Chapter 4. Satellite Position Estimation and Satellite Clock Error Analysis

Chapter 4 Satellite Position Estimation and Satellite Clock Error Analysis 4.1 Introduction In satellite based navigation system, the receiver position accuracy relies on the precise knowledge of the satellite orbits and time. Determination of accurate satellite position, its orbit is important and will always have an impact on the accuracy of receiver`s position. By measuring the time taken for the signal to travel from the satellite to the user, the distance between the user and the satellite can be calculated (Ivan A, 1993). However, in this range measurement there are several errors such as receiver and satellite clock errors, relativistic error apart from the atmospheric propagation errors which results in uncertainty around the GPS receiver position. Each GPS satellite carries an atomic clock to provide precise timing information for the signals transmitted by the satellites (Leo A. M, 2007). The satellite clock time and the true time differ from each other both in scale and in origin. The typical error in GPS positioning, due to the non synchronisation of satellite clock time to Coordinated Universal Time (UTC) is 100ns and the corresponding pseudorange error is 30m. The GPS satellites revolve around the earth with a velocity of 3.874km/s at an altitude of 20,200km. Thus on account of its velocity, a satellite clock appears to run slow by 7µs per day when compared to a clock on the earth s surface. But on account of the difference in gravitational potential, the satellite clock appears to run fast by 45µs per day. Therefore, the clock appears to run fast by 38µs per day. This is an enormous rate difference for an atomic clock with a precision of a few nanoseconds. In this chapter, satellite orbital parameters, orbital parameter estimation, computation of satellite position are presented and the satellite clock error and the relativistic error effect on the navigation solution are carried out by collecting data during several days using the dual frequency GPS receiver (NovAtel make

DL-V3) located at Andhra University College of Engineering, Visakhapatnam (Lat/long: 17.73 o N/83.319 o E, Height: 91.6m) and IISc Bangalore (Lat/long: 13.021 o N/ 77.57 o E, Height: 843.702m). 4.2 Six Keplerian Elements In order to define uniquely the position and velocity of satellite at any instant of time, six Keplerian parameters are required to be estimated. They are: the semi-major axis (a), the eccentricity (e), the inclination angle (i 0 ), the argument of perigee (ω 0 ), the right ascension of ascending node (Ω 0 ) and the mean anomaly (M 0 ). These elements uniquely decide the position and velocity at a given time which is commonly known as an epoch time. In addition to these parameters, there are other forms of orbit representation which have more geometrical significance. Five of them describe the size and shape of the orbit and its orientation in space. The sixth element describes the position of the satellite at a particular time instant or epoch. Given the six elements, satellite position can be computed at any other epoch. The satellite position in its orbit is characterized by Keplerian elements. The satellite position is described by four important factors (Figure 4.1). The shape and size of the orbit: semi - major axis (a) and eccentricity (e) The orientation of orbital plane relative to the fixed stars is specified by the following two parameters. Inclination angle (i 0 ): This is defined as the angle between the orbital plane and Earth s equatorial plane. Right ascension of the ascending node or RAAN (Ω 0 ): It is defined as the angle measured in earth s equatorial plane between the reference direction in space pointing to the vernal equinox (i.e., the direction determined by the intersection of the equatorial plane of earth with the plane of the earth s orbit around the sun), and ascending node.

The orientation of the ellipse in the orbital plane is characterized by the argument of perigee (ω 0 ). Argument of perigee is defined as the angle in the plane of the orbit between the ascending node and the perigee. The position of the satellite in the orbit at a given epoch is specified by True Anomaly, ( ). It is defined as the angle measured in the orbital plane between the perigee and the satellite position at a specified instant. According to Kepler s laws, the path of a satellite orbiting about the earth is an ellipse with one of its foci coincident with the earth s center of mass. Given the position and velocity of a satellite at some arbitrary time, the laws can be used to predict a future position of the satellite. The main advantage of using the orbital elements is that they are related to the equations of motion of a satellite. This advantage allows extrapolating a single state vector to determine its position and velocity at next time interval. z 1 Satellite Perigee 0 Equatorial plane To words the vernal equinox Figure 4.1 Characterization of an ideal orbit and satellite position by Keplerian elements 4.3 Satellite Orbital parameters x 1 Orbital plane 0 i 0 Ascending node Kepler s laws describe the motion of a GPS satellite considering two factors. They are: i) The earth is spherical in shape and uniform in composition, and ii) Only the force acting on the satellite is the gravitational pull of the earth. y 1

But, earth is not a uniform sphere, and there are other forces, beside gravity. The net result of these perturbing forces is that the orbit of the GPS satellite changes with time, and it should be characterized with an appropriate set of time dependent parameters. The main forces changing the motion of a GPS satellite are: 1. Non-central gravitational force field 2. Solar radiation pressure 3. Gravitational fields of the Sun and Moon 4. Equatorial bulge Non-central gravitational force field: Shape of the earth resembles an ellipsoid with equatorial radius about 20km larger than the polar radius. The density of earth is not uniform, and the gravitational force of the earth varies with latitude and longitude, in addition to the radial distance. The gravitational potential function is modeled as a spherical harmonic expansion. It is expressed as 2 GM 5 a 2 V ( r, ', ) 1 C(1 3sin ' ) r 2 r (4.1) where, G = universal gravitational constant M = mass of the earth a = mean equatorial radius of the Earth r = geocentric radius C = normalized spherical harmonic coefficient The first term GM / r is the potential for a spherical earth of uniform density. Its gradient corresponds to the central force for ideal Keplerian motion, giving a centripetal acceleration of 0.56m/s 2 for GPS orbits. The second term with coefficient, C, is the harmonic coefficient of second and zero degree and models the ellipsoidal shape of the earth. Its gradient gives the

major non-central component of the force on a satellite. This term is also referred as equatorial bulge. Equatorial Bulge: Equatorial bulge of earth produces two effects; a torque is produced due to non-radial component of the force on the satellite which results in rotation of the orbital plane. Due to this, line of nodes rotates in the inertial coordinate frame. This effect depends on the inclination of the orbit and is zero for a polar orbit and maximum for an equatorial orbit. This amounts to a change of about 1.2 0 per month for a GPS satellite. Equatorial bulge also produces a twice-per-harmonic perturbation (period 6h). Each time the satellite goes to the equatorial plane, it experiences a greater gravitational force and speeds up. As the satellite goes farther away from the equatorial plane, it slows down. This results in rotation of major axis in the orbital plane. Inclination of a GPS satellite orbit is 55 0 and therefore, < 550 and the corresponding acceleration is less than 5 x 10-5 m/s 2. Gravitational fields of the Sun and Moon: As the sun is much farther away, the effect of solar gravity is lesser even though it is much more massive. Gravitational forces of Sun and Moon produces tides, deforming the shape and gravitational potential on the earth. This effect is negligible for GPS Orbits. Solar Radiation Pressure: Minute pressure is exerted on the satellite due to collision of photons on it. Acceleration due to such force depends upon the mass of the satellite and its surface area exposed to sun (Hofman-Wellenhof, 1994). When the satellite is in earth s shadow the solar radiation pressure is zero. Equation of the motion of the satellite can now be written as, (4.2) GM r r F( r, r, t) 3 r where, r = geocentric radius r = Second order differential of position vector relative to the earth

t = time F( r, r, t) = the perturbation forces GPS accounts for these perturbations with an expanded orbital parameter set to retain the Keplerian look. The expanded set of quasi-keplerian parameters consists of fifteen elements whose values are specified relative to the reference epoch. So, there are sixteen ephemeris parameters in all. The ephemeris parameters broadcasted by each GPS satellite are given in Table 4.1(Rao, 2010). In normal operations, the fit interval is four hours. By demodulating and extracting the navigation data, the user can calculate the satellite position with respect to time. Each satellite broadcasts its own ephemeris. In addition to that, each satellite transmits ephemeris of all satellites in the constellation in the form of an almanac in its navigation message. The almanac is the subset of clock and ephemeris data with reduced precision. To obtain the satellite position in World Geodetic System 1984 (WGS-84) system using the above ephemeris parameters, the following procedure is followed. Each satellite transmits a navigation message containing its orbital elements, clock behavior, system time and status messages. In addition, an almanac is provided that gives the approximate data for each active satellite. This allows the user set to find all satellites once the first has been acquired. No. Parameter Description Units 1 M 0 Mean Anomaly at reference time semi-circles 2 n Mean motion differences from computed value semi-circles/sec 3 E Eccentricity 4 a Square root of the semi major axis m ½ 5 Ω 0 Right ascension of ascending node of orbit plane at weekly epoch semi-circles 6 i 0 Inclination angle at reference time semi-circles 7 ω 0 Argument of perigee semi-circles 8 Rate of right ascension semi-circles/sec 9 di/dt Rate of inclination angle semi-circles/sec 10 C uc Amplitude of the cosine harmonic correction term to the argument of latitude rad

11 C us Amplitude of the sine harmonic correction term to the argument of latitude 12 C rc Amplitude of the cosine harmonic correction term to the orbit radius 13 C rs Amplitude of the sine harmonic correction term to the orbit radius m 14 C ic Amplitude of the cosine harmonic correction term to the angle of inclination 15 C is Amplitude of the sine harmonic correction term to the angle of inclination 16 t oe Reference time for ephemeris computation sec 17 a f0 Satellite clock offset sec 18 a f1 Satellite clock drift sec/sec 19 a f2 Satellite clock frequency drift sec/sec.sec Table 4.1 Ephemeris data definitions and clock correction terms 4.3.1 GPS Satellite Orbit Description Figure 4.1 shows a single GPS satellite in orbit around the earth. The angles, rates, and times indicated are the primary information needed to solve the satellite position. In addition to this information there are second- and third-order terms. These terms are not shown in Figure 4.1. They are used to correct out perturbations in the orbit from a variety of sources. Table 4.1 lists the complete set of sixteen orbital parameters and clock correction terms that are sent down to the GPS receiver. The GPS receiver must perform a series of calculations in order to solve the position of the satellite in Earth-Centered Earth-Fixed (ECEF) coordinates (Sirola, 2002). 4.4 Satellite position determination GPS does not send the satellite position in convenient format such as x-, y-, z-. Instead, it sends ephemeris data that describes each satellite orbit about the earth. Satellite position is computed using the six Keplerian elements (a, e, i o, o, o rad m rad rad, M ) which describe a smooth, elliptical orbit with the satellite position being a function of time, t o. In order to compute the satellite positions, position algorithm makes use of the Keplerian elements along o

with the additional parameters which describe the deviation of an actual satellite motion from this smooth ellipse (Kaplan, E., 1996). 4.4.1 Satellite clock correction terms The satellite sends a correction term to GPS receiver that enables the GPS receiver to correct its replica clocks to the GPS clock time scale. The correction term includes three parameters that model the satellite clock s static offset, drift and rate of drift with respect to the GPS clock. The static offset term is called a f0. This term is used to correct the time indicated on the receivers replica clock(s). The impact of this correction term on computing the satellite position is small because a f0 is small, less than a millisecond. Even if the satellite is moving, the error induced in computed satellite position by a 1ms time error is not large. In 1ms, the satellite moves about 4m. But this is the radial distance where the satellite moves. The change in distance from the satellite to the user would be far less. In short it is optional to correct the time sent data from the receiver s replica clock(s) to compute the satellite position. The effect on the computed path delay (T rec T sent ) of the satellite can be far larger due to high speed involved. A 1ms error in path delay translates to an error in distance of 300km. For this reason, when the path delay is computed it is quite necessary to include the satellite clock error term. 4.4.2 Ephemeris time reference variables t oe and t k The orbits described by the ephemeris data are referenced to time variables t oe and t k, which are used to obtain the satellite position. In other words, one cannot use the time sent information which is received from the satellite. Instead, computations and the current ephemeris data are used to arrive at the values of t oe and t k that are required to obtain satellite position. Ephemeris reference time, t oe: This is one of the sixteen ephemeris parameters sent by the satellite. The ephemeris data is based on a single point of reference in space and time. This

reference occurs at a particular point in the satellite orbit at a particular point in time, t oe. When calculating the satellite position at other points on the orbit, the method used is to extrapolate other satellite positions from this reference position time pair. Therefore as the satellite moves away from this reference point in space (or equivalently time), the accuracy of prediction of the satellite position will degrade. This reference time, t oe, can be used in the future or the past from current GPS time. Typically, it is updated hourly so that errors of prediction do not grow out of tolerance. Another way to explain what is occurring with the orbit parameters is to realize that, even though the parameters sent describe a complete orbit, typically only a small section of that particular orbit is used. This small section, whose center is located in time by t oe, is further refined by correction terms that essentially modify the predicted path from the original primary orbital path. Approximately after an hour, a whole new orbit definition is sent up from the ground control segment of GPS. The GPS receiver then repeats the satellite position estimation process with new ephemeris parameters and value of t oe. The GPS receiver uses the updated orbital data available for maximum satellite position accuracy. Delta Time t k : A term t k is used for the direct measurement of the amount of time between the current satellite position at time t and the ephemeris reference time at t oe. Time t would be usually the time sent information recorded on the receiver replica clock as received from the satellite. So the ephemeris data define an orbit path specified by t oe. The correction terms are used to modify the path a bit for times other than t oe. Those correction terms and the related expressions that incorporate them use t k to improve the satellite position estimation. Hence t k is a computed value from information gathered by the receiver. It is not sent as a part of the ephemeris data. Start Convert time sent to GPS time scale: t = T sent a f0 To obtain satellite positions for this transmit time, t.

Figure 4.2 Flow chart for determining t k Computing t k for any given time sent: t k is determined to calculate the satellite position from difference in the time sent and t oe. Let, t = T sent, then t k = t - t oe. If t k = 0, the satellite will be at the reference position. The calculation of t k is shown in the flow chart of Figure 4.2. The value computed for t k is subject to a mid week value. Furthermore, t k shall be the actual total time difference between the time t and the epoch time t oe and must account for beginning or end of week crossovers. That is, if t k is greater than 302400sec, subtract 604800sec from t k. If t k is less than 302400sec, add 604800sec to t k. 4.4.3 Second- and third-order correction terms in the broadcast ephemeris The parameters of i,, and Δ n are second-order terms that primarily account for variations in the orbit due to variations in the earth s gravity field due to non-spherical and other effects. These effects, if not compensated for, would lead to large errors (thousands of meters) in computed satellite position once move away from the reference time, t oe. The remaining parameters fine-tune the calculations to account for other perturbations to the orbit. Without the six terms C uc, C us, C rc, C rs, C ic and C is, the error in computed position of the SV

could be in the hundreds of meters. To get 100m accuracy ignore these last terms, certainly if the present time is close to the reference time t oe. 4.4.4 Age/Issue of Data Terms The data stream contains a number of terms that give the GPS receiver information about how old the ephemeris and satellite clock error data are or if it has changed in the middle of a data frame. These terms are IODE, IOCD, AODO, and AODE. The Issue of Data Ephemeris (IODE) gives the user to know if the data have changed in the middle of a data frame. A change in the value of IODE between two successive transmissions indicates that the ephemeris parameter set has been updated. The clock correction data have a related term named Issue of Clock Data (IOCD). The term Age of Data Ephemeris (AODE) gives information about how old the ephemeris data is without the knowledge of current GPS time. The Age of data offset (AODO) term gives information about the age of the almanac data. 4.4.5 Satellite Clock Reference Time (t oc ) The clock error terms a f0, a f1, and a f2 have a reference time just like the ephemeris data do. The time t oc is sent in sub-frame 1 with clock error terms and is reference to the GPS time scale. To obtain the position to ±100m and one more time variable increases the error. If time t is equal to t oc then the error of the satellite clock is the term a f0. As GPS time moves away from the t oc point, the satellite clock will drift at the drift rate given by a f1. This drift rate is small. The satellite clock drift rate, a f1, is typically below 3.0 10-12 seconds per second. In 1hour this drift rate would accumulate to (3.0 10-12 ) 3,600~10 ns. This would be the unaccounted satellite clock error after 1hour, the a f0 term is used to correct the satellite time. It is approximately 3m of path delay error per hour (Langley, 1991d). The point of doing this calculation is to show that safely ignore the higher-order clock corrections as the error introduced by just using static model is small. The clock error terms are typically updated with the ephemeris data, which is approximately on an hourly basis.

4.5 Example calculations of various parameters in the SV position algorithm Using the real time navigation data corresponding to 21 st December 2011, the satellite position is found using the following steps. The physical constants used in the calculations are: = 3.986005 10 14 [m 3 /sec 2 ]-WGS 84 value of the earth's universal gravitation constant 5 7.2921151467X10 rad s -WGS 84 value of the earth's rotation rate [rad/sec] e / = 3.1415926535898 - WGS 84 value for. Step 1: Find the semi-major axis of elliptical orbit a a 2 (meter 2 ) Example value: 26559222.5030016m Step 2: The GPS time of transmission, t, corrected for transit time t t sv t sv Example value: 439200sec Step 3: Calculate the time difference (t k ) between the time (t) and the epoch time (t 0e ) and must account for the beginning or end of the week * k t t oe t If tk t toe 302400 then tk tk 604800 If tk t toe 302400 then tk tk 604800 where, t represents the coarse GPS system time and the value can be obtained from Time Of Week (TOW). In addition, t 0e can be obtained from the ephemeris data, 302400 is the time of half a week in seconds. Example value: 0sec Step 4: Calculate the mean motion,

n n0 n n (rad/sec) 3 a Example value: 0.000145867847274776rad/sec Step 5: From this value, the mean anomaly can be found as, M k M0 nt k Example value: 5.45439183237659 0 Step 6: The eccentric anomaly E k can be found from Kepler s equation (M = E e sine) through iteration as E M esin E with, E o = M k k k k Example value: 5.45056359095357 0 Step 7: Once E k is obtained, the value of concise expression of the orbital radius, r k can be found from, r a(1 ecos E ) Example value: 26466723.7877529m Step 8: Determine the true anomaly,. The value can be found from, k k 2 1sin k 1 1 e sin Ek /(1 ecos Ek ) k tan tan cos k (cos Ek e) /(1 ecos Ek ) Example value: -0.836456656676363deg Step 9: The argument of perigee (ω 0 ) can be found from the ephemeris data. Using the definition of argument of latitude, the value of k v k 0 Example value: -3.41119387619636 0 Step 10: Calculate the correction terms of argument of latitude, orbital radius and inclination angle;

uk Cus sin 2k Cuc cos 2k rk Crs sin 2k Crc cos 2k ik Cis sin 2k Cic cos 2k Example: uk = -6.6952762909894710-06 deg, rk = 201.828709519285m, i k = 1.2057262950328710-07 deg Step 11: Calculate the correction terms as follows, uk k uk rk rk r k i i i it k 0 k k Example value: u k =-3.41120057147265deg, r k = 26466925.6164624m and i k = 0.96051508448163deg Step 12: Compute the longitude of the ascending node k by adding the right ascension parameter, 0 and the mean earth rotation rate, 0 ( ) t t k e k e oe Example value: -0.514363534889964deg Step 13: Once all the necessary parameters are obtained, the position of the satellite can be found by applying the three rotations (through ϕ, i and Ω 0 ) described previously. The satellite position calculated using following equation is in the ECEF frame. xk rk cosuk cos k rk sin uk cos ik sin k y k rk cos uk sin r rk sin uk cos ik cos k z k rk sin uk sin i k Example value: x k =-20222226.5285829m, y k = 16068158.9113958m, z k = 577021.29234133m. 4.6 GPS Satellite Clocks and Time

GPS works by using a nominal 24 satellites constellation. These satellites orbit around the earth and relay precise timing information from onboard rubidium and cesium atomic clocks down to earth. Atomic clocks: Atomic clocks are critical equipment for the satellite based navigation systems. The difference between a standard clock and an atomic clock is that the oscillation in an atomic clock is between the nucleus of an atom and the surrounding electronics. Atomic clock uses the electromagnetic waves emitted by the atoms. The most commonly used atomic clocks are Cesium, Hydrogen maser and Rubidium. Cesium clock has high accuracy and good long term stability. The rubidium clock is least expensive, compact and has good short term stability. The cesium atomic clock is used for the purpose of establishing coordinated universal time (UTC) standard. The UTC standard is maintained by over 250 atomic stations around the world. The first three GPS satellites used rubidium clocks. The Block II/IIA GPS satellites carry 2 rubidium and 2 cesium clocks onboard. The different atomic clock standards are given in Table 4.2. Atomic clock Type of standard Accuracy Stability Remarks Cesium Primary 5 10-13 1 10-14 Excellent stability Rubidium Secondary 5 10-11 3 10-11 Good stability Crystal Ternary 1 10-9 1 10-12 Cheap and poor accuracy Table 4.2 Different atomic clock standards The GPS receiver located on or above the earth s surface will pick up this time signal from at least four satellites and computes its exact position using triangulation method. The receiver calculates the distance between each satellite to its antenna phase center by considering how long each timing signal takes to reach the receiver. The time taken by the signal to travel from the satellite to the receiver is known as travel time or transit time of the signal. The velocity of the light is multiplied by the travel time to get the distance between the satellite and the receiver. This computed distance is not the true range and is known as pseudorange. Because of this, the pseudorange measurement includes several errors such as

satellite clock, atmospheric errors, multipath and receiver noise. The pseudo part of the pseudorange is mainly dependent on both the satellite and receiver clock errors. Among these two clock errors, satellite clock error is precisely known, and is broadcasted in the navigation message data. The receiver clock offset is computed as part of the navigation solution algorithm. GPS satellites use cesium and rubidium atomic clocks onboard. These are kept within a millisecond of the master clocks at the GPS master control station located in Colorado Springs, Colorado. The master control station in turn keeps the master clocks synchronized to the Coordinated Universal Time (UTC), except that GPS time is continuous and has no leap seconds. GPS time is derived from an ensemble of Cesium atomic clocks maintained at a very safe place in Colorado. The GPS clock time ensemble is compared with the UTC time scale maintained at the United States Naval Observatory (USNO) in Washington, D.C. GPS time differs from UTC by the integer number of leap seconds that have occurred since the GPS time scale began on 5 th /6 th midnight of January, 1980. This difference is equal to 15sec by the year 2011. 4.7 Satellite Clock Error The satellite clock error is expressed by clock bias, drift, and drift-rate as clock error coefficients. These coefficients are broadcasted in the navigation message. Satellite clock error is caused by satellite oscillator not synchronized to the GPS time which is a true time. This error represents the difference between the time reported by the satellite and the GPS system time. The observation equation for such satellite biased range can be written as P m t sv c (4.3) where P m = measured range = true range sv t = satellite clock error (sec)

c = velocity of light The time at which the signal started from the satellite is called time of transmission (t x_raw ). The time of transmission has to be corrected to get the correct GPS time of transmission (dt) and is given by dt _ raw t x t oc (4.4) where t oc = time of the clock read from the ephemeris As the satellite clocks use atomic clocks and significantly have better long term drift characteristics than the receiver clocks, the clock error can be modeled using the second order polynomial as (Rao, 2010) t sv a f0 a f 1 dt a f 2 dt 2 t r (4.5) where a f 0 = clock bias term (s) a f = clock drift term (s/s) 1 a = clock drift rate (s/s 2 ) f 2 t = satellite clock time (s) t r = correction due to relativistic effects (s) Clock bias is the difference between the clocks indicated time and the universal time. The GPS requires all the transmitter clocks to be synchronized. In reality the GPS satellite clocks are slowly but steadily drifting away from each other. The GPS satellite clock bias (a f0 ), drift (a f1 ) and drift rate (a f0 ) are explicitly determined in the same procedure as the estimation of the satellite orbital parameters. The behavior of each GPS satellite clock is monitored with respect to GPS time, as maintained by an ensemble of atomic clocks at the GPS master control station. The clock bias, drift and drift rate of the satellite clocks are

available to all GPS users as clock error coefficients broadcast in the navigation message at the rate of 50bps. 4.7.1 Relativistic effects All clocks will have a different frequency in GPS orbit compared to the frequency of an identical clock on the earth because of relativity effects. In the day-to-day life, they are quite unaware of the omnipresence of the theory of relativity. However this phenomenon has influence on proper functioning of the GPS system. The clock ticks from the GPS satellites must be known to an accuracy of 20-30ns for precise estimation of user position. However, the GPS satellites are constantly moving with a speed of 3.874km/s relative to observers on the earth. Hence the effects predicted by the Special and General theories of relativity must be taken into account to achieve the 20-30ns accuracy (Frank van Diggelen, 2007). Special relativity predicts that moving satellite clocks will appear to tick slower than non-moving ones. Because of the slower ticking rate due to the time dilation effect of their relative motion, the special relativity predicts that the on-board atomic clocks on the satellites fall behind clocks on the ground by about 7µs per day. General relativity predicts that clocks experiencing strong gravitational field will tick at a slower rate. As such, when viewed from on or near the surface of the earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General relativity predicts that the clocks onboard a GPS satellite should get ahead of ground-based clocks by 45µs per day (David L.M., 2003). This second effect is six times stronger than the time dilation experienced above. The combination of these two relativistic effects results that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38µs/day (45µs -7µs =38µs). If these effects are not properly taken into account, a navigational fix obtained based on the GPS constellation would be false after only 2minutes, and errors in global positions

would continue to accumulate at a rate of about 10km each day. Relativity is not just some abstract mathematical theory. Understanding it is absolutely essential for the global navigation system to work properly. When the satellite is at perigee point, the satellite velocity is higher and the gravitational potential is lower because of which the satellite clocks run slower. In contrast, when the satellite is at apogee, the satellite velocity is lower and the gravitational potential is higher, so the satellite clocks run faster. The relativistic error can be modeled as t Fe r a sin( E k ) (4.6) where, Δt r = relativistic error F 2 C 2 µ = earth s gravitational constant=3.986005 10 14 m 3 /s 2 C = speed of light F = - 4.442807633 10-10 s/ m 1/2 e = satellite orbital eccentricity a = semi major axis of the satellite orbit and E k = eccentric anomaly of the satellite orbit The GPS measures the receiver position and the velocity with respect to satellite constellation. However the receiver position needs to be obtained with respect to the earth. In order to reference the satellite position to a receiver on the earth, the rotation of the earth must be considered. Using the coordinate system, reference position of the satellite can be transformed from one coordinate system to another. The satellite position is calculated in Earth Center Earth Effect (ECEF) coordinate system and transformed to Earth Centered Inertial (ECI) coordinate system. The conversion of ECEF coordinates to Geodetic

coordinates is given in Appendix-A. Correlating the satellite clock for miss-relativistic effect will result in a more accurate estimation of the time of transmission by the user. Due to rotation of the earth during the time of signal transmission, a relativistic error is introduced, this phenomenon is known as the Sagnac effect During the propagation time of the SV signal, a clock on the surface of the earth will experience a finite rotation with respect to the resting reference frame at the geocentric. If the GPS receiver experiences a net rotation away from SV, the propagation time will increase and vice-versa (Boubeker, 2005). If left uncorrected the relativistic error (R) can need to position error in the order of 30m. The satellite correction parameters are estimated using a curve fit to the predicted estimates of the actual satellite clock errors, but during this computation, some residual error remains. This residual clock error ( t ) results in ranging errors that typically vary from 0.3 to 4.2m, depending on the type of GPS satellite and age of the broadcast data. The range errors on the pseudorange measurement due to residual clock errors are generally the smallest following a control segment uploads to a satellite, and they sv slowly degrade over time until the next upload (Akim, 2002). The satellite clock error ( t ), symmetrically affects all the measurements made to satellite, by any GPS receiver making a measurement at the same time. Hence, satellite clock error ( sv t ) is spatially correlated at an epoch and this property can be exploited to overcome the effect of this bias (Boubeker, 2005). 4.8 Results The GPS data required for investigating the satellite clock and relativistic error impact on the proposed navigation solution was collected from a newly installed dual frequency GPS receiver (NovaTel make DL-V3) at Andhra University College of Engineering, Visakhapatnam. The data corresponds to 21 st December 2011. The actual position coordinates of the receiver located at Visakhapatnam are x = 706970.909m, y =

Mean anomaly in Radians 6035941.022m and z = 1930009.582m. The variation of mean anomaly for SV PRN 31 over a day is shown in Figure 4.3. The variation of eccentric anomaly for the same satellite is shown in Figure 4.4. The ephemeris required for the computation of mean anomaly and eccentric anomaly values are transmitted by the satellite in the navigation data for every two hours. The mathematical formulae used in the computation of mean anomaly and eccentric anomaly are given below. M k M0 nt k (4.7) (4.8) E M esin E with, E o = M k k k k -1-1.5-2 -2.5 SV PRN No. 31 Min.: -4.191rad at 10:00:00hours Max.: -1.04rad at 16:00:30hours Mean: -3.088rad σ = 0.91rad σ 2 = 0.828rad 2-3 -3.5-4 -4.5 10 11 12 13 14 15 16 17 GPS Time in Hours Figure 4.3 GPS time vs. Mean anomaly (M k )

Satellite clock bias in Seconds Eccentric anomaly in Radians 5.5 5 4.5 4 SV PRN No. 31 Min.: 2.097rad at 10:00:00hours Max.: 5.236rad at 16:00:30hours Mean: 3.194rad σ = 0.905rad σ 2 = 0.82rad 2 3.5 3 2.5 2 10 11 12 13 14 15 16 17 GPS Time in Hours Figure 4.4 GPS time vs. Eccentric anomaly (E k ) -5.1325 x 10-5 -5.133-5.1335-5.134 SV PRN No. 31 Min.: -5.136 10-05 s at 10:00:00hours Max.: -5.132 10-05 s at 16:00:30hours Mean: -5.135 10-05 s σ = 9.875 10-09 s σ 2 = 9.753 10-17 s 2-5.1345-5.135-5.1355-5.136-5.1365 10 11 12 13 14 15 16 17 GPS Time in Hours Figure 4.5 GPS time vs. satellite clock bias

Relativistic error in Seconds Satellite clock drift in Seconds 0.8 0.6 0.4 0.2 1 x 10-11 SV PRN No. 31 Min.: 1.592 10-12 s/s at 10:00:00hours Max.: 1.592 10-12 s/s at 10:00:00hours Mean: 1.592 10-12 s/s σ = 7.275 10-27 s/s σ 2 = 5.292 10-53 (s/s) 2 0-0.2-0.4-0.6-0.8-1 10 11 12 13 14 15 16 GPS Time in Hours Figure 4.6 GPS time vs. satellite clock drift 1.5 1 0.5 2 x 10-8 SV PRN No. 31 Min.: -1.564 10-08 s at 14:00:30hours Max.: 1.737 10-08 s at 16:00:30hours Mean: -4.054 10-09 s σ = 1.364 10-08 s σ 2 = 1.863 10-16 s 2 0-0.5-1 -1.5-2 10 11 12 13 14 15 16 17 GPS Time in Hours Figure 4.7 GPS time vs. relativistic error

Satellite clock error in Seconds From Figure 4.3 and Figure 4.4 it is clear that the mean anomaly (mean value=- 3.088rad) and the eccentric anomaly (mean value=3.194rad) remains same for two hours and they change for every two hours. Figure 4.5 shows the satellite clock bias variation over a day for the same satellite. This value (-5.13610-05 s) also remains same for two hours and changes every two hours. Variation of satellite clock drift over a day is shown in Figure 4.6. From Figure 4.6, it is clear that the satellite clock drift value (1.59210-12 s/s) remains same for the entire day for SV PRN 31. The satellite drift rate value in the navigation data for SV PRN 31 is zero. Figure 4.7 shows the variation of relativistic error for 17hours duration of GPS time for the same day. This also remains same for two hours and changes every two hours. The mean relativistic error observed is -4.05410-09 s and the minimum and maximum values observed are -1.56410-08 s and 1.73710-08 s respectively. -5.13 x 10-5 -5.131-5.132-5.133 SV PRN No. 31 Min.: -5.1363 10-05 s at 10:00:00hours Max.: -5.1328 10-05 s at 16:09:30hours Mean: -5.134 10-05 s σ = 1.015 10-08 s σ 2 = 1.032 10-16 s 2-5.134-5.135-5.136-5.137 10 11 12 13 14 15 16 17 GPS Time in Hours Figure 4.8 GPS time vs. satellite clock error The variation of the satellite clock error over a day is shown in Figure 4.8. This satellite clock error is used in correcting the pseudoranges to the satellites from the user. From Figure 4.8 it is observed that unlike other parameters it varies continuously over the day

Corrected pseudorange observed on L1 due to C/A code (m) Pseudorange observed on L1 due to C/A code (m) because it is not the parameter which is transmitted by the satellite. The satellite clock error is estimated using Eq. (4.5). The pseudorange of SV PRN 31 observed on L1 signal due to C/A code over a day is shown in Figure 4.9. The corrected pseudorange of SV PRN 31 observed on L1 signal due to C/A code over a day is shown in Figure 4.10. The number of satellites visible, satellite clock bias, satellite clock drift, satellite clock drift rate and satellite clock errors estimated at a particular epoch 10:00:00 hours are presented in Table 4.3. The pseudorange observed on L1 signal due to C/A code and the corrected pseudorange observed on L1 signal due to C/A code (taking satellite clock error into consideration) and the error in range (corrected pseudorange observed pseudorange) due to satellite clock error are also presented in Table 4.3. 2.6 x 107 2.5 2.4 SV PRN No. 31 Min.: 20231100m at 11:54:00hours Max.: 25426700m at 16:09:30hours Mean: 2.215 10 +07 m σ = 1.6312 10 +06 m σ 2 = 2.6609 10 +12 m 2 2.3 2.2 2.6 x 107 2.1 2.5 2 10 11 12 13 14 15 16 17 GPS Time in Hours Figure 4.9 vs. pseudorange on L1 signal C/A code 2.4 2.3 GPS time observed due to 2.2 2.1 SV PRN No. 31 Min.: 20215700m at 11:54:00hours Max.: 25411300m at 16:09:30hours Mean: 2.214 10 +07 m σ = 1.6312 10 +06 m 2 10 11 12 13 14 15 16 17 GPS Time in Hours σ 2 = 2.661 10 +12 m 2

Figure 4.10 GPS time vs. corrected pseudorange observed on L1 signal due to C/A code S. No. SV PRN No. Satellite clock bias, af 0 (s) Satellite clock drift, af 1 (s//s) Satellite clock drift rate, af 2 (s/s 2 ) Satellite clock error (s) Pseudorange observed on L1 signal due to C/A code (m) Corrected Pseudorange observed on L1 signal due to C/A code (m) Error in range due to satellite clock error (m) 1 3 0.00051607 5.23e-12 0 0.00051607 24146014.0 24300734.3 154720.3 2 6 0.00024922-5.23e-12 0 0.000249185 23551751.6 23626454.1 74702.5 3 14 1.1967e-05 4.32e-12 0 1.19982e-05 21099178.4 21102774.7 3596.3 4 19-1.57985e-5-2.387e- 0-1.57985e-05 24675000.3 24670266.6-4733.6 12 5 21-4.07342e-5-2.501e- 0-4.07522e-05 20907167.9 20894947.3-12220.5 12 6 22 0.00017909-9.09e-13 0 0.000179092 21637020.8 21690710.2 53689.3 9 7 24 0.00026256 3.297e-12 0 0.000262592 23222544.8 23301267.1 78722.3 8 8 26-1.87554e-5-3.752e- 0-1.87825e-05 21008229.4 21002594.4-5635.0 12 9 27 0.00012635 3.411e-12 0 0.000126377 25537658.8 25575542.8 37884.0 2 10 30 0.00021973 3.411e-12 0 0.000219732 24438821.2 24504700.9 65879.6 2 11 31-5.13638e-5 1.592e-12 0-5.13638e-05 22753809.5 22738414.7-15394.8 Table 4.3 SV PRN numbers with corresponding satellite clock bias, clock drift, clock drift rate and pseudorange observed on L1 signal (1575.42MHz) due to C/A code

S. No. SV PRN No. Satellite clock bias, af 0 (s) Satellite clock drift, af 1 (s//s) Satellite clock drift rate, af 2 (s/s 2 ) Satellite clock error (s) Pseudorange observed on L2 signal due to C/A code (m) Corrected Pseudorange observed on L2 signal due to C/A code (m) Error in range due to satellite clock error (m) 1 3 0.00051607 5.23e-12 0 0.00051607 24146014.0 24300734.3 154720.3 2 6 0.00024922-5.23e-12 0 0.000249185 23551751.6 23626454.1 74702.5 3 14 1.1967e-05 4.32e-12 0 1.19982e-05 21099178.4 21102774.7 3596.3 4 19-1.57985e-5-2.387e-12 0-1.57985e-05 24675000.3 24670266.6-4733.6 5 21-4.07342e-5-2.501e-12 0-4.07522e-05 20907167.9 20894947.3-12220.5 6 22 0.00017909-9.09e-13 0 0.000179092 21637020.8 21690710.2 53689.3 9 7 24 0.00026256 3.297e-12 0 0.000262592 23222544.8 23301267.1 78722.3 8 8 26-1.87554e-5-3.752e-12 0-1.87825e-05 21008229.4 21002594.4-5635.0 9 27 0.00012635 3.411e-12 0 0.000126377 25537658.8 25575542.8 37884.0 2 10 30 0.00021973 3.411e-12 0 0.000219732 24438821.2 24504700.9 65879.6 2 11 31-5.13638e-5 1.592e-12 0-5.13638e-05 22753809.5 22738414.7-15394.8 Table 4.4 SV PRN numbers with corresponding satellite clock bias, clock drift, clock drift rate and pseudorange observed on L2 signal (1227.6MHz) due to P-code Similarly the number of satellites visible, satellite clock bias, satellite clock drift, satellite clock drift rate and satellite clock estimated at the same epoch along with the pseudorange observed on L2 signal due to P-code and the corrected pseudorange observed on L2 signal due to P-code are presented in Table 4.4. The analysis of different parameters corresponding to SV PRN 31 is presented in Table 4.5. The maximum satellite clock error is found to be 5.132810-05 s which corresponds to a range error of 15398.4 m. S. No. Parameter Minimum value Maximum value Mean Standard deviation (σ) Variance (σ 2 ) 1 Eccentric anomaly (radians) 2.097 5.236 3.194 0.905 0.820 2 Mean anomaly (radians) -4.191-1.040-3.088 0.910 0.828 3 Satellite clock bias (s) -5.136e-05-5.132e-05-5.135e-05 9.875e-09 9.753e-17 4 Satellite clock drift (s/s) 1.592e-12 1.592e-12 1.592e-12 7.275e-27 5.292e-53 5 Relativistic error (s) -1.564e-08 1.737e-08-4.054e-09 1.364e-08 1.863e-16 6 Satellite clock error (s) -5.1363e-05-5.1328e-05-5.134e-05 1.015e-08 1.032e-16 7 Pseudorange observed on L1 20231100 25426700 2.215e+07 1.6312e+06 2.6609e+12

signal due to C/A code (m) 8 Corrected pseudorange observed on L1 signal due to C/A code (m) 20215700 25411300 2.214e+07 1.6312e+06 2.661e+12 4.9 Conclusions Table 4.5 Analysis of different parameters corresponding to SV PRN 31 Determination of precise position of a location anywhere on the earth s surface has become a very important piece of information not only in the military but also in the civilian sector. The critical application of GPS in civil aviation sector is the aircraft landing phase which requires the statistical analysis of GPS error measurements (Kibe S. V, 2003). The error is considered as a deviation of an estimate from a reference value, so it is possible to determine individual errors as a function of time. In this chapter, the behavior of satellite clock errors is studied and their impact on timing and positioning accuracy is analyzed. From this analysis, it is found that due to maximum satellite clock error (51.328µs), the maximum pseudorange can go up to 15.398km and this will translate into the position domain. GPS satellite position error analysis done in this chapter helps to improve the receiver operation in the poor navigation data detection environments such as in urban canyons and indoor environment. The pseudorange observed on L1 signal due to C/A code and pseudorange observed on L2 signal due to P-code measurements are processed and analyzed to obtain the statistical performances of the GPS satellite errors (ephemeris and satellite clock).