Chapter 2: Fundamentals of Satellite Navigation and Orbital Mechanics

The Principle of Position Determination

GNSS is a passive, one-way radio navigation system, meaning signals travel only from the satellite to the receiver. The fundamental principle is trilateration (more accurately, multilateration).

1.

Measuring Distance: A receiver determines its distance to a satellite by measuring the time it takes for the signal to travel from the satellite to the receiver. This travel time (Δt) is multiplied by the speed of light (c) to get a distance.

2.

Pseudorange: This calculated distance is called a pseudorange because it's not the true geometric range. The measurement is contaminated by a clock synchronization error (t_bias) between the satellite's highly accurate atomic clock and the receiver's much less accurate quartz clock. This timing bias results in a range bias (r_bias = c * t_bias).

3.

Solving for Position: To determine its 3D position (x, y, z), a receiver needs to solve for three unknown position coordinates. However, there is a fourth unknown: the receiver's clock bias. Therefore, a receiver must simultaneously measure the pseudorange to a minimum of four satellites to solve for all four unknowns.

GNSS Signal Processing and Measurements

The signal transmitted by a GNSS satellite is extremely weak by the time it reaches Earth, often near the level of background radio noise. It can only be detected because the receiver knows the structure of the signal's code in advance.

Correlation: The receiver generates a local copy of the satellite's unique Pseudo Random Noise (PRN) code. It then correlates this local copy with the incoming satellite signal, shifting its own code in time until it finds a perfect match (maximum correlation).

Time of Flight: The time shift needed to align the codes represents the signal's propagation time (plus the clock bias).

There are two primary types of measurements a receiver makes:

Code-Phase Measurement: This uses the PRN code to determine the pseudorange. It is robust but relatively imprecise.

Carrier-Phase Measurement: This measures the phase of the underlying carrier wave of the GNSS signal. This measurement is about 100 times more precise than the code measurement but has a significant challenge known as integer ambiguity. The receiver can measure the fractional part of the wave's cycle, but it doesn't know the initial whole number of wavelengths between it and the satellite at the moment of signal lock. Resolving this ambiguity is key to high-precision techniques like RTK and PPP.

Factors Influencing Signal Propagation: Atmospheric Effects

A GNSS signal's journey from the satellite to the receiver is affected by several factors that can introduce errors into the position calculation. These disturbances are handled either by estimating them with models or by eliminating them through differential techniques

The signal is delayed as it passes through Earth's atmosphere.

Ionospheric Effect: This occurs in the upper atmosphere (80 km to 1000 km) where solar radiation creates a layer of free electrons. This layer slows down the code part of the signal and advances the carrier phase. The effect is frequency-dependent, meaning different frequency signals (like L1 and L2) are delayed by different amounts. This property allows dual-frequency receivers to measure and largely eliminate the ionospheric error, which can range from 2 to 50 meters.

Tropospheric Effect: This occurs in the lower, non-ionized part of the atmosphere (below ~80 km) where all weather events happen. The delay is caused by changes in pressure, temperature, and water vapor. Unlike the ionosphere, this effect is not frequency-dependent and must be corrected using a numerical model. The delay is typically 2 meters for a satellite directly overhead (zenith) and increases to about 20 meters for a satellite near the horizon.

Factors Influencing Signal Propagation : Multipath Effect

Multipath occurs when the receiver antenna receives signals reflected from surfaces like buildings or the ground. Because reflected signals travel a longer path than the direct signal, they cause interference and range errors. It can be mitigated using choke ring antennas or proper site selection.

Factors Influencing Signal Propagation Relativistic Effects

According to Einstein's theories of relativity, time passes at different rates depending on velocity and gravitational potential.

Special Relativity: Due to their high speed, satellite clocks run slower than clocks on Earth.

General Relativity: Because they are in a weaker gravitational field, satellite clocks run faster than clocks on Earth.

The effect from General Relativity is stronger, causing the satellite clocks to run faster by about 38 microseconds per day. This predictable error is corrected by intentionally building the satellite clocks to run slightly slower on the ground, ensuring they operate at the correct frequency once in orbit.

Kepler's Laws of Planetary Motion

Johannes Kepler described the motion of planets, which also applies to satellites orbiting the Earth:

First Law: The orbit of a satellite is an ellipse, with the center of the Earth at one of the two foci.

Second Law: A line joining the satellite and the Earth sweeps out equal areas in equal intervals of time. This means the satellite moves fastest when it is closest to Earth (perigee) and slowest when it is farthest away (apogee).

Third Law: The square of the orbital period is proportional to the cube of the semi-major axis of its orbit.

Orbital Parameters (Keplerian Elements)

The precise shape, size, and orientation of an orbit, as well as the satellite's position within it, are defined by six parameters:

Semi-Major Axis (a): Defines the size of the orbit.

Eccentricity (e): Defines the shape of the orbit (e=0 for a circle, 0<e<1 for an ellipse).

Inclination (i): The angle between the orbital plane and the Earth's equatorial plane.

Right Ascension of the Ascending Node (Ω): Defines the orientation of the orbital plane with respect to the vernal equinox direction.

Argument of Perigee (ω): Defines the orientation of the ellipse within the orbital plane.

True Anomaly (θ): Defines the position of the satellite along the elliptical orbit at a specific time.

Orbital Maneuvers

To move a satellite from one orbit to another, its velocity must be changed by firing its thrusters. This change in velocity is called delta-v (Δv).

Hohmann Transfer:

An efficient two-burn maneuver to move between two circular orbits. The first burn places the satellite into an elliptical "transfer orbit," and the second burn circularizes the orbit at the new altitude.

Non-Planar Maneuvers:

Changing the inclination of an orbit is extremely "expensive" in terms of fuel, requiring a large Δv. These maneuvers are most efficiently performed where the satellite is moving slowest (at apogee) or at the intersection points of the two orbital planes (the nodes).