How does GPS work exactly?

Question

For a GPS receiver to estimate its position, it first receives signals from at least 4 satellites. Does the receiver calculate the distance that separates it from each satellite, or does it just calculate the distance difference between each pair of satellites?

In other words, does the GPS system use TOF (time of flight) or TDOA (time difference of arrival, aka multilateration) technique to calculate position and how are these techniques applied?

Answer 1

I think GPS receivers do not exclusively need at least 4 satellites, but rather 3 as a minimum for trilateration.

A fourth satellite signal is necessary to synchronize the receiver clock with the satellite clocks.

As for TOF/TOA vs. TDOA, I believe the difference relies whether on if the GPS receiver has an internal clock synchronized with the satellite transmitters or not. In other words, whether if you know the time on the satellite or not (in which case, you measure the range differences).

So, user-end GPS systems probably use TDOA as they wouldn't have a synchronized clock (would be too expensive).

Details on the GPS positioning calculations:

The GPS calculation in the receiver uses four equations in the four unknowns x, y, z, t_c, where x, y, z are the receiver's coordinates, and t_c is the time correction for the GPS receiver's clock.

The equation is:

\begin{equation} \ d_n = c(t_{t,n} - t_{r,n} + t_c) = \sqrt[]{(x_n-x)^2+(y_n-y)^2+(z_n-z)^2} \end{equation}

Where:

• n would be: GPS 1,2,3,4 (for each of the 4 equations respectively)
• c = speed of light (3x10⁸ m/s)
• t_t,1, t_t,2, t_t,3, t_t,4 = times that GPS satellites 1, 2, 3, and 4, respectively, transmitted their signals (these times are provided to the receiver as part of the information that is transmitted).
• t_r,1, t_r,2, t_r,3, t_r,4 = times that the signals from GPS satellites 1, 2, 3, and 4, respectively, are received (according to the inaccurate GPS receiver's clock)
• x₁, y₁, z₁ = coordinates of GPS satellite 1 (these coordinates are provided to the receiver as part of the information that is transmitted); similar meaning for x₂, y₂, z₂, etc.

---

TOA = Time of arrival, TOF = Time of flight, TDOA = Time differences of arrival

Sources:
https://www.courses.psu.edu/aersp/aersp055_r81/satellites/gps_details.html
http://www.adv-radio-sci.net/9/203/2011/ars-9-203-2011.pdf
https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/Kalman.pdf

Answer 2

GPS satellites can be used to determine the attitude of other satellites also. I am not an expert in this field, but it might work the same way for tracking of people/cars on Earth.

Anyway, this is how it works for satellites.

GPS sensors can be used to determine attitude provided that at least 3 antennas are vailable. The 4th is needed for timing.

Slave and master are 2 antennas mounted on the s/c. If the distance $b$ is known we can detect the s/c orientation in space. Taking the projection of $\underline{b}$ on the direction $\underline{S}$ of the incoming GPS signal we obtain the path difference of the signal received by the 2 antennas as $\underline{S}^T \underline{b}$.

The position of the GPS satellite is known, therefore $\underline{S}$ is known in the geocentric reference frame if we know the position of the master antenna.

Measuring the path difference

$\Delta r = \underline{S}^T \underline{b}$

and transforming the vector $\underline{S}$ from geocentric to body frame

$\underline{S} = AS$

$\Delta r = S^T A^T \underline{b} $

the unknown is the rotation matrix A.

Having 3 independent measurements:

$\Delta r_{11} = S_1^T A^T \underline{b}_1 $ baseline 1 GPS satellite 1

$\Delta r_{21} = S_2^T A^T \underline{b}_1 $ baseline 1 GPS satellite 2

$\Delta r_{12} = S_1^T A^T \underline{b}_2 $ baseline 2 GPS satellite 1

$\Delta r_{22} = S_2^T A^T \underline{b}_2 $ baseline 2 GPS satellite 2

...

2 baselines are required, otherwise if you have a single baseline you cannot determine the rotation around that baseline.

To determine attitude minimize the cost function:

$J = \sum_{i = 1}^{N_S} \sum_{j = 1}^{N_b} (\Delta r_{ij} - S_i^T A^T b_j)$