATERNIONS

8. EULER ANGLE

9. EULER ANGLE
Example: 3-2-1 (z-y-x) rotation.
x
x1
z
y
z1

y1

z2 z1
y2
y1
x1
x2

x2
x3
y2
y3
z2
z3
   
































z
y
x
AAA
z
y
x
Z
Y
X
3
3
3


10. QUATERNION
Quaternion: More computational efficient.
Euler’s Theorem : any finite rotation of a rigid body can be represented
by the rotation through a definite angle (Euler-angle, )
around a definite axis (Euler-axis, e)
the simplest way to describe the quaternion is using the Euler axis e and Euler angle
Φ
YB
ZB
XB

e
XLVLH
YLVLH
ZLVLH

11. QUATERNIONS (SYMMETRIC EULER PARAMETERS)
• Representation of Euler-Axis and Euler-Angle by a 4-dimensional vector
• Interpretation of this vector as ‘Quaternion’ (=hypercomplex number)
• Quaternion algebra is applicable for attitude kinematics computations





 






 






 






 

2
cos
2
sin
2
sin
2
sin
4
33
22
11
q
eq
eq
eq
 T
qqqq 4321q==>

12. QUATERNION ALGEBRA
• If Frame AB   = 0 
• Determination of Euler axis/angle from a quaternion q
q4  0;
0<<180 deg (direction of rotation included in e)
q4 = 0;
 = 180
4
2
3
2
2
2
1
arctan2
q
qqq 














1
0
0
0
q
 
 
 
3
2
2
2
2
1
43
42
41
1
qqq
qsignq
qsignq
qsignq
e
e
e
e
z
y
x



























 T
qqqe 321

13. QUATERNION ALGEBRA
• For transformation of vectors between two coordinate systems such as the
LVLH and satellite’s body coordinate systems, the equation can be related
as
LVLH B
qLVLH/B
 
LVLHB
B LVLH/ B LVLH
B LVLH
XX
Y A q Y
Z Z
  
      
     
 
  
  
2 2 2 2
1 2 3 4 1 2 3 4 1 3 2 4
2 2 2 2
LVLH/B 1 2 3 4 1 2 3 4 2 3 1 4
2 2 2 2
1 3 2 4 2 3 1 4 1 2 3 4
q q q q 2(q q q q ) 2(q q q q )
A(q ) 2(q q q q ) q q q q 2(q q q q )
2(q q q q ) 2(q q q q ) q q q q
     
 
        
 
       
Direction Cosine Matrix computed from a quaternion

14. QUATERNION SUCCESSIVE ROTATIONS
By multiplying two known values of attitude quaternions, the desired unknown attitude
quaternion can be found.
• Quaternion multiplication
{
1
LVLH/B IE/LVLH IE/B
4x14x1
q q q
 e
14 2 43
1
LVLH/ B IE / B IE / LVLHq S(q ) q
 
 
4 3 2 1
3 4 1 2
IE/B
2 1 4 3
1 2 3 4
q q q q
q q q q
S q
q q q q
q q q q
 
  
 
 
   

15. QUATERNION SUCCESSIVE ROTATIONS
Inertial Earth LVLH
Satellite’s Body
XLVLH
YLVLH
ZLVLH
ZIE
XIE
YIE
XB
ZB
YB
IE / B LVLH / B IE / LVLHq S(q ) q 
1
IE / LVLH LVLH / B IE / Bq S(q ) q
 
1
LVLH / B IE / B IE / LVLHq S(q ) q
 
1
IE / LVLHq
1
IE / Bq 1
LVLH / Bq

16. QUATERNION TO EULER ANGLES
Transformation from quaternions to Euler angles
 
1 4 2 3
2 2
1 2
4 2 3 1
4 3 1 2
2 2
2 3
2(q q q q )
arctan
1 2(q q )
arcsin 2(q q q q )
2(q q q q )
arctan
1 2(q q )



  
  
     
       
      
  
    

17. ORBITAL ELEMENTS
Ω
ω
i
XIE
ZIE
YIE
Ascending Node
Descending Node
Vernal Equinox
Direction γ
θ
Equatorial plane
Perigee
Orbital plane
Earth Pointing
Satellite
XLVLH
YLVLHZLVLH

18. ORBITAL PARAMETERS
For circular orbit
• orbital period
• Using the define
value of RAAN
and inclination,
the initial
reference
quaternion can
be known
 
3
e
o
h R
T 2



o
o
2
T

 
IE/LVLH
i i
sin cos sin cos cos sin
2 2 2 2 2 2
i i
sin cos sin cos cos sin
2 2 2 2 2 2
i i
sin cos sin cos cos sin
2 2 2 2 2 2
i
sin cos sin cos
2 2 2
          
        
          
         

          
      
   
    
   
q
i
cos sin
2 2 2
 
 
 
 
 
 
 
 
 
     
    
   o ot
instantaneous angle of satellite position
Earth’s orbital frequency

19. EXTERNAL DISTURBANCES
• The major source of external disturbance torques:
• Gravity Gradient Torque, T gg
Exist form the variation of the Earth’s gravitational force over the
asymmetric body that orbiting the Earth
• Aerodynamic Torque, T Aero
Caused by the interaction between the upper atmosphere with
the satellite surface
• Magnetic Torque, T Mag
Caused by the interaction between the satellite’s residual
magnetic field and the geomagnetic field
• Solar Radiation Torque, T Solar
Exist from the solar radiation particle that hit the satellite’s
surface

20. EXTERNAL DISTURBANCES

21. Axis Disturbance Torques
Roll
(solar)
Pitch
(aero+solar)
Yaw
(aero)
 5
dx oT 8 10 sin t Nm
 
   6 5 5
dy o oT 8 10 5 10 cos t 8 10 sin t Nm   
     
 6 5
dz oT 8 10 5 10 cos t Nm 
   
0 1 2 3 4 5
-1.5
-1
-0.5
0
0.5
1
1.5
x 10
-4
Time[Orbits]
Td[Nm]
T
dx
T
dy
T
dz
Worst Case Torque Condition:
 Solar radiation torques act along the roll
and pitch axis.(Solar torque parallel to
yaw axis)
 Aeodynamic torques act along the pitch
and yaw axis.

22. DYNAMICS EQUATIONS OF MOTION
• Angular momentum at Body Coordinate System
ThI 
………(20)
………(17)
zxyyxz
yxzzxy
xyzzyx
Thhh
Thhh
Thhh










hhh BI   ………(18)
ThhB  ………(19)Euler’s Moment Equation 
 
 
  zxyyxzz
yzxzxyy
xyzyzxx
TIII
TIII
TIII









………(21)

23. DYNAMICS EQUATIONS OF MOTION
IE/B IE/B IE/B Iω T ω Iω& 
B w h h h
 w w   h T h ω Iω h& & 
s b s w w d= - ×( + ) - h ω h h h T& &
hs : Satellite’s angular momentum
ωb: Satellite’s body Angular velocity
w.r.t Inertial Earth
hw : Wheel’s angular momentum.
Td : External disturbances torques
x
y
z
I 0 0
0 I 0
0 0 I
 
   
  
I
With reaction wheels:
 
 
 
x x x y z z y wz y wy z
y y y z x x z wx z wz x
z z z x y y x wy x wx y
I T I I h h
I T I I h h
I T I I h h
    
    
    
    
    
    
&
&
&

24. LINEARIZED EQUATIONS OF MOTION
Angular velocity vector of a rotating vector
LVLH/B  Angular velocity vector of the body frame w.r.t LVLH frame
I/LVLH  angular velocity vector of the LVLH frame w.r.t Inertial frame
I/LVLH/B  I/LVLH w.r.t body frame
I/B  angular velocity vector of the body frame w.r.t Inertial frame
BLVLHIBLVLHBI ////   ………(2)
 
 
  zxyyxzz
yzxzxyy
xyzyzxx
TIII
TIII
TIII









Euler’s Moment Equation
………(1)

25. LINEARIZED EQUATIONS OF MOTION
• For small Euler Angle
• The angular velocity of LVLH coordinate system w.r.t Inertial coordinate
system














1
1
1
/



BLVLHA











0
0
0/  LVLHI







































0
0
0
0
////
0
0
1
1
1







 LVLHIBLVLHBLVLHI A


















BLVLH /
Because of small Euler Angles
………(3)
…(5)
…(4)

26. LINEARIZED EQUATIONS OF MOTION
• Insert equation (4) and (5) into equation (2)














0
0
0
/







BI














00
0
0
/







 BI
………(6)
………(7)

27. LINEARIZED EQUATIONS OF MOTION
• Insert equation (6) and (7) into equation (1)
   
    zzyxxyz
yy
xzyxzyx
TIIIIII
TI
TIIIIII









0
2
0
0
2
0

28. SATELLITE ATTITUDE KINEMATICS
LVLH/B
1
( )
2
 q ω q&
z y x
z x y
LVLH/B
y x z
x y z
0
0
( )
0
0
  
  
  
  
 
   
 
 
    
ω
Since quaternion is used for attitude representation, the derivatives of the
Euler parametes can be updated using the kinematics equation as follows:
differential equation, 1st order, dimension 4
ADVANTAGE: no trigonometric functions

29. SATELLITE ATTITUDE CONTROL

30. ATTITUDE PERFORMANCES
simple, cheap
cheap, slow, lightweigh
LEO only
inertially oriented
RWs: Expensive, precise, faster slew, Momentum Unloading
CMG: Expensive, heavy, quick, for fast slew, 3-axes
Thrusters: Expensive, quick response, consumables
GG: Long booms-Restricted maneuverability

31. GRAVITY GRADIENT
• An elongated object in a gravity field tends to align its longitudinal axis to the Eart’s
center.
• Earth oriented
• Requires stable inertia – limited accuracy
• No Yaw stability (can add momentum wheel)
• Only effective in LEO – because gravity varies with the square of the distance.
Gravity-Gradient
X
Y
Z
Gravity-gradient satellite with momentum
wheel
-Momentum wheel for yaw stability
-Satellite body rotates along Y-axis
-at one revolution per orbit

32. GRAVITY GRADIENT
Example - UoSAT
Satellite mass : 70 kg
Satellite moment of inertia : (120, 120, 1) kgm2
Satellite body : 40 x 40 x 60 cm
Boom : 8 m

33. SINGLE SPIN STABILIZED SATELLITE
• Make use of physical principles/elements for s/c attitude control.
• Entire s/c rotates so that its angular momentum vector remains fixed in
inertial space.
• An advantage of this technique is the capacity achieve a relatively long
operational life. The typical disadvantages are the poor attitude accuracy
and the dependence of the environmental elements
• Because single spin stabilized satellites have a fixed pointing w.r.t inertial
space, they are not a good choice for Earth-pointing missions.
H

H

H


34. DUAL SPIN STABILIZED SATELLITE
Stowed
(during launch)
In orbit

• One way to avoid Earth-pointing limitations of spin stabilization is to use a
dual-spin system. These systems consists of an inner cylinder called the ‘de-
spun’ section, surrounded by an outer cylinder that is spinning at a high
rate.
de-spun section :
stays pointed at the
Earth
spun section :
provides stiffness

35. DUAL SPIN STABILIZED
Example
TACSAT 1
• Launched in 1969 and was the dual spin stabilized
satellite.
• The antenna is the platform, and is intended to
point continuously at the Earth, spinning at one
revolution per orbit.
• The cylindrical body is the rotor, providing
gyroscopic stability through its 60 RPM spin.
H


36. THREE AXIS CONTROL TECHNIQUE
Actuators – require continuous feedback and adjustment:
• Thrusters,
• Magnetic Torquers
• Momentum-control devices
• Biased momentum systems
• Zero-bias systems
• Control-moment gyroscopes
• Fast; continuous feedback control
• Relatively high power, weight and cost
Active Control Systems directly sense spacecraft attitude
and supply a torque command to alter it as required. This
technique require energy consuming attitude actuators.
Good attitude accuracies can be achieved

37. ADCS BLOCK DIAGRAM
SpacecraftActuatorsController
Sensors
Physical output:
the current
attitude, realcommands
Difference:
error signal,
error
System Input:
desired attitude,
desired +
Measured Output:
the measured
attitude, measured
-
+
+
Disturbance
Torques
-Gyros & Accelerometer
-Sun Sensors
-Star Sensors
-Horizon Sensors
-Magnetometer
-Thrusters
-Reaction Wheels
-Momentum Wheels
-Control Moment Gyros
-Magnetic Torquers

38. ACTUATORS : MOMENTUM-CONTROL
DEVICES
Biased momentum system
“momentum wheel” with a large fixed momentum to
provide gyroscopic stiffness. The wheel’s speed
gradually increases to absorb disturbance torques
Zero-bias system
“reaction wheel” with little or no initial momentum.
Each wheel spins independently to rotate the
spacecraft and absorb disturbance torques
Control-moment gyroscope
“wheel” with a large fixed momentum. The wheel is
mounted on gimbals, rotating the wheels about their
gimbals changes the satellite orientation

39. MOMENTUM BIASED PRINCIPLE
• The same concept used by spin-stabilized spacecraft. Only in this case,
instead of spinning the whole spacecraft, only a small wheel (momentum
wheel) inside the spacecraft is spinning providing a gyroscopic stiffness.
• Momentum vector (momentum wheel)
perpendicular to orbit plane (parallel to satellite pitch axis)
• Pitch Axis : continuous control through change of wheel speed
• Roll/Yaw Axis : improved passive
stabilization due to increased
momentum stiffness through pitch
bias momentum X
Y
Z
h


40. ACTUATORS : MAGNETIC TORQUERS
The interaction between the Earth’s geomagnetic field and magnetic dipole
moment within the satellite that normally comes from electrical equipments
onboard will generate a magnetic disturbance torque. Fortunately, this torque can
be used for controlling purposes when it is generated in desirable amount and
direction. This is done by generating a controllable value of magnetic dipole
moment within the satellite using an electromagnetic based device called
magnetic torquer.
MBT


-Often used for LEO satellites
-Useful for initial acquisition maneuvers
- Also commonly use for momentum
desaturation - (“dumping”) in wheel-based
system

41. 3-AXES CONTROL VIA REACTION WHEELS
• The reaction wheel
concept relies on the
principle angular
momentum
conservation.
• When a satellite
rotates one way due
to the disturbance
torque, the reaction
wheel will be
counter rotated to
produce a same
magnitude reaction
torque in order to
correct the attitude

42. 3-AXES CONTROL VIA REACTION WHEELS

43. BASIC CONTROL LAWS






dzEpzcz
dyEpycy
dxEpxcx
KKT
KKT
KKT



Control command for Euler
Angle Errors















E
E
E
E
E
q
q
q
q
q
4
3
2
1
rKqqKT
qKqqKT
pKqqKT
dzEEpycz
dyEEpycy
dxEEpxcx



43
42
41
2
2
2
Control command for
Quaternion Error Vector

44. MOMENTUM DUMPING
• By controlling the satellite’s attitude using the reaction wheels,
the change in the angular momentum of the satellite will be
transferred to the wheels and vice versa in order to
compensate for the external disturbance torques.
• The constant disturbance torques can cause the reaction wheel
angular momentum to constantly increase or decrease, hence
induces a build-up of the angular momentums.
• Since the reaction wheels lack of the ability to remove the
excess angular momentums and that the wheels have a limited
capacity to store angular momentum.
• The angular momentum of the wheels will be accumulated and
saturated over time thus preventing the application of any
further wheel control torques.

45. MOMENTUM DUMPING

46. MOMENTUM DUMPING
 2
k
B
 m = B h
m BT = m × B
   
 
   
sin cos
cos
2 sin sin
LVLH
x o
y o
oz
B B i
B B i
B iB


   
   
    
     
B
Δh : excess momentum to be removed
k : unloading control gain. (PI Controller)
Magnetic Control Equation
Wheel Unloading law Simple Dipole Model
c k  T h
k  h m×B

47. MOMENTUM DUMPING
m
dT
wh
 w2
k
B
  m = B h
m bT m×B
Magnetic Dipole
Moment
B
Magnetic
Control Torquers
Dipole Saturation
Limit
Disturbance
Torques
Simplified
Magnetic Model
Reaction Wheels
Satellite
Dynamics