GNSS: Global Navigation Satellite Systems

GNSS: Global Navigation Satellite Systems Global: today the American GPS (Global Positioning Service), http://gps.losangeles.af.mil/index.html the Russian GLONASS, http://www.glonass-center.ru/frame_e.html In future Chinese Compass (already operating at the regional scale), European GALILEO

GLONASS, born almost at the same time as GPS, due to the USSR crisis in the 90, seemed to be pushed aside. Only in 2001 the Russian government has officially declared they are going to make the system fully operational: now the system is composed of 24 operational satellites. Therefore there has not yet been a systematic and widespread development of the GLONASS signal treatment devoted to geodetic applications. GLONASS data are to be used together with the GPS ones and hopefully with those of the European system GALILEO, which should be fully operational in?tbd?

The GPS satellites and milestones Satellites and milestones are grouped as follows 1978: first satellite launch Block I: experimental satellites launched from 1978 to 1985, today out of work...block II......1995: fully operational system...2005-2010: Block IIR(M), 7 satellites with L2C + M codes...2010-now: Block IIF, now 10 satellites, L5

Weight 1500 kg Several (Block IIF: 3 rubidium) oscillators. Solar panels to supply energy and retro-rockets to correct their orbits. Expected life of a satellite: 7.5 years. Requirements Communicate their position and GPS time with the highest possible accuracy.

The GPS satellites position and orbits The satellite speed is about 4 km/sec, therefore it cannot communicate its instantaneous position. It transmits a set of parameters which let us compute its position (ephemerides). General law of motion for a punctual body in a 3D space x P (t) = 1 m P i f i (x P, x P,t) P x is the body position vector, x P the velocity vector, x P the acceleration vector, t the time.

Forces acting on a generic orbiting satellite f ( x x ) = GM m + δf x x P G T P 3 P P G GM E: earth gravitational constant; G geocenter x : position of the Earth Earth gravitational attraction, central component, is 3 orders of magnitude higher than the other forces.

In a central force field approximation the orbit can easily described: maintaining the central field approximation, artificial satellites move along elliptic orbits predictable in time by Kepler laws The orbit is elliptic, with the Earth center of mass at one focus apogee: the furthest point of the orbit from the Earth, perigee: the nearest point to the Earth. Apogee and perigee are stationary points with respect to an inertial reference frame.

The position vector spans equal areas in equal times: the satellite is faster at the perigee and slower at the apogee. T a 2 2 3 = 4 π µ where T is the orbital period, a is the semi-major axis, µ depends 4 3 2 on the Earth mass ( µ 3.986 10 m / s ).

A keplerian orbit is stationary. It is completely defined by 6 parameters. i, orbit inclination: angle between the orbital plane and the reference equatorial plane. Ω, right ascension of the ascending node: angle, on the reference equatorial plane, between the equinox node and the ascending node. ω, perigee argument: angle between the intersection of the orbital plane with the equatorial plane and the orbit perigee direction. T 0, epoch of passage at the perigee. a and e: major semi-axis and orbit eccentricity. f (t), true anomaly: angle with vertex in the orbit focus between the perigee and the satellite position.

In the Keplerian orbit, given T 0, a, e it is possible to compute f (t) (satellite position on the orbital plane). By i, ω, Ω, the position in the orbital plane can be transformed to the inertial reference frame. By considering the Earth rotation, the satellite position can be computed in a Earth fixed reference frame.

The other forces Earth, not central gravitational force. Earth, temporal perturbations of gravitational force. Gravitational attraction of moon, sun, other planets. Solar radiation pressure on the solar panels. Atmospheric drag and magnetic forces. Their practical effects cause variations in time of all the Keplerian parameters. A set of Keplerian parameters cannot describe the satellite orbit in a sufficiently accurate (better than 10 m) way: a slightly more complex parameterization (that we do not treat here) is needed.

Perturbative force effect on the orbit in 24h Earth non-homogeneity 10.0 km Moon attraction 3.0 km Sun attraction 0.8 km Sun pressure 0.2 km

Satellite ephemerides The set of parameters necessary and sufficient to compute the satellite position in time. For the GPS satellites they can be Broadcast ephemerides (predicted by US National Geospatial Intelligence Agency (NGA, https://www1.nga.mil/, formerly NIMA) and transmitted by the satellites by quasi-keplerian parameters). Precise ephemerides (a posteriori estimated ephemerides, delivered via web by IGS, in a tabular format).

The system control network and the broadcast ephemerides A network of GPS PSs of the United States Army, to predict satellites orbits and clocks. It is also called Control Segment of the GPS system.

The observations of each control station to each satellite are sent to the Control center located at the Master station (Colorado Springs). By complex modeling applied to the received observations, the control center predicts the orbit and clock offset of each satellite for the next 24 hours. The predicted ephemerides and clock offsets are transmitted to the satellite, which distribute them to the users. The broadcast ephemerides are in WGS84 realization of ITRF and have an accuracy of about 1m.

Precise ephemerides Precise ephemerides are a posteriori computed by IGS Rapid ephemerides, available within 1 day, Final ephemerides: available within 2 weeks. They consist of the a posteriori estimates of the satellites positions in the IGS reference frame and are distributed via web. Final ephemerides have an accuracy better than 2.5 cm.

... SKIPPED HEADER * 2007 9 30 0 0 0.00000000 PG01 2667.489628 14512.617647 22299.283328 155.952112 PG02-12204.536387-23228.393961 4878.362107 140.117083... PG32-2811.094639 22575.212168 14287.586567 99999.999999 * 2007 9 30 0 15 0.00000000 PG01 173.660695 14683.334461 22339.970226 155.952820... * 2007 9 30 23 45 0.00000000... EOF

The GPS satellite system At present, 30 operational satellites, displaced on 6 different orbital planes. i = 55, e 0, a 26000km; T = 12h Each satellite moves at about 4 km/sec.

GPS satellites tracks on the Earth surface

Visibility of satellites from an observer The system has been designed to always guarantee the visibility of at least 4 satellites everywhere on the Earth surface. A typical visibility table in the 24 hours: typically 6-7 satellites are visible, each satellite is visible for a period of 1-6 hours.

Satellites elevation: η The angle between the signal direction and the horizontal plane. Only signals coming from satellites whose elevations are η > 10 or 15 are commonly used, in order to reduce the atmospheric delays.

Skyplot For a given time span, shows the polar (azimuth and elevation) trajectory of all the satellites in view. The above skyplot, for a 24 hours period, is common at our latitudes: no tracks toward North

GPS time and satellites clock Positioning is based on the hypothesis that we know exactly the satellite emission epoch of the signal. This would imply either a perfect synchronism between the satellites clocks and the GPS time, which is practically impossible, or to know the offset of each clock with respect to GPS time. The offset between satellites clocks ( t S ) and GPS time ( t GPS ) is defined as follows S S dt = t t GPS

The offset of satellites clocks can be accurately enough described by a polynomial of 2nd order in time dt ( t ) = dt + a t + b t S S S S 2 GPS 0 GPS GPS Satellites transmit broadcast clock parameters More accurate clock offsets are a posteriori estimated by IGS and distributed via web Accuracies - broadcast offsets: 3 ns, corresponding to 1 m - precise estimates: better than 0.15 ns, corresponding to 5 cm.

The signal The on-board oscillators produce a signal, whose nominal frequency 13 14 f is equal to 10.23 MHz, very stable in time ( Δ f / f = 10 /10 ). 0 From 0 f : three sinusoidal carrier phases (L1, L2 and L5) five binary codes: C/A (Coarse Acquisition Code), P (Precise Code), nowadays transformed in Y (EncrYpted), L2C (Block II-R), M (Block IIR-M), navigational message D (Navigation Data).

The carrier phases for an ideal oscillator Example of a cyclic oscillating phenomenon and the corresponding behavior in time.

An oscillating phenomenon, which repeats itself cyclically in time (sinusoidal) is described by the following equation: At () = Asin( ωt+ ϕ ) = Asin( ϕ()) t 0 0 0 A 0: signal amplitude, ω: angular velocity (rad/s), 0 (rad), ωt + ϕ instantaneous phase. 0 ϕ : initial phase ϕ () t is the state of the phenomenon at epoch t, can be expressed in radians, or, taking φ() t = ϕ()/2 t π in part of a cycle.

The period T of the signal is the time needed to complete one full revolution T = 2 π / ω indeed At ( + T) = Asin( ω( t+ 2 π / ω) + ϕ ) = Asin( ωt+ 2 π + ϕ ) = At ( ) 0 0 0 0 The frequency is defined as follows f = 1/ T = ω/2π

f can be defined also as the first derivative of the phase with respect to time: in fact, to a time interval Δ t a phase variation Δ φ = fδ t corresponds f =Δφ / Δ t taking the limit as Δ t 0, one gets f = dφ / dt Let consider an oscillating phenomenon that propagates in space (e.g. the sea waves): it is characterized by a law which depends both on time (t) and space (x). The propagation law is given by

Axt (, ) = A0sin( ω ( t x ) + 0) A0sin(2 ( t x ) 0) c ϕ = π + T λ ϕ c: signal space propagation velocity, λ = ct = c / f : wavelength, the distance in space between two successive repeating units.

The main characteristics of L1, L2 and L5 carrier phases (sinusoidal signals propagating in space) Name f (MHz) λ (cm) L1 1575.42 19 L2 1227.60 24 L5 1176 26

Binary Codes A binary code is a sequence of pulses equal to +1 and -1. The pulses transmission sequence, according to a proper decoding key, represents the signal content.

Code duration: time needed to transmit the whole sequence, Code frequency: the inverse of one pulse lasting, Code wavelength: length of one pulse. Name f (MHz) λ (m) Pulses number T All satellites codes C/A 1.023 293.0 1023 1 ms P(Y) 10.230 29.3 3.2703264 10 16 37 weeks Block IIR-M and II-F new codes L2C 1.023 293.0 10230 10 ms M 10.230 29.3??

C/A (Coarse acquisition) code 1023 pulses (1 ms.), one different C/A for each satellite used to identify the satellite and for pseudo-range observations. P (Precise) or Y (EncrYpted P) military code 37 weeks long, common to all the satellites. P(Y) guarantees better pseudo-range precisions than C/A code. At the beginning it was public, since August 1994 it has been encrypted in Y code, that, theoretically can be exploited only by the US Army receivers.

L2C code New civilian code Counterpart of C/A code on L2 carrier, 10230 (10 ms) pulses. M code New military code All these codes are called Pseudo Random (PR) because, their pulses follow an apparently random order. On the contrary, they have a well-known sequence.

Navigational message D (Navigation Data) A further binary code is generated, called navigational message D (f=50 Hz): it consists of 25 blocks, each 30 s long, with a resulting duration of 12.5 min. Each block contains satellite ephemerides and clock offsets, ionospheric model, cyclic information about the state of the other orbiting satellites (almanacs).

Combination of the signals: carrier phase modulation by a binary code. Carrier phases are pure oscillating signals, codes are sequences of square pulses. By multiplying them: a signal equal to the carrier phase but for slips equal to 180 corresponding to code state transitions.

Why a so complex signal? Originally two frequencies (carriers) allow to compute ionospheric free observables. two separate codes (C/A and P(Y)) for civilian and military users. C/A, less precise, only one frequency, public, P(Y): more precise, two frequencies, public in experimental phase (up to the full operability), encrypted in Y and restricted to US army receivers after full operability.

In the last 40 years, changes of US strategies, design of concurrent GNSS constellations: introduction of double frequency codes (C/A+L2C) for civilian users, liberalization of Y code to some civilian high cost receiver, need of a new, more refined, restricted code: M. a third carrier to vehicle new messages for (in future) high accuracy positioning. The navigation message is needed to communicate broacast ephemerides, clocks, ionospheric models and other constellation information.

Frequency, phase and time of a real oscillator For each epoch t, phase and frequency of an oscillator are given by dφ() t f() t =, φ() t = f( τ) dτ + φ0 dt t t 0 If i is a clock based on an atomic oscillator with nominal frequency f, the time t () t of the clock is computed according to the 0 i t t φ () t φ ( t ) φ () t φ ( t ) i i 0 i i 0 i () = = f0 f0 f0

1 ti() t = fi( τ) dτ f 0 t t 0 The frequency of an oscillator can be written as f () t = f + δ f () t i 0 i δ f () t is the frequency fluctuation. i The clock time can be written as t 1 1 ti() t = f dτ + δ fi( τ) dτ f 0 0 f t 0 0 0 t t

t () t = t+ dt () t i where t is GPS time, i dt () t is called the clock offset or error. i The phase at the epoch t can be written as φ() t = f t () t + φ( t ) i 0 i 0 φ() t = f [ t+ dt ()] t + φ( t ) = f t+ f dt () t + φ( t ) i 0 i 0 0 0 i 0

Observation equations Code (or pseudo-range) observation

The satellite is identified by its own C/A code After the identification the receiver performs a correlation between its internally generated C/A code and the one received from the satellite. These observations can be done with both C/A and P(Y) codes The delay between the received signal and the internal one is electronically measured. Δ T S R () t

The starting epoch is recorded by the satellite clock, The receiving epoch is recorded by the receiver clock: Therefore, the epochs are related to the local time of the two clocks: they are affected by the two clocks offsets.

Let t be the observation epoch; S let Δ T () t be the delay observed between the receiver R and the R satellite S signals. The observation equation is given by: S S S Δ T () t = t () t t ( t τ ) R R R where: t t is the receiving epoch recorded by the receiver R clock, () R S S t ( t τ R ) is the starting epoch recorded by the satellite S clock. τ is the signal traveling time from the satellite to the receiver;

The satellite clock is not synchronized with the GPS time; we have: S S S S S t ( t τ ) = t τ + dt ( t τ ) R R R in τ 66ms clock offset doesn t change: S S S dt ( t τ ) dt ( t) R for the receiver clock t () t = t+ dt () t R R

Therefore, we have S S S S S Δ T () t = t t + dt () t dt () t = τ + dt () t dt () t R R R R R Multiplying by the propagation signal velocity in vacuum, the so called code (or pseudo-range) observation is obtained S S S S P () t = cδ T () t = cτ + c( dt () t dt ()) t R R R R Observation noise P(Y): 10-30 cm C/A & L2C: 20-200 cm (depending on the receiver quality)

Phase observations Basic concepts An observation of the difference in phase between the received carrier (L1 or L2) and a sinusoid of the same frequency internally generated by the receiver can be done.

The observation equation at the epoch t is given by S S φ () t = φ () t φ ( t τ ) R R S R S φ () t is the phase difference observation at the epoch t R φ () t is the receiver internally generated phase at the epoch t R S φ ( t τ ) is the phase of the satellite S signal generated at the S R emission epoch. taking into account the previous formulas, φ () t = f t f ( t τ ) + f dt () t f dt ( t τ ) + φ φ S S S S S R 0 0 R 0 R 0 R 0R 0 = f τ + f ( dt ( t) dt ( t)) + φ φ S S S 0 R 0 R 0R 0

The integer ambiguity

The integer ambiguity The receiver can observe the fractional part of the incoming carrier, but not the integer number of cycles from its emission at the satellite: a integer unknown has to be added to the observation equation φ () t = f τ + f ( dt () t dt ()) t + φ φ + N () t S S S S S R 0 R 0 R 0R 0 R N S R () t is the integer number of cycles passed from the satellite signal phase at the emission epoch and the satellite signal phase at the receiving epoch: N cannot be observed.

The observation can be multiplied by the wavelength of the carrier: Thus the so-called phase observation is obtained L () t = λφ () t = cτ () t + c( dt () t dt ()) t + λ( N () t + φ φ ) S S S S S S R R R R R 0R 0 Observation noise: about 1 mm, both for L1 and for L2. S φ, φ, N ( t) represent completely different physical quantities. S R0 0 R φ, φ are fractional values, constant in time: represent the initial S R0 0 phases of receiver and satellite respectively.

S NR ( t ) is an integer, time varying, value: represents the integer part 1 of the distance between satellite and receiver. When needed, they will be expressed separately in the observation equation, when the separation is useless they will be represented by the η R S (t 1 ) = N R S (t 1 ) + φ 0 R φ 0 S S S S S S L () t = λφ () t = cτ () t + c( dt () t dt ()) t + λη () t R R R R R