In-orbit magnetic disturbance compensationusing feed forward control in Nano-JASMINE mission

Takaya InamoriFaculty adviser: Prof. Shinichi Nakasuka

Dept. of Aeronautics and Astronautics, The University of Tokyo7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN

Keywords; Nano-satellite, Magnetic disturbance, Feed forward, Attitude control

Abstract

Nano-JASMINE is planned as a nano astrometry satellite at the ISSL lab, University of Tokyo incooperation with the National Astronomical Observatory of Japan (NAOJ). This research presents themethod for controlling the attitude using a feed forward controller. During Nano-JASMINE’soperation in space, magnetic disturbances are dominant and must be canceled to ensure astrometricobservations and measurements. This paper concludes that the use of magnetic disturbancecompensation is indispensable for the Nano-JASMINE mission and it demonstrates a useful method forachieving the strict attitude control demands for the nano-satellite.

Nomenclature

v :The star velocity on CCD arrays (pixel/s)p :Velocity coefficient on CCD arrays (pixel/(rad/s))T :Exposure time(s)xccd,yccd :x-axis coordinate on CCD (pixel) and y-axis coordinate on CCD (pixel) respectivelyCx :x-axis LSF variance on CCD coordinate system when the satellite attitude is completely stabilized.Cy :y-axis LSF variance on CCD coordinate system when the satellite attitude is completely stabilized.ω :Angular velocity in body frame (rad/s)f(x) :The point spread function(PSF) for unit exposure timeIx, Iy, Iz :Moment of inertia in x,y,z axis respectivelyhx,hy,hz :Angular momentum in x,y,z axis respectivelyω0 :satellite spin angular velocity

1 Nano-JASMINE outline

Nano-JASMINE (Nano-Japan Astrometry Satel-lite Mission for INfrared Exploration) is a nano-satellite currently underdevelopment at the Intel-ligent Space System Laboratory (ISSL) Univ. ofTokyo in cooperation with National AstronomicalObservatory of Japan (NAOJ). Fig 1.1 shows anoverview of the Nano-JASMINE satellite. To limitthe disturbance torques from sources such as airdrag, solar pressure and gravity gradient, Nano-JASMINE has a symmetric shape and momentumof inertia.

The Nano-JASMINE mission objective is to mea-sure the 3D positions of stars to an accuracy of1.8 mas(milli- arcsecond) by triangulation and ac-quires information about the stars. Astrometry,which provides precise information about the po-sition and motion of astronomical objects, is im-

Figure 1.1: Nano-JASMINE

portant to clarify the structure and the evolutionof our galaxy and universe. However it is difficultto measure stars on Earth because of blurring anddistortion effects from the atmosphere. There for

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more accurate star position data can be acquired inorbit and this information can be used for updat-ing current star catalogues such as the Hipparcosecatalogue. (Ref.1, Ref.2)

Figure 1.2: Triangulation method to observe thedistance from solar system to the star

In order to acquire astrometry data for the en-tire celestial globe, Nano-JASMINE must operatein observation mode.for more than 90 percent ofthe mission duration. The sequence of operationalmodes for the Nano-JASMINE mission is shown inFig 1.3.

Figure 1.3: Mission sequence

Table 1 shows the basic properties for the Nano-JASMINE satellite, and Fig1.4 shows the distri-bution of the mission and bus hardware devices.As shown in Fig1.5, optical bench is installed inthe center of the structure and the sensors and ac-tuators which have strict tolerances for alignmentduring the mission are attached to this bench. Sev-eral of the different coordinate systems used by theADCS are referenced to the optics bench. Dur-ing the initial mission phase, the satellite is con-

trolled using the star tracker (STT) coordinate sys-tems, and during the observation phase, it is con-trolled using the mission telescope coordinate sys-tems. Both the STT and the mission telescope areimportant sensors for attitude control and both areattached to the optical bench. Thus the alignmentbetween these sensors and the bench must be guar-anteed to sub degree accuracy. This makes thetransition from the initial mission phase to observ-ing phase easier. Fig1.4 shows the Block Diagramof the bus system. During the Nano-JASMINEmission, a typical centralized information process-ing style is adopted. Attitude control logic is in-stalled to the main OBC which collects data frommission componentsand then sends commands tothe mission OBC and actuators.

Figure 1.4: Bus system

In order to fulfill the mission demands, namely,measuring star position to 1.8 mas accuracy, twoproblems must be considered. The first problem isthe thermal stability of the mission telescope. Tosolve this problem a thermal shield and a heat radi-ation panel are introduced as shown in Figure 1.5.The second problem is to achieve the high accuracy

Table 1: Nano-JASMINE specification

Size 485*485*412mmMass 14kgMission Infrared AstrometryMission Life two yearOrbit Sun-synchronous Orbit

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Figure 1.5: Mission and bus devices’ distribution

attitude control required by the nanosatellite. Thisresearch focuses on the second problem.

2 Attitude determination andcontrol system overview

In order to obtain accurate data from the missiontelescope, satellite must be stabilized to less than740 mas/8.8s (equivalent to 4e-7rad/s) during ob-servation. Currently there are not sensors availablefor a nano-satellite to achieve the accurate attitudeinformation required. To solve this problem, the at-titude will be determined by assessing the qualityof the star image based on how blurred it appearsfrom the mission telescope. During observation, thesatellite is controlled to rotate with at the orbitalangular velocity rate in the z axis with no rotationabout the x and y axis. This means that the starimage which is acquired from mission telescope willbe slightly blurred even though the satellite is sta-bilized completely. Time Delay Integration (TDI)is adopted to solve the problem and to observe thestars for a long exposure time. In this mode, therate that the electric change is transferred to thenext Pixel is synchronized to the spin angular ve-locity.

To determine attitude from mission telescopeimage, Nano-JASMINE has to be stabilized toless than 5e-5rad/s, and this means the satelliteneeds other attitude determination sensors prior tothe observation phase. Fig2.2 shows the controlstrategy and required stability for various controlmodes. FOGs, STTs and silicone gyros are in-stalled to determine attitude in the initial missionphase and the unloading phase. As shown in Fig1.4

Figure 2.1: Observation using Time Delay Integra-tion

there are everal sensors and actuators. Magnetictorquers (MTQ) are used during the coarse controlphase and for magnetic moment compensation, andreaction wheels (RW) are used for controlling atti-tude during observation.

Figure 2.2: Control Phase

3 Residual Magnetic MomentEstimation using KalmanFilter

The disturbances, which should be considered inLEO orbit satellite, are gravity gradient torque,air disturbance torque, solar pressure disturbancetorque, and magnetic disturbance torque. In Nano-JASMINE mission, gravity gradient torque is notthe dominant disturbance, because momentum ofinertia is adjusted to be symmetric using a rotat-ing table and placing dummy masses . Air and so-

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lar pressure disturbance torques have a small effecton satellite attitude, because offset between cen-ter of gravity and the center of air pressure arealso adjusted on the ground prior to launch. Onthe other hand, magnetic disturbance which comefrom current loops in electric devices, the antennaand batteries, can not be passively compensatedfor. Fig3.1 shows the concept of the magnetic dis-turbance compensation.

Figure 3.1: Magnetic disturbance compensation

The residual magnetic moment is the cause ofthe magnetic disturbance and can be compensatedwith a MTQ. The magnetic disturbance torque isexpressed as follows,

T = M × B (3.1)

Where B is magnetic field around earth and Mis the residual magnetic moment of the satellite. Ifthe residual magnetic moment can be estimated, itsmagnetic disturbance can be canceled by applyinga magnetic torque. The method for estimation ofthe residual magnetic disturbance is presented inthis section. The dynamics of the satellite can beexpressed as follows,

Iω = −[B×]M + N − ω × (Iω + h) − h (3.2)

Where [B×] is defined as,

[B×] =

⎛⎝ 0 −Bz By

Bz 0 −Bx

−By Bx 0

⎞⎠ (3.3)

Where I is the momentum of the inertia, h is theangular momentum of the reaction wheel, N is thedisturbance, M is the residual magnetic momentof the satellite, and B is the magnetic field of theearth. The state vector is expressed as follows,

x =

⎛⎜⎜⎜⎜⎜⎜⎝

Δωx

Δωy

Δωz

Mx

My

Mz

⎞⎟⎟⎟⎟⎟⎟⎠

(3.4)

This vector consists of angular velocity ! andresidual magnetic moment M. The state equationwhich is obtained by linearizing Eq is expressed asfollows,

x = Ax + Bw (3.5)

Where,

x =(

ΔωMd

)(3.6)

A =(

I−1[(Iω + h)×]3×3 β3×3

03×3 03×3

)(3.7)

B =(

E3×3

03×3

)(3.8)

w =

⎛⎜⎜⎝

hx+Nx

Ix

hy+Ny

Iy

hz+Nz

Iz

⎞⎟⎟⎠ (3.9)

β =

⎛⎜⎝

0 Bz

Ix−By

Ix

−Bz

Iy0 Bx

IyBy

Iz−Bx

Iz0

⎞⎟⎠ (3.10)

The propagation equation is represented as fol-lows,

xk = Φk−1xk−1 + Γk−1wk−1 (3.11)

Where Φ is defined as follow,

Φ ∼ I + (∂f/∂x)/δt (3.12)

The observation matrix is expressed as follows,

H =(

E3×3 03×3

)(3.13)

The state vector is updated as follows,

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xk = ¯xk−1 + Kk(zk − Hkxk) (3.14)

Fig3.2 shows simulation results of the residualmagnetic moment estimation. In this case the mag-netic moment of the satellite was set to 0.1Am2,0.01Am2, and −0.01Am2 in x, y, and z axis respec-tively. The estimated value becomes more accurateas the estimation time increases.

Figure 3.2: The simulation result of the residualmagnetic moment estimation

Fig3.3 shows the error of the estimation. Inthis case, the residual magnetic moment is es-timated using FOG measurements with a 1 ×106rad/saccuracy, and the estimation accuracy isless than 1 × 104Am2. The satellite ’s attitude iscontrolled with feed forward control using the esti-mated value of residual magnetic moment.

Figure 3.3: The simulation result of the residualmagnetic moment estimation error

A comparison between Fig3.4 ad Fig3.5 showsthe result of the magnetic disturbance compensa-tion. In this case, the satellite’s attitude controllertried to stabilize to a spin with angular velocityonly in the z axis. In Fig3.4, without disturbancecompensation, the attitude can not be stabilized;

however in Fig3.5 the satellite is effectively stabi-lized. For the Nano-JASMINE mission, which de-mands high accuracy attitude control, feed forwardcompensation of the magnetic disturbance is the in-dispensable.

Figure 3.4: The simulation result of the attitudecontrol using only PD feed back controller

Figure 3.5: The simulation result of the attitudecontrol using both PD feed back controller and feedfoward magnetic disturbance compensation

4 Sensor Bias noises and Dy-namics model error effect

The previous section shows how to use a Kalmanfilter to estimate the residual magnetic Moment,which is the cause of the most significant distur-bance torque in orbit. . But for the Nano- JAS-MINE mission, this method is not sufficient for highaccuracy residual magnetic moment estimation as-sumed there are no bias noises such as alignmenterror in the sensor data. Momentum of inertia andangular momentum can also cause of estimation er-ror. Because this estimation method uses the Eu-ler’s equation, dynamic modeling error causes the

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bias noise and error. Fig4.1 shows the simulationresult of the sensor bias noise for gyros. The resultsof the RMM estimation using true angular velosityare the almost same to the result of estimation us-ing FOG data which is canceled bias noises, on theother hand, the result of the estimation using FOGraw data is pretty different from the other two re-sult. The strategy for estimating the residual mag-netic moment when sensor data include bias noisesin discussed in this section.

Figure 4.1: The FOG bias noise effect on the accu-racy of RMM estimation

In order to accurately estimate the residual mag-netic moment, sensor alignment and bias have to beeliminated from the sensor data. Fig 4.3 shows thestrategy to estimate several sensors ’noises. Atfirst FOG and magnetic sensor bias and alignmentnoises are estimated from the telemetry data on theonboard computer.(Ref5) With this estimated in-formation, MTQ magnetic moment scale factor ,off-set, and alignment are calculated from the Kalmanfilter. Residual magnetic moment can be canceledto a high accuracy with these estimated actuatorand sensors parameters.

At first FOG alignment error is estimated as fol-lows using the onboard computer,

x = Ax + Bw (4.1)

˙⎛⎝ Δω

mq

⎞⎠

=

⎛⎝ I−1[(Iω + h)×] 0 0

0 0 0Q2 0 0

⎞⎠

⎛⎝ Δω

mq

⎞⎠

+

⎛⎝ I−1(T − h)

00

⎞⎠ +

⎛⎝ wω

wm

wq

⎞⎠ (4.2)

Figure 4.2: Calibration strategy before residualmagnetic moemnt estimation

Where m, ω, and q are the FOG alignment fac-tor, angular velocity, and quaternion respectively.Q is defined as follows,

Q =

⎛⎜⎜⎝

q4 −q3 q2

q3 q4 −q1

−q2 q1 q4

−q1 −q2 −q3

⎞⎟⎟⎠ (4.3)

The observed value from the FOG is expressed asfollows,

ωFOG = ω + Δω + v (4.4)

Where,

Δω = H(ω)m(4.5)

=

⎛⎝ ωy ωz 0 0 0 0

0 0 ωx ωz 0 00 0 0 0 ωx ωy

⎞⎠

⎛⎜⎜⎜⎜⎜⎜⎝

mxy

mxz

myx

myz

mzx

mzy

⎞⎟⎟⎟⎟⎟⎟⎠

(4.6)

Where mij is the alignment effect of the‘ i’axison the‘ j’axis. The observation matrix is repre-sented as follows,

ω =(

I H(ω) 0)⎛⎝ ω

mq

⎞⎠ + v (4.7)

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Figure 4.3: The simulation result of the FOG align-ment estimation

Fig4.3 shows the result of the simulation usingon board computer.

As can be seen in Figfig4-3, FOG bias noise isestimated in the next step. This Kalman Filter isrun at the same time as RMM observation, and biasnoises including the error of the misalignment esti-mation are canceled simultaneaously. Fig4.4 showsthe estimation result of the estimation bias noises.The red line, blue points, and green points, showthe true value, observed value and estimated valuerespectively. Bias noise which change with time isestimated in Fig4.5.

Figure 4.4: The result of ths on-line FOG bias noiseestimation

During residual magnetic moment estimation,the magnetic sensor calibration is also importantfor high accuracy estimation. There are severalcalibration methods for magnetic sensors and mostof these methods can be considered attitude inde-pendent or attitude dependent. (Ref6, Ref7) Inthis mission, an attitude independent method isadopted because of eliminating FOG alignment er-ror or bias noises Magnetic sensor bias and mis-alignment error are estimated from a comparison

between IGRF model and sensor telemetry data onthe ground.

With these sensors ’parameters, residual mag-netic moment is more accurately estimated, thoughMTQ parameters have to be also estimated in or-der to cancel residual magnetic moment to a highaccuracy. MTQ alignment and scale factor are es-timated in a similar way to the method to estimateresidual magnetic moment. During parameter es-timation, one axis MTQ should be considered asthe output, then one axis MTQ parameter can beestimated.

Fig4.5 shows the estimation result of the MTQalignment.

Figure 4.5: The simulation result of the MTQ align-ment estimation

5 Alternate current ResidualMagnetic Moment estima-tion

In the previous section, the estimation method ofthe direct current residual magnetic moment isstated. In practice residual magnetic moment willchange with time because of changes from the cur-rent loop’s devices and solar battery. For the Nano-JASMINE mission, Alternate current residual mag-netic moment should be considered because of thestrict attitude control demand. In this section, themethod to estimate alternate current residual mag-netic moment is presented. Initially, the alternatemagnetic moment equation is expressed as follows,

M = −1τM + η (5.1)

Where τ is the time constant M is the magneticmoment, η is the white noise.

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This equation rewritten in the form of the fol-lowing state equation,

x = Ax + Bw (5.2)

Where,

A =(

I−1[(Iω + h)×]3×3 β3×3

γ3×3 03×3

)(5.3)

B =(

E3×3 03×3

03×3 E3×3

)(5.4)

Where,

γ =

⎛⎝ − 1

τ 0 00 − 1

τ 00 0 − 1

τ

⎞⎠ (5.5)

β =

⎛⎜⎝

0 Bz

Ix−By

Ix

−Bz

Iy0 Bx

IyBy

Iz−Bx

Iz0

⎞⎟⎠ (5.6)

The propagation equation is represented as fol-lows,

xk = Φk−1xk−1 + Γk−1wk−1 (5.7)

The observation matrix is presented as follows,

H =(

E3×3 03×3

)(5.8)

The state vector is updated as in the estimationmethod presented in section 3. Fig5.1 shows thesimulation result of the alternate current RMMestimation. In this simulation, alternate currentRMM which is the orbital period is assumed. Inthis simulation, alternate current RMM at the or-bital period is assumed. For the Nano-JASMINEmission, RMM is expected to change with orbitalperiod, because it spins at the orbital rate and de-vices such as solar batteries current loop changewith the orbital period. The results shows thatestimated RMM value is delayed by several hun-dreds to one thousand seconds. This means highfrequency RMM estimation is more difficult and ahigher frequency kalman filter update is required.The estimation error is shown in 5.2. The accuracyis less than 1 × 10−3Am2.

A rough estimate of the Kalman filter accuracycan be obtained from the one dimensional dynamicequation and the Riccati equation.

The linearized one dimensional dynamic equa-tion can be represented as follows,

d

dt

(ωM

)=

(0 −B

I0 − 1

τ

)(ω

Md

)

+(

1 00 1

)(η1

η2

)(5.9)

Figure 5.1: Simulation results for alternate currentRMM estimation

Figure 5.2: Simulation results for alternate currentRMM estimation error

The Riccati equation can be represented as fol-lows,

AP(t) + P(t)AT + BwQ(t)BTw

−P(t)CTR−1(t)CP(t) = 0 (5.10)

The observation matrix is modified as follows,

y =(

1 0) (

ωM

)+ n (5.11)

The observation accuracy matrix and noise co-variance matrix can be represented as follows,

R = σ2ome (5.12)

Q =(

σ21 00 σ2

2

)(5.13)

The estimation covariance matrix can be repre-sented as follows,

P =(

p11 p12

p21 p22

)(5.14)

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The estimate variance is calculates as follow,

p11 = −r +√

(q1r + r2 − 2√

q1r3 + q2r3) (5.15)

p22 =12(−2q1 − 2r − 4

√(q1 + q2)r

−√

q1r + r2 + 2r√

(q1 + q2)r)

+q1

r

√q1r + r2 + 2r

√(q1 + q2)r

+1r2

(q1r + r2 + 2r√

(q1 + q2)r))1.5 (5.16)

This result is plotted as Fig 5.3,

Figure 5.3: A rough estimate of the alternate cur-rent

6 Magnetic Disturbance com-pensation using star imagefrom mission telescope

For the Nano-JASMINE mission, satellite attitudeis determined with high precision using star imagesfrom the mission telescope in observation phase.This is possible because the blurred quality of starimage contains accurate information. Ref1 andRef2 show the accuracy of this method in deter-mining angular velocity is less than 1e − 7rad/s.This section will show that the star’s blurred qual-ities are very useful for estimation of the residualmagnetic moment.

In the simulation, Nano-JASMINE is fed syn-thesized star images or PSF (Point Spread Func-tion) data as depicted in Fig6.1. As shown inFig6.2, each star image contains attitude informa-tion about two axes. If the satellite is unstable inXbody-or Ybody − axis directions, the PSF will be

extracted in the Yccd − axis direction of the CCDpixel coordinate. If unstable in the Zbody − axis,the image will be extracted in the Xccd − axis.The satellite will obtain star image from two di-rections which are both captured on a single CCDimager (Xbody − axis and Ybody − axis in the satel-lite body-frame). This means the observation di-rection determines which axes’information is con-tained within a star image: for Ybody − axis obser-vations the XZbody − plane is the focus, and in theXbody−axis, information about the Y Zbody−planeis obtained. The detailed information can beeaned in Ref.8, Ref.9, Ref.10. The angular velocitymagnitude can be obtained from the LSF (Linearspread function) which can be calculated from com-pressed PSF information as shown in Fig6.2.

Figure 6.1: Star image from mission telescope ob-tained by simulation

Figure 6.2: Trimmed star image obtained by LSFcalculation method

The relationship between LSF variance and an-gular velocity is important for the angular velocitydetermination method. The variance of LSF is de-fined as follows in a continuous model.

V ar(xccd) = E(xccd − μx)2 (6.1)= E(x2

ccd) − μ2x (6.2)

Where μx is the average of LSF and Var(xccd) isthe variance of LSF.

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E(x) is defined as the expected value of an arbi-trary function of X,g(X), with respect to the prob-ability density function f(x) as follows,

E(g(X)) =∫ ∞

−∞g(x)f(x)dx (6.3)

The LSF is defined as follows,

∫ ∞

−∞

∫ T

0

f(x)dtdx = 1 (6.4)∫ ∞

−∞f(x)dx =

1T

(6.5)

Where T is the exposure time and f(x) is thePSF (point Spread Function) of the continuousmodel in the case of a one second exposure time.Nano-JASMINE will get star images whose appar-ent magnitudes are different. As shown in Eq.6.4,Eq.6.5, the LSF is normalized to be treated easily.If the satellite is unstable, the center of the star im-age moves to one side on the CCD arrays and theLSF becomes broader. The variance of LSF can becalculated as follows,

V ar(x) =∫ ∞

−∞x2

ccd

∫ T

0

f(xccd − vt)dtdx

− (∫ ∞

−∞xccd

∫ T

0

f(xccd − vt)dtdx)2(6.6)

Where p is the pixel velocity coefficient ((pixel/(rad/s))). The relationship between angularvelocity and LSF variance is calculated as follows

∫ ∞

−∞(x2

ccdT )f(xccd)dx + ω2p2T 2/12 (6.7)

Eq.6.7 shows that relationship between angularvelocity and LSF variance is a Quadratic functionand represented as follows,

V ar(x) = Av2x + Cx (6.8)

Where coefficients A and Cx are defined as fol-lows,

Cx =∫ ∞

−∞(x2T )f(xccd)dx (6.9)

A = p2T 2/12 (6.10)

In the case of LSFy variance can be calculatedas follows,

∫ ∞

−∞(y2

ccdT )f(yccd)dy + ω2p2T 2/12 (6.11)

Eq.6.11 represents the relationship between an-gular velocity about the z axis and LSFy variance.

V ar(y) = Av2y + Cy (6.12)

Where,

Cz =∫ ∞

−∞(y2T )f(yccd)dy (6.13)

A = p2T 2/12 (6.14)

The coefficient A is determined only from ob-servation time. The coefficients Cxy and Cz whichare the variance of the LSF is determined from PSFshape. Fig.8 shows the theoretical relationship ofLSFx variance and angular velocity. ([6])

Figure 6.3: The relationship between LSF varianceand angular velocity

Kalman Filter is adopted to estimate residualmagnetic moment.

Satellite attitude dynamics is represented as fol-low,

Iω = N − ω × (Iω + h) − h (6.15)

Where I,N,h are the momentum of inertia ,distur-bance, angular momentum respectively.

And, ω is represented as follow,

ωx = Δωx (6.16)ωy = Δωy (6.17)

ωz = ω0 + Δωz (6.18)

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Where ω0 is spin angular velocity andΔωx, Δωy,Δωz are the perturbation from theequilibrium. Spin rate is synchronized to theorbital period. Eq is linearized and state equationis given as follow,

x = Ax + Bv (6.19)

A =(

α β3×3

03×3 03×3

)(6.20)

α =⎛⎜⎝

0 ( Iy

Ixωz− Iz

Ixωz− hz

Ix) hy

Ix

( Iz

Iyωz− Ix

Iyωz+ hz

Iy) 0 −hx

Iy

−hy

Iz

hx

Iz0

⎞⎟⎠(6.21)

B =(

E3×3

03×3

)(6.22)

w =

⎛⎜⎜⎝

hx+Nx

Ix

hy+Ny

Iy

hz+Nz

Iz

⎞⎟⎟⎠(6.23)

β =

⎛⎜⎝

0 Bz

Ix−By

Ix

−Bz

Iy0 Bx

IyBy

Iz−Bx

Iz0

⎞⎟⎠ (6.24)

The spacecraft attitude state vector is given bythe angular velocity.

x =

⎛⎜⎜⎜⎜⎜⎜⎝

Δωx

Δωy

Δωz

Mx

My

Mz

⎞⎟⎟⎟⎟⎟⎟⎠

(6.25)

(6.26)

The Kalman Filter is calculated as Ref.[6] how-ever several points should be considered.

Firstly, when the star image is obtained from themission telescope, the state vector is updated asfollow,

xk(+) = xk(−) + Kk(zk − Hkxk(−)) (6.27)Pk(+) = Pk(−) − KkHkPk (6.28)

Where K is the Kalman gain and P is an estimateof the covariance matrix. In this situation, the ob-servation matrix H has to be changed depending on

direction which the star image comes from. If theimage comes from the x direction the matrix H is

H =(

1 0 00 0 1

)(6.29)

If the direction which star comes from is Y, thematrix H is

H =(

0 1 00 0 1

)(6.30)

ondly, R which is observation covariance matrixshould be changed in particular star, because obser-vation accuracy depends on the magnitude of starsas Fig6.4 . R should be calculated based on starmagnitude.

Figure 6.4: The accuracy of the observation

Figure 6.5: The simulation result of estimation us-ing Kalman Filter

Fig6.6 shows the star density in particular posi-tion on sky. This result proves that step time thatstar image is earned is arbitrary and attitude infor-mation can not be updated 10s in the worst case.

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Figure 6.6: The star density map

The simulation result of the RMM estimation isshown in Fig6.5, the convergence time is shorterand higher accuracy. Update time is arbitrary inposition on sky and this cause the plot be indent.

7 Conclusion

This research considered the Nano-JASMINE at-titude control system, focusing on measuring andresponding to magnetic disturbances. For successof the 10kg satellite attitude control system, man-aging the magnetic disturbance is critical. This re-search considers the method to estimate residualmagnetic moment using a Kalman filter in orbitusing simulation results. In this case, sensors’andactuators’alignment and offset error is critical forhigh accuracy estimation. For the Nano-JASMINEmission, these parameters are estimated and can-celed before controlling the residual magnetic mo-ment. This research also considered the alternatecurrent residual magnetic moment. This magneticmoment can also be estimated using a Kalman fil-ter and then canceled. During observation blurredstar images from The mission telescope are impor-tant for attitude determination system This imageinformation is utilized for more accurate attitudedetermination and it is also useful for residual mag-netic moment estimation. This research proposes anew method to estimate the residual magnetic mo-ment from the mission telescope image data. Basedon simulation results, the method is successful dur-ing operation of the Nano-JASMINE satellite.

References

[1] Nobutada Sako, Yoichi Hatsutori, TakashiTanaka, Takaya Inamori, Shinichi Nakasuka

”Nano-JASMINE: A Small Infrared Astrom-etry Satellite ” 21st Annual Conference onSmall Satellites

[2] Dr.W.H. Steyn and Y. Hashida ”In-Orbit At-titude Performance of the 3-Axis StabilisedSNAP-1 Nanosatellite” 15th AIAA/USU Con-ference on Small Satellite

[3] Takaya. Inamori, N. Sako, Y. Hatsutori,T.Tanaka, S. Nakasuga 　”Method to stabi-lize a nano-satellite,using the blurred qualityof star image” the 17th JAXA Workshop onAstrodynamics and Flight Mechanics

[4] E.J.Leffers F.L.Makley ,M.D.Shuster”Kalman filtering for spacecraftattitude estimation” journal ofGuidance,Vol.5,No.5,pp.417-429,Sep-Oct1982

[5] Itzhack Y.Bar-Itzhack ”Impricit and ExplicitSpacecraft Gyo Calibration” AIAA Guidance,Navigation, and Control Conference and Ex-hibit 16-19 August 2004, Providence, RhodeIsland.

[6] John L. Crassidis, Kok-Lam Lai ”Real-TimeAttitude -Independent Three-Axis Magne-tometer Caribration” Journal of Guidance,Control, and Dynamics Vol28, No.1, January-February 2005

[7] Roberto Alonso, Malcolm D. Shuser ”Com-pleate Linear Attitude Independent Mage-tometer Calibration” The Jounal of the As-tronautical Sciences, Vol 50, No.4, October-December 2002,pp477-490

[8] Nobutada Sako, Yoichi Hatsutori, TakashiTanaka, Takaya Inamori ,Shinichi Nakasuka”About small infrared astrometry satellite“Nano-JASMINE”” 2007 Space Sciences andTechnology Conference, The Japan Society forAeronautical and Space Science.

[9] Nobutada Sako, Yoichi Hatsutori, TakashiTanaka, Takaya Inamori, Shinichi Nakasuka,”Nano-satellite attitude stabilization methodusing star images”,IFAC 2007

[10] Takaya Inamori, Shinichi Nakasuka ”Methodto stabilize a nano-satellite, using star image”2007 Space Sciences and Technology Confer-ence, The Japan Society for Aeronautical andSpace Science.

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