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Satellite dynamic and control

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.

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SPACECRAFT DYNAMIC AND CONTROLS By Zuliana Ismail
INTRODUCTION Modes of Operation Reference Frame Satellite Attitude Representation Orbital Elements External Disturbances Dynamics Kinematics Satellite Attitude Control
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.
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)
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. 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 
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                                 2 cos 2 sin 2 sin 2 sin 4 33 22 11 q eq eq eq  T qqqq 4321q==>
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
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
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             
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
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 )                                      
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
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
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)  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.
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)
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                               & & &
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)
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)
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          0 2 0 0 2 0
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
SATELLITE ATTITUDE CONTROL
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
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 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
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 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 
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
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
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 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
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 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
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

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In this document
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Introduction to spacecraft dynamics covering operational modes, reference frames, orbital elements, disturbances, and control mechanisms.

Attitude Control Systems stabilize spacecraft orientation against external disturbances during missions.

Details typical modes such as Attitude Acquisition, Safe Attitude, and Nominal Attitude necessary for mission objectives.

Describes the necessary reference frames (Inertial Earth, Satellite’s Body, LVLH) used in satellite motion description.

Satellite attitude defined by roll, pitch, yaw in LVLH frame using methods like Euler Angles, DCM, and Quaternions.

Covers quaternion fundamentals, their algebra, and advantages in computational efficiency over traditional Euler angles.

Illustrates orbital elements and parameters relating to satellite motion and orbital period calculations.

Major external torque influences: Gravity Gradient, Aerodynamic, Magnetic, and Solar Radiation Torques affecting satellite behavior.

Equations representing the angular momentum and dynamic responses of the satellite in different coordinate systems.

Linearized equations providing angular velocities of satellite frames regarding external and internal forces.

Various spacecraft attitude control methods including reaction wheels, magnetic torquers, and momentum dumping processes. Overview of 3-axis control, actuator types, and basic control laws for maintaining and adjusting satellite orientations.

Satellite dynamic and control

  • 1.
  • 2.
    INTRODUCTION Modes of Operation ReferenceFrame Satellite Attitude Representation Orbital Elements External Disturbances Dynamics Kinematics Satellite Attitude Control
  • 3.
    INTRODUCTION Purpose of AttitudeControl 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: Acquiredefinite 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 todescribe 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.
  • 8.
  • 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 computationalefficient. 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 EULERPARAMETERS) • 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 • IfFrame 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 • Fortransformation 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 Bymultiplying 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 InertialEarth 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 EULERANGLES 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 DescendingNode Vernal Equinox Direction γ θ Equatorial plane Perigee Orbital plane Earth Pointing Satellite XLVLH YLVLHZLVLH
  • 18.
    ORBITAL PARAMETERS For circularorbit • 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 • Themajor 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.
  • 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 OFMOTION • 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 OFMOTION 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 OFMOTION 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 OFMOTION • 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 OFMOTION • Insert equation (4) and (5) into equation (2)               0 0 0 /        BI               00 0 0 /         BI ………(6) ………(7)
  • 27.
    LINEARIZED EQUATIONS OFMOTION • 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.
  • 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 • Anelongated 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 STABILIZEDSATELLITE • 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 STABILIZEDSATELLITE 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 TACSAT1 • 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 CONTROLTECHNIQUE 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 Physicaloutput: 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 Biasedmomentum 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 : MAGNETICTORQUERS 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 VIAREACTION 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 VIAREACTION WHEELS
  • 43.
    BASIC CONTROL LAWS       dzEpzcz dyEpycy dxEpxcx KKT KKT KKT    Controlcommand 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 • Bycontrolling 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.
  • 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