Autonomous GEO Satellite Navigation with

Autonomous GEO Satellite Navigation with Multiple GNSS Measurements Li Qiao, Nanjing University of Aeronautics and Astronautics, China Samsung Lim，Chris Rizos University of New South Wales, Australia Jianye Liu, Nanjing University of Aeronautics and Astronautics, China

finished and is keeping great interest in New Inertial Devices, New Integrated Navigation Theory and Application, Mini Low Cost Navigation System for Vehicle, Wireless sensor positioning system and GIS application. Now, he is the syndic of Chinese Society of Inertial Technology and vice president of Nanjing Society of Inertial Technology.

BIOGRAPHY Li Qiao is a PhD student at Navigation Research Center (NRC http://www.nuaanrc.com/), College of Automation Engineering, Nanjing University of Aeronautics and Astronautics (NUAA), China since 2006. Now she is a visiting student at School of Surveying and Spatial Information System in the University of New South Wales. Her focus is on the satellite autonomous navigation method. She received a BSc in electric engineering and automation from NUAA in 2004.

ABSTRACT The poor satellite visibility and the weak signal power of the Global Positioning System (GPS) signal reception in side lobes are the main constraint on the Geostationary Earth Orbit (GEO) determination, however, the situation will improve when multi-constellation Global Navigation Satellite Systems (GNSS) become available in the near future. In this study, a navigation algorithm is developed in order to determine the GEO state vector using multiconstellation GNSS measurements. The proposed navigation algorithm is based on a Kalman filter where the estimated state includes position and velocity corrections to the nominal reference trajectory and clock biases. The tests with simulated multi-GNSS measurements indicate that this algorithm meets the accuracy requirement for autonomous GEO satellite navigation.

Samsung Lim is a Senior Lecturer in the School of Surveying and Spatial Information Systems, The University of New South Wales (UNSW), Sydney, Australia. For the past fifteen years his research has been focused on the area of GNSS and GIS. Samsung's research interests are in theoretical problems related to network-based RTK and applying geo-spatial information technologies to real-world problems. In 2005, Samsung developed an address-based search tool in conjunction with contemporary web-map services such as Google Earth. Samsung received his B.A. and M.A. in Mathematics from Seoul National University and his Ph.D. in Aerospace Engineering and Engineering Mechanics from the University of Texas at Austin. Chris Rizos is a graduate of the School of Surveying and Spatial Information Systems, UNSW; obtaining a Bachelor of Surveying in 1975, and a Doctor of Philosophy in 1980. Chris is currently Professor and Head of School. Chris has been researching the technology and high precision applications of GPS since 1985, and has published over 200 journal and conference papers. He is a Fellow of the Australian Institute of Navigation and a Fellow of the International Association of Geodesy (IAG). He is currently the Vice President of the IAG and a member of the Governing Board of the International GNSS Service.

1 INTRODUCTION Advances in autonomous satellite navigation technologies are essential in order to minimize the cost of operating satellites missions. The use of Global Navigation Satellite System (GNSS) signals is one of the most promising autonomous satellite navigation methods, being able to generate high accuracy ephemerides of Geostationary Earth Orbit (GEO) satellites trajectories when equipped with a GNSS receiver. Research on space-based applications of GNSS has been growing rapidly ever since the feasibility of using a spaceborne GPS receiver to autonomously determine a

Jianye Liu is the dean and the professor in CAE of NUAA. As the head of NRC, Dr Liu developed the research area based on the INS and GNSS system. He has

22nd International Meeting of the Satellite Division of The Institute of Navigation, Savannah, GA, September 22-25, 2009

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GLONASS satellites, 30 GALILEO satellites and 27 COMPASS satellites. Moreover, a multi-GNSS spaceborne receiver is feasible because of the planned interoperability of the different GNSSs, at least for some of the signals/frequencies. Hence the next generation GNSS receiver will likely be capable of processing not only GPS signals but also GLONASS, GALILEO and COMPASS signals [6]. It is assumed in this paper that a multi-GNSS receiver receives signals from all the systems and makes use of the pseudo-range measurements.

spacecraft orbit in real-time was demonstrated [1, 2]. Over the last decade, satellite navigation using spaceborne GPS receivers has become a common technique for determining the ephemerides of Low Earth Orbit (LEO) satellites. Numerous spaceborne GPS receivers have been developed around the world, especially in the U.S. and in the European Union [3-6]. After its successful use on LEO satellite applications there is a widespread interest in expanding the use of GPS/GNSS to on-orbit navigation of satellites in highly eccentric or geostationary orbits. By several space experiments, the use of GPS signals has been explored for different types of orbits, from LEO to Highly Elliptical Orbit (HEO) and Geostationary Satellite Orbit (GEO) [3, 4].

This paper presents a navigation algorithm to determine the GEO state vector using multi-constellation GNSS pseudo-range measurements. A conventional approach for solving the GEO navigation problem is to take into account a satellite motion model and to perform the differential orbit correction and clock model parameter estimation. Experimental results are presented in this paper for the performance analysis. An improved navigation of the GEO satellite can be achieved using a dynamical model and sequential updates by a Kalman filter [3, 7, 8].

However, there are two main, and closely connected, problems with GEO navigation using GNSS[5]. Firstly, a GNSS signal received at the GEO satellite has very weak signal power. The second problem is the poor geometry of the GEO-GNSS satellites, as illustrated in Figure 1. This implies that the difference in the reception of GNSS signals by GEO satellite-borne receivers in comparison to LEO satellites lies in the geometric distance between a GNSS satellite and the navigating satellite vehicle (NSV) at high altitudes, where the GNSS satellites transmit signals towards the Earth. Satellites at GEO orbits are located above the altitudes of the GNSS satellites, and their reception of the signals is only possible from GNSS satellites on the other side of the Earth.

2 ORBIT DETERMINATION STRATEGY 2.1 Measurement Models GNSS pseudo-range measurements are related to the satellite centre as the receiver location is assumed to be with respect to the satellite centre of mass in the satellite body-fixed frame.

GEO

The measurement equation at each epoch is:

Navigation Sat. Orbit

(1)

ρ j = R j + δ tu + vρ j

NAV Sat

where GNSS satellite index pseudo-range from the GEO satellite to the navigation satellite S j Rj geometrical distance from the GEO satellite to the navigation satellite S j δ tu range error equivalent to the receiver clock offset with respect to GNSS time frame vρ j random measurement noise When n satellites are available, the filter measure model can be expressed as: Z = h( X ) + V (2) where Z = [ ρ1 ρ2 L ρn ]T is the measurement vector, composed of measured pseudo-ranges ρj ,

21.3°

j

ρj

Main lobe GEO satellite

Figure 1: Geometry of earth, GNSS satellites and GEO satellite. This GEO geometric situation will improve when multiconstellation GNSSs are fully operational in the future. This is because it is expected that there will be more than 100 GNSS satellites in orbit for positioning, in particular the satellites of the four GNSSs developed by U.S. (GPS), Russia (GLONASS), Europe Union (GALILEO) and China (COMPASS). This paper assumes a multi-GNSS constellation consisting of 31 GPS satellites, 24


h( X ) = [ r1 + δ tu

r2 + δ tu L rn + δ tu ]

is a function defining the relationship between the measurements and the state T vector, and V = ⎣⎡vρ1 vρ 2 L vρ n ⎦⎤ is the receiver noise. All T

the measurement perturbations not included in the model are considered as Gaussian errors with E[V ] = 0 and

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and ri = ( x − xNSi )2 + ( y − yNSi )2 + ( z − z NSi )2 is the true distance between satellite and navigation satellite. E[VV T ] = R

a Earth

2.2 Dynamic Models For the deterministic dynamic model for a GEO satellite, the satellite state dynamics are established for the state equation of the system with respect to the J2000.0 Earth Centered Inertial (ECI) coordinate system. The X and Z axes point toward the mean vernal equinox and mean rotation axes of the Earth at January 1, 2000 at 12:00:00.00 UTC. J2000.0 = 2000 January 1.5 = JD 2451545.0 TDT (Terrestrial Dynamical Time). Assuming that the position vector is r = [ x y z ] and the velocity vector is v = [v , v , v ] , the orbit state vector can be written as:

2 3 4 ⎡ μx⎡ ⎤ z2 ⎞ z2 ⎞ z z2 z 4 ⎞⎤ ⎛R ⎞ 3⎛ ⎛R ⎞ 5⎛ ⎛R ⎞ 5⎛ ⎢ − e3 ⎢1 + J 2 ⎜ e ⎟ ⎜⎜ 1 − 5 2 ⎟⎟ + J 3 ⎜ e ⎟ ⎜⎜ 3 − 7 2 ⎟⎟ − J 4 ⎜ e ⎟ ⎜⎜ 3 − 42 2 + 63 4 ⎟⎟ ⎥ ⎥ r ⎠ r ⎠r r r ⎠ ⎦⎥ ⎢ r ⎣⎢ ⎥ ⎝ r ⎠ 2⎝ ⎝ r ⎠ 2⎝ ⎝ r ⎠ 8⎝ ⎢ ⎥ 2 3 4 2 2 2 4 ⎤ ⎢ μe y ⎡ ⎥ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ R 3 z R 5 z z R 5 z z ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = ⎢ − 3 ⎢1 + J 2 ⎜ e ⎟ ⎜⎜ 1 − 5 2 ⎟⎟ + J 3 ⎜ e ⎟ ⎜⎜ 3 − 7 2 ⎟⎟ − J 4 ⎜ e ⎟ ⎜⎜ 3 − 42 2 + 63 4 ⎟⎟ ⎥ ⎥ r ⎠ r ⎠r r r ⎠ ⎥⎦ ⎝ r ⎠ 2⎝ ⎝ r ⎠ 2⎝ ⎝ r ⎠ 8⎝ ⎢ r ⎢⎣ ⎥ ⎢ ⎥ 2 3 2 4 2 2 2 4 ⎢ μe z ⎡ z ⎞ z ⎞ z ⎤ μe ⎛ Re ⎞ 3 z z ⎞⎥ ⎛ Re ⎞ 3 ⎛ ⎛ Re ⎞ 5 ⎛ ⎛ Re ⎞ 5 ⎛ ⎢ − 3 ⎢1 + J 2 ⎜ ⎟ ⎜⎜ 3 − 5 2 ⎟⎟ + J 3 ⎜ ⎟ ⎜⎜ 6 − 7 2 ⎟⎟ ⎥ + 2 J 3 ⎜ ⎟ − J 4 ⎜ ⎟ ⎜⎜ 15 − 70 2 + 63 4 ⎟⎟ ⎥ r ⎠ r ⎠ r ⎦⎥ r r r ⎠ ⎦⎥ ⎝ r ⎠ 2⎝ ⎝ r ⎠ 2⎝ ⎝ r ⎠ 2 ⎝ r ⎠ 8⎝ ⎣⎢ r ⎣⎢

where distance between the geocentre of the earth and the GEO satellite Re the earth mean radius (based onWGS84 reference frame) μe the gravitational parameter of the earth J 2 J 3 J 4 coefficients are referred to as zonal harmonics, related to the terrestrial model. r = x2 + y 2 + z 2

T

T

x

y

z

T ⎡r ⎤ Xorbit = ⎢ ⎥ = ⎡⎣ x, y, z , vx , v y , vz ⎤⎦ v ⎣ ⎦

The luni-solar effect can be represented as[10]: ⎛ r −r r ⎞ a Sun = μ s ⎜ s − s ⎟ ⎜ r −r 3 r 3 ⎟ s ⎝ s ⎠

(3)

⎛ r −r r ⎞ − m ⎟ a Moon = μm ⎜ m ⎜ r −r 3 r 3 ⎟ m ⎝ m ⎠

In a general form, the dynamical equation of the satellite state can be presented as: & X (4) orbit = f orbit ( X, t ) + Worbit (t ) where f orbit is the function that describes the satellite state dynamics, and Worbit (t ) is the process noise of the state equation. Ignoring the process noise Worbit (t ) , the first derivative of the state vector X therefore is: ⎡r& ⎤ ⎡ v ⎤ & X orbit = f orbit ( X orbit , t ) = ⎢ ⎥ = ⎢ ⎥ ⎣ v& ⎦ ⎣a ⎦

where μs and μm are the gravitational parameters of the sun and moon, respectively, and rs = [ xs , ys , zs ]T and T rm = [ xm , ym , zm ] are position vectors of the sun and moon in the ECI coordinate frame respectively. The receiver clock is also modeled. The state of the clock part X clock is composed of clock bias and bias rates, i.e. T X clock = [δ tu δ tru ] . An exponentially correlated clock model driven by a zero mean white noise is integrated in the process model; referred to as first order Markov processes. It is assumed the four different GNSSs have unified time, hence the clock error is:

(5)

where a is the acceleration of the satellite. The main components of the satellite acceleration a in the dynamical equation depend on the altitude of the satellite orbit. For a GEO satellite whose orbit altitude is about 36,000km, the acceleration will be perturbed by the two-body acceleration effect, the non-spherical earth gravitational potential effect, and the luni-solar gravitational effect. Thus, the total acceleration of a GEO satellite can be written as:

T

⎡ ⎤ 1 T X& clock = f clock ( X orbit , t ) + Wclock = ⎢δ tu − δ tru ⎥ + [ wtu wtu ] T ru ⎣ ⎦ The parameter Tru is the clock model correlation

time. The resulting navigation state is a eight-state vector composed of six spacecraft position and velocity components and the receiver clock bias and bias rates. The overall process model can be expressed as: (8) X& = f ( X ) + W where the state vector X , the state model f ( X ) and the measurement noise, in the ECI frame, are:

a = a Earth + a Sun + a Moon + a w

(6) where a Sun and a Moon are the third-body effects of the sun and moon respectively, a Earth includes the two-body effect and the non-spherical gravitational potential effect from the earth, and aw represents the high-order terms that are considered to be noise of the system model. Other perturbations (such as atmospheric drag, solar radiation, earth tidal, etc.) are negligible compared to the remaining effects in aw .

⎡X ⎤ X = ⎢ orbit ⎥ ⎣ X clock ⎦

⎡ f orbit ( X orbit , t ) ⎤ ⎡Worbit ⎤ ⎥ , W = ⎢W ⎥ . ⎣ f clock ( X clock , t ) ⎦ ⎣ clock ⎦

, f (X ) = ⎢

All perturbations W not included in the process model are described as Gaussian errors with E[W ] = 0 and E[WW T ] = Q .

The main zonal harmonics are J 2 , J 3 and J 4 in the nonspherical perturbation. The non-spherical gravitational potential a Earth can be represented in the ECI coordinate system as[9]:


(7)

The error covariance matrices Q and R have to be diagonal, an important hypothesis using Kalman filter.

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That means the perturbations and noises in the process and measurement filter models must be uncorrelated.

3 SIMULATION AND ORBIT DETERMINATION STRATEGY 3.1 Simulation Description

2.3 The Extended Kalman Filter A theoretical approach to the GEO determination has been described in the previous section. Now the proposed algorithm is implemented and its performance is analyzed using a Monte-Carlo simulation. The test scenario given in Figure 2 is used to analyze the described algorithm which includes the following modules: GNSS Satellite Database, GEO User Satellite Orbit Dynamics, measurement computation, Kalman filter process.

The blending of GEO satellite state dynamics and pseudorange measurements has been implemented within a Kalman filter technique. The typical navigation solution consists of estimates of spacecraft position, velocity and model parameters of the receiver clock behavior. In this case, the estimation is implemented as an Extended Kalman Filter (EKF), due to the non-linear state dynamics listed above. The main advantages of the Kalman filter approach with respect to other algorithms are: 1) it is a statistical filter that increases the accuracy of the solution in the presence of random perturbations; and 2) the implemented dynamic model makes the filter able to predict position and velocity at future time, even in the absence of measurements [11].

GNSS Satellite Database

GNSS Positon Computation

H=

f k = f ( X% k )

Fk = F ( X% k )

F=

P, x

Kalman Filter Update

Figure 2: Orbit determination algorithm architecture. The Satellite Tool Kit (STK, v5.0) software (www.stk.com) is used to produce satellite ephemeris to establish the GNSS satellite database. The initial almanac (i.e. osculating orbit parameters) of GPS and GLONASS constellations was adopted from the web sources (see [13]). The almanac of GALILEO was obtained from [14], and that of COMPASS from [15].

∂X

X%

In the simulation, the antenna half-cone is set as follows: • Navigation satellite transmit antenna half-cone angle: 21.3°[16]（Only consider receiving the main lobe） • User satellite receiver antenna half-cone angle: 90°（ main lobe） The above tow angle is considered only for the main lobe angle. The test simulations for the algorithms have been conducted for a 24-hour dataset. The test epoch starts on 30 April 2009 at 12:00:00.00 UTCG, and stops on 1 May 2009 at 12:00:00 UTCG.

ˆ (k + 1 / k ) = X ˆ (k / k ) + f [ X(k )] ⋅ T X ˆ ˆ ˆ (k + 1) X(k + 1) = X(k + 1 / k ) + δ X ˆ (k + 1) = K (k + 1){Z(k + 1) − H[ X ˆ (k + 1 / k )]} δX

P (k + 1 / k ) = Φ(k + 1 / k )P(k / k )Φ(k + 1 / k )T + Q K (k + 1) = P (k + 1 / k )H (k + 1)T [H (k + 1)P (k + 1 / k )H (k + 1)T + R (k + 1)]−1

P (k + 1) = [I − K (k + 1)H (k + 1)]P(k + 1 / k )[I − K (k + 1)H (k + 1)]T + K (k + 1)R (k + 1)K (k + 1)T

where P is the covariance matrix of the state estimation uncertainty, K is the Kalman gain matrix, Φ is the state transition matrix of the discrete dynamic system Φ ≈ I + F ⋅ T , and T is the integration interval of the filter.


State Vector and Covariance Propagation

H, z

where and are the estimated and the predicted state vectors respectively. From the previous definitions, the correction process consists of the following steps [12]. The prediction and filtering equations of the EKF consist of the following steps: Xˆ

Transition Marix and State Noise Marix Computation

φ, Q

Measrement Processing and Observation Matrix Computation

∂h( X% ) ∂X ∂f ( Xˆ )

H k = H ( X% k )

Access Computation

rGNSS

The filter measurement model and the system dynamic model are Equations (2) and (4). In an EKF implementation, the partial derivative matrices of the predicted state vector rate f ( X ) and of the predicted measurement vector h( X ) at the k-th step must be considered: hk = h( X% k )

User Satellite Orbit Dynamics

3.2 GEO Tracking Performance Initial tests were conducted for several GEO scenarios to assess the feasibility of tracking GNSS satellites. The scenario assumes the GNSS receiver has a down-looking antenna to the earth. Here the impact of the weak signal tracking capabilities is not discussed. Satellites are considered as visible and available if their line-of-sight is

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inside both the transmission cone and the reception cone, and does not intersect the earth. This section will present the results from the tests conducted for four geostationary orbits, where the longitude of the four satellites are 0, 90, 180 and 270 degrees.

5

GPS

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270°GEO Figure 3: Comparison of single GNSS satellites in view versus multi-GNSS system for four GEO satellite configurations.

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Time(s)

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Figure 3 shows that there are zero satellites tracked using only a single GNSS. Because the number of visible satellites using a single system is less than four, classical least squares single-epoch solutions are not possible. The number of available satellites increases using multi-GNSS system, and there are no satellite gaps.

0°GEO 5 4

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270°GEO Figure 4: Comparison of the multi-GNSS visibility conditions for four GEO satellite configurations.

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90°GEO 20 18

Figure 4 shows a comparison of the number of satellites tracked in the four different GEO satellite configurations. The availability is different depending on the longitude. From these four plots, it can be noted that the receiver always tracked at least one satellite, and there are many instances when there were four or more satellites being tracked. The percentages of four or more satellites that were tracked continuously are shown in Table 1. It indicates that in 90°GEO case, the GEO satellite can receive four or more GNSS signals continuously; the longest time occupied being about 11% of the whole running time. This continuity ensures the convergence of the filter.

16

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0

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180°GEO 18 16 14

Longest(%) 36.806 11.042 11.806 19.653

12

Percentage

Minimum (%) 3.264 0.069 0.069 4.861

10

6

Table 1 Percentage of four or more satellites continuously tracking time GEO 0 90 180 270

12

10 8 6 4 2 0


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270°GEO Figure 5: Comparison of the percentage of N satellites tracked for four GEO satellite configurations.

The errors of position and velocity are assumed to be: δ r = ( xˆ − x)2 + ( yˆ − y )2 + ( zˆ − z )2 δ v = (vˆx − vx )2 + (vˆ y − v y )2 + (vˆz − vz )2

where

Table 2 Percentage of n satellites were tracked Number of Satellite 0 1 2 3 4 5 6 7 8 9 10

0°

90°

180°

270°

0 0 0.694 2.500 8.750 18.611 15.764 12.222 18.750 16.319 6.389

0.278 4.375 11.736 13.958 10.069 13.472 18.056 15.069 9.583 1.458 1.944

0.278 1.875 11.042 17.639 19.028 10.416 15.347 13.264 7.847 2.222 1.042

0 0 0 2.708 10.208 17.778 16.944 16.528 12.778 14.931 8.125

x, y , z , v x , v y , v z

are the true values of the satellite

position and velocity, and

xˆ , yˆ , zˆ, vˆx , vˆ y , vˆz

are the estimated

values. If the value of the position error δ r is recorded for N different samples (δ r1 , δ r2 ,Kδ rN ) , then the root mean square (RMS) of δ r is defined as: 1 N ∑ δ ri2 N i =1

RMS (δ r ) = E (δ r 2 ) ≅

The RMS of δ v can also be defined as: 1 N ∑δ vi2 N i =1

RMS (δ v) = E (δ v 2 ) ≅

Two settings were tested: • Using four or less visible navigation satellite signals. • Using all the visible navigation satellite signals.

Figure 5 shows histograms indicating the percentage of time one, two, three, etc., satellites tracked. The statistical data is given in Table 2. Figure 5 and Table 2 indicate that less than 4 satellites were tracked for less than 31% of the time, while four or more satellites were tracked for more than 70% of the time.

1000 Position Error Estimation Standard Deviation

900 800 700 Position Error(m)

2.4 Filtering Results Choosing the 90° GEO configuration to implement the Kalman filter, its elements are presented in Table 3.

600 500 400 300

Compared to any single GNSS, the number and the duration are improved due to the multi-GNSSs adopted. Therefore it is more suitable for using a navigation filter for orbit determination in multi-GNSSs scenario.

200 100 0

Orbit type

Geostationary

0.5

Cartesian (J2000)

Keplerian (J2000)

0.45

-26,214,220,335.009

Y(m) Z(m)

33,024,715,796.160 0

Vx(m/s)

-2,408.201

Vy(m/s)

-1,911.571

Vz(m/s)

0.000

Semi-major Axis(m) Eccentricity Inclination(°) RA Ascn. Node(°) Long. Ascn. Node(°) True Anomaly(°)

42,164,169.637 1.42579e-15 0 0 90 128.441696

W(k )

3

4 5 Time(s)

6

7

8

9 4

x 10

Velocity Error Estimation Standard Deviation

0.35 0.3 0.25 0.2 0.15

0.05 0

is:

0

1

2

3

4 5 Time(s)

6

7

8

9 4

x 10

Figure 6: Convergence of Kalman filter (four or less the visible navigation satellites)

Q(k ) = diag ([(0.02m)2 , (0.02m)2 , (0.02m)2 , (2 × 10-4 m / s) 2 , (2 × 10-4 m / s)2 , (2 × 10-4 m / s)2 ,(0.5m)2 ,(1 × 10-6 m)2 ])

The initial errors of the satellite position and velocity are: δ X(0) = [1km, 1km, 1km, 10m / s, 10m / s, 10m / s,300m, 0.0003m ]

The initial covariance matrix of the satellite position and velocity is: Pg (0)=diag ([(1km) 2 , (1km) 2 , (1km) 2 ,( 10m / s) 2 , ( 10m / s) 2 , ( 10m / s) 2 , ( 300m) 2 , ( 0.0003m) 2 ])


2

0.1

Table 3: Orbital elements for 90° GEO configuration orbits. The covariance matrix of process noise

1

0.4 Velocity Error Estimation(m/s)

X(m)

0

2175

1000

In this paper, the navigation performance using the extended Kalman filter with the hypothesis that all the four GNSSs are available was investigated. Even when using the main lobe only, the accuracy remains within an acceptable range. Therefore it is concluded that this algorithm meets the requirement for a GEO satellite’s autonomous navigation. Although the algorithm is designed for a GEO satellite, it is expected to be also applicable to any other HEO satellites. However this algorithm still requires further investigation and testing.

Position Error Estimation Standard Deviation

900 800

Position Error(m)

700 600 500 400 300 200 100 0 0

1

2

3

4

Time(s)

5

6

7

8

9

ACKNOWLEDGMENTS

4

x 10

0.5

The authors wish to thank their colleagues from the School of Surveying and Spatial Information Systems, University of New South Wales, and the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics. The first author wishes to thank the China Scholarship Council for the scholarship support during her joint training.

Velocity Error Estimation Standard Deviation

0.45

Velocity Error Estimation(m/s)

0.4 0.35 0.3 0.25 0.2 0.15 0.1

REFERENCES

0.05 0 0

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Time(s)

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[1].Melbourne W, Yunck T, Bertiger W, et al., Scientific Applications of GPS on Low Earth Orbiters. 1994, JPL Technical Report Server, DSpace at Jet Propulsion Laboratory, Pasadena, United States: Stuttgart, Germany. [2].Martin-Mur T, Dow J, and Bondarenco N. Use of GPS for Precise and Operational Orbit Determination at ESOC. in ION GPS-95. 1995. Palm Springs, California.619-626 [3].Kronman J. Experience Using GPS for Orbit Determination of a Geosynchronous Satellite. in ION GPS 2000. 2000. Salt Lake City, Utah.1622-1626 [4].Eissfeller B, Balbach O, Rossbach U, et al. GPS Navigation on the HEO Satellite Mission EQUATORS - Results of the Feasibility Study. in ION GPS-96. 1996. Kansas City, Missouri: Institute of Navigation.1331-1340 [5].Vasilyev M. Real time autonomous orbit determination of GEO satellite using GPS. in ION GPS-99. 1999. Nashiville,Tennesee.451-457 [6].Krauss PA, Kühl C, Mittnacht M, et al. Modernized Spaceborne GNSS Receivers. in Proceedings of the 17th World Congress The International Federation of Automatic Control. 2008. Seoul, Korea.4707-4712 [7].Mittnacht M and Fichter W. Real Time On-board Orbit Determination of GEO Satellites Using Software or Hardware Correlation. in ION GPS 2000. 2000. Salt Lake City,UT.1976-1984 [8].Mehlen C and Laurichesse D, Real-time GEO Orbit Determination Using TOPSTAR 3000 GPS Receiver. Navigation, 2001. 48(3): p. 169-179.

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Figure 7: Convergence of Kalman filter (all visible navigation satellites) The P matrix is convergent during the elapsed hours, which represents the covariance of the estimate errors[17]. The rebound in the lower branch of the orbit is due to the loss of orbit dynamics propagation accuracy. However in the second setting test, the convergence is process is quicker. From the plot (see Figure 6), the RMS errors of the satellite position and velocity of the integrated navigation solution are 185.776m (1σ) and 0.073m/s (1σ) for the GEO orbit. In Figure 7, an accuracy of 157.767m (1σ) and 0.096m/s (1σ) is reached. 3. CONCLUDING REMARKS The exploitation of multi-GNSS signals for precise orbit determination for GEO satellites is not only possible but also practical. No single GNSS can solve the problem of GEO satellites' line-of-sight limitations, but multi-GNSS improves the navigation performance. For orbit determination, the Extend Kalman filter is a well-known technique, but it requires accurate orbit dynamics modeling. Using multi-GNSS with an orbit determination filter is a viable option for GEO missions to sequentially process sparsely available pseudorange measurements.


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[9].Vallado DA and McClain WD, Fundamentals of astrodynamics and applications. Second ed. Boston MA: Space Technology Library, Kluwer Academic Publishers, 2001. [10].Montenbruck O and Eberhard G, Satellite orbits: models, methods, and applications. First ed. Berlin: Springer Verlag, 2000. [11].Campana R and Marradi L. GPS-based Space Navigation: Comparison of Kalman Filtering Schemes. in ION GPS 2000. 2000. Salt Lake City, UT.1636-1645 [12].Minkler G and Minkler J, Theory and Application of Kalman Filtering: Magellan Book Company, Palm Bay, Florida, USA. 608, 1993. [13].Russian Space Agency ,Information Analytical Center. GLONASS constellation status. 2009; Available from: http://www.glonass-ianc.rsa.ru. [14].European Space Agency. How to build up a constellation of 30 navigation satellites. 2007 17 July 2007; Available from: http://www.esa.int/esaNA/galileo.html. [15].Tan S, Development and Though of Compass Navigation Satellite System. Journal of Astronautics, 2008. 29(2): p. 391-396. [16].Moreau MC, Axelrad P, Garrison JL, et al. Test Results of the PiVoT Receiver in High Earth Orbits using a GSS GPS Simulator. in ION GPS 2001. 2001. Salt Lake City, Utah.2316-2326 [17].Qin Y, Zhang H, and Wang S, Kalman Filtering and Principle of Integrated Navigation. Xian, China: Northwestern Polytechnical University Press. 344, 1998.


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