SPACECRAFT DYNAMIC AND CONTROLSByZuliana Ismail
INTRODUCTIONModes of OperationReference FrameSatellite Attitude RepresentationOrbital ElementsExternal DisturbancesDynamicsKinematicsSatellite Attitude Control
INTRODUCTIONPurpose 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.
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 conditionsNominal Attitude:Steady state earth pointing (roll/pitch bias capability) supplying the mission objectives (e.g. telecommunication)
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
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.
QUATERNIONS
EULER ANGLE
EULER ANGLE
Example: 3-2-1 (z-y-x) rotation.
xx1
z
y
z1
y1
z2 z1
y2
y1
x1
x2
x2
x3
y2
y3
z2
z3
zyx
AAAzyx
ZYX
3
3
3
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
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
2cos
2sin
2sin
2sin
4
33
22
11
q
eq
eq
eq
Tqqqq 4321q==>
QUATERNION ALGEBRA
• If Frame AB = 0
• Determination of Euler axis/angle from a quaternion qq4 0;
0<<180 deg (direction of rotation included in e)
q4 = 0; = 180
4
23
22
21arctan2
qqqq
1000
q
32
22
21
43
42
41 1
qqqqsignqqsignqqsignq
eee
e
z
y
x
Tqqqe 321
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 BqLVLH/B
LVLHB
B LVLH / B LVLH
B LVLH
XXY A q YZ Z
2 2 2 21 2 3 4 1 2 3 4 1 3 2 4
2 2 2 2LVLH / B 1 2 3 4 1 2 3 4 2 3 1 4
2 2 2 21 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
QUATERNION SUCCESSIVE ROTATIONSBy 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
1LVLH / B IE / B IE / LVLHq S(q ) q
4 3 2 1
3 4 1 2IE / B
2 1 4 3
1 2 3 4
q q q qq q q q
S qq q q qq q q q
QUATERNION SUCCESSIVE ROTATIONS
Inertial Earth LVLH
Satellite’s Body
XLVLH
YLVLH
ZLVLH
ZIE
XIEYIE
XB
ZB
YB
IE / B LVLH / B IE / LVLHq S(q ) q
1IE / LVLH LVLH / B IE / Bq S(q ) q
1LVLH / B IE / B IE / LVLHq S(q ) q
1IE / LVLHq
1IE / Bq 1
LVLH / Bq
QUATERNION TO EULER ANGLESTransformation from quaternions to Euler angles
1 4 2 32 2
1 2
4 2 3 1
4 3 1 22 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 )
ORBITAL ELEMENTS
Ω
ωi
XIE
ZIE
YIE
Ascending Node
Descending Node
Vernal Equinox Direction γ
θ
Equatorial plane
Perigee
Orbital plane
Earth Pointing Satellite
XLVLH
YLVLHZLVLH
ORBITAL PARAMETERSFor circular orbit• orbital period
• Using the define value of RAAN and inclination, the initial reference quaternion can be known
3eo
h RT 2
oo
2T
IE / LVLH
i isin cos sin cos cos sin2 2 2 2 2 2i isin cos sin cos cos sin2 2 2 2 2 2i isin cos sin cos cos sin2 2 2 2 2 2isin cos sin cos2 2 2
q
i cos sin2 2 2
o otinstantaneous angle of satellite position
Earth’s orbital frequency
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
EXTERNAL DISTURBANCES
Axis Disturbance Torques
Roll (solar)
Pitch(aero+solar)
Yaw(aero)
5dx oT 8 10 sin t Nm
6 5 5dy o oT 8 10 5 10 cos t 8 10 sin t Nm
6 5dz oT 8 10 5 10 cos t Nm
0 1 2 3 4 5-1.5
-1
-0.5
0
0.5
1
1.5x 10
-4
Time[Orbits]
Td[N
m]
Tdx
Tdy
Tdz
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.
DYNAMICS EQUATIONS OF MOTION
• Angular momentum at Body Coordinate System
Th I
………(20)
………(17)
zxyyxz
yxzzxy
xyzzyx
Thhh
Thhh
Thhh
hhh BI ………(18)
ThhB ………(19)Euler’s Moment Equation
zxyyxzz
yzxzxyy
xyzyzxx
TIII
TIII
TIII
………(21)
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 momentumωb: Satellite’s body Angular velocity
w.r.t Inertial Earthhw : Wheel’s angular momentum.Td : External disturbances torques
x
y
z
I 0 00 I 00 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
LINEARIZED EQUATIONS OF MOTION
Angular velocity vector of a rotating vector
LVLH/B Angular velocity vector of the body frame w.r.t LVLH frameI/LVLH angular velocity vector of the LVLH frame w.r.t Inertial frameI/LVLH/B I/LVLH w.r.t body frameI/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)
LINEARIZED EQUATIONS OF MOTION
• For small Euler Angle
• The angular velocity of LVLH coordinate system w.r.t Inertial coordinate system
11
1
/
BLVLHA
0
0
0/ LVLHI
0
0
0
0
////
0
0
11
1
LVLHIBLVLHBLVLHI A
BLVLH /
Because of small Euler Angles
………(3)
…(5)
…(4)
LINEARIZED EQUATIONS OF MOTION
• Insert equation (4) and (5) into equation (2)
0
0
0
/
BI
00
0
0
/
BI
………(6)
………(7)
LINEARIZED EQUATIONS OF MOTION
• Insert equation (6) and (7) into equation (1)
zzyxxyz
yy
xzyxzyx
TIIIIII
TI
TIIIIII
020
020
SATELLITE ATTITUDE KINEMATICS
LVLH / B1 ( )2
q ω q
z y x
z x yLVLH / B
y x z
x y z
00
( )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 4ADVANTAGE: no trigonometric functions
SATELLITE ATTITUDE CONTROL
ATTITUDE PERFORMANCES
simple, cheap
cheap, slow, lightweightLEO only
inertially oriented
RWs: Expensive, precise, faster slew, Momentum Unloading
CMG: Expensive, heavy, quick, for fast slew, 3-axesThrusters: Expensive, quick response, consumables
GG: Long booms-Restricted maneuverability
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
GRAVITY GRADIENTExample - UoSAT
Satellite mass : 70 kgSatellite moment of inertia : (120, 120, 1) kgm2
Satellite body : 40 x 40 x 60 cmBoom : 8 m
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
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
DUAL SPIN STABILIZEDExampleTACSAT 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
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 attitudeand supply a torque command to alter it as required. This technique require energy consuming attitude actuators.
Good attitude accuracies can be achieved
ADCS BLOCK DIAGRAM
SpacecraftActuatorsController
Sensors
Physical output: the current attitude, real
commands
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
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
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
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
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
3-AXES CONTROL VIA REACTION WHEELS
BASIC CONTROL LAWS
dzEpzcz
dyEpycy
dxEpxcx
KKT
KKT
KKT
Control command for Euler Angle Errors
E
E
E
E
E
qqqq
q
4
3
2
1
rKqqKT
qKqqKT
pKqqKT
dzEEpycz
dyEEpycy
dxEEpxcx
43
42
41
2
2
2
Control command for Quaternion Error Vector
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.
MOMENTUM DUMPING
MOMENTUM DUMPING
2
kB m = B h
m BT = m×B
sin coscos
2 sin sinLVLH
x o
y o
oz
B B iB 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
MOMENTUM DUMPING
m
dT
wh
w2
kB m = B h
m bT m×B
Magnetic Dipole Moment
B
Magnetic Control Torquers
Dipole Saturation Limit
Disturbance Torques
Simplified Magnetic Model
Reaction Wheels
SatelliteDynamics
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