Attitude Determination and Control System (ADCS)€¦ · Time Coordinates Orbital elements Satellite position from orbital elements Attitude determination, description of attitude

Attitude Determination and Control System

(ADCS)

11 March 2019 Kyushu Institute of Technology 1

Sangkyun Kim

Contents


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


• Rotate Body frame to match with Target frame

©Juliana Ismail

Time


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


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


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


© 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


© 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


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

https://celestrak.com/software/tskelso-sw.asp




TLE


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


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


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


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


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


• Rotate Body frame to match with Target frame

©Juliana Ismail


Schematic of Attitude Control

Passive Attitude Control


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


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


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

https://www.ngdc.noaa.gov/geomag/WMM/calculators.shtml

https://www.ngdc.noaa.gov/geomag/WMM/calculators.shtml

https://en.wikipedia.org/wiki/Dipole_model_of_the_Earth's_magnetic_field

https://en.wikipedia.org/wiki/Dipole_model_of_the_Earth's_magnetic_field

Sensors


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


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


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


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

Attitude Determination and Control System (ADCS)€¦ · Time Coordinates Orbital elements Satellite position from orbital elements Attitude determination, description of attitude

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