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Attitude Determination and Control System
(ADCS)
11 March 2019 Kyushu Institute of Technology 1
Sangkyun Kim
Contents
11 March 2019 Kyushu Institute of Technology 2
Time
Coordinates
Orbital elements
Satellite position from orbital elements
Attitude determination, description of attitude
Attitude control
Sensors
Actuators
Test of ADCS
Conclusions for the test of ADCS
Attitude Control
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• Rotate Body frame to match with Target frame
©Juliana Ismail
Time
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Solar time, based on Sun GMT(Greenwich Mean Time) : Based on hypothetical circular orbit around
Sun
Sidereal time, based on Stars GMST(Greenwich Mean Sidereal Time)
GAST(Greenwich Apparent Sidereal Time)
Atomic clock time Based on the resonance frequency of the cesium atom, SI(System of unit)
time
TAI(International Atomic Time), Julian century = 36525 [SI days]
Universal time(Zulu time) UT0 : Mean solar time, Based on the data from observatory
UT1 : Compensate UT0 for polar motion of Earth
UTC : Coordinated Universal Time, Based on TAI, but compensated by leap
second, keep UTC with UT1 within 0.9[sec]
© Wikipedia
© Microsemi
Equations between times
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TT, Terrestrial time : theoretical time TT = TAI + 32.184[sec]
TAI = UTC + dAT, dAT can be checked on IERS or USNO(http://www.usno.navy.mil/USNO)
UTC and UT1 UT1 = UTC + dUT1, dUT1 can be checked on USNO or IERS(www.iers.org)
Julian date, JD Continuous time count from 12h UT on 1 January 4713BC
JD = 367(year) – INT(7(year+INT((month+9)/12))/4) + INT(275month/9) + day + 1721013.5
+(((sec/60)+min)/60 + hour)/24
For January 1st, 2000, 12:00 >> J2000.0 = 2451545.0
Modified Julian Date, MJD MJD = JD – 2400000.5
Julian centuries, J2000.0 T = (JD – 2451545.0)/36525
Coordinates
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Celestial coordinates, ECI(Earth Centered Inertial coordinates) Origin at the center of mass of the earth
Z-axis along the axis of rotation (Instantaneous pole)
X-axis in the equatorial plane pointing toward the vernal equinox
Y-axis is defined by right-handed system
GCRF(Geocentric Celestial Reference Frame)
Terrestrial coordinates, ECEF(Earth Centered Earth Fixed coordinates) Origin at the center of mass of the earth
Z-axis through CTP(Conventional Terrestrial Pole, mean pole)
X-axis passing through a reference meridian(Mean Greenwich meridian)
Y-axis is defined by right-handed system
ITRF(International Terrestrial Reference Frame)
WGS84 of DoD for GPS
Satellite body coordinates
© Sondays Y Satellites
Transforming Celestial and Terrestrial
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© Navipedia
𝑟𝐸𝐶𝐼 = 𝑃(𝑡 𝑁(𝑡 𝑅(𝑡 𝑊(𝑡 𝑟𝐸𝐶𝐸𝐹
𝑃 𝑡 ∶ Precession matrix
𝑁 𝑡 ∶ Nutation matrix
𝑅 𝑡 ∶ Earth rotation matrix
𝑊 𝑡 ∶ Polar motion matrix
𝑅 𝑡 = 𝑅𝑜𝑡𝑧(−𝜃𝐸𝑅𝐴
𝜃𝐸𝑅𝐴 = 2𝜋(0.7790572732640 + 1.00273781191135448 𝐽𝐷𝑈𝑇1 − 2451545.0
© gmat.sourceforge.net 𝑟𝐸𝐶𝐼 ≈ 𝑅(𝑡 𝑟𝐸𝐶𝐸𝐹
Orbit
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© Wikipedia
Orbital elements, 6 elements Eccentricity, e : Shape of the ellipse
Semimajor axis, a : The sum of the periapsis and apoapsis distance divide by two
Incilination, i : Vertical tilt of the ellipse with respect to the reference plane
Longitude of the ascending node, Ω : horizontal orient of the ascending node with vernal point
Argument of periapsis, ω : Angle measured from the ascending node to the periapsis
Anomaly, υ : Position of the body at a specific time(epoch)
Anomaly
Mean anomaly, M : Hypothetical circular orbit
Eccentric anomaly, E : From the center of ellipse
True anomaly, υ or f : From the focus of ellipse
© Wikipedia
tan 𝜈 =sin 𝐸 1 − 𝑒2
cos𝐸 − 𝑒
𝑀 = 𝐸 − 𝑒 sin𝐸
Satellite Position
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From the orbital 6 elements Calculate true anomaly
Calculate the distance r
Calculate the position using angle conditions of the orbital elements
SGP4 Using TLE(Two Line Element), 6 elements + additional information
Better accuracy than 6 elements, usually 1km initial error, error is increasing 1 – 3 km per day
Sample code is available (https://celestrak.com/software/tskelso-sw.asp)
Heavier than orbital 6 elements on the orbit calculation
GPS 10 – 100 [m] accuracy
Space model needs wide Doppler shift tracking, and accurate time management
Continuous output is not available without attitude control
Power consumption and its price are decreasing now, but still not easy for small satellite
TLE
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Provided by NORAD https://www.celestrak.com/NORAD/elements/
𝑛 =2𝜋
𝑇
𝑛2 =𝜇
𝑎3
𝜇 = 3.98600442 × 1014[𝑚3𝑠𝑒𝑐−2]
𝑎 =𝜇
𝑛2
3
𝑀 = 𝑀0 + 𝑛∆𝑡
© Wikipedia
Satellite position from 6 elements
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𝐸𝑡 = 𝐸𝑡−1 −𝐸𝑡−1 − 𝑒 sin 𝐸𝑡−1 − 𝑀
1 − 𝑒 cos𝐸𝑡−1, 𝑢𝑛𝑡𝑖𝑙 𝐸𝑡 − 𝐸𝑡−1𝑖𝑠 𝑠𝑚𝑎𝑙𝑙 𝑒𝑛𝑜𝑢𝑔ℎ
𝜐 = 𝑎𝑡𝑎𝑛2(sin 𝐸 1 − 𝑒2, cos 𝐸 − 𝑒
𝑟 = (𝑎(cos 𝐸 − 𝑒 2+(𝑎 sin𝐸 1 − 𝑒2 2
𝑟𝑥𝑟𝑦𝑟𝑧
=
𝑟(cosΩ cos 𝜔 + 𝜐 − sinΩ cos 𝑖 sin 𝜔 + 𝜐
𝑟(sinΩ cos 𝜔 + 𝜐 + cosΩ cos 𝑖 sin 𝜔 + 𝜐 𝑟 sin 𝑖 sin(𝜔 + 𝜐
Flow of calculation Calculate eccentric anomaly from mean anomaly by iterative method
Calculate true anomaly using the eccentric anomaly and eccentricity
Calculate the distance from focus of ellipse using the eccentric anomaly, eccentricity, and major radius
Calculate the position using the angle condition of orbital elements
Attitude determination
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Define body coordinate based on reference coordinate ECI or ECEF coordinate is reference coordinate usually
Euler angle, Direction Cosine Matrix, and Quaternion are usually used for the definition
© NTU © Sondays Y Satellites
Attitude description
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Eurler Angle Sequence of angle rotation on three different axes in order to align
coordinates
3 angle parameters are used, easy to understand
Even angle values are same, sequence makes different attitude description
Singularity problem
Direction Cosine Matrix, Rotation Matrix Three unit vector basis describes the rotation between two
coordinates
The rotation angles define the unit vector of basis
9 parameters are used
Quaternion One vector(axis of rotation) + One scalar(amount of rotation)
4 parameters are used
Computationally convenient
𝑅𝑥𝑦𝑧𝑢𝑣𝑤 = 𝑢 𝑟𝑒𝑓 𝑣 𝑟𝑒𝑓 𝑤 𝑟𝑒𝑓
𝑅 𝜃𝑧 =cos 𝜃 − sin 𝜃 0sin 𝜃 cos 𝜃 00 0 1
𝑞 = [𝑞1, 𝑞2, 𝑞4, 𝑞4]
𝑞1 = 𝑒 𝑥 sin𝜃
2
𝑞2 = 𝑒 𝑦 sin𝜃
2
𝑞3 = 𝑒 𝑧 sin𝜃
2
𝑞4 = cos𝜃
2
© Wikipedia
Attitude Determination Algorithm
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TRIAD method Two vectors are available in body coordinate and reference coordinate
v1, v2 : vectors in reference coordinate, w1, w2 : vectors in body coordinate
w = Av, A : Direction Cosine Matrix from reference to body
Simple and Fast, but not minimum error
𝑟 1 = 𝑣 1, 𝑟 2 =𝑣 1 × 𝑣 2𝑣 1 × 𝑣 2
, 𝑟 3 = 𝑟 1 × 𝑟 2 =𝑣 1 × (𝑣 1 × 𝑣 2
𝑣 1 × 𝑣 2
𝑠 1 = 𝑤1, 𝑠 2 =𝑤1 × 𝑤2
𝑤1 × 𝑤2, 𝑠 3 = 𝑠 1 × 𝑠 2 =
𝑤1 × (𝑤1 × 𝑤2
𝑤1 × 𝑤2
𝐴 = 𝑠𝑖𝑟𝑖𝑇
3
𝑖=1
q-method Two vectors are available in body coordinate and reference coordinate
Make K matrix using two vectors, and solve it for eigenvector of maximum eigenvalue
Determination with minimum error
QUEST or ESOQ2 algorithm for fast calculation
Attitude Control
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• Rotate Body frame to match with Target frame
©Juliana Ismail
11 March 2019 Kyushu Institute of Technology 16
Schematic of Attitude Control
Passive Attitude Control
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Permanent Magnet Provide torques to align the magnet to the earth magnetic field
Hysteresis Damper
Provide damping torques by its hysteresis characteristic
© University of Michigan © University of Colorado
𝜏𝑚𝑎𝑔𝑛𝑒𝑡 = 𝑀𝑚𝑎𝑔𝑛𝑒𝑡 × 𝐵𝑒𝑎𝑟𝑡ℎ
© Wikipedia
Sensors
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Sun sensor Find sun vector in Body coordinate, Vsun_body
Coarse sun sensor, Fine sun sensor
Solar cell can be used as coarse sun sensor
Sun position model in ECI Find sun vector in ECI coordinate, Vsun_eci
DE405 or Approximation equation
© MDPI.com
© Wikipedia
𝜆𝑠 = 282.94𝑜 + 𝑀 + 6892" sin𝑀 + 72" sin 2𝑀 − (0.002652𝑂 − 1250.09115"𝑇𝑈𝑇1
𝑀 = 357.5256𝑜 + 35999.045𝑇𝑈𝑇1
𝑟𝑠[𝑘𝑚] = 149619000 − 2499000 cos𝑀 − 21000 cos 2𝑀
𝑟𝑠 = 𝑟𝑠
cos 𝜆𝑠
sin 𝜆𝑠 cos 𝜀sin 𝜆𝑠 sin 𝜀
𝜀 = 23.43929111𝑜
Sensors
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Magnetic sensor Earth magnetic field vector in Body frame, Vmag_body
Magnetic resistor sensor
Flux gate type sensor
Earth magnetic field Earth magnetic field vector in ECEF coordinate, Vmag_ecef
IGRF announce the accurate model every 5 years, https://www.ngdc.noaa.gov/geomag/WMM/calculators.shtml
Dipole model is available with less accuracy, https://en.wikipedia.org/wiki/Dipole_model_of_the_Earth%27s_magnetic_field
© Hamamatsu © www.sensorsmag.com
Sensors
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Gyroscope Output of angular velocity
Rapid accumulation of bias error
The bias usually estimated with KF(Kalman Filter) using reference sensor
Reference sensor example A : Sun sensor + Magnteic sensor
Reference sensor example B : Star sensor
MEMS gyroscope
Optical fiber gyroscope(FOG)
Gyroscope mathmatical model
© EDN © Neubrex Co.
𝜔𝑚 = 𝜔 + 𝑏 + 𝑛𝑣
𝑏 = 𝑛𝑢
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Sensors Star sensor
Use the star position in inertia frame
Take a photo of stars, and identify the stars with the information of star catalogue
Calculate the attitude with the star position data
The most accurate attitude sensor usually
Expensive and High power consumption
Popular star catalogue Hipparcos catalogue
Star2000 catalogue
© MDPI © Jena Optronik GmbH © Berlin Space Technologies
11 March 2019 Kyushu Institute of Technology 22
Actuators
Magnetic torquer Basically current loop with coil, generate magnetic moment by itself
Generate torque combined with Earth magnetic field
Very simple and Robust, Low power consumption
Torque on the perpendicular plane of Earth magnetic field only, No torque on the direction of Earth magnetic field
Type of magnetic torquer Coreless type : Relatively weak torque, No risk of residual magnetic moment
Core type : Relatively strong torque, Risk of residual magnetic moment
𝜏𝑚𝑡𝑞 = 𝑚𝑚𝑡𝑞 × 𝐵𝑒𝑎𝑟𝑡ℎ
© Princeton Satellite Systems © VECTRONIC Aerospace
𝜏𝑚𝑡𝑞
𝐵𝑒𝑎𝑟𝑡ℎ
𝑚𝑚𝑡𝑞
© University of
Colorado
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Actuators
Reaction wheel Basically rotating wheel by electrical motor
Generate torque by the rotating acceleration of wheel
Torque by itself
Some complex structure
Limit of maximum rotating speed, Saturation of momentum
Momentum dumping strategy is required
© Maxon Motor ag © NanoAvionics
Software in the loop test
24
Verify control activity on software simulator Easy to make and use
Just PC is required to make the simulator
Mainly algorithm check
Any software tool can be used for the development of simulator
Specific software is usually used because of convenience, Matlab, Scilab, Octave and so on
Actual code of software, hardware can not be tested
Processor in the loop test
25
Verify control activity on processor with simulator Simulator emulates orbit environment, dynamics of craft, sensor data, actuators
OBC board or Evaluation board is usually used for the controller
Attitude determination, error and desire torques calculation are performed on controller
Usually serial interfaces connect Simulator and Controller
Major part of actual code can be tested, Part of OBC hardware can be tested
Hardware in the loop test
26
Verify control activity on OBC with simulator and emulator Simulator emulates dynamics of craft, actuators
Usually emulator generate sensor data and give it to OBC
OBC is used for the controller, takes sensor information and give the command torques of each actuator using same interface of craft
Almost same code can be tested, OBC hardware can be tested
Attitude control test bed
27
Verify attitude control function on test bed Install craft on the platform of test bed
Usually platform uses air bearing for the extremely low friction condition
Test bed emulate orbit environment(Sun, Earth magnetic field, Stars)
Very close to End to End test
Very complex
Very expensive
Maintenance work is heavy
© Astro Und Feinwerktechnik Adlershof GmbH
1. Air bearing
2. Pneumatic actuator
3. Air tank
© Astro Und Feinwerktechnik Adlershof GmbH
Conclusion
28
Target of attitude control test on the ground
Verify its function, and minimize the risk of failure as much as possible
Performance such as accuracy is not target usually
Difficulties of attitude control test
Basically, End to End test is not possible
Test for attitude control function Several loop test methods are available for the verification work
Loop test system is very flexible, you can design your own system structure
Simple one tests algorithm only, usually not enough for the verification
Verification with test bed is close to end to end test, but it needs much resources. Not easy to use
Hardware in the loop test is commonly used for the verification work in small satellites development
Still needs some experience and imagination capability for developer
Reprogramming capability reduces the risk dramatically