Attitude Determination And Control (ADCS)

Attitude Determination and Control(ADCS)16.684 Space Systems Product DevelopmentSpring 2001Olivier L. de WeckDepartment of Aeronautics and AstronauticsMassachusetts Institute of Technology

ADCS Motivation Motivation—In order to point and slew opticalsystems, spacecraft attitude controlprovides coarse pointing whileoptics control provides finepointing—— Spacecraft Slew Maneuvers— Spacecraft Control——Spacecraft Stabilization— Spin Stabilization— Gravity Gradient— Three-Axis Control— Formation FlightActuators— Reaction Wheel Assemblies(RWAs)— Control Moment Gyros(CMGs)— Magnetic Torque Rods— ThrustersSensors: GPS, star trackers, limbsensors, rate gyros, inertialmeasurement unitsControl Laws—Euler AnglesQuaternionsKey Question:What are the pointingrequirements for satellite ?NEED expendable propellant: On-board fuel often determines life Failing gyros are critical (e.g. HST)

Outline Definitions and TerminologyCoordinate Systems and Mathematical Attitude RepresentationsRigid Body DynamicsDisturbance Torques in SpacePassive Attitude Control SchemesActuatorsSensorsActive Attitude Control ConceptsADCS Performance and Stability MeasuresEstimation and Filtering in Attitude DeterminationManeuversOther System Consideration, Control/Structure interactionTechnological Trends and Advanced Concepts

Opening Remarks Nearly all ADCS Design and Performance can be viewed interms of RIGID BODY dynamicsTypically a Major spacecraft systemFor large, light-weight structures with low fundamentalfrequencies the flexibility needs to be taken into accountADCS requirements often drive overall S/C designComponents are cumbersome, massive and power-consumingField-of-View requirements and specific orientation are keyDesign, analysis and testing are typically the mostchallenging of all subsystems with the exception of payloaddesignNeed a true “systems orientation” to be successful atdesigning and implementing an ADCS

TerminologyATTITUDE :Orientation of a defined spacecraft body coordinatesystem with respect to a defined external frame (GCI,HCI)ATTITUDE DETERMINATION: Real-Timeor Post-Facto knowledge,within a given tolerance, of the spacecraft attitudeMaintenance of a desired, specified attitudewithin a given toleranceATTITUDE CONTROL:“Low Frequency” spacecraft misalignment;usually the intended topic of attitude controlATTITUDE ERROR:“High Frequency” spacecraft misalignment;usually ignored by ADCS; reduced by good design or finepointing/optical control.ATTITUDE JITTER:

Pointing Control easkcdesired pointing directionactual pointing direction (mean)estimate of true (instantaneous)pointing accuracy (long-term)stability (peak-peak motion)knowledge errorcontrol errora pointing accuracy attitude errors stability attitude jitterSource:G. MosierNASA GSFC

Attitude Coordinate Systems(North Celestial Pole) ZGCI: Geocentric Inertial CoordinatesCross productGeometry: Celestial Sphere Y ZxXdihedralthgnelArcDVERNALEQUINOXG XD : Right AscensionG : DeclinationInertial CoordinateSystem YX and Y arein the plane of the ecliptic

Attitude Description Notations{ } Coordinate system*P VectorA*P Position vector w.r.t. { A}Ẑ APzA*PPyPx Px A* P Py P z Ŷ AX̂ A 1 0 0 Unit vectors of { A} Xˆ A YˆA Zˆ A 0 1 0 0 0 1 []Describe the orientation of a body:(1) Attach a coordinate system to the body(2) Describe a coordinate system relative to aninertial reference frame

Rotation Matrix{ A} Reference coordinate systemẐ AJeffersonMemorial ŶẐ B{ B} Body coordinate systemBŶ AX̂ AẐABX̂ A X̂Bθ0 1A B R 0 cos 0 sin]Special properties of rotation matrices:(1) Orthogonal:JeffersonMemorialŶẐθ[A ˆ AˆAA ˆR XZBBB YBX̂ BBRotation matrix from {B} to {A}RT R I , RT R 1ŶA - sin cos 0(2) Orthonormal:R 1(3) Not commutativeA BB AB R C R C R BR

Euler Angles (1)Euler angles describe a sequence of three rotations about differentaxes in order to align one coord. system with a second coord. system.Rotate about Ẑ A by αẐ AẐ BRotate about ŶB by βẐ BẐCβŶBαX̂ AαŶAX̂ B cosα - sinαA cosαB R sinα0 0X̂ BẐ DŶBβẐCγγŶCX̂ C0 0 1 Rotate about X̂ C by γ cosβB CR 0 - sinβŶCX̂ C010AA B CD R B R C R D Rsinβ 0 cosβ ŶDX̂ D00 1C D R 0 cosγ - sinγ 0 sinγ cosγ

Euler Angles (2) Zi (parallel to r)θ about YiConcept used in rotationalφ about X’kinematics to describe bodyYaworientation w.r.t. inertial frame ψ about ZbRollSequence of three angles andBodyXiprescription for rotating oneCM(parallelreference frame into anotherto v)PitchCan be defined as a transformation(r x v direction)matrix body/inertial as shown: TB/IYiEuler angles are non-unique andrnadirexact sequence is criticalNote:Goal: Describe kinematics of body-fixedframe with respect to rotating local verticalTB /1I TI / B TBT/ I cosψTransformation -sinψfrom Body to T B/I “Inertial” frame: 0(Pitch, Roll, Yaw) (T I \)sinψcosψ0YAW0 100 0 cosφ1 0 -sinφROLLEuler Angles0 cosθsinφ 0cosφ sinθ0 -sinθ 10 0 cosθ PITCH

Quaternions *q A vector describes theMain problem computationally is q1 axis of rotation. q q*the existence of a singularity 2Q q4 A scalar describes theProblem can be avoided by an q3 q4 amount of rotation. application of Euler’s theorem:q 4 A ˆθKẐ AEULER’S THEOREMJefferson MemorialThe Orientation of a body is uniquelyspecified by a vector giving the directionof a body axis and a scalar specifying arotation angle about the axis. Definition introduces a redundantfourth element, which eliminatesthe singularity.This is the “quaternion” conceptQuaternions have no intuitivelyinterpretable meaning to the humanmind, but are computationallyconvenientẐ BX̂ AA: InertialB: Body k x AK̂ k y k z ŶBŶ AX̂ B θ q1 k x sin 2 θ q2 k y sin 2 θ q3 k z sin 2 θ q4 cos 2

Quaternion Demo (MATLAB)

Comparison of Attitude arVelocity ZQuaternionsPlusesIf given φ,ψ,θthen a uniqueorientation isdefinedOrientationdefines aunique dir-cosmatrix RVectorproperties,commutes w.r.tadditionMinusesGiven orientthen Eulernon-uniqueSingularity6 constraintsmust be met,non-intuitiveIntegration w.r.t Not Intuitivetime does notNeed transformsgive orientationNeeds transformBest foranalytical andACS design workMust storeinitial conditionComputationallyrobustIdeal for digitalcontrol implementBest fordigital controlimplementation

Rigid Body KinematicsZTime Derivatives:(non-inertial)BodyCMr k KZ Angular velocityof Body FrameU iRInertialFrame jRotatingBody FrameYJ IXApplied toposition vector r:r R ρBASIC RULE:Positionr R ρ BODY ω ρρ INERTIAL ρ BODY ω ρExpressed inthe Inertial FrameRate( ρ ρ ω ω ρ2ωρω r RBODYBODYInertialrelative accelaccel of CMw.r.t. CMcoriolisangularaccel)centripetalAcceleration

Angular Momentum (I).rnAngular MomentumH total n r i mi r ii 1mnZSystem inmotion relativeto Inertial Frame.rimirnrir1m1.r1YIf we assume thatX(a)(b)Then :Origin of Rotating Frame in Body CMFixed Position Vectors ri in Body Frame(Rigid Body)Angular Momentum DecompositionH total mi R R i 1 n ANGULAR MOMENTUMOF TOTAL MASS W.R.TINERTIAL ORIGIN n mi ρ i ρ ii 1H BODYBODY ANGULARMOMENTUM ABOUTCENTER OFMASSCollection of pointmasses mi at riNote that Ui ismeasured in theinertial frame

Angular Momentum (II)For a RIGID BODYwe can write:ρ i ρ i ,BODY ω ρi ω ρiRELATIVEMOTION IN BODYAnd we are able to write:H IωRIIGID BODY, CM COORDINATESH and Z are resolved in BODY FRAME“The vector of angular momentum in the body frame is the productof the 3x3 Inertia matrix and the 3x1 vector of angular velocities.”Inertia MatrixProperties: I11I I 21 I 31I12I 22I 32Real Symmetric ; 3x3 Tensor ; coordinate dependentnI13 I 23 I 33 ()I12 I 21 mi ρi 2 ρi1()I13 I 31 mi ρi1 ρi 3()I 23 I 32 mi ρi 2 ρi 3I11 mi ρi22 ρi23i 1nI 22 mi ρi21 ρi23i 1nI 33 mi ρi21 ρi22i 1ni 1ni 1ni 1

Kinetic Energy and Euler EquationsKineticEnergyEtotal 2 1 n1 n mi R mi ρ i22 i 1 2 i 1E-TRANSE-ROT11EROT ω H ω T I ω22For a RIGID BODY, CM Coordinateswith Z resolved in body axis frameH T ω I ω Sum of external and internal torquesIn a BODY-FIXED, PRINCIPAL AXES CM FRAME:H 1 I1ω 1 T1 ( I 22 I 33 )ω 2ω3H 2 I 2ω 2 T2 ( I 33 I11 )ω 3ω1H I ω T ( I I )ω ω33 3311221 2Euler EquationsNo general solution exists.Particular solutions exist forsimple torques. Computersimulation usually required.

Torque Free Solutions of Euler’s Eq.TORQUE-FREECASE:An important special case is the torque-free motion of a (nearly)symmetric body spinning primarily about its symmetry axisω x , ω y ω z ΩBy these assumptions:The components of angular velocitythen become:And the Euler equations become:ω x (t ) ω xo cos ω n tω y (t ) ω yo cos ω n tThe Zn is defined as the “natural”or “nutation” frequency of the body:ZZω x ω n2 K x K y Ω 2ZHI xx I yyQHZI zz I yyI xxΩω yKxI zz I xxΩω yω y I yyeConIz Ix I yyBodneCodyBoQSpaceConeKyQ : nutationangleSpaceConeIz Ix I yω z 0H and Z never alignunless spun abouta principal axis !

Spin Stabilized SpacecraftUTILIZED TO STABILIZE SPINNERSZb:Perfect CylinderI xx I yyAntennadespun at1 RPOI zzm L2 R 4 3 2mR 2 2YbXbDUAL SPIN :BODY Two bodies rotating at different ratesabout a common axisBehaves like simple spinner, but partis despun (antennas, sensors)requires torquers (jets, magnets) formomentum control and nutationdampers for stabilityallows relaxation of major axis rule

Disturbance TorquesAssessment of expected disturbance torques is an essential partof rigorous spacecraft attitude control designTypical Disturbances Gravity Gradient: “Tidal” Force due to 1/r2 gravitational field variationfor long, extended bodies (e.g. Space Shuttle, Tethered vehicles)Aerodynamic Drag: “Weathervane” Effect due to an offset between theCM and the drag center of Pressure (CP). Only a factor in LEO.Magnetic Torques: Induced by residual magnetic moment. Model thespacecraft as a magnetic dipole. Only within magnetosphere.Solar Radiation: Torques induced by CM and solar CP offset. Cancompensate with differential reflectivity or reaction wheels.Mass Expulsion: Torques induced by leaks or jettisoned objectsInternal: On-board Equipment (machinery, wheels, cryocoolers, pumpsetc ). No net effect, but internal momentum exchange affects attitude.

Gravity Gradientn µ / a 3 ORBITAL RATE1) Local vertical2) 0 for symmetric spacecraftGravity Gradient:3) proportional torˆ sin θZb rT 3n 2 rˆ I rˆ Gravity GradientTorquesIn Body Frame 1/r3SmallangleapproximationTXbsin φ 1 sin θ sin φ [ θ22T- sin Tφ 1]TEarthResulting torque in BODY FRAME:Typical Values:I 1000 kgm2n 0.001 s-1T 6.7 x 10-5 Nm/deg ( I zz I yy )φ T 3n 2 ( I zz I xx )θ 0Pitch Libration freq.:ωlib n3 ( I xx I zz )I yy

Aerodynamic TorqueT r Far Vector from body CMto Aerodynamic CP1Fa ρV 2 SCD2AerodynamicDrag CoefficientFa Aerodynamic Drag Vectorin Body coordinates1 CD 2Typically in this Range forFree Molecular FlowS Frontal projected AreaV Orbital VelocityTypical Values:Cd 2.0S 5 m2r 0.1 mr 4 x 10-12 kg/m3T 1.2 x 10-4 NmNotes(1) r varies with Attitude(2) U varies by factor of 5-10 ata given altitude(3) CD is uncertain by 50 %U Atmospheric Density2 x 10-9 kg/m3 (150 km)3 x 10-10 kg/m3 (200 km)7 x 10-11 kg/m3 (250 km)4 x 10-12 kg/m3 (400 km)Exponential Density Model

Magnetic TorqueT M BM Spacecraft residual dipolein AMPERE-TURN-m2 (SI)or POLE-CM (CGS)M is due to current loops andresidual magnetization, and willbe on the order of 100 POLE-CMor more for small spacecraft.Typical Values:B 3 x 10-5 TESLAM 0.1 Atm2T 3 x 10-6 NmB Earth magnetic field vector inspacecraft coordinates (BODY FRAME)in TESLA (SI) or Gauss (CGS) units.B varies as 1/r3, with its directionalong local magnetic field lines.Conversions:1 Atm2 1000 POLE-CM , 1 TESLA 104 GaussB 0.3 Gaussat 200 km orbit