14 - Positioning

14.1 - Geometric interpretation

The GPS pseudorange measurement relates to the geometric range(distance) from satellite to receiver but is also caused by the receiver clock offset. The receiver clock offset is the same for all pseudoranges measured by the receiver at a specific time.

In two dimensions, we would need to solve for two receiver position coordinates and one receiver clock error, hence in total three unknown parameters, so we need at least three pseudorange measurements:

\[\underline{\varphi}_r^s = \sqrt{(x^s - x_r)^2 + (y^s - y_r)^2 + (z^s - z_r)^2} + b_r + d_r^s + \lambda N_r^s + \underline{e}_r^s\]

The measured pseudoranges must be reduced or enlarged with exactly the same amount to meet at one physical position. The amount to make that happen is the receiver clock offset.

14.2 Pseudorange observation equation

\(br\): is positive if the receiver clock is ahead of GPS system time, and the measured pseudoranges are ‘too long’.
ers: unavoidable random measurement error

14.3 Positioning: parameter estimation

GPS positioning employs the principle of least squares estimation. Since the GPS observation model is nonlinear, this involves a linearisation for the unknown parameters, around an approximate position. The linearized model of observation equations reads:

Figure 2: Linearized model of observation equations

Next, a leastsquares algorithm is used to solve this linearized model, presented in matrix-vector form, where a Best Linear Unbiased Estimation solution can be obtained.

NMEA is a well-known and widely used format for storing and exchanging GPS (GNSS) Position, Velocity and Time (PVT) solutions.

14.4 - Reference systems

By default, GPS positioning yields Cartesian coordinates (𝑥, 𝑦, 𝑧) in WGS84.

In differential mode, the position coordinates:

Are generally provided in a local or regional reference system(e.g. ETRS89 in Europe).
For the user receiver is in the same reference system as the position coordinates of the base, or reference station.

14.5 - GPS accuracy and error sources

The quality of the GPS position solution is largely dependent on:

The number of available satellites (enough satellites are visible)
Their geometry with respect to the user (On all sides of the receiver at high and low elevation angles)

No satellites are visible beneath the receiver (below the local horizon): vertical position accuracy is generally poorer than horizontal accuracy by about a factor of 1.5.

The accuracy of standalone positioning with GPS is in the order of 5-15 meters under reasonable satellite visibility.

The GPS pseudorange measurements contain errors due to:

Inaccurate satellite orbit and clock information,
delays along the path of the radio signal:
- atmospheric delays (ionosphere and troposphere),
- local effects including multipath,
- and measurement noise

\[\underline{p}_r^s = \sqrt{(x^s - x_r)^2 + (y^s - y_r)^2 + (z^s - z_r)^2} + b_r + \underline{e}_r^s\]

Local effects:

Shadowing: one or more satellite signals are blocked by surrounding obstacles.
Signal reflections: signals arrive at the receiver after bouncing off an object.
Multipath: both the direct and reflected signals arrive at the receiver

14.6 - Standalone positioning: example

Nothing interesting.