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Accepted Manuscript

DOA estimation for attitude determination on communication satellites

Yang Bin, He Feng, Jin Jin, Xiong Huagang, Xu Guanghan

PII: DOI:

Reference:

S1000-9361(14)00080-6 http://dx.doi.org/10.1016/j.cja.2014.04.010 CJA 283

Received Date: 13 May 2013

Revised Date: 4 November 2013

Accepted Date: 10 December 2013

Please cite this article as: Y. Bin, H. Feng, J. Jin, X. Huagang, X. Guanghan, DOA estimation for attitude determination on communication satellites, (2014), doi: http://dx.doi.org/10.1016/j.cja.2014.04.010

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DOA estimation for attitude determination on communication satellites

Yang Bina, He Fenga, Jin Jinb, Xiong Huaganga, Xu Guanghana'*

a School of Electronic and Information Engineering, Beihang University, Beijing 100191, China b School of Aerospace, Tsinghua University, Beijing 100084, China

Received 13 May 2013; revised 4 November 2013; accepted 10 December 2013

Abstract

In order to determine an appropriate attitude of three-axis stabilized communication satellites, this papei describes a novel attitude determination method using direction of arrival (DOA) estimation of a ground signal source. It differs from optical measurement, magnetic field measurement, inertial measurement, and global positioning system (GPS) attitude determination. The proposed method is characterized by taking the ground signal source as the attitude reference and acquiring attitude information from DOA estimation. Firstly, an attitude measurement equation with DOA estimation is derived in detail. Then, the error of the measurement equation is analyzed. Finally, an attitude determination algorithm is presented using a dynamic model, the attitude measurement equation, and measurement errors. A developing low Earth orbit (LEO) satellite which tests mobile communication technology with smart antennas can be stabilized in three axes by corporately using a magnetometer, reaction wheels, and three-axis magnetorquer rods. Based on the communication satellite, simulation results demonstrate the effectiveness of the method. The method could be a backup of attitude determination to prevent a system failure on the satellite. Its precision depends on the number of snapshots and the input signal-to-noise ratio (SNR) with DOA estimation.

-paper

Keywords: Attitude determination; Direction of arrival (DOA) estimation; Smart antennas; Low Earth orbit (LEO); Communication satellite

1. Introduction1

sigr rep< esti

Attitude of communication satellites, which should be as accurate as possible to obtain a better data link, is determined by measurements from attitude sensors [1-3]. Attitude sensors output the projection of the reference vector in the sensitive direction of attitude [3-5]. Commonly used vectors are defined by stars, the Sun, the Earth, geomagnetic field, inertial space, and GPS satellites [3-6]. Owing to distinguishing features of the above sensors, most of communication satellites achieve attitude determination by corporately using different attitude references to prevent a system failure [1-2]. For example, a three-axis gyroscope, a three-axis magnetometer, and a coarse horizon sensor are used for attitude references of Iridium satellites [1]. The attitude of Globalstar satellites is determined by four Sun sensors, a horizon sensor, a three-axis magnetometer, and GPS [2].

The work in this paper is based on a developing low Earth orbit (LEO) satellite, which tests mobile communication technology with fixed smart antennas. The transient space information between a ground signal source and the smart antennas can be obtained by array signal processing. The direction of arrival (DOA) estimation uses the data received by the smart array to estimate the direction of the signal source [7-8]. At present, DOA estimation algorithms have been applied in direction estimation of signal sources [9-12]. A new method of attitude estimation using a dipole triad antenna on an aircraft was ported in Ref. [13]. The attitude angle of the aircraft can be determined by corporately using DOA imation of a ground source and an electric ellipse orientation angle in Ref. [13]. However, the method is base on a dipole triad antenna on an aircraft, not on smart antennas. The DOA of a ground signal has never been used in the attitude determination on communication satellites. The DOA contains attitude information of smart antennas relative to a ground signal source. Therefore, the ground signal source of a known location can be used for extra attitude reference of a satellite.

This paper is interested in using the DOA estimation of a ground signal source (a ground station or a ground mobile user) as the observation of the extended Kalman filter (EKF) for attitude determination of a communication satellite. A DOA attitude measurement equation is derived on the basis of the ground signal source location, the satellite orbit position, and the space geometric information between the signal source and the smart antennas. The error of the attitude measurement equation with DOA is

»Corresponding author. Tel.: +86 10 62802002. E-mail address: guanghanxu@hotmail.com

analyzed in detail. Then, an attitude determination algorithm is presented using the dynamics of the satellite, the attitude measurement equation with DOA estimation, and the measurement errors. Finally, the proposed method of attitude determination is analyzed in terms of stability, convergence time, and estimation accuracy by exploring the influences of the number of snapshots and input signal-to-noise ratio (SNR).

This paper is organized as follows. The basic model of satellite attitude is introduced in Section 2. The deduction steps of attitude measurement with DOA estimation are presented in Section 3. The error of the attitude measurement equation with DOA is analyzed in Section 4. The attitude determination algorithm with DOA is derived in Section 5. Simulation results of the method are shown and analyzed in Section 6. Conclusions and future work are put forward in Section 7.

2. The basic model

.. -. In this pai

To address the attitude determination problem of a satellite with smart antennas, Fig. . shows the relation between the satellite and the attitude reference of the ground signal source Sd . In this paper, we only consider the case of a single ground source for attitude determination, and assume that a line-of-sight (LOS) signal component from the ground to the satellite is available at the smart antennas. Three reference frames are illustrated in Fig .1, all being right-hand orthogonal triads. The orbital frame is denoted as Fo, where O is the mass center of the satellite, £ axis points to the center of the Earth, and V axis points along the flight direction. The satellite body fixed frame is denoted as Fb, and the receiving smart antennas frame Rxyz is denoted as Fr, where x axis is parallel to x body axis and y to y body axis, and R is the center of the smart antennas which is located in the plane.

Orbital plane

Satellite

;. 1. DOA and attitude of reference frames for the communication satellite In Fig. 1, the DOA vector is defined by (#, <p) in Fr, where 0 is the polar angle which is

measured from the signal source vector vRSi to z axis, and 0 is the azimuth angle that is to 0 in the same spherical coordinate system. Then, the unit vector of the reference or vRsd denoted as Sdl = [S^, Sdy, S^ ]T in Fr is calculated as

Sd = [sin d cos 0, sin d sin 0, cos# ]T (1)

The satellite attitude is defined by the orientation of the satellite body fixed frame Fb with respect to the orbital frame Fo . The attitude matrix Cbo is defined as the transfer matrix from Fo to Fb and Cob = Cbo is the attitude matrix from Fb to Fo . Since the smart antennas frame Fr parallels to Fb , the attitude matrix Cro from Fo to Fr is given by

Cro = Cbo (2)

The attitude from Fo to Fb is described by quaternion parameterization

qbo = ^bo^ 9bo) (3)

?bo = [?bo,> qbo2' qbo3]T (4)

The quaternion satisfies the following normalization constraint

qbo02+ibo,2+qbo22+qbo32 =1 (5)

Cbo can be obtained by

Cbo = q 2 - qboqbo)£,3x3+2?bo?bo - 2?bo0 Iqx] (6)

where E3x3 is the identity matrix and [qboX] is the cross-product matrix of the vector qbo, which is defined by

0 -qbo3 qbo2

faboX] = ibo3 0 -qbo

_~qbo2 qbo1 0

3. Attitude measurement equation with DOA

3.1. The basic model of DOA estimation

The DOA estimation is based on the relative arrival times of a source signal at the array elements and the second-order statistics [7,9]. Assume that there are M elements in the antenna array located at the x-yplane. A steering vector «(0,0) characterizes the relative phase with the DOA (#,0). The

received input data vector X(t) = [x1(t), x2(t), ... ,xM(t)]T <can be expressed as

X (t) = «(0, <p)s(t) + N (t) (8)

where s(t) is the received signal and N(t) = [^(t),n2(t),...,nM(t)]T is the noise vector. The spatial covariance matrix Rxx is defined in

Rxx = E[ X (t) X (t)H] (9)

where X(t)H is the complex conjugate transpose of X(t)[9]. In practice, this matrix is estimated by L snapshots of the actual antenna array output, as shown in

Rxx - J i X(IT)X(IT)H

where T is the sampling interval [9].

An eigenvalue decomposition of Rxx can be used to form a noise subspace matrix VM containing the noise eigenvectors [7'9]. Multiple signal classification (MUSIC) is a high-resolution estimation method of DOA using the steering vector [7' 14'15]. The DOA (0,0) of the signal can be estimated by locating the peaks of an MUSIC spatial spectrum [14]

W^) - ^ H(WM VM a (0,0

The precision of the DOA (i?^) is limited by the input SNR and the number of snapshots on condition that the difference between the elements and the mutual coupling between the antennas array ompensated [9' 16'17].

sing the DOA estimated from Eq. (11), the estimated unit vector is written as

cond are c Us

sSdr = [sin Q cos <j>, sin 9 sin </>, cos ff]

where W = [Wx,Wy,Wz] is the error vector of reference vector estimation in Fr, which depends on the DOA (0,0).

3.2. Attitude measurement equation with DOA

In the on-board system, the satellite location vector vOG denoted as Go = [G^G^G^ can be

calculated from the orbit position in Fo . Since the location information of the ground signal source is uploaded by the uplink with the smart antennas, the ground signal source vector vSio denoted as Sdo = [Sdo„, S^, S°iC ]T in Fo is known in the on-board system. The vector vOSd, which is the direction from the satellite to the ground signal source, denoted as Do = [D°, D°°, D°- ]T in Fo is then given by

D = G - S

Moreover, the vector vOR, which is the direction from the satellite's mass point to the center of the

Therefore, vOR denoted as

smart antennas, denoted as Rb = [Rb, Rb, R? ]T is known in Fb

Ro = [R°,R|,R0°]T in Fo is calculated as

Ro = C To Rb

The unit vector vRSd denoted as So = [S°, S|, S^ ] in Fo is calculated as

S - Do ~ R

So = fDT^Roii

Using Eq. (2), in the smart antennas frame Fr, Sdr can be given by

Sdr = Cro So = Cbo So

Compared with the altitude of the satellite and the location of the could be treated as infinitesimal (i.e., ||Rb||«||Do||). Then, u can be rewritten as

Therefore, using Eq. (12), Eq. (16), and the appri with DOA estimation is determined as

ignal source, the satellite size 13), Eq. (14), and Eq. (15), So

[sin 9 cos 0, sin 6 sin <p, cos #]T

tion above, the attitude measurement equation

. G - S„,

Go - sd

4. Error analysis of attitude measurement

4.1. Smart antennas model

In our study, the LEO satellite we are developing will run in a Sun-synchronous circular orbit with an altitude of 800 km and a local time of descending node of nominal 8:00 AM [18]. The LEO satellite tests mobile communication technology with a uniform circular array (UCA) which is composed of 12 quadrifilar helix antennas (QHAs). The uplink signal wavelength is A = 150 mm. Figure 2

represents the smart antennas model of the satellite.

Fig.2 The smart antennas model In Fig.2, the UCA radius is r = 150 mm. d and dm are the distance from the signal source Sd to center of array R and the element . The element phase angle is <pm = 2k( m—1) /12 for m = 1,2,...,12 . The unit vector of vRm denoted as Mr = [Mrx,Mx Mzr]Tin Fr is given by

Mr = [cos (pm, sin (pm ,0]T (19)

Using Eq. (1) and Eq. (19), the distance difference between different locations from far-field electromagnetic wave Ad is given by

Ad = d - dm = rSdr M r

= r sin 0cos(0-pm ) (20)

The difference between the elements and the mutual coupling of the antenna array can be compensated by array calibration methods [9,16,17]. From this, the steering vector is given by

a(0,$) =

exp(j—r sin Q cos(0---0)

exp( j—r sinöcos(0---1)

exp(j—r sin 6>cos(0---11)

4.2. Error analysis

Regarding to the satellite orbit and smart antennas model, we use an orbital simulation package to generate 8 ground stations. The minimum elevation angle for ground stations is 10 degrees (i.e., 0o <0< 62o , 0o <$< 360o ). In addition, at least one ground signal is available at the smart antennas for 3000s orbit plane time in this simulation. In the simulation case, the ground station informs the satellite to prepare to hand off because it knows both the station's location and the satellite's location. The online processing step size of the satellite is 250 ms. Using Eq. (11) and Eq. (21), the results of the DOA (6,0) are given. Fig.3 and Fig.4 show the theoretical values of the DOA (#,0) and the

DOA () in terms of the number of snapshots L and the input SNR.

Q of the DOA estimation

350 - 65

300 - 60

—-Theoretical valut

Estimated val ue: L= 100. SN R=0 dB — Estimated value: L=SO, SNR—¿¿4dB

S00 820 840 860 880 900

Fig.4 0 of the DOA estimation

It is seen that the precision of the DOA (0,0) is limited by the number of snapshots and the input SNR. In addition, the DOA (^0) can be more accurate as the increases of the number of snapshots and the input SNR. After that, the statistical properties analysis of W is plotted in Fig.5 using Eq. (12).

4QÛ r

200 ■

Numbers of data Normal fit

^ä.....Iltit^

Numbers of dam Normal fit

°0.02

t-jj— [V Uli na I ill

1.02 -0,01 0 0,01

200 ■

Numbers of data Normal fit

(a) ¿«100 SNR-OdB

(b)f=50SNR=-4dB

Fig.5 Statistical properties analysis of W As can be seen in Fig.5, the noise vector W is independent and an identically distributed Gaussian random process with zero-mean. In addition, the variance of W becomes bigger with the decreases of the number of snapshots and the input SNR. Then, the covariance RW of W can be chosen as follows:

' 2 aWx 0 0

RW - 0 2 0Wr 0 (22)

0 0 _2 °wz _

where (JW

and (?W are the variances of x axis, y axis, and z axis, respectively, which can be calculated with the above simulations.

5. Attitude determination algorithm

5.1. The satellite dynamic model

The satellite can be stabilized in three axes by corporately using reaction wheels and three-axis magnetorquer rods [18]. Therefore, the dynamic equation is given by

la^ (t) = -[fflbi (tM/3K3^. (t) + h(t)] - h(t) + T (t) + T (t) (23)

fflbo =®bi - Cbo®o (24)

where ©b is the inertial referenced angular velocity vector; h is the total angular momentum of wheels; Tmt is the torque produced by magnetorquer rods, Td is the disturbance torques; fflbo is the attitude angular rate in Fo ; fi>o is the satellite orbit angular rate; I3x3 is the inertia matrix of the satellite defined by

" 6.6199 0.01636 0.2363" 73x3 = 0.01636 8.95 0.2247 kgXm2 (25)

0.2363 0.2247 9.4968

5.2. EKF algorithm

The EKF algorithm is employed to determine the attitude using the DOA information of the ground signal during the three-axis stabilized phase of the satellite, and the estimation state vector contains six elements

x(k) = [^T(k), ^oW Then, using Eq. (6), Eq. (23), and Eq. (24), the state dynamic equation is given by

,«bo(f ).

®bi(t ) .9bo(t ).

: {-«bi (t) X [/3^ (t) + h(t)] - [h(t) - h(t - At)] / At + Td (t) -

1(9bon (t)EiA3 + [qbo (t)]*} [®bi (t) - Cbo(t)©o(t)]

Tmt(t)}

itude aylor

The EKF is widely used to estimate state variables with the state dynamic equation and the atti measurement equation [19,20]. The state dynamic equation Eq. (27) is expanded as the first-order Taylor series about the estimate X(k / k) using the EKF algorithm [18-20]. Therefore, the transitio: Eq. (27) denoted as #(k) is given by

&(k +1/k) « E6;

where At = 250 ms is the step size of the discretization on the communicatio calculated as

" I-l {[(/^ (k / k) + h(k) )x] - [fflbi (k / k )x]7}

0.5E,S

n satellite

H®bi(k/

ition matrix of

(28) and A is

The noise of the state dynamic equation is uncorrelated observation noise with covariance Q = 10~9 E6y6 , due to the disturbance torque and linearization error of the state dynamic equation Eq.

(27)[18].

In addition, using Eq. (16.b) and Eq. (26), the attitude rewritten as

[sin 9 cos 0, sin 6 sin 0, cos #]T

surement equation Eq. (18) can be

(t ) ibo(t).

In a similar process of the state dynamic equation, the attitude measurement equation Eq. (30) is expanded about X(k +1 / k) and is truncated at the first order. Thus, using Eq. (6) and Eq. (7), the measurement matrix denoted as H(k +1) is given by

-i "lT

H (k +1) =

x=;c(k+1/k )

dà)T(k +1/k )

dqboT(k +1/ k )

Sdr(k + 1)]T

S^(k + 1)]T

dâT(k +1/k )

S^k +1) =

dqboT(k +1/ k )

Sdr(k +1) =

[2Sdr (k +1/ k )x]A?bo

Cbo(k +1/k )

Go(k +1) - Sdo(k +1) ||Go(k +1) - Sdo(k +1)||

6. Simulation results

In this section, two simulations are presented: 1) the attitude determination with DOA is simulated first to show the efficiency of the proposed method; 2) the Monte Carlo simulation is then presented to show the influences of the input SNR and the number of snapshots. The flight missions in those simulations are the same as the case in Section 4.2 and at least one ground signal is available at the satellite for 5000 s orbit plane time. The initial Euler angles are yaw = 50o, roll = 50o , and pitch = 50o , and the initial inertial referenced angular velocity ®bi is [0.5,0.5,0.5]T ° /s . In addition, R w and Q are chosen the same as in Section 4.2 and Section

The first simulation considers that the number of snapshots is L = 100 and the input SNR is 0 dB. The total simulation time is about 5000 s (one orbit cycle). Fig. 6 and Fig. 7 depict the estimated results of the Euler angles and the inertial referenced angular velocity fl^ using the DOA estimation in this simulation case. Simulation results show that the errors of the attitude angles converge to a value of 0.5° within 1500 s and the errors of the angular velocity take almost 1000 s to reach 0.1° /s. It is acceptable in the Earth pointing estimation accuracy of the communication satellite. In addition, the period for observation of each station or mobile user is about 10 min (i.e., 600 s) for the LEO satellite. Therefore, the method is available on condition at least 3 ground stations or mobile users continuously communicate with the satellite for an attit determination which needs 1500 s for the convergence. The results demonstrate that the pro method using the DOA (i?,0) can be used to estimate the satellite attitude.

100 50 0 -50

50 0 -50

100 so 0 -so

-Simulation - Estimation

4Ü2H -mo

-Simulation - Estimation

— Simulation

- Estimation

Ml_im_4840

1500 2000 2500 3000

Fig.6 Estimation errors of attitude angles

-Simulation - Estimation

-Simulation -Estimai ion

4800 4820

« 0 a .

1000 -1-

— Simulation ■ ■ Estimation

2000 -1-

2500 -1-

3000 -1-

4000 I

4500 "1-

ám_um_¿Mi.

Fig.7 Estimation errors of angular velocity

The second simulation investigates the influences of the number of snapshots and the input SNR on performance. The precision of the DOA (0,0) and the attitude determination can be described with the average root mean square error (RMSE) of estimates from 200 Monte Carlo trials

RMSE =

i=1 j=1

true \2

x - xj )

where J is the total step number in each trial; xj™e is the actual value; and xj is the estimation of

in the i th Monte Carlo trial. The RMSEs of the DOA estimation and the attitude

determination versus the input SNR with L = 100 are shown in Table 1. The RMSEs of the DOA estimation and the attitude determination versus the number of snapshots with an SNR of 0 dB are shown in Table 2.

Table 1 RMSE of the attitude determination versus the input SNR

Number of snapshots SNR (dB) RMSE of G ( ° ) RMSE of 0 ( ° ) RMSE of yaw ( ) RMSE of roll ( ° ) RMSE of pitch ( ° )

100 - 14 11.118 68.128 7.397 3.662 1.732

100 - 12 4.106 17.337 0.652 0.177 0.457

100 - 10 1.518 1.307 0.130 0.079 0.215

100 - 8 0.921 0.791 0.109 0.078 0.157

100 - 6 0.595 0.534 0.094 0.071 0.111

100 - 2 0.354 0.334 0.084 0.060 0.094 0.094

100 0 0.315 0.308 0.083 0.061 0.060

100 14 0.287 0.289 0.084 0.093

Table 2 RMSE of the attitude determination versus the

: number of snaps

napshots

Number of snapshots

SNR (dB)

RMSE of

G ( ° )

RMSE of 0 ( ° )

RMSE of

SE of roll ( ° )

20 50 100 1000

8.354 0.781 0.408 0.342 0.315 0.287

51.089 0.684 0.385 0.326 0.308 0.289

99 .086 0.084 0.083 0.084

10.071 0.070 0.064 0.062 0.061 0.062

6.852 0.146 0.100 0.091 0.094 0.093

RMSE of pitch ( ° )

Obviously as shown in Table 1 and Table 2, the DOA estimation is more accurate as the increases of the number of snapshots and the input SNR. It is consistent with the results in Section 4.2. In the same way, the performance of the attitude determination is much better. Especially, the RMSE of the attitude determination is better than 0.1° when the input SNR is greater than _ 6 dB or the number of snapshots is more than 50.

In addition, we can see that the attitude determination accuracy is higher than that of the DOA estimation especially in the case of lower input SNRs and less number of snapshots in Table 1 and Table 2 (e.g., the case of one snapshot or _ 10 dB). The error of the DOA estimation W is injected into the attitude measurement equation Eq. (18) and its statistical properties are analyzed in Section 4.2. As a result of the attitude determination using the measurements as well as a prior knowledge about the time evolution of attitude and the statistical properties of the measurement error W , the precision of the attitude determination is improved. The simulation results also demonstrate the validity of the measurement error W analysis in Section 4.2.

To sum up, the proposed method of attitude determination using DOAs estimation of ground terminals or stations could be an available method for communication satellites with smart antennas. Its tccuracy depends on the input SNR and the number of snapshots. Thus, the above factors should be comprehensively considered for different situations in practical applications.

7. Conclusions and future work

This paper introduces a novel approach to determine satellite attitude with DOA estimation. Based on the geometrical relationship between a signal source (a ground station or a ground mobile user) and an antenna array, an attitude measurement equation is derived and the error of the attitude measurement equation is analyzed in detail. By using this equation, the algorithm of attitude determination is obtained for communication satellites with smart antennas. Simulation results validate that this algorithm can estimate the attitude of a satellite effectively. The method could be a backup method of attitude determination to prevent a system failure. In addition, the influences of the SNR and the number of snapshots are explored on the performance of attitude determination.

For future studies, on conditions of several reference sources on the Earth for attitude determination and DOA estimation in presence of mutual coupling, the effects of attitude accuracy will be discussed in great detail in another paper.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (No.61073012) and the Aviation Science Foundation of China (No.20111951015).

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Yang Bin is a Ph.D. student in the School of Electronic and Information Engineering at Beihang University. He received his B.S. and M.S. degrees in signal and information processing from North University of China in 2003 and 2007, respectively. His main research interests are statistical signal processing and multi-antenna communications. E-mail: yangbin@ee.buaa.edu.cn

He Feng is a lecturer in the School of Electronics and Information Engineering at Beihang University in China. He received his PhD degree in Communication and Information System in March 2009. His research interests cover airborne communication, real-time system, and distributed computing. E-mail: robinleo@buaa.edu.cn

Jin Jin received his B.S. and Ph.D. degrees in Navigation, Guidance, and Control from Beijing Institute of Technology in 2006 and 2011, respectively, and then became a postdoctor at Tsinghua University till now. His main research interests are spacecraft attitude dynamics and control, aerospace engineering. E-mail: guleixibian@gmail.com

Xiong Huagang is a full professor in the School of Electronics and Information Engineering at Beihang University in China. His research interests include digital communication and avionics system. E-mail: hgxiong@buaa.edu.cn

Xu Guanghan is a professor and Ph.D. advisor in the School of Electronics and Information Engineering at Beihang University in China. He received his Ph.D. degree from Stanford University in 1991. His current research interests are array signal processing and multi-antenna communication. E-mail: guanghanxu@hotmail.com