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Acta Astronautica

Acta Astronautica 98 (2014) 133–137

http://d0094-57

n Corrmobile:

E-mdaniliva

journal homepage: www.elsevier.com/locate/actaastro

Academy Transactions Note

Approach to study satellite attitude determination algorithms

Michael Ovchinnikov, Danil Ivanov n

Keldysh Institute of Applied Mathematics of RAS, Miusskaya Sq. 4, 125047 Moscow, Russian Federation

a r t i c l e i n f o

Article history:Received 30 May 2013Received in revised form17 December 2013Accepted 27 January 2014Available online 4 February 2014

Key words:Attitude determinationKalman filterMethod of accuracy study

x.doi.org/10.1016/j.actaastro.2014.01.02465 & 2014 IAA. Published by Elsevier Ltd. A

esponding author. Tel.: þ7 499 250 79 29,þ7 963 729 58 53.ail addresses: ovchinni@keldysh.ru (M. Ovchnovs@gmail.com (D. Ivanov).

a b s t r a c t

The Kalman filter accuracy study method is proposed. Application of the method isdemonstrated by a satellite attitude determination algorithm which uses sun-sensor andmagnetometer measurements. The algorithm was implemented on board of a micro-satellite Chibis-M. The attitude determination algorithm study method validated using in-flight measurements.

& 2014 IAA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Microsatellite attitude control requires determinationof the attitude motion state vector. The state vector isdetermined by processing attitude sensors measurementsby on-board computer which has computational restric-tions because of the power limit of microsatellites. That iswhy recursive algorithms based on the Kalman filter areoften applied. The Kalman filter uses linearized satelliteattitude motion equations and sensor measurements toestimate state vector by mean square criteria. Due tocomputational restrictions, the motion equations cannottake into account all perturbations affecting the satelliteand caused by environment and by actuators errors. Itleads to attitude determination accuracy decreasing. So,there appears a problem of, so called, unaccounted per-turbations influence.

Selection of motion model error and measurementnoise statistics, commonly referred to as filter tuning,is a major implementation issue for the Kalman filter.This process can make a significant impact on the filter

ll rights reserved.

innikov),

performance. Gelb [1] described the sensitivity of thesteady-state covariance of a scalar Kalman filter to Kalmangain selection which illustrates the effect of filter tuning.

In practice, Kalman filter tuning is often an ad hocprocess involving a considerable amount of trial-and-errorto obtain a filter with desirable performance characteris-tics. Maybeck [2,3] and others suggested to tune a Kalmanfilter with a numerical minimization technique. Oshmanet al. used a Monte Carlo simulation technique to statisti-cally assess the performance of the algorithm appliedand to demonstrate its viability [4]. Oshman also usedthe genetic algorithms for the filter tuning [5]. Algorithmsresult in objective function minimization by varying set offilter parameters in simulations. However, the simula-tion takes a plenty of time, its results are stochastic andobtained minimum is local only. The numerical minimiza-tion technique applied in [6] is the downhill simplexmethod which is a function optimization algorithm avail-able in several programming languages. It uses onlyfunction evaluations to fix a local minimum of the objec-tive function. Here the objective function is the root meansquare of the state estimation errors (estimate minustruth) which assumes “true” states are available.

Another approach to Kalman filter accuracy determina-tion can be applied to steady-state motion. It does notrequire the filter work simulations. In [7,8] it is shown that

M. Ovchinnikov, D. Ivanov / Acta Astronautica 98 (2014) 133–137134

filter covariance matrix asymptotical value can be obtainedfrom the solution of a matrix quadratic equation. For thesingle-axis attitude estimator this equation is solvedanalytically in the explicit formulas [9,10]. In more com-mon case the equation can be solved only numerically.Nevertheless, the considered approach cannot provide anestimation of unaccounted perturbation influence on thefilter accuracy.

In the paper we introduce the analytical approach thatallows us to study the filter performance and it can beapplied for quasi-stationary motion. The approach basedon a computation of filter covariance matrix after conver-gence, and it allows estimating the influence of unac-counted perturbations on motion determination accuracy.The main advantages of this approach are that it does notrequire the simulation of Kalman filter, consequently doesnot require a lot of computational time, the estimation offilter accuracy is an asymptotic value after convergence.Accuracy dependences on filter parameters and perturba-tion are derived analytically, and more reliable than onesderived by approaches described above.

Kalman filters have been widely used for spacecraftattitude determination. Star sensors [11,12], sun sensors[13], gyros [14,15] and magnetometers [16,17] have beenused for filter inputs. Attitude estimation algorithms usingvector observation are presented in [18,19]. For magnet-ometer and sun sensor use there arises a problem whenmagnetic field and Sun directions become collinear. In thatcase the three-axis attitude is non-observable. Actually,Kalman filter accuracy decreases when the angle betweentwo vectors is getting small. In the paper this dependenceis obtained analytically and compared to in-flight dataexperiments of the Chibis-M microsatellite which wasdeveloped and built by the Space Research Institute ofRAS [20].

2. Attitude determination algorithm based on theKalman filter

Consider briefly a Kalman filter processing the mea-surements of sun sensor and magnetometer. It is a masteralgorithm for the attitude determination of the microsa-tellite Chibis-M in the sunlit segment of orbit [21,22]. Inorder to use the Kalman filter the mathematical model ofthe satellite attitude motion must be available. The equa-tion of the satellite attitude motion (including the controltorque from reaction wheels) for the Kalman filter is asfollows:

J _ω¼ �kaq�kωωþ 3μr3

ðη� JηÞ; ð1Þ

where q is a vector part of quaternion Λ¼ ½q q4� repre-senting the transition from the local-vertical-local-horizontal (LVLH) frame to the body-fixed frame, ω is avector of angular velocity of body-fixed frame relativelyEarth centered inertial (ECI) frame, η is the local unitvector normal to the plane of orbit in the body-fixedframe, kω40, ka40 are control parameters of reactionwheels, μ is Earth's gravitational constant and r is thedistance from the Earth center to the satellite center ofmass. The last term in (1) represents the gravitational

torque. Let us write the kinematic relations as

_Λ¼ 12ΩΛ: ð2Þ

Here Ω is a skew symmetric matrix of angular velocity.For the filter which uses the measurements of sun

sensor and magnetometer, the vector part of the quater-nion and the angular velocity of the body-fixed frame withrespect to the ECI frame x¼ ½qT ωT �T are taken as thevector of estimated values. Let us linearize the equations ofmotion in the vicinity of the current motion xðtÞ. RewriteEqs. (1) and (2) as

ddt

δxðtÞ ¼ FðtÞδxðtÞ;

where δxðtÞ is a small increment of the state vector and FðtÞis a linearized matrix of the motion equations in thevicinity of xðtÞ.

The predicted value of the state vector x̂�k at the step k

of the filter operation can be obtained by integrationof Eqs. (1) and (2) with initial conditions x̂þ

k�1. Ф is atransition matrix approximately calculated as [2]

Фk�1 � EþФðx̂þk�1; tk�1Þ½tk�tk�1�

þðФðx̂þk�1; tk�1Þ½tk�tk�1�Þ2=2!þ…;

where E is the identity matrix. To save computationalresources, often just one first term of the expansion istaken into account. The propagated value of the errormatrix P�

k at the kth step is calculated as

P�k ¼Фk�1P

þk�1Ф

Tk�1þQ ; ð3Þ

where Q is the error matrix of the motion model assumedto be constant and diagonal, Pþ

k�1 is a posteriori estimationof the error matrix at the previous step.

The a posteriori estimation is a priori estimation correctedby the measurements sampled. In our case the measurementvector nonlinearly depends on the state vector,

zk ¼ hðx̂�k ; tkÞþυk:

Here zk is the measurement vector obtained at the kth step,hðx̂�

k ; tkÞ is the measurement model, υk is the measurementnoise vector with the covariance matrix R. The measurementvector consists of the magnetic field b and Sun direction svectors in the body-fixed frame,

zk ¼ ½bk sk�T :The vector h can be written as

h¼ ½ðAðq̂�k ÞboÞT ðAðq̂�

k ÞsoÞT �T ;

where A is the direction cosine matrix for the transitionfrom the body-fixed to the LVLH frame written in termsof the quaternion estimation, bo and so are vectors of thegeomagnetic field and the Sun direction written in theLVLH frame, respectively.

Linearize the measurement model,

δzðtÞ ¼HðtÞδxðtÞ;where δzðtÞ is the small increment of measurements incase of small change of the state vector δxðtÞ at the time t,HðtÞ is the linearized matrix of measurements.

Table 1Kalman filter and “truth model” equations.

Kalman filter equations “Truth model” equations

x�kþ1 ¼Фkxþ

k þqk ;zk ¼Hkxkþrk

x�kþ1 ¼Фkxþ

k þyI;k ;yI;kþ1 ¼ ГI;kykþθI;k;zk ¼HkxkþyII;k ;

yII;kþ1 ¼ ГII;kyII;kþθII;k

M. Ovchinnikov, D. Ivanov / Acta Astronautica 98 (2014) 133–137 135

Since the sensitivity matrix is constructed, the gainmatrix Kk can be written as

Kk ¼ P�k HT

k ½HkP�k HT

k þR��1: ð4ÞThe a posteriori estimation of the Kalman filter has the

form

x̂þk ¼ x̂�

k þKk½zk�hðx̂�k ; kÞ�:

The a posteriori estimation for the error matrix has theform

Pþk ¼ ðE�KkHkÞP�

k :

3. Investigation of a filter performance

The Kalman filter covariance matrix of errors Р is aqualitative criterion of the state vector estimation. If thequantity of matrix PðtkÞ at the time tk is known, one cancalculate the state vector x̂ðtkÞ determination accuracy.However, the value of PðtkÞ depends on a number of factorslike initial state vector xðt0Þ, initial value Рðt0Þ, covariancematrix of motion model error Q , measurement errors R,system dynamics (changing Fðx; tÞ on the time interval½t0; tk�). In addition, motion equation used by the Kalmanfilter does not include some disturbance torques with acomplex mathematical model because it is not easy toimplement it to on-board computer. Usually the influenceof unaccounted perturbation on accuracy estimation isinvestigated by simulation of filter work. Such a Kalmanfilter study approach takes a lot of time for computing andits results are correct with a certain probability only.

Sometimes another Kalman filter study approach canbe applied. If the satellite attitude motion is sufficientlyslow (or the measurement sampling frequency is highenough) let us consider it as a quasi-stationary and allowto be Фk ¼ФCconst, Hk ¼HCconst. Then let us deter-mine the quasi-stationary motion as such a motion whenthe acting forces and the measurement model are close tobe a constant during the time between sequential mea-surements. For the discreet extended Kalman filter one cancalculate the covariance error matrix Р1 value after con-vergence. So, the filter performance quality after transientprocess is investigated [8]. For this matrix values on twosequential steps should be equal,

Р1 ¼ Pk ¼ Pk�1:

Therefore, the following matrix equation:

Р1 ¼ ½E�ðФР1ФT þQ ÞHT ½HðФР1ФT

þQ ÞHT þR��1H�ðФР1ФT þQ Þ; ð5Þis valid. Note that all matrices in this equation are to beconstant. Taking into account matrix Р1 is a symmetricalone, the considered nonlinear matrix equation can berewritten as nonlinear equations with n unknown vari-ables (i.e. elements of matrix Р1). These equations can besolved, for example, by the Newton method.

Consider how to estimate influence of disturbances onthe filter accuracy. Let the real satellite dynamics and realmeasurements have color noise yI ; yII correspondinglywith equations like given in a right column of Table 1.

For such a dynamical system a new state vector:

ξ¼ ðxTyTI yTIIÞT ;

can be constructed with the propagated value of the errormatrix P�

ξ;jþ1 (see [7])

P�ξ;jþ1 ¼Фξ;jP

þξ;jФ

Tξ;jþQ ξ;j;

where

Фξ;j ¼Фj E 00 ГI;j 00 0 ГII;j

0B@

1CA; Q ξ;j ¼

0 0 00 ΘI;j 00 0 ΘII;j

0B@

1CA;

and ΘI;j ¼MðθI;jθTI;jÞ, ΘII;j ¼MðθII;jθTII;jÞ. The a posteriori esti-mation for the error matrix has the form

Pþξ;j ¼ CjP

�ξ;j C

Tj ;

where

Сj ¼E�KjHj 0 �Kj

0 E 00 0 E

0B@

1CA;

Kj is a gain matrix of the Kalman filter that is valid forthe equations described in the left column of the Table 1(calculated in (4)). Assuming the system is stationary, onecan calculate Pξ;1 from (5) using K1. The part of the matrixPξ;1 corresponding to initial state vector x is an estimationof the Kalman filter accuracy.

So, concluding, in this study approach a high-fidelity“truth model” from right column of the Table 1 is used forcalculating the error matrix Pξ;1 which diagonal elementscorrespond to primary Kalman filter accuracy.

4. “Chibis-M” Kalman filter tuning and investigation

Consider performance study of the Kalman filter basedon magnetometer and sun sensor measurements imple-mented on board the Chibis-M microsatellite [21]. Its sunsensors measurement root-mean-square error is 0.11,magnetometer measurement error is 250 nT (taking intoaccount errors of an ordinary Earth magnetic field model).The goal of the Chibis-M mission is the upper atmospherelightning research and its main control mode is three-axisstabilization with respect to the LVLH frame by reactionwheels. Its mass is about 42 kg and tensor of inertia J isclose to diagonal one, i.e. J ¼ diag 1:03; 1:54; 1:82ð Þ kg m2:

Consider the dependence of the attitude determinationprecision of the angle between the Sun and magneticfield directions. Since satellite angular motion is negli-gible in the stabilized position with respect to the LVLH,

09:30:00 09:35:00 09:40:00 09:45:000

20

40

60

TimeAng

le b

etw

een

b o and

so,

deg

09:30:00 09:35:00−100

−50

0

50

Atti

tude

est

imat

ions

, deg

Data 22.12.2012

09:40:00 09:45:00−2

−1

0

1

2

Atti

tude

est

imat

ions

, deg

αβγ

Fig. 3. Attitude estimation of Euler angles (above), angle between bo andso(below) during stabilization mode. For better resolution at time 9:37the graph scale is changed (scales at the right).

M. Ovchinnikov, D. Ivanov / Acta Astronautica 98 (2014) 133–137136

assume the motion to be a quasi-stationary andapply the proposed method for accuracy investigation.Assume the system noise matrix to be diagonal, i.e.Q ¼ diagðs2q ; s2q ; s2q ; s2ω; s2ω; s2ωÞ. The disturbances causedby residual magnetic moment, control errors let beconstants. First, consider how accuracy depends onparameters sq; sω when bo perpendicular to so anddisturbance is of 10�6 N m order. The attitude determi-nation accuracy dependence on parameters sq; sω ispresented in Fig. 1 where the lines correspond to Eulerangles accuracy levels given in degree, i.e. the worstaccuracy among three Euler angles. The Euler anglescorrespond to quaternion representing the transi-tion from the LVLH frame to the body-fixed frame.The highest accuracy which can be achieved is about0.081 (1�s).

However, the accuracy decreases when angle betweenbo and so differs from 901 (see Fig. 2). It is caused by thefact that the three-axis attitude has no observability whentwo measured vectors became collinear. The closer twovectors to be collinear the worse the filter estimation

0.09

0.09

0.1

0.1

0.12

0.120.12

0.150.15

0.150.15

0.2 0.2

0.20.2

0 25

0.250.25

0.3

0.30.50.7

σ q, s

−1

σω, s−2

10−3 10−2

10−3

10−2

Fig. 1. Attitude determination accuracy dependence on parameterssq; sω under bo ? so . Contours correspond to Euler angles accuracylevels (in deg).

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Angle between so and bo , deg

Atti

tude

det

erm

inat

ion

accu

racy

, deg

Fig. 2. Attitude determination accuracy dependence on angle between bo

and so .

accuracy. Fig. 2 shows this dependence obtained by theproposed approach of the filter study.

The Kalman filter described above was implemented onthe Chibis-M microsatellite. The example of satellite sta-bilization in the LVLH frame is presented in Fig. 3 wherethe satellite angles estimation is shown in above graph andthe angle between bo and so is presented below. Theattitude angles converge to zero since the reaction wheelsforce the satellite to coincide body-fixed frame with LVLHone. After the convergence one can estimate the attitudedetermination accuracy by considering the deviation ofthe Euler angles from zero values. As one can see fromFig. 3 the attitude estimation decreases to 11 (1�s) whenthe angle between bo and so is of about 101. This value is alittle bit more then presented in Fig. 2. It can be explained,that the values in Fig. 2 are asymptotic and algorithm doesnot converge yet in Fig. 3 at angle of 101 between bo and so.The attitude accuracy increases to 0.11 when the anglebecomes more than 301. This value is in a good correspon-dence with the given in Fig. 2.

If during the satellite exploitation the angle between bo

and so is less than 101 or more then 1701 the accuracy ofEuler angles estimation is unsatisfactory, and the Chibis-Mswitches its attitude control mode to a one-axis orienta-tion to the Sun (for battery charging) which uses sunsensor measurements only.

5. Conclusion

The proposed method of Kalman filter performance studyis an effective instrument for accuracy analysis and filtertuning in quasi-stationary satellite motion case. The approachallows estimating the influence of unaccounted perturbationon motion determination accuracy. The main advantages ofthis approach are that it does not require the simulation ofKalman filter, consequently does not require a lot of compu-tational time, the estimation of filter accuracy is an asympto-tic value after convergence. Accuracy dependence on filterparameters and perturbations is derived analytically, and ismore reliable than one derived by known investigation

M. Ovchinnikov, D. Ivanov / Acta Astronautica 98 (2014) 133–137 137

approaches. It should be emphasized that the method can beapplied only to the quasi-stationary motion when theacting forces and the measurement model are close to be aconstant during the time interval between sequentialmeasurements.

The proposed method is verified and it shows corre-spondence with in-flight data of the microsatelliteChibis-M. The dependence of the attitude estimation accu-racy on angle between geomagnetic field vector and Sundirection is obtained by the method and compared with theactual estimated satellite accuracy. Its highest accuracy is of0.11 (1�s) and decreases when the angle between bo and sodiffers from 901.

Acknowledgments

This study was conducted in the framework of theContract 1226/11-1 with SputniX Ltd., supported by theRussian Foundation for Basic Research (Grants 12-01-33045,13-01-00665 and 14-01-31313) and the Ministry of Educa-tion and Science of the Russian Federation.

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[15] M.E. Pittelkau, Kalman Filtering for spacecraft system alignmentcalibration, J. Guid. Control Dyn. 24 (2001) 6.

[16] J.D. Searcy, H.J. Pernicka, Magnetometer-only attitude determinationusing novel two-step Kalman filter approach, J. Guid. Control Dyn.35 (6) (2012) 1693–1701.

[17] M.L. Psiaki, F. Martel, P.K. Pal, Three-axis attitude determination viaKalman filtering of magnetometer data, J. Guid. Control Dyn. 13 (3)(1990) 506–514.

[18] J. Guidance, et al., Recursive attitude determination from vectorobservations: Euler angle estimation, J. Guid. Control Dyn. 10 (2)(1987) 152–157.

[19] I.Y. Bar-Itzhack, Y. Oshman, Attitude determination from vectorobservations: quaternion estimation, IEEE Trans. Aerosp. Electron.Syst. 21 (1) (1985) 128–135.

[20] L.M. Zeleny, et al., Transient luminous event phenomena andenergetic particles impacting the upper atmosphere: Russian spaceexperiment programs, J. Geophys. Res. 115 (6) (2010) A00E33.

[21] D.S. Ivanov, et al., Testing of attitude control algorithms for micro-satellite “Chibis-M” at laboratory facility, J. Comput. Syst. Sci. Int. 51(1) (2012) 106–125.

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Michael Ovchinnikov is Head of the AttitudeControl System & Orientation Division at theKeldysh Institute of Applied Mathematics,Russian Academy of Sciences, Professor ofthe Theoretical Mechanics, Chair at the Mos-cow Institute for Physics & Technology, Doc-tor of Science (in Physics and Mathematics),and IAA Member (Section 2).

Area of his professional interests lies amongnatural mechanics, spaceflight dynamics, devel-opment of attitude control for small satellite,theory and algorithms for formation flying

development and maintenance.

Danil Ivanov is Ph.D. (in Physics and Mathe-matics), Junior Researcher at the KeldyshInstitute of Applied Mathematics of RussianAcademy of Science, Assistant Professor atthe Theoretical Mechanics, and Chair of theMoscow Institute of Physics and Technology.

Area of professional interest: spaceflightdynamics, attitude motion, attitude determi-nation, satellite formation flying control algo-rithms, computer and laboratory methods ofsimulation of attitude dynamics and forma-tion flying motion.