Attitude Determination and Control (ADCS) Olivier L. de Weck Department of Aeronautics and Astronautics Massachusetts Institute of Technology 16.684 Space Systems Product Development Spring 2001 ADCS Motivation G129 Motivation — In order to point and slew optical systems, spacecraft attitude control provides coarse pointing while optics control provides fine pointing G129 Spacecraft Control — Spacecraft Stabilization — Spin Stabilization — Gravity Gradient — Three-Axis Control — Formation Flight — Actuators — Reaction Wheel Assemblies (RWAs) — Control Moment Gyros (CMGs) — Magnetic Torque Rods — Thrusters — Sensors: GPS, star trackers, limb sensors, rate gyros, inertial measurement units — Control Laws G129 Spacecraft Slew Maneuvers — Euler Angles — Quaternions Key Question: What are the pointing requirements for satellite ? NEED expendable propellant: On-board fuel often determines life Failing gyros are critical (e.g. HST) Outline G129 Definitions and Terminology G129 Coordinate Systems and Mathematical Attitude Representations G129 Rigid Body Dynamics G129 Disturbance Torques in Space G129 Passive Attitude Control Schemes G129 Actuators G129 Sensors G129 Active Attitude Control Concepts G129 ADCS Performance and Stability Measures G129 Estimation and Filtering in Attitude Determination G129 Maneuvers G129 Other System Consideration, Control/Structure interaction G129 Technological Trends and Advanced Concepts Opening Remarks G129 Nearly all ADCS Design and Performance can be viewed in terms of RIGID BODY dynamics G129 Typically a Major spacecraft system G129 For large, light-weight structures with low fundamental frequencies the flexibility needs to be taken into account G129 ADCS requirements often drive overall S/C design G129 Components are cumbersome, massive and power-consuming G129 Field-of-View requirements and specific orientation are key G129 Design, analysis and testing are typically the most challenging of all subsystems with the exception of payload design G129 Need a true “systems orientation” to be successful at designing and implementing an ADCS Terminology ATTITUDE : Orientation of a defined spacecraft body coordinate system with respect to a defined external frame (GCI,HCI) ATTITUDE DETERMINATION: Real-Time or Post-Facto knowledge, within a given tolerance, of the spacecraft attitude ATTITUDE CONTROL: Maintenance of a desired, specified attitude within a given tolerance ATTITUDE ERROR: “Low Frequency” spacecraft misalignment; usually the intended topic of attitude control ATTITUDE JITTER: “High Frequency” spacecraft misalignment; usually ignored by ADCS; reduced by good design or fine pointing/optical control. Pointing Control Definitions target desired pointing direction true actual pointing direction (mean) estimate estimate of true (instantaneous) a pointing accuracy (long-term) s stability (peak-peak motion) k knowledge error c control error target estimate true c k a s Source: G. Mosier NASA GSFC a = pointing accuracy = attitude error s = stability = attitude jitter Attitude Coordinate Systems X Z Y ^ ^ ^ Y = Z x X Cross product ^^^ Geometry: Celestial Sphere G68G3: Right Ascension G71 : Declination (North Celestial Pole) G68 G71 A r c le n g t h dihedral Inertial Coordinate System GCI: Geocentric Inertial Coordinates VERNAL EQUINOX X and Y are in the plane of the ecliptic Attitude Description Notations Describe the orientation of a body: (1) Attach a coordinate system to the body (2) Describe a coordinate system relative to an inertial reference frame A Z ? A X ? A Y ? }{ w.r.t.vector Position Vector system Coordinate }{ AP P A = = =? G42 G42 P A G42 y P x P z P ? ? ? ? ? ? ? ? ? ? = z y x A P P P P G42 [] ? ? ? ? ? ? ? ? ? ? == 1 0 0 0 1 0 0 0 1 }{ of vectorsUnit AAA ZYXA ??? Rotation Matrix Rotation matrix from {B} to {A} Jefferson Memorial A Z ? A X ? A Y ? system coordinate Reference }{ =A B X ? B Y ? B Z ? system coordinate Body }{ =B [ ] BBB AA B ZYXR ??? AA = Special properties of rotation matrices: 1 , ? == RRIRR TT 1=R (1) Orthogonal: RRRR AB C B C A B B ≠ Jefferson Memorial A Z ? A X ? A Y ? B X ? B Y ? B Z ? θ θ ? ? ? ? ? ? ? ? ? ? =R A B cos sin 0 sin- cos 0 0 0 1 (2) Orthonormal: (3) Not commutative Euler Angles (1) Euler angles describe a sequence of three rotations about different axes in order to align one coord. system with a second coord. system. ? ? ? ? ? ? ? ? ? ? = 1 0 0 0 cos sin 0 sin- cos αα αα R A B α by about Rotate A Z ? β by about Rotate B Y ? γ by about Rotate C X ? A Z ? A X ? A Y ? B X ? B Y ? B Z ? α α B Z ? B X ? B Y ? C X ? C Y ? C Z ? β β C Z ? C X ? D Y ? D X ? C Y ? D Z ? γ γ ? ? ? ? ? ? ? ? ? ? = ββ ββ cos 0 sin- 0 1 0 sin 0 cos R B C ? ? ? ? ? ? ? ? ? ? = γγ γγ cos sin 0 sin- cos 0 0 0 1 R C D RRRR C D B C A B A D = Euler Angles (2) G129 Concept used in rotational kinematics to describe body orientation w.r.t. inertial frame G129 Sequence of three angles and prescription for rotating one reference frame into another G129 Can be defined as a transformation matrix body/inertial as shown: TB/I G129 Euler angles are non-unique and exact sequence is critical Zi (parallel to r) Yaw Pitch Roll Xi (parallel to v) (r x v direction) Body CM Goal: Describe kinematics of body-fixed frame with respect to rotating local vertical Yi nadir r / YAW ROLL PITCH cos sin 0 1 0 0 cos 0 -sin -sin cos 0 0 cos sin 0 1 0 0 0 1 0 -sin cos sin 0 cos BI T ψ ψθθ ψψ φφ φ φθ θ ?????? ?????? =?? ?????? G9G12G12G12G10G12G12G12G11G9G12G12G12G10G12G12G12G11G9G12G12G12G10G12G12G12G11 Note: about Yi about X’ about Zb θ φ ψ 1 // / T BI IB BI TTT ? == Transformation from Body to “Inertial” frame: (Pitch, Roll, Yaw) = (G84G15G73G15G92)EulerAngles Quaternions G129 Main problem computationally is the existence of a singularity G129 Problem can be avoided by an application of Euler’s theorem: The Orientation of a body is uniquely specified by a vector giving the direction of a body axis and a scalar specifying a rotation angle about the axis. EULER’S THEOREM G129 Definition introduces a redundant fourth element, which eliminates the singularity. G129 This is the “quaternion” concept G129 Quaternions have no intuitively interpretable meaning to the human mind, but are computationally convenient ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? = 4 4 3 2 1 q q q q q q Q G42 Jefferson Memorial A Z ? A X ? A Y ? B X ? B Y ? B Z ? θ K A ? ? ? ? ? ? ? ? ? ? ? = z y x A k k k K ? ? ? ? ? ? ? = ? ? ? ? ? ? = ? ? ? ? ? ? = ? ? ? ? ? ? = 2 cos 2 sin 2 sin 2 sin 4 3 2 1 θ θ θ θ q kq kq kq z y x rotation. of axis the describesvector A=q G42 rotation. ofamount the describesscalar A= 4 q A: Inertial B: Body Quaternion Demo (MATLAB) Comparison of Attitude Descriptions Method Euler Angles Direction Cosines Angular Velocity G90 Quaternions Pluses If given φ,ψ,θ then a unique orientation is defined Orientation defines a unique dir-cos matrix R Vector properties, commutes w.r.t addition Computationally robust Ideal for digital control implement Minuses Given orient then Euler non-unique Singularity 6 constraints must be met, non-intuitive Integration w.r.t time does not give orientation Needs transform Not Intuitive Need transforms Best for analytical and ACS design work Best for digital control implementation Must store initial condition Rigid Body Kinematics Inertial Frame Time Derivatives: (non-inertial) X Y Z Body CM Rotating Body Frame i J K ^ ^ ^ ^ ^ ^ j k I r R G85 G90 = Angular velocity of Body Frame BASIC RULE: INERTIAL BODY ρρωρ=+× G6G6 Applied to position vector r: () BODY BODY BODY 2 rR rR rR ρ ρωρ ρ ωρ ωρωωρ =+ =+ +× =+ + × +×+×× G6 G6G6 G6G6 G6G6 G6G6G6 G6 Position Rate Acceleration Inertial accel of CM relative accel w.r.t. CM centripetal coriolis angular accel Expressed in the Inertial Frame Angular Momentum (I) Angular Momentum total 1 n iii i Hrmr = =× ∑ G6 m1 mn mi X Y Z Collection of point masses mi at ri ri r1 rn rn ri r1 . . . System in motion relative to Inertial Frame If we assume that (a) Origin of Rotating Frame in Body CM (b) Fixed Position Vectors ri in Body Frame (Rigid Body) Then : BODY total 11 ANGULAR MOMENTUM OF TOTAL MASS W.R.T BODY ANGULAR INERTIAL ORIGIN MOMENTUM ABOUT CENTER OFMASS nn ii ii H HmR mρρ == ?? =×+× ?? ?? ∑∑ G0 G6 G6 G9G12G12G10G12G12G11 G9G12G12G10G12G12G11 Note that G85 i is measured in the inertial frame Angular Momentum Decomposition Angular Momentum (II) For a RIGID BODY we can write: ,BODY RELATIVE MOTION IN BODY ii i i ρ ρωρωρ=+×=× G6G6 G9G12G10G12G11 And we are able to write: HIω= “The vector of angular momentum in the body frame is the product of the 3x3 Inertia matrix and the 3x1 vector of angular velocities.” RIIGID BODY, CM COORDINATES H and G90 are resolved in BODY FRAME Inertia Matrix Properties: 11 12 13 21 22 23 31 32 33 III II I I III ?? ?? = ?? Real Symmetric ; 3x3 Tensor ; coordinate dependent () () () 22 11 2 3 12 21 2 1 22 22 1 3 13 31 1 3 22 33 1 2 23 32 2 3 11 nn ii i iii ii i iii nn ii i iii Im II m Im II m Im II m ρ ρρρ ρ ρ ρρρ == == =+ ==? =+ ==? =+ ==? ∑∑ ∑∑ Kinetic Energy and Euler Equations 22 total 11 E-ROT E-TRANS 22 nn iii EmRmρ == ?? =+ ?? ?? ∑∑ G6 G6 G9G12G10G12G11 G9G12G12G10G12G12G11 Kinetic Energy For a RIGID BODY, CM Coordinates with G90 resolved in body axis frame ROT 11 22 T EHIωωω=?= HT Iωω ?? =?× ?? G6 Sum of external and internal torques In a BODY-FIXED, PRINCIPAL AXES CM FRAME: 1111 2 323 2 2 2 2 33 11 3 1 3333 1 212 () () () HI TI I HI TI I HI TII ωωω ωωω ==+? ==+? ==+? G6 G6 G6 G6 G6 G6 Euler Equations No general solution exists. Particular solutions exist for simple torques. Computer simulation usually required. Torque Free Solutions of Euler’s Eq. TORQUE-FREE CASE: An important special case is the torque-free motion of a (nearly) symmetric body spinning primarily about its symmetry axis By these assumptions: , xy z ωω ω<< =? xx yy II? And the Euler equations become: 0 x y zz yy xy xx K zz xx yy yy K z II I II I ωω ωω ω ? =? ? ? =? = G6 G9G12G10G12G11 G6 G9G12G10G12G11 G6 The components of angular velocity then become: () cos () cos xxn yyn ttωωω ωωω = = The G90 n is defined as the “natural” or “nutation” frequency of the body: 22 nxy KKω =? B o d y C o n e S p a c e C o n e Z H G90 zxy III<= H Z G90 B od y C on e Space Cone zxy III>= G81 : nutation angle G81 G81 H and G90 never align unless spun about a principal axis ! Spin Stabilized Spacecraft UTILIZED TO STABILIZE SPINNERS G58 Xb Yb Zb G129 Two bodies rotating at different rates about a common axis G129 Behaves like simple spinner, but part is despun (antennas, sensors) G129 requires torquers (jets, magnets) for momentum control and nutation dampers for stability G129 allows relaxation of major axis rule DUAL SPIN Perfect Cylinder BODY G58 Antenna despun at 1 RPO 2 2 2 43 2 xx yy zz mL II R mR I ?? == + ?? ?? = Disturbance Torques Assessment of expected disturbance torques is an essential part of rigorous spacecraft attitude control design G129 Gravity Gradient: “Tidal” Force due to 1/r2 gravitational field variation for long, extended bodies (e.g. Space Shuttle, Tethered vehicles) G129 Aerodynamic Drag: “Weathervane” Effect due to an offset between the CM and the drag center of Pressure (CP). Only a factor in LEO. G129 Magnetic Torques: Induced by residual magnetic moment. Model the spacecraft as a magnetic dipole. Only within magnetosphere. G129 Solar Radiation: Torques induced by CM and solar CP offset. Can compensate with differential reflectivity or reaction wheels. G129 Mass Expulsion: Torques induced by leaks or jettisoned objects G129 Internal: On-board Equipment (machinery, wheels, cryocoolers, pumps etc…). No net effect, but internal momentum exchange affects attitude. Typical Disturbances Gravity Gradient Gravity Gradient: 1) ⊥ Local vertical 2) 0 for symmetric spacecraft 3) proportional to ∝ 1/r 3 Earth r ^ -sinG84 Zb Xb G84 3 / ORBITAL RATEnaμ== 2 ?? 3TnrIr ?? =?× ?? G0 Gravity Gradient Torques In Body Frame [] 22 ? sin sin 1 sin sin 1 T T r θφ θ φ θφ ?? =? ? ? ?? ?? Small angle approximation Typical Values: I=1000 kgm 2 n=0.001 s -1 T= 6.7 x 10 -5 Nm/deg Resulting torque in BODY FRAME: 2 () 3( ) 0 zz yy zz xx II TnII φ θ ? ?? ?? ∴? ? ?? ?? () 3 xx zz lib yy II n I ω ? = Pitch Libration freq.: Aerodynamic Torque a TrF=× r = Vector from body CM to Aerodynamic CP Fa = Aerodynamic Drag Vector in Body coordinates 2 1 2 aD FVSCρ= 12 D C≤≤ Aerodynamic Drag Coefficient Typically in this Range for Free Molecular Flow S = Frontal projected Area V = Orbital Velocity G85 = Atmospheric Density Exponential Density Model 2 x 10 -9 kg/m 3 (150 km) 3 x 10 -10 kg/m 3 (200 km) 7 x 10 -11 kg/m 3 (250 km) 4 x 10 -12 kg/m 3 (400 km) Typical Values: Cd = 2.0 S = 5 m 2 r = 0.1 m r = 4 x 10 -12 kg/m 3 T = 1.2 x 10 -4 Nm Notes (1) r varies with Attitude (2) G85 varies by factor of 5-10 at a given altitude (3) C D is uncertain by 50 % Magnetic Torque TMB=× B varies as 1/r3, with its direction along local magnetic field lines. B = Earth magnetic field vector in spacecraft coordinates (BODY FRAME) in TESLA (SI) or Gauss (CGS) units. M = Spacecraft residual dipole in AMPERE-TURN-m2 (SI) or POLE-CM (CGS) M = is due to current loops and residual magnetization, and will be on the order of 100 POLE-CM or more for small spacecraft. Typical Values: B= 3 x 10 -5 TESLA M = 0.1 Atm 2 T = 3 x 10 -6 Nm Conversions: 1 Atm2 = 1000 POLE-CM , 1 TESLA = 104 Gauss B ~ 0.3 Gauss at 200 km orbit Solar Radiation Torque s TrF=× r = Vector from Body CM to optical Center-of-Pressure (CP) Fs = Solar Radiation pressure in BODY FRAME coordinates () 1 ss FKPS=+ K = Reflectivity , 0 < K <1 S = Frontal Area / ss PIc= I s = Solar constant, depends on heliocentric altitude 2 1400 W/m @ 1 A.U. s I = Significant for spacecraft with large frontal area (e.g. NGST) SUN Typical Values: K = 0.5 S =5 m 2 r =0.1 m T = 3.5 x 10 -6 Nm Notes: (a) Torque is always ⊥ to sun line (b) Independent of position or velocity as long as in sunlight Mass Expulsion and Internal Torques Mass Expulsion Torque: TrF=× Notes: (1) May be deliberate (Jets, Gas venting) or accidental (Leaks) (2) Wide Range of r, F possible; torques can dominate others (3) Also due to jettisoning of parts (covers, cannisters) Internal Torque: Notes: (1) Momentum exchange between moving parts has no effect on System H, but will affect attitude control loops (2) Typically due to antenna, solar array, scanner motion or to deployable booms and appendages Disturbance Torque for CDIO ground Air Bearing Body CM Pivot Point Air Bearing G39 offset Expect residual gravity torque to be largest disturbance Initial Assumption: 0.001 100 9.81 1 [Nm]Trmg=× ? ? ? ? r mg Important to balance precisely ! Passive Attitude Control (1) G129 Requires Stable Inertia Ratio: Iz > Iy =Ix G129 Requires Nutation damper: Eddy Current, Ball-in- Tube, Viscous Ring, Active Damping G129 Requires Torquers to control precession (spin axis drift) magnetically or with jets G129 Inertially oriented Passive control techniques take advantage of basic physical principles and/or naturally occurring forces by designing the spacecraft so as to enhance the effect of one force, while reducing the effect of others. G90 Precession: G39H H G39G84 r TrF=× F into page HTrF== G6 SPIN STABILIZED dH H H dt t ? =? ? G6 HrFt∴? ? ? 2sin 2 HH H I θ θωθ ? ?= ??= ?? rF t rF t HI θ ω ? ?? = ? Large G90 = gyroscopic stability F Passive Attitude Control (2) GRAVITY GRADIENT G129 Requires stable Inertias: I z << I x , I y G129 Requires Libration Damper: Eddy Current, Hysteresis Rods G129 Requires no Torquers G129 Earth oriented G129 No Yaw Stability (can add momentum wheel) Gravity Gradient with Momentum wheel: nadir d o w n f o r w a r d Wheel spins at rate G58 BODY rotates at one RPO (rev per orbit) O.N. Gravity Gradient Configuration with momentum wheel for yaw stability “DUAL SPIN” with GG torque providing momentum control Active Attitude Control G129 Reaction Wheels most common actuator G129 Fast; continuous feedback control G129 Moving Parts G129 Internal Torque only; external still required for “momentum dumping” G129 Relatively high power, weight, cost G129 Control logic simple for independent axes (can get complicated with redundancy) Active Control Systems directly sense spacecraft attitude and supply a torque command to alter it as required. This is the basic concept of feedback control. Typical Reaction (Momentum) Wheel Data: Operating Range: 0 +/- 6000 RPM Angular Momentum @ 2000 RPM: 1.3 Nms Angular Momentum @ 6000 RPM: 4.0 Nms Reaction Torque: 0.020 - 0.3 Nm Actuators: Reaction Wheels G129 One creates torques on a spacecraft by creating equal but opposite torques on Reaction Wheels (flywheels on motors). — For three-axes of torque, three wheels are necessary. Usually use four wheels for redundancy (use wheel speed biasing equation) — If external torques exist, wheels will angularly accelerate to counteract these torques. They will eventually reach an RPM limit (~3000-6000 RPM) at which time they must be desaturated. — Static & dynamic imbalances can induce vibrations (mount on isolators) — Usually operate around some nominal spin rate to avoid stiction effects. Needs to be carefully balanced ! Ithaco RWA’s (www.ithaco.com /products.html) Waterfall plot: Actuators: Magnetic Torquers G129 Often used for Low Earth Orbit (LEO) satellites G129 Useful for initial acquisition maneuvers G129 Commonly use for momentum desaturation (“dumping”) in reaction wheel systems G129 May cause harmful influence on star trackers Magnetic Torquers G129 Can be used — for attitude control — to de-saturate reaction wheels G129 Torque Rods and Coils — Torque rods are long helical coils — Use current to generate magnetic field — This field will try to align with the Earth’s magnetic field, thereby creating a torque on the spacecraft — Can also be used to sense attitude as well as orbital location ACS Actuators: Jets / Thrusters G129 Thrusters / Jets — Thrust can be used to control attitude but at the cost of consuming fuel — Calculate required fuel using “Rocket Equation” — Advances in micro-propulsion make this approach more feasible. Typically want I sp > 1000 sec G129 Use consumables such as Cold Gas (Freon, N2) or Hydrazine (N2H4) G129 Must be ON/OFF operated; proportional control usually not feasible: pulse width modulation (PWM) G129 Redundancy usually required, makes the system more complex and expensive G129 Fast, powerful G129 Often introduces attitude/translation coupling G129 Standard equipment on manned spacecraft G129 May be used to “unload” accumulated angular momentum on reaction-wheel controlled spacecraft. ACS Sensors: GPS and Magnetometers G129 Global Positioning System (GPS) — Currently 27 Satellites — 12hr Orbits — Accurate Ephemeris — Accurate Timing — Stand-Alone 100m — DGPS 5m — Carrier-smoothed DGPS 1-2m G129 Magnetometers — Measure components Bx, By, Bz of ambient magnetic field B — Sensitive to field from spacecraft (electronics), mounted on boom — Get attitude information by comparing measured B to modeled B — Tilted dipole model of earth’s field: 3 29900 6378 0190 22 2 530 north east km down BCSCS BSC r CS ??λ?λ λλ ??λ?λ ?? ?? ?? ? ? ?? ?? ?? ? ? =?? ?? ?? ? ? ?? ?? ? ? ?? ? ?? ? ? ?? Where: C=cos , S=sin, φ=latitude, λ=longitude Units: nTesla +Y +Z flux lines +X Me ACS Sensors: Rate Gyros and IMUs G129 Rate Gyros (Gyroscopes) — Measure the angular rate of a spacecraft relative to inertial space — Need at least three. Usually use more for redundancy. — Can integrate to get angle. However, — DC bias errors in electronics will cause the output of the integrator to ramp and eventually saturate (drift) — Thus, need inertial update G129 Inertial Measurement Unit (IMU) — Integrated unit with sensors, mounting hardware,electronics and software — measure rotation of spacecraft with rate gyros — measure translation of spacecraft with accelerometers — often mounted on gimbaled platform (fixed in inertial space) — Performance 1: gyro drift rate (range: 0 .003 deg/hr to 1 deg/hr) — Performance 2: linearity (range: 1 to 5E-06 g/g^2 over range 20-60 g — Typically frequently updated with external measurement (Star Trackers, Sun sensors) via a Kalman Filter G129 Mechanical gyros (accurate, heavy) G129 Ring Laser (RLG) G129 MEMS-gyros Courtesy of Silicon Sensing Systems, Ltd. Used with permission. ACS Sensor Performance Summary Reference Typical Accuracy Remarks Sun 1 min Simple, reliable, low cost, not always visible Earth 0.1 deg Orbit dependent; usually requires scan; relatively expensive Magnetic Field 1 deg Economical; orbit dependent; low altitude only; low accuracy Stars 0.001 deg Heavy, complex, expensive, most accurate Inertial Space 0.01 deg/hour Rate only; good short term reference; can be heavy, power, cost CDIO Attitude Sensing Will not be able to use/afford STAR TRACKERS ! From where do we get an attitude estimate for inertial updates ? Potential Solution: Electronic Compass, Magnetometer and Tilt Sensor Module Problem: Accuracy insufficient to meet requirements alone, will need FINE POINTING mode Specifications: Heading accuracy: +/- 1.0 deg RMS @ +/- 20 deg tilt Resolution 0.1 deg, repeatability: +/- 0.3 deg Tilt accuracy: +/- 0.4 deg, Resolution 0.3 deg Sampling rate: 1-30 Hz Spacecraft Attitude Schemes G129 Spin Stabilized Satellites — Spin the satellite to give it gyroscopic stability in inertial space — Body mount the solar arrays to guarantee partial illumination by sun at all times — EX: early communication satellites, stabilization for orbit changes — Torques are applied to precess the angular momentum vector G129 De-Spun Stages — Some sensor and antenna systems require inertial or Earth referenced pointing — Place on de-spun stage — EX: Galileo instrument platform G129 Gravity Gradient Stabilization — “Long” satellites will tend to point towards Earth since closer portion feels slightly more gravitational force. — Good for Earth-referenced pointing — EX: Shuttle gravity gradient mode minimizes ACS thruster firings G129 Three-Axis Stabilization — For inertial or Earth-referenced pointing — Requires active control — EX: Modern communications satellites, International Space Station, MIR, Hubble Space Telescope ADCS Performance Comparison Method Typical Accuracy Remarks Spin Stabilized 0.1 deg Passive, simple; single axis inertial, low cost, need slip rings Gravity Gradient 1-3 deg Passive, simple; central body oriented; low cost Jets 0.1 deg Consumables required, fast; high cost Magnetic 1 deg Near Earth; slow ; low weight, low cost Reaction Wheels 0.01 deg Internal torque; requires other momentum control; high power, cost 3-axis stabilized, active control most common choice for precision spacecraft ACS Block Diagram (1) Feedback Control Concept: + - error signal gain K Spacecraft Control Actuators Actual Pointing Direction Attitude Measurement c TKθ=?? Correction torque = gain x error desired attitude G84 G39G84 T c G84 a Force or torque is proportional to deflection. This is the equation, which governs a simple linear or rotational “spring” system. If the spacecraft responds “quickly we can estimate the required gain and system bandwidth. Gain and Bandwidth Assume control saturation half-width θ sat at torque command T sat , then sat sat T K θ ? hence 0 sat K I θθ ?? +? ?? ?? G6G6 Recall the oscillator frequency of a simple linear, torsional spring: [rad/sec] K I ω= I = moment of inertia This natural frequency is approximately equal to the system bandwidth. Also, 12 [Hz] = 2f f ωπ τ πω =? Is approximately the system time constant G87. Note: we can choose any two of the set: ,, sat θθ ω G6G6 EXAMPLE: 2 10 [rad] sat θ ? = 10 [Nm] sat T = 2 1000 [kgm ]I = 1000 [Nm/rad]K∴= 1 [rad/sec]ω= 0.16 [Hz]f = 6.3 [sec]τ= Feedback Control Example Pitch Control with a single reaction wheel Rigid Body Dynamics B O D Y wext ITT I Hθω=+== G6G6 G6 G6 ? Wheel Dynamics () w JThθ?+ =? = G6G6G6G6 Feedback Law, Choose G44 G44 wpr TKKθθ=? ? G6 Position feedback Rate feedback Then: () () () () 2 22 //0 //0 20 rp rp K I K I Laplace Transform sKIsKI ss θθ ζω ω ++=→ ++= ++= G6G6 G6 Characteristic Equation r / =K / 2 pp KI KIωζ= Nat. frequency damping Stabilize RIGID BODY Re Im Jet Control Example (1) T c F F G84 l l Introduce control torque T c via force couple from jet thrust: c ITθ= G6G6 Only three possible values for T c : 0 c Fl T Fl ? ? = ? ? ? ? Can stabilize (drive G84 to zero) by feedback law: On/Off Control only () sgn c TFlθτθ=? ? + G6 prediction term Where () sgn x x x = G87 = time constant G84 G84 . START “PHASE PLANE” SWITCH LINE “Chatter” due to minimum on-time of jets. Problem c TFl=? c TFl= Jet Control Example (2) “Chatter” leads to a “limit cycle”, quickly wasting fuel Solution: Eliminate “Chatter” by “Dead Zone” ; with Hysteresis: G84 G84 . “PHASE PLANE” c TFl=? c TFl= At Switch Line: 0θτθ+= G6 SL c θ C T 2 1 Is 1 sτ+ ε θτθ=+ G6 + - E 1 E 2 ε? Results in the following motion: G84 G84 . DEAD ZONE 1 ε? 2 ε? 1 ε 2 ε max θ max θ G6 Low Frequency Limit Cycle Mostly Coasting Low Fuel Usage G84 and G84 bounded . ACS Block Diagram (2) Spacecraft + + + dynamic disturbances sensor noise, misalignment target estimate true accuracy + stability knowledge error control error Controller Estimator Sensors In the “REAL WORLD” things are somewhat more complicated: G129 Spacecraft not a RIGID body, sensor , actuator & avionics dynamics G129 Digital implementation: work in the z-domain G129 Time delay (lag) introduced by digital controller G129 A/D and D/A conversions take time and introduce errors: 8-bit, 12-bit, 16-bit electronics, sensor noise present (e.g rate gyro @ DC) G129 Filtering and estimation of attitude, never get q directly Attitude Determination G129 Attitude Determination (AD) is the process of of deriving estimates of spacecraft attitude from (sensor) measurement data. Exact determination is NOT POSSIBLE, always have some error. G129 Single Axis AD: Determine orientation of a single spacecraft axis in space (usually spin axis) G129 Three Axis AD: Complete Orientation; single axis (Euler axis, when using Quaternions) plus rotation about that axis 2 filtered/corrected rate 1 estimated quaternion Wc comp rates Switch1 Switch NOT Logical Kalman Fixed Gain KALMAN Constant 2 inertial update 1 raw gyro rate Example: Attitude Estimator for NEXUS Single-Axis Attitude Determination G129 Utilizes sensors that yield an arc- length measurement between sensor boresight and known reference point (e.g. sun, nadir) G129 Requires at least two independent measurements and a scheme to choose between the true and false solution G129 Total lack of a priori estimate requires three measurements G129 Cone angles only are measured, not full 3-component vectors. The reference (e.g. sun, earth) vectors are known in the reference frame, but only partially so in the body frame. X Y Z ^ ^ ^ true solution a priori estimate false solution Earth nadir sun Locus of possible S/C attitude from sun cone angle measurement with error band Locus of possible attitudes from earth cone with error band Three-Axis Attitude Determination G129 Need two vectors (u,v) measured in the spacecraft frame and known in reference frame (e.g. star position on the celestial sphere) G129 Generally there is redundant data available; can extend the calculations on this chart to include a least-squares estimate for the attitude G129 Do generally not need to know absolute values () ? / / ? ?? iuu juvuv kij = =× × =× Define: Want Attitude Matrix T: ?? ?? ?? BBB RRR MN ijk Tijk ???? =? ???? G9G12G12G10G12G12G11G9G12G12G10G12G12G11 So: 1 TMN ? = Note: N must be non-singular (= full rank) ,uv Effects of Flexibility (Spinners) The previous solutions for Euler’s equations were only valid for a RIGID BODY. When flexibility exists, energy dissipation will occur. HIω= CONSTANT Conservation of Angular Momentum ROT 1 2 T EIωω= DECREASING ∴ Spin goes to maximum I and minimum G90 CONCLUSION: Stable Spin is only possible about the axis of maximum inertia. Classical Example: EXPLORER 1 initial spin axis energy dissipation Controls/Structure Interaction G84 Spacecraft Sensor Flexibility G129 Can’t always neglect flexible modes (solar arrays, sunshield) G129 Sensor on flexible structure, modes introduce phase loss G129 Feedback signal “corrupted” by flexible deflections; can become unstable G129 Increasingly more important as spacecraft become larger and pointing goals become tighter -2000 -1500 -1000 -500 0 500 1000 -200 0 200 NM axis 1 to NM axis 1 Gain [dB] Phase [deg] Loop Gain Function: Nichols Plot (NGST) Flexible modes Stable no encirclements of critical point Other System Considerations (1) G129 Need on-board COMPUTER — Increasing need for on-board performance and autonomy — Typical performance (somewhat outdated: early 1990’s) — 35 pounds, 15 Watts, 200K words, 100 Kflops/sec, CMOS — Rapidly expanding technology in real-time space-based computing — Nowadays get smaller computers, rad-hard, more MIPS — Software development and testing, e.g. SIMULINK Real Time Workshop, compilation from development environment MATLAB C, C++ to target processor is getting easier every year. Increased attention on software. G129 Ground Processing — Typical ground tasks: Data Formatting, control functions, data analysis — Don’t neglect; can be a large program element (operations) G129 Testing — Design must be such that it can be tested — Several levels of tests: (1) benchtop/component level, (2) environmental testing (vibration,thermal, vacuum), (3) ACS tests: air bearing, hybrid simulation with part hardware, part simulated Other System Considerations (2) G129 Maneuvers — Typically: Attitude and Position Hold,Tracking/Slewing, SAFE mode — Initial Acquisition maneuvers frequently required — Impacts control logic, operations, software — Sometimes constrains system design — Maneuver design must consider other systems, I.e.: solar arrays pointed towards sun, radiators pointed toward space, antennas toward Earth G129 Attitude/Translation Coupling — G11G20G12 G39v from thrusters can affect attitude — (2) Attitude thrusters can perturb the orbit G129 Simulation — Numerical integration of dynamic equations of motion — Very useful for predicting and verifying attitude performance — Can also be used as “surrogate” data generator — “Hybrid” simulation: use some or all of actual hardware, digitally simulate the spacecraft dynamics (plant) — can be expensive, but save money later in the program CM F l T T (1) (2) F 1 F 1 = F 2 G39F H/W A/D D/A sim Future Trends in ACS Design G129 Lower Cost — Standardized Spacecraft, Modularity — Smaller spacecraft, smaller Inertias — Technological progress: laser gyros, MEMS, magnetic wheel bearings — Greater on-board autonomy — Simpler spacecraft design G129 Integration of GPS (LEO) — Allows spacecraft to perform on-board navigation; functions independently from ground station control — Potential use for attitude sensing (large spacecraft only) G129 Very large, evolving systems — Space station ACS requirements change with each added module/phase — Large spacecraft up to 1km under study (e.g. TPF Able “kilotruss”) — Attitude control increasingly dominated by controls/structure interaction — Spacecraft shape sensing/distributed sensors and actuators -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 y/R o (ve locity ve ctor) Circular Paraboloid E llips e Optimal Focus (p/R o =2.2076) Projected Circle z/R o (Cros s axis ) Hyperbola (Foci) x/ R o (Z e n i t h N a d i r) Advanced ACS concepts G129 No ?V required for collector spacecraft G129 Only need ?V to hold combiner spacecraft at paraboloid’s focus Visible Earth Imager using a Distributed Satellite System Exploit natural orbital dynamics to synthesize sparse aperture arrays using formation flying Hill’s equations exhibit closed “free- orbit ellipse” solutions 2 x y 2 z x 2yn 3n x a y 2xn a z n z a ??= += += G6G6 G6 G6G6 G6 G6G6 Formation Flying in Space TPF ACS Model of NGST (large, flexible S/C) gyro Wt true rate WheelsStructural Filters Qt true attitude Qt prop PID Controllers K Estimated Inertia Tensor KF Flag Attitude Determination K ACS Rate Matrix Command Rate Command Position 72 DOF 72 4 3 3 3 4 4 3 63 3 3 6x1 Forces & Torques PID bandwidth is 0.025 Hz 3rd order LP elliptic filters for flexible mode gain suppression Kalman Filter blends 10 Hz IRU and 2 Hz ST data to provide optimal attitude estimate; option exists to disable the KF and inject white noise, with amplitude given by steady-state KF covariance into the controller position channel Wheel model includes non-linearities and imbalance disturbances FEM “Open” telescope (no external baffling) OTA allows passive cooling to ~50K Deployable secondary Mirror (SM) Beryllium Primary mirror (PM) Spacecraft support module SSM (attitude control, communications, power, data handling) arm side Science Instruments (ISIM) Large (200m 2 ) deployable sunshield protects from sun, earth and moon IR radiation(ISS) Isolation truss cold side NGST ACS Design Attitude Jitter and Image Stability Guider Camera * * roll about boresight produces image rotation (roll axis shown to be the camera boresight) “pure” LOS error from uncompensated high-frequency disturbances plus guider NEA total LOS error at target is the RSS of these terms FSM rotation while guiding on a star at one field point produces image smear at all other field points Target Guide Star Important to assess impact of attitude jitter (“stability”) on image quality. Can compensate with fine pointing system. Use a guider camera as sensor and a 2-axis FSM as actuator. Source: G. Mosier NASA GSFC Rule of thumb: Pointing Jitter RMS LOS < FWHM/10 E.g. HST: RMS LOS = 0.007 arc-seconds References G129 James French: AIAA Short Course: “Spacecraft Systems Design and Engineering”, Washington D.C.,1995 G129 Prof. Walter Hollister: 16.851 “Satellite Engineering” Course Notes, Fall 1997 G129 James R. Wertz and Wiley J. Larson: “Space Mission Analysis and Design”, Second Edition, Space Technology Series, Space Technology Library, Microcosm Inc, Kluwer Academic Publishers