Autonomous prediction of GPS and GLONASS satellite orbits

Transcription

1 Autonomous prediction of GPS and GLONASS satellite orbits Mari Seppänen*, Juha Ala-Luhtala, Robert Piché, Simo Martikainen, Simo Ali-Löytty Tampere University of Technology *Corresponding author: ABSTRACT A method to predict satellite orbits in a stand-alone consumer-grade GNSS navigation device is presented. The motivation for this work was to reduce the startup time of a navigation device without the use of network assistance. The presented orbit prediction method works for both GPS and GLONASS, achieving median accuracies of 58 and 23 meters in satellite s position, respectively, for prediction up to four days ahead. A simple method for prediction of the satellite s clock offsets is also discussed. A basic analysis indicates that the method gives a line of sight range error of 15 meters, for both GPS and GLONASS, with most of the error due to the clock offset prediction. INTRODUCTION When standalone GNSS navigation devices are turned on, there is a delay of at least 30 seconds before they begin to provide position information. This delay occurs because of the time needed for signal acquisition and tracking and after that one has to wait until the navigation data is acquired. It takes 18 seconds in GPS to send the three first subframes [1, p ] and 8 seconds in GLONASS to send the first four strings [2, p ], which contain the immediate or minimum data required for positioning. Furthermore, in both systems, this essential information is repeated only once every 30 seconds [1; 2]. If the receiver does not have a direct view to the sky, for example because of trees or buildings in the way, the signal acquisition and tracking slows down and the time until the first positioning result can take longer, even several minutes [3]. This is annoying for typical users, and could have more serious consequences in emergency or other special situations. Therefore there is a demand for methods which could decrease the time from turning on a device to the first position estimate, known as the Time To First Fix (TTFF). The minimum information needed for positioning in GNSS includes the satellite s ephemeris parameters and the clock correction terms, which model the difference between onboard time scale and GNSS s system time. Because one

2 of the reasons for long TTFF is the time it takes to receive the satellite s ephemeris broadcast, alternative ways of obtaining satellite s position, velocity and clock information can be used to reduce it. Then the satellite would be needed only for receiving the time of the satellites clock or a pseudorange measurement. This is faster because the satellite sends the time stamp once every six seconds in GPS and every two seconds in GLONASS [1; 2]. In addition, the information about satellite s position and velocity can be used to speed up the signal acquisition in the GNSS receiver, which is another cause of the startup delay, in two ways. First, assuming that a crude estimate about the receiver s location and approximate time are known, the satellite s position coordinates can be used to identify the visible satellites. Secondly, with the information about satellite s velocity the range of possible Doppler frequency shifts can be reduced. Altogether, by predicting satellite s kinematic state i.e. position and velocity as well as clock correction terms, it would be possible to get the first positioning result in about 5 seconds [3] after turning on the device. Besides reducing TTFF, satellite orbit prediction can also widen the capability of GNSS devices. In some navigation cases, the user is indoors or in urban canyons where the signal is strong enough to be detected and for getting pseudorange measurements, but too weak or fragmental for reading the whole ephemeris. Then the predicted orbit can make it possible to both compute a position and reduce TTFF. A widely used alternative to the navigation message broadcast by the satellite is the use of assistance data servers that send data to the navigation device that enable the computation of satellite s position, velocity and satellite s clock offset. However, there are problems associated with such assistance data: Connection to the assistance server may fail or the user may consider the assistance data connection to be too expensive. Furthermore, many navigation devices are not even equipped to make a network connection. There is therefore interest in methods that can be implemented entirely in the navigation device, without network connection. In self-assisted GNSS the aim is to attain properties similar to assisted GNSS, without a network connection. This is accomplished by generating within the device the information that would have been received as assistance data. In this paper, satellite orbits are predicted using only the satellite s broadcast data that was received the previous times the device was in operation. This technique has been implemented in commercial products and is outlined in the literature [3; 4], but these publications do not give a detailed description of the algorithms. We have developed a related method and believe its accuracy is competitive compared to the other methods. In our previous paper [5] we gave a detailed presentation of the ephemeris prediction algorithm for GPS satellites. The aim of this paper is to present an extended version of the algorithm that also covers the GLONASS satellites. Moreover, we present a study of satellite s clock offset prediction and of the positioning error that can be expected with the position and clock offset predictions. The paper begins with a description of the satellite s equation of motion. It includes only the forces that have the 2

3 greatest effect on the satellite s orbit. Then, the reference frames used in the computation are introduced and the ephemerides structure and accuracy in GNSS is discussed. After these introductory sections the actual algorithm is presented and its performance in GPS and GLONASS is evaluated. The next section discusses the prediction of the clock offsets. Finally the errors due to the satellite s predicted orbit and predicted clock offsets are combined in order to consider the total error in the range measurement. The paper concludes with a short summary. FORCE MODEL The orbit prediction algorithm introduced in this work is based on the satellite s equation of motion r = a(t,r), (1) where r is the second time derivative of the satellite s position vector and a is a vector valued function which gives the satellite s acceleration vector as a function of time and satellite s position. Given the satellite s position and velocity at instant t 0, say r(t 0 ) = r 0 and v(t 0 ) = v 0, we can compute the satellite s position at any other instant t by doubleintegrating equation (1) with respect to time. Usually, in order to predict GNSS satellite orbits with very high accuracy, a large number of different forces have to be included to the model. However, we have taken into account only the four forces that have the greatest influence on a satellite s orbit at GNSS satellite altitude, and write the acceleration as a(t,r) = a g + a moon + a sun + a srp, (2) where a g, a moon, a sun and a srp are the accelerations due to Earth gravitation (taking into account the unsymmetrical mass distribution of the Earth), lunar gravitation, solar gravitation and solar radiation pressure, respectively. If the Earth was a uniform sphere, the gravitational acceleration would depend on the satellite s radial distance r only, and the gravity potential U would be of the form U(r) = GM E, r where G is the gravitational constant and M E is the mass of the Earth. When the unsymmetrical mass distribution of the Earth is taken into account, the gravity potential U can be written in the form of the spherical harmonics expansion 3

4 [6, p. 57] U(r, λ, ϕ) = GM E r n=0 m=0 [ n (RE ) n P nm(sin ϕ) r ( C nm cos(mλ) + S nm sin(mλ)) ]. (3) Here the potential U is not only a function of satellite s radius r, but also the longitude λ and latitude ϕ. The constant R E in this formula is the Earth s radius and the terms P nm are the associated Legendre polynomials of degree n and order m. The coefficients S nm and C nm in the formula are experimentally determined constants whose magnitude decreases very fast with increasing n and m. Therefore, the potential can be approximated by taking into account only the first few terms. At GNSS satellite altitude, the terms up to the degree and order 4 are significant [7, p. 54], but we use terms up to the degree and order 8. The values for the geopotential coefficients C nm and S nm are obtained from the EGM2008 model [8]. The acceleration due to Earth gravitation can be computed as a gradient of the gravity potential U. We have used a recursive algorithm introduced by Cunningham [9] and later extended by Métris et al.[10] to compute the derivatives. The algorithm takes the partial derivatives with respect to the Earth centered Earth fixed (ECEF) position vector r e, which is related to the satellite s inertial position by the transformationr e = Rr, where R is the transformation matrix. The gradient of the potential U gives the acceleration in ECEF coordinates, which can be transformed to the inertial reference frame by the formula [6, p. 68] a g (r) = R 1 U(r e ). (4) After the Earth s gravitation the second biggest acceleration components in the satellite s equation of motion are caused by the gravitational forces of the Moon and the Sun. To compute the acceleration acting on the satellite because of the gravitational force of any celestial body, one can use the form ( rcb r a cb = GM r cb r 3 r ) cb r cb 3, (5) where M is the mass of the celestial body, r cb is its position in Earth centered inertial reference frame and r is the position of the satellite in the same reference frame. Besides the Moon and the Sun this formula can be used also for computing the planetary accelerations, but we have ignored these because their influence to the satellite s orbit is negligible. The orbits of the Sun and the Moon have to be known in order to compute the gravitational acceleration with the formula (5). In our model we use simple models presented in [6, p ] to compute the lunar and the solar 4

5 coordinates. The coordinates are accurate to about 0.1-1% [6]. The last acceleration component in equation (2) arises when a satellite reflects and absorbs photons emitted by the Sun. Taking into account only the Sunwards component of radiation we obtain the formula a srp = α λ P 0 (1 + ǫ) AU2 r 2 sun A m e sun. (6) This kind of model for solar radiation pressure (SRP) is called also the Cannonball model [11]. In formula (6) the factor r sun is the distance from satellite to Sun and e sun is a unit vector from satellite to Sun. The factor λ is a shadow function, whose value equals one when satellite is in sunlight, zero when it is in umbra and something between when it is in penumbra. We have used a conical shadow model described in the book [6, p ]. The remaining factors are constants: AU is the astronomical unit, P 0 is the solar radiation pressure at the distance of 1AU from the Sun, ǫ is the reflectivity coefficient of the satellite, m is the mass of the satellite and A is the satellite s surface area. For these constants we have used values shown in the Table 1. Table 1: Constants in the solar radiation pressure formula P 0 [Nm 2 ] ǫ AU [km] A [m 2 ] m [kg] It is hard to know the exact mass, surface area or reflectivity of the satellite. These numbers are also different for different satellite types. For these reasons, the acceleration formula (6) is multiplied with an additional parameter α. The value of this parameter is estimated based on real GPS and GLONASS orbital data. We estimated α separately for each satellite using an extended Kalman filter, in which the measurement model is discrete-time and the state model is continuous-time. This kind of filter is presented in [12, p. 278] and [13, p. 405]. Details of the state and measurement models for this case are presented in [7]. As measurement data we used precise ephemeris positions published by the National Geospatial-Intelligence Agency (NGA) [14] and the International GNSS Service (IGS) [15]. The estimation of the parameter α was done several times using different periods during the GPS weeks The mean RMS value for a 7 day arc was 8.0 m for GPS satellites and 1.8 m for GLONASS satellites. The RMS values show that our model works significantly better for GLONASS satellites. The possible reasons for this are discussed in the test section. The resulted values were varying a little bit as a function of time, especially during times when the satellite is in umbra. However we wanted a constant parameter for each satellite and therefore took the satellite-specific median. Results of this estimation process are shown in the Tables 2 and 3. As mentioned, the α parameters were estimated for the constellation which was in operation during GPS weeks When the satellite constellation is changed, the parameters should be updated or else the prediction accuracy will be slightly decreased. Fortunately new satellites are added quite seldom. Of course, changes in the constellation 5

6 Table 2: Solar radiation pressure parameters for GPS satellites PRN α Table 3: Solar radiation pressure parameters for GLONASS satellites PRN α can be handled by updating the α parameters a few times per year, for instance, as a part of a software update. To summarize, in this section we presented a motion model for GNSS satellites. We can now introduce a vector valued function ϑ which, given satellite s position and velocity at instant t 0, returns the satellite s position and velocity at another instant t f. The definition for this kind of function is ϑ(t 0, t f,r 0,v 0 ) = r(t f) v(t f ) = r(t 0) + ( tf t 0 v(t 0 ) + ) t f t 0 a(t,r)dt dt v(t 0 ) + t f t 0 a(t,r)dt. (7) We have used the Runge-Kutta-Nyström -method [16, p. 284; 6, p. 124] to solve the equation of motion. The algorithm is an efficient integration method for second order differential equations in which the second derivative (the acceleration a) is independent on the first derivative (the velocity). REFERENCE FRAMES The International Earth Rotation and Reference Systems Service (IERS) maintains two important reference systems: a Celestial Reference System (CRS) and a Terrestrial Reference System (TRS). CRS is a reference system whose coordinate axes maintain their orientation with respect to distant stars. The origin of this reference frame is in the center of the Earth and Earth is in an accelerated motion while orbiting around the sun. Therefore CRS is not precisely inertial, but is an adequate approximation of an inertial reference frame for our purposes. TRS is an Earth fixed reference frame. Its origin is the Earth s centre of mass and its z-axis is the mean rotational axis of the Earth. This mean pole of rotation was defined, because the Earth s instantaneous rotation pole moves with respect to Earth s crust whereas in Earth fixed reference frame the axes must be pointing at a fixed point on the Earth s surface. The coordinate 6

7 transformation between these two reference systems can be written as r TRS (t) = W(t)G(t)N(t)P(t)r CRS, (8) where the matrices W, G, N and P describe polar motion, Earth rotation, nutation and precession, respectively. The transformation matrices are time dependent and thus the vector r CRS, being constant in CRS, is time dependent after transformation to TRS. We follow the book [6] and use IAU76 theory when computing the precession matrix P and nutation theory IAU80 for matrix N, although more recent models IAU2000A and IAU2000B are available and can be found from [17]. The third matrix G describes the rotation of the Earth. In order to compute it, we need Greenwich Mean Sidereal Time (GMST), which can be computed as follows. Starting from GPS time, we first compute the corresponding Julian date as JD GPS = t/86400 s, where t is the number of seconds elapsed since the beginning of GPS time, We express the Julian date (JD) in Universal Time UT1 by subtracting the leap seconds τ that have been added to Coordinated Universal Time (UTC) since the beginning of GPS time, and adding the difference dut1 = UT1-UTC. That is JD UT1 = JD GPS + dut1 τ s. The number of leap seconds is typically known by the GPS device, because the current number of leap seconds is part of the broadcast message and new leap seconds are added quite seldom. However the time difference dut1, which is one of the Earth orientation parameters (EOP), is not necessarily known when starting the prediction. This time difference is small ( dut1 < 0.9 s) and it does not cause noticeable error if we neglect it. Next JD UT1 is divided into two parts, such that the first part, JD 0 UT1, is the JD at the beginning of the current day and h the second part, UT1, is the rest of JD UT1. When computing these, one has to notice that the Julian date integer part changes at noon, but 0 h universal time is at midnight. Also the UT1 must be given in seconds. Taking these facts into account we can apportion as follows: JD 0 h UT1 = JD UT UT1 = (JD UT1 JD 0h UT1) s. Now we can compute the Greenwich Mean Sidereal Time in seconds with the formula [6; 17] GMST = UT T T T 3, (9) 7

8 where the time arguments T and T 0 are T = JD UT T 0 = JD 0 h UT , i.e the number of Julian centuries of Universal Time elapsed since 2000 Jan. 1.5 UT1 at the current time and at the beginning of the day, respectively. Furthermore the equation of equinoxes [6; 17] GAST = GMST + ψ cosε sinω cos Ω (10) is used to compute Greenwich Apparent Sidereal Time (GAST). In this equation the parameters ψ, ε and Ω come from the nutation theory and. denotes that the number is presented in arcseconds. Finally, we can compute the transformation matrix G(t) = R z (GAST). In this equation R z is a rotation around the z-axis i.e R z (γ) = cosγ sin γ 0 sinγ cosγ (11) After multiplying the vectorr CRS with precession, nutation and Earth rotation matrices, it is in a Terrestial Intermediate Reference System (TIRS), whose z-axis points to the Celestial Ephemeris Pole (CEP). We assume that the orientation of Earth s rotation axis is the same as the orientation of this CEP pole. CEP is not fixed with respect to the surface of the Earth, but performs a periodic motion around its mean position, called polar motion. The motion is small, having a radius of under 10 m, but it is important to take it into account. The rotation matrix describing the polar motion is W(t) = R y ( x p )R x ( y p ). where x p and y p are the polar motion parameters and R x and R y are simple rotation matrices around the x- and y-axes. Together with dut1 they are called also Earth Orientation Parameters (EOP). The daily values for these parameters can be found from the homepage of IERS [18]. IERS reports the observed values for EOP and quite accurate short term predictions. However the Earth orientation parameters are not long-term predictable, which causes some problems while trying to do the prediction in a device 8

9 without any network connection. We can not form a prediction model that would be valid for the life of the device i.e. years. Neither are the EOP parameters part of the broadcast message yet, though this fact will be changed in the future as the new L1C signal comes into use [19]. However, later in section Initial value improvement we will show how to infer these parameters based on the collected broadcast ephemeris data. When transforming the position vector r CRS to the TRS one matrix at a time according to equation (8), every intermediate step is also a reference frame. For example when multiplying with matrix P we get a mean of date (mod) system and after multiplied also with nutation matrix N we get a true of date (tod) system. Figure 1 illustrates the connection between CRS and TRS, as well as the intermediate reference frames between them. Next we will present the reference frames we have used, and illustrate how the transformation matrices between them are compounded of those four matrices connecting CRS and TRS. Figure 1: Connection between the IERS reference systems CRS and TRS. Transformation matrices and intermediate reference frames. The inverse of a transformation matrix is its transpose. The broadcast position and velocity we get from broadcast ephemeris are in Earth fixed reference in the beginning. In GPS the reference frame is WGS84, which is so close to TRS that we assume these two to be equivalent. In GLONASS the received ephemeris is in Earth fixed PZ90.02 reference frame [20] but it can be transformed to WGS84 or TRS by 9

10 a translation of origin [21] r WGS84 = r PZ m 0.08 m 0.18 m. (12) Now for numerical integration we need an inertial reference frame and we choose to use the TIRS system at epoch t 0, the initial epoch for the equation of motion. Before starting to predict satellite s position according to the model (7), we need to transform the position and velocity vectors to this inertial reference frame, denoted by subscript IN. For the position vector, the transformation from TRS at an arbitrary time t to the TIRS system at epoch t 0 is r IN = r TIRS(t0 ) (13) = G(t 0)N(t 0)P(t 0)P T (t)n T (t)g T (t)w T (t)r TRS. However, when we start the prediction we have t = t 0 and the transformation reduces to r IN = W T (t 0 )r TRS(t0). (14) For velocity the transformation from TRS to IN can be computed by differentiating equation (13) with respect to time. When differentiating, the other matrices are treated as constants and the time dependence of the Earth rotation matrix G T (t) is taken into account [5]. Denoting the transformation matrix from reference frame A to B as R B A, we have R TIRS CRS = GNP and RCRS TIRS = (RTIRS CRS )T = P T N T G T. With these notations the velocity transformation is [5] v IN = R TIRS CRS (t 0)R CRS TIRS (t) (W T (t)v TRS + ω ( W T (t)r TRS ) ). (15) where ω = [0 0 ω] T is the angular velocity vector of the Earth s rotation. After predicting the satellite s position and velocity at time t in the future, we have to do the transformations the other way around. Solving r TRS and v TRS from equations (13) and (15) we get r TRS v TRS = W(t)R TIRS CRS (t)r CRS TIRS(t 0 )r IN (16) ( = W(t) R TIRS CRS (t)r CRS TIRS(t 0 )v IN ω ( R TIRS CRS (t)r CRS TIRS(t 0 )r IN ) ). (17) When the satellite s orbit is predicted, we actually do not know the exact matrix W(t). However, we might know 10

11 the polar motion parameters x p and y p at the beginning of the prediction and because they do not change much in a few days long prediction, we can approximate the matrix by W(t 0 ) = W(x p (t 0 ), y p (t 0 )). Similarly the third Earth orientation parameter dut1 is needed to compute the rotation matrices R TIRS CRS (t 0) and R TIRS CRS (t). If we set dut1 = 0 when computing these matrices, we do two approximations: setting dut1(t 0 ) = 0 causes a constant offset and setting dut1(t) = dut1(t 0 ) neglects the variation of dut1 during the prediction. The former approximation does not lead to significant errors, but neglecting the change in dut1 results in median error of 4.2 meters in the satellite s position after a 4 days long prediction. However, this approximation is necessary because the value of dut1 may, in general, be unknown to the device and its evolution is very difficult to predict. By making these approximations, we can introduce a transformation function ξ, which depends both on time and the polar motion parameter values but is independent of dut1 which is set to zero. The transformation function transforms the state of the satellite from TRS to the inertial reference frame IN, that is r v IN = ξ r v TRS, t,p = R TIRS CRS (t 0)R CRS TIRS (t)wt (p)r TRS R TIRS CRS (t 0)R CRS TIRS (t) (W T (p)v TRS + ω ( W T (p)r TRS ) ). Here p = [x p y p ] T is the vector of polar motion parameters. The inverse transformation, from IN to TRS, is denoted ξ 1. With these notations, we can write down the prediction function (7) in the TRS system. We end up with the. formula ϑ TRS (t 0, t f,r TRS (t 0 ),v TRS (t 0 ),p) = r(t f) v(t f ) TRS = ξ 1 ϑ t 0, t f, ξ r(t 0) v(t 0 ) TRS, t 0,p, t f,p. (18) In addition to the transformations ξ and ξ 1, which are done at the beginning and at the end of the prediction, we need to do some transformations during the prediction. In the previous section it was mentioned that the satellite s position vector has to be represented in an Earth fixed reference frame to calculate the acceleration due to the Earth s gravitation. TRS is an Earth fixed reference frame, so we can do the transformation to it using equation (16). Therefore the transformation matrix R presented in equation (4) is W(t)R TIRS CRS (t)rcrs TIRS (t 0). To speed up the numerical integration, we approximate the matrix R. If the length of prediction is only some days, the precession and nutation matrices remain almost unchanged. We can write PP T I and NN T I and thereby get the following approximation: W(t)G(t)N(t)P(t)P T (t 0 )N T (t 0 )G T (t 0 ) W(t)G(t)G T (t 0 ). 11

12 Multiplication with the matrix G rotates the reference frame according to the Earth s rotation. Mainly it is a simple rotation around the z-axis, with the angular speed of the Earth. If the x-axis points to certain meridian at the initial time t 0, then at the time t it points to the direction we get by rotating the x-axis around the z-axis with an angle of (t t 0 )ω. Thus W(t)G(t)G T (t 0 ) = W(t)R z ((t t 0 )ω) is the matrix R used to transform the inertial vector to an Earth fixed reference frame when computing the Earth gravitational acceleration. Its transpose is then used to transform the acceleration vector to the IN reference frame. Again, the matrix W(t) is replaced with W(t 0 ) while predicting. EPHEMERIS REPRESENTATION IN GPS AND GLONASS In GPS the satellites s position is calculated using the 16 ephemeris parameters that are broadcast by the satellite. One of these parameters is called the time of ephemeris (TOE). With one received navigation message i.e. one set of ephemeris parameters the satellite s position can be calculated at any instant within ±2 h from the TOE. Going outside of this range the accuracy of the ephemeris deteriorates rapidly, as illustrated in Figure 2. Fortunately, the satellite starts to send a new ephemeris every second hour. The GPS satellite s velocity can be computed by differentiating the ephemeris parameters with respect to time. The equations for computing GPS satellite s broadcast velocity are presented for instance in [22]. The obtained position and velocity coordinates are in WGS84 reference frame which is very close the IERS s TRS, and in this paper we neglect the difference. In GLONASS the broadcast ephemeris is given in the form of satellite s position and velocity at the TOE instant, which is denoted t b in the GLONASS ICD [2]. In addition, the current value for the acceleration originating from the gravitational interactions with the Sun and the Moon is given. This acceleration is part of a simple motion model that can be used to solve the satellite s ephemeris at any other time instant within about 15 minutes from t b. A new navigation message is broadcast every half hour. In addition, the state coordinates in the GLONASS broadcast are represented in a different reference frame, PZ90.02, so we have to do the transformation (12). The GLONASS broadcast ephemeris is less accurate than that of GPS, as illustrated in Figure 2. In this Figure the error is computed using the IGS precise orbits [15] as a reference trajectory. INITIAL VALUE IMPROVEMENT We have observed that the dynamic model gave good predictions when using precise ephemeris position and velocity as an initial condition but the predictions were about 10 times worse when using the less accurate broadcast position and velocity. Therefore it is necessary to find means to improve the satellite s initial state. 12

13 5 [m] GLONASS GPS Time with respect to TOE [h] 3 Figure 2: The median error in the position coordinates (r x, r y and r z) of satellite broadcast ephemerides In the previous sections we have presented how to predict the satellite s orbit by solving its equation of motion using satellite s positions and velocity at t 0 as initial conditions. In addition we need to know the two Earth orientation parameters: x p and y p in order to be able to calculate the needed transformation matrix W(t 0 ). The third Earth orientation parameter, dut1, is also needed when computing the matrices R TIRS CRS (t), but we set it zero when doing the prediction. We do not have an analytical nor an empirical model for computing parameters x p and y p accurately enough for the lifetime of the device and according to our assumptions we do not have a network connection to get updated information related to these parameters. Therefore the observation that the polar motion parameters can be solved from the broadcast ephemeris is an important detail. Otherwise this algorithm would not be suitable for a totally self-assisted GNSS device. We now call the satellite s initial state and polar motion parameters as initial conditions, and present how the values of these initial conditions are computed. The first thing we do to improve the initial state accuracy is to add an antenna correction to the broadcast ephemeris position. The antenna correction is the difference between the satellite s center of mass and its antenna phase center. When positioning with GNSS, the receiver measures the pseudoranges to the satellites using the signal transmitted from the satellites antennas, so the broadcast ephemeris position describes the position of the satellite s antenna. However, the dynamic model used in orbit prediction describes the motion of the satellite s center of mass. For GPS satellites we have provided antenna offsets used by NGA [23], and for GLONASS we use the values from [24]. For instance, the offset for GPS block IIA satellites is [ δ = m m m ] T. The offset vectors δ are given in a satellite body fixed reference frame. We now let r be the satellite s position in an Earth centered reference frame, for example TRS, and let e sun be a unit vector from satellite to sun in the same reference frame. Then the unit vectors pointing to the x, y and z directions of satellite body fixed frame can be written 13

14 as u z = r esun uz uy uz, uy = and ux = r e sun u z u y u z in TRS. The unit vector u z is pointing towards the Earth, u y is chosen such that it is perpendicular to the plane containing both the Earth and the Sun, and u x is defined such that it completes the right-handed system. Using these, the antenna offset in TRS can be written as δ = δ x u x + δ y u y + δ z u z. The antenna offset vector δ gives the antenna s position coordinates with respect to the center of mass of the satellite. The center of mass r com is then given by r com = r δ. The opposite correction, from center of mass to antenna phase center, could be done after the prediction. However, in our tests we do not do it, because we compare the predicted satellite positions to the precise ephemeris, which is given in terms of the center of mass. Like the solar radiation pressure parameters α, the antenna offset values should also be updated when a new satellite is added to the constellation or when an old satellite is removed. This update can be done as a part of a software update. After applying the antenna correction, the satellites position coordinates can be used together with the motion model to find estimates for the satellite s velocity and polar motion parameters by fitting them to the broadcast data. Let t 0 denote the time instant related to the latest received broadcast ephemeris. For GLONASS it is the instant of the position and velocity coordinates in the ephemeris. For GPS the time instant can be chosen to be anything within ±2 h from the time of ephemeris (TOE) of the latest broadcast ephemeris, because the accuracy inside this range is uniform. For prediction, it is convenient to choose t 0 to be later than TOE, for instance TOE+1.5 h. After t 0 is chosen the antenna corrected broadcast ephemeris position r 0 = r(t 0 ) can be computed as explained above. From equation (18) we see that in order to predict satellites states, we still need the satellite s initial velocity and the polar motion parameters at instant t 0. Therefore, if there is available ephemeris data from n satellites, the vector of unknowns is x = v 1 0. v0 n p 0 = vall 0 p 0, (19) where v i 0 is the velocity of the i:th satellite and p 0 = [x p (t 0 ) y p (t 0 )] T are the polar motion parameters. 14

15 We introduce a function f which, starting from the desired initial instant t 0, computes satellite s state at t k i.e. ri (t k ) v i (t k ) = f i k(x) = ϑ TRS (t 0, t k,r i 0,v i 0,p 0 ), i = 1,... n. (20) Here ϑ TRS, defined in equation (18), is the function that predicts satellite states by integrating a nonlinear differential equation and carrying out the required transformations between the TRS and the inertial reference frame. Therefore, it represent the connection between a satellite s two TRS states, for instance broadcast states. Moreover, the function ϑ TRS models the effect of the polar motion parameters p 0, which are buried in the TRS state representation. Because the initial moment t 0 and antenna corrected position r 0 are assumed to be fixed, they are left out of the arguments of the function f. Furthermore, in the simulations of this paper we use five satellites broadcasts, which all have the same initial instant t 0. Identical instants were assumed for simplicity and that the algorithm can equally well handle broadcasts having slightly different instant t 0. Indeed, this is an important notion as in real applications the t 0 differ from broadcast to broadcast, depending on what data the receiver has collected. However one should not use too old, say over a week old broadcasts. This is because the polar motion parameters p are changing with respect to time and this is not taken into account in our algorithm. The least square fitting needs some observation data. Assume that there is some broadcast data available, i.e. positions and velocities of n satellites at the instants t k. We might have one or more observations for each satellite and the time indexes of the observations might differ from one satellite to another. Therefore, the set of time indexes k would actually be depending on the satellite in question. However, we again simplify a little bit and present the algorithm in the form, where the number of observations m and time instants k = 1,...,m related to these observations are the same for every satellite. To conclude, we have a set of satellites states yk i as measurements and these can be described as yk i = fi k (x) + εi k, i = 1,...,n k = 1,..., m. (21) Here the vectors ε i k are the differences between the measured state at instant t k and the one which was predicted using the nonlinear function fk i defined in equation (20). All the observation vectors from different satellites at different time instants can be combined into one measurement vector y. The corresponding combined nonlinear function is denoted f and the residual vector ε. With these notations, the nonlinear weighted least squares problem is defined as finding the state ˆx that minimizes the quadratic loss function ε T Dε = (y f(x)) T D(y f(x)). 15

16 y m t m t k t 2 t 1 t 0 time f m (x) ^ f k (x) ^ y 2 f 1 (x) ^ y k f 2 (x) ^ y 1 Figure 3: The nonlinear function f k predicts or connects satellites states from t 0 to t k. This connection depends on the vector x, which includes satellites velocities and the polar motion parameters. As measurements we have the satellites state vectors y k, which are obtained from the broadcast ephemeris. At ˆx the predicted states f k (ˆx) fit the measurements best in the least squares sense. Here the diagonal matrix D is a weight matrix that has a value of (1000) 2 in those elements corresponding to velocity components and ones elsewhere. These weights are empirical and we have found that they work well in this case. Furthermore, to use only satellites positions as measurements instead of position and velocity, the weights corresponding to the velocity components can be set to zero. A nonlinear least squares problem (LSQ) ˆx = argmin x (y f(x)) T D(y f(x)) can be solved with the Levenberg-Marquardt method. The algorithm requires an initial iterate for x. An initial iterate near the true value speeds up the convergence of the least squares fitting algorithm. For the initial velocities v all 0 we use the broadcast velocity v BE (t 0 ). For polar motion parameters we use x p = 0.05 arcseconds and y p = 0.35 arcseconds, which is the approximate center of the polar motion spiral during the years In practice the initial iterate for polar motion could be taken from the results of a previous LSQ solution. From the LSQ solution we obtain estimates for the polar motion parameters ˆp 0 and for the velocities ˆv all 0 at the time instant t 0. The orbit prediction for future times then can be computed using the methods described in previous sections. TESTS For GPS satellites the least squares fitting can be done already with one received navigation message. As illustrated in Figure 2, GPS s broadcast ephemeris accuracy is uniform within ±2 h from the time of ephemeris. The same holds true also for the velocity, except the accuracy of velocity deteriorates a bit earlier. Therefore, we can compute satellite s position and velocity at any instant inside this range, and use them as initial conditions or observations. However, we have found that the algorithm works better if the initial instant t 0 of the prediction and the moment or moments of observations t k are chosen apart from each other. In paper [5] we chose the initial moment of prediction as TOE h and as the measurements was the satellite s state at t 1 = TOE 1.5 h. Now in this paper, we do not 16

17 use the whole state, but only the position components as observations. Doing this improves the accuracy of the solved polar motion parameters, and does not significantly affect the prediction accuracy. Using only the position components as measurements, we need more than one measurement instant in order to keep the least squares problem overdetermined. As one can see from the equation (19), the number of variables is 3n + 2, where n is the number of received satellites. Therefore, one position measurement from each satellite, or 3n measurements, is not enough. However, two position measurements for each satellite is already enough to make the system overdetermined. Therefore, in the tests of this paper we made the following choices: the initial instant is t 0 = TOE h and the measurement instants are t 1 = TOE and t 2 = TOE 1.5 h. For GLONASS the structure of the broadcast ephemeris is different. The position and velocity are obtained only at one time instant, which can be used to integrate the orbit with the simple motion model from GLONASS ICD [2] about 15 minutes to each direction. Compared to GPS the GLONASS broadcast ephemeris is accurate for only a very short time (Figure 2). Therefore, in order to achieve good prediction results also with GLONASS we have to use more than one received ephemeris parameter set. For this work it was decided to use the position and velocity obtained without integration from two broadcast ephemerides. When two ephemeris parameter sets are used, the best separation between the initial and measurement instant would be about 6 hours (see Figure 4). However this kind of situation is never achieved in practice because the satellite is on the other side of the Earth after 6 hours. A more feasible time difference is about 12 hours and the algorithm works pretty well for this situation also. The disadvantage of using as long separation as 12 hours is that the required computational time increases in comparison with the GPS. 200 GLONASS prediction errors error [m] t=3h 24h 12h 6h Length of prediction [days] Figure 4: The effect of time difference t between broadcast ephemerides to the prediction errors for GLONASS satellites The tests were done using broadcast ephemerides from GPS weeks The predicted positions were compared to the NGA precise ephemeris positions for GPS satellites and to the IGS precise ephemeris positions for GLONASS satellites. The norms of the prediction errors for 4 day long interval are shown in Figure 5. Here the time difference 17

18 between broadcast ephemerides for GLONASS satellites is 12 hours. The median error is shown by the line inside the box. The upper and lower edges of the box show the 75% and 25% quantiles of the errors respectively. The upper and lower whiskers extending from the box show the 95% and 5% quantiles of the errors respectively. For one day long prediction the results are almost the same for GPS and GLONASS satellites. For longer prediction intervals we see that the errors get larger for GPS satellites. This difference in the prediction results is not because of poorer accuracy of initial values for GPS, in fact, for GPS the initial values, after doing the least squares fitting, are more accurate than the least squares fitted initial values for GLONASS. This indicates that our force model works better for GLONASS satellites. 200 Prediction errors error [m] GLONASS GPS Length of prediction [days] Figure 5: Prediction errors for GPS and GLONASS satellites More information can be obtained by looking at the errors in the RTN (Radial, Transverse, Normal) coordinate system. Let r ECI and v ECI be the position and velocity of the satellite in ECI coordinate system. Transformation from ECI to RTN is achieved using the transformation equation [25] r RTN = [ e R e T e N ]r ECI, (22) where the unit vectors e R, e T and e N are calculated using equations e R = r ECI r ECI (23) e T = e N e R (24) e N = r ECI v ECI r ECI v ECI (25) Here the directions e T and e N may be referred also to as along-track direction and cross-track direction respectively. Prediction errors in RTN coordinate system are shown in Figure 6. We see that for both GPS and GLONASS the errors 18

19 are mostly in the tangential or along-track direction. Prediction errors in the radial and normal directions are small. For positioning applications most important is the radial error, since this has the largest effect to the pseudorange error (see next section). For four day long prediction 95% of the radial errors are under 6 meters for GPS and under 4 meters for GLONASS R T N GPS prediction errors in RTN (95% quantile) R T N GLONASS prediction errors in RTN (95% quantile) error [m] 100 error [m] Length of prediction [days] Length of prediction [days] Figure 6: 95% quantile of prediction errors in RTN coordinate system for (left) GPS satellites and (right) GLONASS satellites From the prediction results it is seen that our model works better for GLONASS satellites than for GPS satellites. A possible explanation for the results may be in the solar radiation pressure model. The solar radiation pressure model used in this work is a simple model including acceleration only from the direct solar radiation pressure. However studies have shown that for GPS satellites there is also an acceleration, often called y-bias, in the direction of the solar panel axis of the satellite [11; 26]. The magnitude of the y-bias acceleration is quite small, in the order of 10 9 m/s 2, but for predictions over several days the effects can be significant. The exact reason for the force is unknown, but one possible explanation is the misalignment of the solar panel axis [11]. The effect of solar radiation pressure on GPS and GLONASS satellites has been studied by Ineichen et al. [27], who found out that the mean values for y-bias acceleration were very close to zero for GLONASS satellites, whereas for GPS satellites the mean values of y-bias were significantly different from zero. This suggests that including the y-bias acceleration to the solar radiation pressure model might increase the accuracy of the prediction results for GPS satellites. In order to estimate the computational complexity we tested the runtime of our algorithm with Nokia N900 mobile phone which has a 600MHz ARM Cortex-A8 CPU processor. The C-code used for testing was automatically generated from our MATLAB implementation, using Matlab Coder TM -tool. We believe that the code could be further optimized for real device implementation. The time needed to perform a 4 days long orbit prediction for one satellite was 1.95 seconds, including the time needed to transform the predicted positions into extended Keplerian form i.e. to the GPS ephemeris format. The time needed to fit the initial velocity of one GPS satellite was 0.36 seconds for known 19

20 polar motion parameters. However, when solving the polar motion parameters together with the initial velocities of 5 satellites, the computational time is 25.0 seconds. This is the bottleneck of the whole algorithm and, as a consequence, the parameters should be solved as seldom as possible. Instead, extrapolation based on earlier solved parameter values should be preferred when possible. We wish to point out that the fitting and subsequent prediction can begun immediately after the previous navigation session is completed. Predictions can be done before the device is turned off, so that they are instantaneously available when the device is opened again. If the device is a mobile phone, that is, the user tends to close the navigation application but keep the phone on, the algorithm can be carried out progressively such that the positions are predicted only slightly ahead of real time. The time needed to integrate the GLONASS orbit is the same as for GPS, but the time needed for the fitting of initial velocity differs, because it depends on the time difference between the two ephemerides used for fitting. For GPS satellites the time difference of the furthermost states used for fitting is 3 hours and therefore each iteration of the algorithm includes the integration of the satellites equation of motion over 3 hours long interval. As a consequence, if the time difference of the 2 received GLONASS ephemerides is 12 hours, the initial velocity fitting will take 4 times longer than for GPS satellites i.e = 1.44 seconds. SATELLITE CLOCK OFFSETS In addition to the satellites ephemeris, GNSS satellite sends information about the satellite s clock offsets. These offsets, which describe the difference between the satellite s own clock and the GNSS system time, are modeled with a low order polynomials whose coefficients are sent in the navigation message. In GPS the offset, denoted δt gps, is approximated with a second order polynomial and in GLONASS the offset δt glo is approximated with a first order polynomial, that is δt gps (t) = a f0 + a f1 (t t oc ) + a f2 (t t oc ) 2 (26) δt glo (t) = τ n + γ n (t t b ). (27) The polynomial coefficients are often denoted with the symbols written in the equations above. In addition to the polynomial coefficients, the equations include two other terms t oc and t b, which are the clock data reference times. For further information about the clock offset polynomials, see the references [28] and [2]. 20

21 When computing the user s position in a standalone GNSS, the pseudorange measurements are of the form ρ = u r + c(t u δt) + ε, (28) where u is the user s position, r is the satellite s position, t u is the receiver s clock bias and δt is the satellite s clock offset and c is the speed of light in vacuum. Furthermore, there is an additional error ε, which among others includes the atmospheric delays, receiver noise and multipath errors. These are sometimes referred to User Equipment Errors (UEE). When considering the positioning calculation without reading the whole navigation message, i.e. with only the time difference from satellite to the receiver, we need to predict the satellites positions r as well as the satellite s clock offsets δt in order to calculate the user s position. As a part of the navigation message the navigation device receives the coefficient for the clock offset models stated in the equations (26) and (27). The simplest way to predict the offsets further is to use the same polynomial models with the latest received coefficients. This approach worked quite well with the GPS satellites. The tests with the GPS broadcast ephemeris data showed that the second order coefficient a f2 was zero, but the first order polynomials were enough to model the offsets in longer term. In contrast, for GLONASS even the first order coefficient, the drift term γ n, was zero for several satellites. Nevertheless, the ephemeris prediction algorithm presented for GLONASS required two earlier receiver broadcasts, so we can use two broadcasts when calculating the clock offsets as well. Computing the future offset at t with the model δt glo (t) = τ n (t b ) + (τ n(t b ) τ n (t b )) t b t (t t b ), (29) b where t b is the reference time of the latest received broadcast and t b of the another broadcast, one can obtain more precise offsets δt glo (t) for the GLONASS satellites. This model, where the coefficient γ n is replaced with the slope computed from two sequential τ n parameters, seems to work better than the parameter γ n that is sometimes given by the satellite. In analogy to the tests of the previous section, the time difference t b t b between the two broadcasts was chosen to be 12 hours when testing the prediction model of the equation (29). Furthermore, the time instants t b of the chosen test sample were equal to those used in the GLONASS ephemeris prediction tests in the previous section. The results are shown in the Figure 7 together with the GPS offset predictions which were based on only one received broadcast. The error in the clock offset cδt is denoted by c t and it is expressed in meters. Again, the boxes show the 75%, 50% and 25% error quantiles, respectively, while the upper and lower whisker show the 95% and 5% quantiles. The individual error c t was calculated as an absolute value between the predicted clock offset and the precise clock 21