Post on 27-Jun-2020
1Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Brian Howley408-743-7367brian.howley@lmco.com
Space Systems Company
AA236: Overview of AA236: Overview of Spacecraft AttitudeSpacecraft Attitude
Determination and ControlDetermination and Control
2Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Course ObjectivesCourse Objectives• Define the Attitude Determination and Control (ACS)
subsystem, its role, and relationship to other spacecraft subsystems
• Introduce ACS fundamental concepts including coordinate systems, and vehicle dynamics & kinematics
• Discuss possible cubesat ACS implementations including passive stabilization and ground based attitude determination
3Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Course OutlineCourse Outline• ADCS introduction and overview • Coordinate systems and transformations• Vehicle dynamics• Environmental forces and torques• Example CubeSat stabilization techniques• Example CubeSat attitude determination techniques• ADCS hardware components (Backup)
4Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
ADC & GNC SubsystemsADC & GNC SubsystemsAttitude Determination and Control • Provides rate stabilization and pointing for payload, power,
communication, and thermal subsystems during normal and safingoperations
• Provides rate and attitude control for transfer orbit, and station keeping maneuvers
• Provides spacecraft attitude knowledge to support mission objectivesGuidance Navigation and Control• Provides spacecraft position and velocity knowledge for antenna and
payload pointing• Provides timing, magnitude, duration, and direction of burns for transfer
orbit and station keeping maneuvers• Provides luni-solar positions for satellite and payload steering
ADC & GNC subsystems are often lumped together and collectively referred to ACS or ADCS
5Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Definitions and TermsDefinitions and Terms• Attitude Determination: Knowledge of spacecraft orientation
with respect to a frame of reference• Navigation: Knowledge of spacecraft position and velocity with
respect to a frame of reference• Attitude Control: The process of achieving and maintaining
desired orientation or attitude rate • Orbit Control: The process of achieving and maintaining the
desired orbit• Guidance: A command sequence from the current attitude or
orbital state to the desired attitude or orbital state- For attitude control the guidance algorithm is often called a
command generator- For orbit control the guidance algorithm is simply referred to as
guidance
ADC & GNC have analogous functions in rotation and translation
6Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Subsystem Functional RelationshipsSubsystem Functional Relationships
ADCSADCSPowerPower
ThermalThermal
FSWFSW
PropulsionPropulsion
Comm.Comm.
C&DHC&DH
PayloadPayloadProtectiveMeasuresProtectiveMeasures
Struct&MechStruct&Mech
S/A Pointing
Power
Thruster Cmnds
∆V, Torque
Pointing&Stab.
SensorData
Radiator Pointing Thermal
Control
StructuralSupport
AlignmentAntenna Pointing
Cmnds
Timing, Sensor/Actuator Data
Sensor/AcutatorData
CmndsSensor Data
Actuator Cmnds
GroundGround
Telem
7Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Example ADCS Block DiagramExample ADCS Block Diagram
TVCAngles
RWATachs
SolarArrays
TVCGimbals
Bus1553
AntennaGimbals
SensorsSensors
RIU
Thrusters
LAE
ControlMode
SpecificLogic
RWAs
Automatic Switching
Logic
SensorProcessing
Logic
IMU
Sun Sensor
RedundancyManagement
Logic
EarthSensor
ADCS Algorithms (Reside in Flight Computer)ADCS Algorithms (Reside in Flight Computer) ActuatorsActuators
Event Recorder
AttitudeCommandProcessing
Maneuver Sequencer andSupport Logic
P I DController
TVC Gimbal Controller
MomentumManagement
Logic
GuidanceLogic
Antenna PntngLogic
Thruster
RWA
TVC Gimbal
Bus
Scheduler
Attitude Determination and Ephemeris Propagation Logic
Antennas
Arrays
S/APots
Payload
Payload1553
AD&CS Component
Star Tracker
AntennaAngles
Non AD&CS Component
RIU
8Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
ADCS AlgorithmsADCS Algorithms
ModeManager
SensorProcessing Actuator
Processing
AttitudeDetermination
CommandGenerator
Navigation
ReactionWheel Control
ThrusterControl
AppendageControl
MomentumManager
Guidance
ProtectiveMeasures
TimeManager
Commands
IMUStar TrackerEarth SensorSun SensorWheel TachsAppendages
Clock
CommandsNavigationGuidanceMoment. Mgmt
GPS
Ephemeris
CommandGenerator
CommandGenerator
HW I/F
9Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Typical ADCS Modes of OperationTypical ADCS Modes of Operation
Sep&Rate Capture
Sep&Rate Capture
Sun AcqSun Acq
Earth AcqEarth Acq
TransferOrbit
TransferOrbit
DeployDeploy
Normal Ops
Normal Ops
Station KeepingStation Keeping
EventRecovery
EventRecovery
ProtectiveMeasuresProtectiveMeasures
Collision Avoidance,Zero Rates
Rotisserie,LAE burns,Thrust VectorControl
1st Maneuver,2nd Maneuver
Earth AcqManeuver,Init. Attitude
Standby
ReactionWheelControl Thruster
and wheelControl
Safe Mode
After launch vehicle separation, spacecraft cycle through a series of modes to maintain safe attitude, reach the desired station, and deploy arrays and antenna for normal missionoperations. Normal mission ops may be interrupted to perform station keeping and momentum dumping or to respond to failures or threat events.
ADCS subsystem must support a variety of operational modes using different hardware suites
10Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Coordinate Systems and Coordinate Systems and TransformationsTransformations
11Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
HelioHelio--Tropic Coordinate SystemsTropic Coordinate Systems
First dayof summer
First dayof spring
First dayof winterFirst day
of autumn
Zc
Yc
XcVernal Equinox
direction
(Seasons are for Northern Hemisphere)
ϒ
SUN
Source: SW833 S/C Att Det & Cntrl Spring 2003
The Helio-Tropic Coordinate system is inertially fixed (fixed with respect to the stars) with originat the center of the sun. It is typically used for interplanetary missions.
The illustration below is a useful way to visualize the seasons and seasonal effects, such aseclipse periods of a satellite in Earth orbit
12Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Inertial Coordinate SystemsInertial Coordinate Systems
North PoleZΙ
First Point of Aries
XΙ YΙ
ASC Ω
DEC
An Earth centered inertially fixed coordinate system is used to describe satellite orbital position andorientation. The origin is at the geometric Earth center, the Z axis is aligned to the North pole, and the X axis points towards the first point in Aries. Since the Earth axis and the stars move slowly over time, the inertial reference is specified with respect to an epoch date, J2000.
The position of stars with respect to ECI is generally specified in spherical angles: ascension and declination. Since star locations are well known, satellite orientation with respect to the ECI frame can be determined from stellar observations.
Source: SW833 S/C Att Det & Cntrl Spring 2003
13Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Earth Fixed Coordinate SystemsEarth Fixed Coordinate Systems
ZE, ZI
XE
YE
XI
YI
First Point of Aries
The Earth Centered Earth Fixed (ECEF) coordinate system (XE, YE, ZE) rotates with the Earth and is related to the ECI primarily by time of day, but also polar axis nutation and precession andsmaller corrections for polar motion with respect to the crust and irregularities in the Earth’s rotationsthese irregularities are measured by astronomic observations and are the reason for leap seconds. The transformation bewteen ECI and ECEF coordinate systems is defined in the WGS 84.
The Z axis of the ECEF system is coincident with the polar axis and the X axis is from Earth center tothe intersection between the prime meridian and the equator. Satellite position and orientation withrespect to ECEF must be known for maintain space to ground communications and for any Earthsensing.
Prime Meridian
Source: SW833 S/C Att Det & Cntrl Spring 2003
14Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Orbit Fixed Coordinate SystemsOrbit Fixed Coordinate Systems
XI
YI
First Point of Aries
ZI
Ω,Right Ascension
i, inclination
ZO YO
XO
An orbit fixed system has an origin fixed with respect to the satellite body and rotates as thesatellite orbits so that one axis points toward (or directly away from) Earth center and anotheraxis is normal to the orbit plane. Often the orbit Z axis points to nadir and the orbit Y axis is normal to the orbit plane.
For Earth pointing spacecraft the satellite body is generally commanded to an orientation or ratewith respect to orbit fixed coordinate system. Spacecraft orientation with respect to the orbitReference system can be described by an Euler sequence of roll, pitch, and yaw angles about theSpacecraft x, y, and z axes respectively.
Orbit Trajectory
Orbit Frame
15Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Body Fixed Coordinate SystemsBody Fixed Coordinate Systems
XB
YB
ZB
XOYO
ZO
Sun
Orbit frame (at 6 pm)
Spacecraft Body Frame
Local noon
6PM
6AM
ZI
XIYI
Earth
Note: S/C shown at 180 deg yaw
The Body frame is fixed with respect to the spacecraft body. The ADCS uses a combination of sensorsand actuators to maintain a desired body frame orientation or rate. The desired orientation depends onmission and spacecraft needs. Examples include the Sun-Nadir-Yaw profile and orbit or ineritial fixed.
16Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Payload/Sensor Fixed Coordinate SystemsPayload/Sensor Fixed Coordinate Systems
XB
YB
ZBZT2
XT2
YT2
ZT1
XT1
YT1ZP
XPYP
ZE
YE
XE
XS
YS
ZS
EarthSensor
Payload
Body Frame Sun Sensor
StarTrackers
Payload and sensor data and commands are parameterized with respect to local coordinate systems.alignment between different reference frames is measured on ground but may shift during launch anddue to gravity unloading and thermal distortions. Precision attitude knowledge requires on-orbitcalibration of these alignment shifts and distortions.
17Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Transformations Between SystemsTransformations Between Systems• Coordinate systems form a reference for position and angular
measurement• Relationships between coordinate systems can be
characterized several ways- Direction Cosine Matrices- Euler Angle Rotation Sequence- Euler Parameters
• Knowledge of the relationship between reference frames is required for attitude determination and control
18Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
A reference frame R, with axes XR, YR, is rotated with respect to reference frame B, with axes XB, YB, by an angle θ.
A vector from point O to point P (vop) can be expressed in either coordinate system in matrix form:
The relationship between coordinate systems can be described by a direction cosine matrix (DCM) thatvaries with θ. The DCM transforms the vector vop from the body fixed to the rotated reference.
Single Axis RotationSingle Axis Rotation
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=
B
B
R
R
yx
xx B
opRop vv ;
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡
B
B
R
R
yx
yx
)cos()sin()sin()cos(
θθθθ
XB
XR
YBYR
O
P
bx
by
rxry
vop
θ
R
B
The transformation matrix aboveis a DCM for planar rotations. Thetransformation from B to R, TR/B is
⎥⎦
⎤⎢⎣
⎡−
=)cos()sin()sin()cos(
θθθθ
R/BT
19Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Direction Cosine MatrixDirection Cosine Matrix
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
•••••••••
=
zzzyzx
yzyyyx
xzxyxx
R/B
rbrbrbrbrbrbrbrbrb
T
In 3 dimensions the direction cosine matrix is a 3x3 transformation matrix. The elements of theDCM correspond to the inner or dot between basis vectors (the dot product between unit vectorsis the cosine of the angle between the two vectors). A general expression for the transformationMatrix from reference B to R, TR/B in terms of basis vector cross products is given below.
Transformations between successive frames can be determined from a series of matrix multiplications.For example the transformation from Inertial to Body frames is the Inertial to Earth fixed transformationpost multiplied by the Earth fixed to Orbit frame transformation, post multiplied again by the Orbit tobody frame transformation
E/IO/EB/OB/I TTTT =
20Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Euler Angle RotationsEuler Angle RotationsEuler angles can be used to define the orientation of one reference frame with respect to another.A sequence of three rotations is sufficient to describe any transformation, however, the order ofRotation and the size of the angles is not unique and is subject to mathematical singularities.
For example a 3-1-3 Euler sequence can be used to describe satellite orbit parameters with respectto ECI. The first rotation is the angle of ascending node, Ω, about the inertial Z axis. The second rotation is the inclination, i, about the line of nodes. The final rotation is the true anomaly, ν, about orbit normal.
A 3-1-2 Euler sequence is often used to describe spacecraft orientation with respect to the Orbit frame.The first rotation is the yaw about nadir. The second rotation is the roll about the spacecraft X axis, andthe final rotation is the pitch.
XI
YI
ZI
Ω,Right Ascension
i, inclination
ν, true anomaly
XI
YI
ZI
ZO
YO
XOXB
YB
ZB
3-1-3 Euler Sequence for Orbit Parameters 3-1-2 Euler Sequence for yaw, roll, and pitch
21Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Euler ParametersEuler Parameters
Eigenvector, e
Rotation angle, θ
Euler Rotation Theorem:Euler Rotation Theorem:The orientation of an object can always be described as a single rotation about a fixed axis
The fixed axis, or eigenvector, e, is a unit vector with the same components in both the original and rotated frames of reference: eR = eB. Thus, four quantities are required to unambiguously describe orientation with respect to a frame of reference: the three components of e and the angle of rotation, θ.
Euler parameters are a combination of these elements arranged in a 4 element vector or quaternion,q. The 4 element quaternion contains the same information as a 9 element direction cosine matrix. Euler parameters are compact and a useful representation of orientation for attitude determination.
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡==
)2cos()2sin()2sin()2sin(
are parametersEuler the
.
reigenvecto For the
3
2
1
3
2
1
θθθθ
eee
eee
q
ee BR
Euler parameters are expressed in a quaternion vector
22Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Vehicle DynamicsVehicle Dynamics
23Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Particle DynamicsParticle Dynamics
( ) ( )[ ] ( )[ ] [ ]( ) ( )[ ][ ] ( ) ( ) ( )[ ]
[ ]EEEEI/E
EEEI/E
EEI/E
II
EEI/E
EI/E
EI/E
II
EI/E
I
arαvωrωωT
vrωrωTvrωTva
vrωTrTrTrv
rTr
+×+×+××=
+×+×++×==
+×=+==
=
2
dtd
dtd
dtd
dtd
dtd
dtd
dtd
dtd
XI XE
YI
YE
ZI, ZE
r
f a
ω
[ ] [ ]
E
E
EEEEIE/I
E
II
vω
rωω
arαvωrωωaTf
af
×
××
+×+×+××==
=
2 :onaccelerati Coriolis
:onaccelerati lcentripeta
2
:becomes law sNewton' frame rotating aIn
:is law sNewton' frame inertialan In
mm
m
Force, velocity, and acceleration are vector quantities described with respect to a frame of reference.
Newton’s law, f=ma, holds for forces and accelerations with respect to an inertial reference frame
Rotating reference frames, such as an Earth fixed frame, introduce additional terms such as centripetal and Coriolis accelerations
Rotating reference frames introduce centripetal and Coriolis acceleration terms to Newton’s Law
Coriolis was a French artillery officer and engineer in the early 19th century
24Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Rigid BodiesRigid Bodies• A rigid body is a collection of mass particles that maintain a fixed
relationship with one another in a reference frame.• A rigid body has properties of mass and inertia.• A rigid body can have both linear and angular rates and accelerations.• For rigid bodies, Newton’s 2nd law is amended somewhat to: the sum of
forces acting on a body is proportional to acceleration at the center of mass.
• In a free body diagram (FBD) rigid bodies are often depicted as “potatoes”.
f1
f2
mg
a
masscenter
Body B
25Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Angular Velocity and AccelerationAngular Velocity and Acceleration• For a rigid body rotating about a fixed axis angular velocity is the time
rate of change of the angle about that axis.• For more general motion angular velocity is better defined in terms of
the time rate of change of the body fixed reference frame basis vectors with respect to another (fixed) set of basis vectors.
• Angular velocity is a vector quantity- Unlike rotations angular velocities can be added
• Angular acceleration is the time rate of change of angular velocity.
Angular velocity is a straight forward concept for an object spinning about a fixed axis but less intuitivefor general 3 axis motion. Unlike linear acceleration and velocity, angular acceleration and velocity is the same at all points of a rigid body.
vA vB
ω=d(θ)/dt
vA = vB
( ) [ ] B/AB/A TωTω ×=dtd::
26Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Forces and TorquesForces and Torques• Force is the action of one body on another. Torque is a force acting
through a distance. For a rigid body without restraint a force will accelerate the center of mass, and a torque will generate a rotation about the mass center.
- A pure couple is a pair of equal and opposite forces acting through a distance. A pure couple generates a torque with no net force.
• Systems of rigid bodies interact by applying equal and opposite forces and torques on one another.
f
a
Line of action
r
α
αIfraf
cm=×= m
A
BC
Body A Body B Body C
fA/B
tA/B
fB/A
fB/CtB/A
tB/C
fC/B
tB/C
fA/B=-fB/AfB/C=-fC/B
tA/B=-tB/AtB/C=-tC/B
Free Body Diagrams can be used toanalyze multiple body systems
27Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Inertia PropertiesInertia Properties• Inertia properties of a rigid body are fully described by the mass, the
location of the mass center, and by the moments and products of inertia about reference frame basis vectors at a specified point.
• All rigid bodies have a set of principal axes with origin at the mass center about which the products of inertia disappear
• The maximum and minimum moments of inertia are found about different principal axes
• Moments and products of inertia about an axis that does not pass through the mass center (such as a pivot point) can be determined using the parallel axis theorem
• Units of inertia are mass length squared (Kg-m2), but in English units inertias are typically given in snails (ft-lb-sec2)
Principal axis(minimum moment)
Principal axis(maximum moment)
x1
y1x2
y2 The inertia matrix for a referencesystem aligned with the principleaxes (x1, y1) is diagonal. Inertiamatrices for other reference framesinclude cross product terms.
28Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Angular MomentumAngular Momentum• Linear momentum of a rigid body is the product of mass velocity at the
mass center, mv. From Newton’s laws the time rate of change of linear momentum is equal to the sum of external forces acting on the body.
• Angular momentum of a particle about a point is the cross product of position and the particle linear momentum
• For a rigid body, the angular momentum is the product of the moments and products of inertia with angular velocity
• The principle of conservation of angular momentum states that the time rate of change of the angular momentum of a system of particles is equal to the sum of the externally applied torques
( )L∑ = dtdForces External
L=mv
r
H=rxL
Η=Ιω
( )H∑ = dtdMoments External
29Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Spinning BodiesSpinning Bodies• Torque free motion – angular momentum is conserved but may not be
aligned with principal axes so that the spinning body wobbles or nutates.
• Torque on a spinning body normal to the angular momentum changes the direction of the momentum causing the body to precess
Prolate
Space cone
Body cone
angular velocity, ω angular momentum, H
Oblate
angular momentum, H
angular velocity, ωSpace cone
Body cone
angular momentum, H
mg
Gravitational torque causes aspinning top to precess at aconstant rate
30Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Attitude KinematicsAttitude Kinematics• Dynamics relate forces and torques to body rates and accelerations.• Kinematics relate angular rates to orientation.
- Straight forward for rotation about a fixed axis- Less intuitive for general motion where the axis of rotation changes
∫+=1
0
0
t
t
dtωθθ
ω=d(θ)/dt( ) [ ] B/AB/A TωTω ×=dt
d::
For a body rotating about a fixed axis,the orientation about that axis can be determined by simply integrating the angular rate:
( ) B/AB/A TT⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=
00
0
xy
xz
yz
dtd
ωωωω
ωω
For a body with general motion, theTime rate of change of orientation (here orientation is parameterized asA DCM) is more complex.
31Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Euler Parameter EquationsEuler Parameter Equations• Like DCMs, Euler parameters can describe multiple rotations by
multiplication operation. - But the be quaternion multiplication must be defined.
• Attitude kinematics can be expressed in terms of differential equations of the Euler parameters.
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−
−−
≡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
=
4321
3412
2143
1234
4
3
2
1
,for
:tionmultiplica Quaternion
qqqqqqqqqqqqqqqq
qqqq
B/AB/A
A/IB/AB/I
qQq
Angular rate, ω
( )
B/IB/I
/BB
B/I/BBB/I/IBB/I
Q
qIQqqq
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−
−−
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
∆−∆−∆−∆∆−∆∆∆∆−∆∆−∆
≡
∆−
=∆−
=
+
+
→∆
+
→∆
00
00
21
11
11
limlim
321
312
213
123
321
221
121
321
121
221
221
121
321
121
221
321
00
ωωωωωωωωωωωω
ωωωωωωωωωωωω
&
&
tttttttttttt
tt tt
Source: “A Tutorial on AttitudeKinematics” – Don Reid
32Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
DCM DCM vsvs Quaternion RepresentationsQuaternion Representations• The DCM contains 9 elements only 3 of which are independent
- There are 6 constraints to maintain a orthonormal matrix• The set of 4 Euler parameters require less memory and computational
throughput.- There is one constraint to maintain a unitary quaternion
• Spacecraft attitude determination algorithms generally use quaternionsor Euler angle sets, but must convert to DCM to transform vectors to different frames of reference.
124
23
22
21 =+++ qqqq
33Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Environmental Forces and Environmental Forces and TorquesTorques
34Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Disturbances Forces Affecting OrbitDisturbances Forces Affecting Orbit
Earth
Earth Atmosphere
Moon
Gravity
Gravity
Gravity (higher-order harmonics; beyond inverse-square)
V
Solar Radiation
Drag
Sun
Source: Stanford/ELDP 2005Orbital Mechanics – Dennis Haas
35Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Environmental Disturbance TorquesEnvironmental Disturbance Torques• Aerodynamic Drag – At altitudes below 400 km, the upper atmosphere generates
forces and torques as spacecraft travel through it. The force is a function of the atmospheric density, satellite velocity, cross sectional area, and satellite geometry, the effect of which is captured by the non-dimensional coefficient of drag. The forces, acting in different directions on different elements of the spacecraft affect the orbit and generate a net torque.
• Solar Pressure – Sun light is reflected and absorbed by spacecraft surfaces. Light has momentum: the change in momentum generates a radiation pressure on the spacecraft that depends on geometry and optical surface properties. If the center of pressure is distant from the center of mass solar pressure generates a disturbance torque.
- Other sources of radiation pressure can include the Earth’s albedo (reflected sunlight) and satellite communications
- Solar pressure is a major torque disturbace at geo-synchronous altitudes
r
Center of mass
Center ofpressure
SunlightSolar panel
S/C body
An asymmetric satellite can have high solartorques requiring frequentmomentum dumps andadditional propellant
36Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Magnetic TorquesMagnetic Torques
ZE
XEN
S
Magnetic Torque – At lower altitudes, the Earth’s magnetic field interacts with current loops within the spacecraft to generate torque.
Magnetic torque rods use this interaction intentionally to generate torques for momentum management.
Spinning spacecraft can develop eddy currents which cause the spin axis to precess and spin rate to decay
Earth’s magnetic field can be modeledas a dipole tilted about 11.5 deg fromthe pole
Earth’s magnetic field, b
Torquer magnetic moment, µ
Control torque, τc = µ X bTorque Rod
Earth’s magnetic field can be beneficial.In addition to a source of control torquesfor momentum management, the directionof the field can be used for coarse attitudedetermination
37Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Gravity Gradient TorqueGravity Gradient TorqueGravity Gradient – results from inverse square gravitational field interacting with a distributed mass spacecraft. Gravitational acceleration is stronger on the portion of the spacecraft near Earth. The gradient generates a torque that can be used to passively control attitude.
Source: Rodden Seminar1985
The Moon is a good example of a gravity gradient controlled satellite
( )
2314
3
sm10986.3
constant nalgravitatioEarth :framebody in r unit vecto:ˆ
radiusorbit :
ˆˆ3
−⋅×=
×=
µ
µ
µ
r
rIrτGG
RR
Gravity Gradient Torque
From Spacecraft Attitude DeterminationAnd Control, page 567
38Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Attitude Stabilization TechniquesAttitude Stabilization Techniques
39Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Example CubeSat Mass/PowerExample CubeSat Mass/Power
l
b
a
Ixx
Iyy
Izz
Mass, m = 5 Kg
Mass center (m): +/- [0.1, 0.1, 0.2]
Length, l = 0.9 m
Width, Height, a = b = 0.3 m
Inertia, Ixx = Iyy = 0.375 Kg-m2
Inertia, Izz = 0.075 Kg-m2
Solar cell area: 0.9x0.3 = .27 m2 per faceSolar cell efficiency = 10%Solar Illumination intensity = 1350 W/m2
Generated power (assuming 1 face fully illuminated) = 36.45 WCubeSat Voltage Supply: 30V
40Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Cube Sat Disturbance TorquesCube Sat Disturbance Torques• Assume 300 Km orbit altitude, velocity = 7,726 m/s• Aerodynamic Torque
- Atmospheric density, ρ = 2.0e-11 Kg/m3
- Coefficient of Drag, CD = 2.0- Area, A = 0.9x0.3 = 0.27 m2
- Force, f = (1/2)CDρAv2 = 3.22e-4 N- Moment, rxf = 6.44e-5 N-m (about X or Y), 3.22e-5 N-m (about Z)
• Gravity gradient- Assume radius vector in the X-Z body plane: r = [sinα, 0, cosα]T
- Inertia matrix, I = diag([0.375, 0.375, 0.075]) Kg-m2
- Cross product, rxIr = 0.3*[0, sinαcosα, 0]T, (max at α = π/4)- Gravity gradient torque, τGG = (3µ/R3)(rxIr) = 6.0e-7 N-m
• Solar Pressure Torque- Momentum flux, P = (solar rad)/(speed of light) = 4.5e-6 kg-m-1-s-2
- Coefficient of specular reflection, CS = 0.9- Solar force, f = -2PCSA = 2.43e-7 N (A = 0.03m2)- Moment, rxf = 4.86e-8 N-m (about X or Y), 2.43e-8 N-m (about Z)
Atmospheric torques are50-100x greater than gravitygradient or solar pressuretorques
41Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Magnetic Torques/Rate CaptureMagnetic Torques/Rate Capture• Earth’s magnetic field at 300 Km altitude ~ 2.6e-5 Tesla• Torquer magnetic moment, µ:
- Air core: (0.5Ax(π/4)(.03m)2x1000 turns, µ = 0.353 A-m2
- Iron core: µ = 20 A-m2 (estimate)• Max torque (air core): τ = µxb = 9.2 e-6 N-m• Max torque (iron core): τ = µxb = 5.2e-4 N-m• Control implementation: assume 0.1 rad/sec separation rates:
- CubeSat angular momentum, h = ω*Ixx = 0.0375 Kg-m2/s- Desired torque (proportional control): τD = -Kph- Cross product control law: µ = bxτD/(bTb)- Control torque: τC = µxb = τD – [(bTτD)/(bTb)]b
- If b is parallel to τD, control torque is zero- If b is perpendiuclar to τD, control torque is max
• Constraints- Requires gyro for rate measurement, magnetometer for field direction- Magnetometer should be far enough removed from magnetic torquers to preclude interference
(or mulitplex magnetometer and torquer operation)- Iron or highly permeable core torquers are non-linear (hysteresis, saturation, residual magnetism)
and difficult to control ZI
XI
r
t
θ
ψ
Earth( )( )30
0
0
50.3
)sin()cos(2
RR
t
r
EeB
BbBb
−=
−==
ψψ
Single Dipole Model of Earths Mag. Field
Ref: Spacecraft Attitude Determination andControl, pg 783
Note that max torque with air core is less then estimated atmoshpericdrag torque (may require ferro-magnetic core)
42Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
MEMS GyroMEMS Gyro
Input Rate
Output Response
MEMs Gyro: solid state tuning fork type sensor• Top half driven to vibrate in plane at a fixed frequency• Rate about input axis generates Coriolis accelerations that cause
out of plane deflections that are sensed by pickup sensors• Sensor outputs are demodulated by the frequency generator and
DC output is proportional to input rateLow accuracy devices with high drift rates (20-30 deg/hour) and unsuitedFor most space applications
Source: http://www.systron.com/tech.asp
• Gyros measure inertial rates required for attitude stabilization
43Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Spin StabilizationSpin Stabilization• Assume spin stabilization about the major principal axis of inertia (IXX)
- Spin axis remains inertially fixed absent external torques
• Considerations:- Limited Earth viewing (maybe OK for observations in space)- Control of angular momentum vector
- Spin up using external torques?- Spin up from deployment mechanism at separation?- Orientation of the angular momentum vector- Nutation damping to keep body axis aligned with momentum vector
ZI
XI
ψ
Earth
h, ωv
v
v
v
h, ωh, ω
h, ω Momentum vector, h,Angular velocity, ω,remain inertially, fixed.
Aero. Forces in direction oppositevelocity change over course of orbit
44Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Momentum BiasMomentum Bias• Spin up momentum wheel along direction normal to orbit plane
- Spin axis remains inertially fixed absent external torques- Torque about spin axis to keep spacecraft Earth pointing
• Considerations:- Spin up of momentum wheel requires external torque- Earth pointing requires an Earth sensor and closed loop control- Need external torques to manage wheel speed and control precession
induced from aerodynamic drag- Vibration from wheel imbalance
ZI
XI
ψ
Earth
v
v
v
vMomentum wheelaxis normal to orbit(out of paper). TorqueAbout wheel axis toMaintain nadir pointing
Aerodynamic torquesConstant over courseof orbit – requires
External torques to Dump wheel momentumAnd maintain yaw
Momentumwheel
45Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Spin Stabilization AnalysisSpin Stabilization Analysis
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
θθθθ
cossin0sincos0001
I/BT
• Average torque over one revolution (inertial frame)− τX = τZ = 0− τY = (1/2)ρCDv2(x)[(1/2)lhsin(2ψ) – (2/π)(wh+lw)sin2ψ]
• Integrated average torque over a quarter orbit (π/2<ψ<π in 1350 sec)− ∆H = (1/2)ρCDv2(x)[(1/2)(lh+wh+lw)]*1350 sec− ∆H = 0.0508 Nms
• Spin rate required to keep libration within +/- 5 deg- H = ∆H/tan(5 deg) = 0.581 Nms- Angular rate, ω = H/IXX = 0.581/0.375 = 1.55 rad/sec
- Almost 90 deg/sec
0.1m
0.2mXB
MassCenter
FX1
FZ1
FZ2
FX2
CP1
CP2
VelocityDirectionψ
ZB
Satelliteat θ = 0XI
ZI
Spinning rectangularSatellite (dimensions:lxwxh) with displaced mass center: pCM = [x, y, z]T
0.1m
0.2mXBFX1
FZ1
FX2
CP2
ZB
Satelliteat θ = π
CP1
FZ2
Note changein CM locationrelative to dragforces
Body to Inertialtransformation
46Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Gravity GradientGravity Gradient• Need to increase rxIr by about 30 Kg-m2:
- Two 1 Kg masses supported on 4m booms could do it• Considerations:
- Deployment mechanism- Vibration of the boom- Thermal distortion of the boom (thermal “snap”)- Limited Earth viewing angle
47Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Summary of CubeSat Attitude Control Summary of CubeSat Attitude Control
Wheel bearings+/- 0.1° to +/- 5°in two axes (rate dependent)
Momentum vector normal to orbit
Best for local vertical
Bias Momentum(1 wheel)
Despin bearingsDepends on rate measurement
Same as aboveDespun platform rotation about inertially fixed
Dual Spin Stabilization
Rate/attitude sensors
+/- 0.1° to +/- 5°in two axes (rate dependent)
Large torques for precession maneuvers
Inertially fixedSpin Stabilization
None+/- 5° (two axis)Very limitedNorth/South (low inclination orbit)
Passive Magnetic
Life of wheel or bearings
+/- 5° (three axis)Very limitedEarth local vertical with fixed yaw
Gravity Gradient & Momentum bias
None+/- 5° (two axis)Very limitedEarth local vertical only
Gravity Gradient
LifetimeLimits
Typical Accuracy
Attitude Maneuverability
Pointing Options
Type
Ref: Larson & Wertz, Space Mission Analysis and Design
48Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Attitude Determination Attitude Determination ApproachesApproaches
49Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Attitude DeterminationAttitude Determination• 3 axis attitude determination requires two or more directional
measurements to well separated objects (e.g. Sun and Earth, 2 stars).- Gyros measure attitude rate but not orientation
• Direct and state estimation techniques:- Direct approaches calculate transformation matrix directly from vector
measurements.- Estimation techniques require an initial (coarse) initial attitude
knowledge and estimate attitude and measurement error sources- Estimation techniques can incorporate measurements from a
variety of sensors including gyros- Multiple measurements may be processed recursively to reduce
computational burden for on orbit processing.- Estimation techniques may incorporate measurement error
statistics and are generally more accurate than direct methods- estimation and algebraic approaches
50Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Attitude from Vector MeasurementsAttitude from Vector Measurements
[ ] [ ]BM
BM
BMB
IR
IR
IRR
BM
BM
BM
BM
BM
BM
BM
BM
BM
BM
IR
IR
IR
ΙΡ
ΙΡ
IR
IR
IR
IR
IR
srqMsrqM
rqs|vu|/vur uq
rqs|vu|/vur uq
==
×=××==
×=××==
;
;;
;;
• Assume two non parallel unit reference vectors uR, vR, and corresponding measurements uM, vM (eg Earth-Sun, 2 stars, etc.)
• Reference vectors are known with respect to a desired reference (eg ECI)
• Measured vectors are measured in the frame of interest (eg Body frame)
• Form an orthogonal basis set and corresponding body and reference matrices:
• The inertial to body direction cosine matrix, TB/I, can be calculated directly from matrix multiplication:
TRB
1RBB/I
RB/IB
MMMMT
MTM
==
=−
Ref: Spacecraft Attitude DeterminationAnd Control, Wertz, 1978
51Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Magnetic Field MeasurementsMagnetic Field Measurements
• Assume a simple dipole model with north pole at 78.5N deg latitude and 69.7W longitude.
- Determine transformation from Earth to Magnetic Field frame, TM/E- Compute magnetic field vector, uM, from magnetic declination, ψ
• If spacecraft position is known, the reference magnetic field vector can be computed from magnetic field model
• Attitude error (single axis) determined by comparing measured and reference vectors.
- Magnetometer must be distanced from magnetic torquers- Note: single vector measurement must be used with another vector
measure (eg sun vector) for 3 axis attitude.• Sources: Applied Physics Systems (Mountain View)
- http://www.appliedphysics.com/
ZM
XM
r
t
θ
ψ
Earth
( )( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+
−−=
ψ
ψψ
2
3
cos10
sincos50.3 R
REeMu
Single Dipole Model of Earths Mag. Field
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
°°°−°°°°
°−°°−°°=
)5.11cos()7.69sin()5.11sin()7.69cos()5.11sin(0)7.69cos()7.69sin(
)5.11sin()7.69sin()5.11cos()7.69cos()5.11cos(
M/ET
52Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Star Trackers/CameraStar Trackers/Camera• Star trackers detect visible light on a CCD focal plane and measure position
relative to the optical boresight. - If the position of the originating star is known this measurement can be used to
update or correct attitude knowledge- Knowledge of star positions and magnitudes is maintained in a catalog
• Star identification is usually dealt with one of two ways- If spacecraft attitude knowledge prior to the measurement is within ½ degree
or so, the expected position of the star is predicted to be within some window on the star tracker focal plane
- If spacecraft attitude knowledge is poor, then for trackers with a wide enough field of view some sort pattern matching can be used to identify stars from a catalog
• Star trackers may require large sun shades to eliminate stray light during operation near the Sun (30-40 degrees)
Sun
Shade
Optics
FocalPlane
CCD Focal Plane Array
CCD Focal Plane Array
Star Windowing Pattern Match
Boresight
53Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Sun sensorsSun sensors• Direction to sun determined from photo or solar cell currents
- Pyramid configuration improves accuracy single cell
Output current varies with cosineof angle to surface normal of cell- Poor sensitivity at small cone angles- Single angle (non-vector) measure
Difference output currentsof cells on opposite sides- Highest sensitivity at apex- Dual angle (vector) measure
Single cell Pyramid configuration
54Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
Horizon SensorsHorizon Sensors• Camera pointed towards Earth horizon
- For 300 Km orbit, 72.7° from nadir- For wide Field of View determine roll and pitch from image processing
from Earths curvature- For narrow field of view roll and pitch angles may require two cameras
• Considerations- Sun obscuration- Earth acquistion maneuvers
6378 KmEarth Radius
300 KmAltitude
6378 Km
Horizon line
Camera Field of view
pitch
roll
Camera Image
55Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
GPS ArrayGPS Array• Determine attitude from signal phase differences between antennas
mounted at different locations.- Minimum of 4 GPS antennas for unambiguous attitude determination- Need carrier phase measurement - Need to resolve integer phase ambiguity
Wavelength
θ
To GPSSatellite
Baseline lengthBetween antennae
GPS signal wavelengthIs about 0.2 meter.
Without fractional phaseDifference measurement,Baseline between antennaeWould need to be 2.3 mFor 5 deg accuracy
References:Cohen, C. E. “Attitude Determination using GPS”PhD Dissertation, Dept. of Aero and Astro, StanfordDec. 1992
Axelrad, P. and Ward, L. M., “Spacecraft AttitudeEstimation using GPS: Methodology and ResultsFor RADCAL”, Journal of Guidance, Control andDynamics, Vol. 19, No 6, 1996. pp 1201-1209
56Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
State Estimation Example (Single Axis)State Estimation Example (Single Axis)
Consider a star tracker with a random centroiding error of 10 arc-seconds at a sample rate of 1 Hz and a gyro with a drift rate of 0.1 deg/hour, one sigma, at turn on and a random walk of 0.001 deg/hr1/2
For a perfectly static situation we could simply average star tracker measurements to smooth out noise. 100 samples in 100 seconds would reduce attitude knowledge error to 1 arc-second, one sigma.
On the other hand, if we started with perfect attitude knowledge and relied solely on the gyro, the drift rate would integrate to a knowledge error of 10 arc-seconds in 100 seconds (1 deg/hour = 1 arc-second/second)
( )N
E
N
T
Tii
N
ii
σθθθ
σννθθθθ
=⎥⎦⎤
⎢⎣⎡
⎟⎠⎞⎜
⎝⎛
+== ∑=
2/12
2i
1
-ˆ is ˆfor deviation standarderror Then the
ceith variangaussian wmean zero is and where1ˆ
Gyro ST θ
( ) ( ) ( )
( ) ( ) tσθ-θE tθ
σtηdtθωdtt
d
/
dgyro
t
tgyro
=⎥⎦⎤
⎢⎣⎡
⎟⎠⎞
⎜⎝⎛
++=+= ∫212
20
ˆis ˆfor deviation standarderror the(t), Neglecting
of variancea hasdrift gyro and whereˆ
0
η
ωθθ &
57Space Systems Company
AA236: Overview ofSpacecraft GN&C Subsystems
Brian Howley
State Estimation Example (Single Axis)State Estimation Example (Single Axis)• To combine gyro and star tracker, use the best estimate of attitude and gyro drift at time
tk, θ(tk) and d(tk). Integrate gyro rate to the time of the next star tracker measurement, tk+1.
• At time tk+1 multiply the difference between the star tracker measurement, θST(tk), and θ(tk+1) by a set of gains to correct our estimate of attitude and drift.
• The set of gains must be chosen carefully to get satisfactory performance. A Kalmanfilter computes the optimal set of gains based on error and process noise statistics and the error dynamics. This is computationally expensive and not always necessary.
• Steady state attitude estimation accuracy for a 10 arc-second star tracker and a MIMU gyro with optimal gains shown below:
-5.804e-6
Optimal Kd
sec-1
0.0297
Optimal Kθ
1.750.005<0.01 (over 8 hours)
110
Steady State Accuracy
arc-sec, 1-σ
Gyro Random Walk
deg/hr1/2, 1-σ
Gyro Bias Stability
deg/hr, 1-σ
Sample Rate
Hz
Star Tracker Noise
arc-sec, 1-σ
( ) ( ) ( ) ( )( )
( ) ( )kk
t
tkgyrokk
tdtd
dstdsttk
k
++
−
+++
−
=
−+= ∫+
ˆˆ
ˆˆˆ
1
1
1
ωθθ
( )( )
( )( )
( ) ( )( )111
1
1
1 ˆˆˆ
ˆˆ
+−
++
−+
−
++
++
−⎥⎦
⎤⎢⎣
⎡+
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
⎥⎥⎦
⎤
⎢⎢⎣
⎡kkST
dk
k
k
k ttKK
tdt
tdt θθθθ θ
Estimator gains and performance are determined through analysis