LM Attitude Determination Control

1Space Systems Company

AA236: Overview ofSpacecraft GN&C Subsystems

Brian Howley

Brian [email protected]

Space Systems Company

AA236: Overview of AA236: Overview of Spacecraft AttitudeSpacecraft Attitude

Determination and ControlDetermination and Control



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



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)



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



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



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



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



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



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



Brian Howley

Coordinate Systems and Coordinate Systems and TransformationsTransformations



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



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.




Brian Howley

Earth Fixed Coordinate SystemsEarth Fixed Coordinate Systems

ZE, ZI

XE

YE

XI

YI


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




Brian Howley

Orbit Fixed Coordinate SystemsOrbit Fixed Coordinate Systems

XI

YI


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



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.



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.



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



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



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 =



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



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



Brian Howley

Vehicle DynamicsVehicle Dynamics



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



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



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::



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



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.



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



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



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.



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

Qq

qQq

Angular rate, ω

( )

B/IB/I

/BB

B/I/BBB/I/IBB/I

qq

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



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



Brian Howley

Environmental Forces and Environmental Forces and TorquesTorques



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



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



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



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



Brian Howley

Attitude Stabilization TechniquesAttitude Stabilization Techniques



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



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



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)



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



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



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



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



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



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



Brian Howley

Attitude Determination Attitude Determination ApproachesApproaches



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



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



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



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



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



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



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



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 &



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

LM Attitude Determination Control

Documents

subsystems

att det amp

space systems

station keeping

spacecraft

earths magnetic

spacecraft

reaction wheel