How do you compute relative position using GNSS?
“GNSS Solutions” is a regular column featuring questions and answers about technical aspects of GNSS. Readers are invited to send their questions to the columnist, Dr. Mark Petovello, Department of Geomatics Engineering, University of Calgary, who will find experts to answer them. email@example.com
This time, Mark Petovello answers your question himself.
Q: How do you compute relative positions with GNSS?
A: GNSS is well recognized as an excellent means of computing position, but many people think that GPS only provides absolute position information. However, GNSS can also provide relative position information. In this column, we will look at some of the details of how this is done.
Definition of Relative
This somewhat pedantic distinction aside, this article focuses on computing the relative positions of GNSS users. To this end, we will consider two situations.
The first involves positioning one object relative to another object. This may include, for example, positioning one receiver relative to a known static receiver. Alternatively, it may include positioning two cars relative to each other to avoid collisions.
The second, and perhaps less well known situation, is positioning one user over time. That is, computing the position of a GNSS receiver relative to its position at some previous point in time. This is useful when the motion of the vehicle is the important factor, not its absolute position.
Pia = ρia + bi + eia
where ρ is the geometric range to the satellite, b is the receiver clock bias (scaled to units of distance), and e is the combination of all measurement errors.
Similarly, the carrier phase measurement, scaled to units of distance by multiplying by the carrier wavelength, can be written as
Φia = ρia + bi + eia + λNia
where λ is the carrier phase wavelength and N is the integer carrier phase ambiguity. We acknowledge, of course, that the measurement errors for the pseudorange and carrier phase cases are not the same. For the time being, however, they can be denoted the same.
Relative Positioning of Two or More Users
. . .
Two assumptions were made in the foregoing development, and we will now address them in the opposite order in which they were introduced. First, we assumed that the initial position estimate was the position of the base receiver. Even if this was not the case, the computed position for receiver i will still be relative to the other point (see Figure 1, inset photo above right). In other words, the process of differencing the measurements between receivers automatically leads to relative positioning solutions.
Second, we assumed that the coordinates for the base receiver were known. In fact, this is not an overly restrictive requirement because we do not require this position to be known with a high degree of accuracy.
. . .
Over-Time Relative Positioning
. . .
In effect, we are approximating the case in Figure 1, except that instead of two receivers, we are considering the same receiver at two epochs (e.g., the base station is the user at the previous epoch and the receiver is the user at the current epoch).
This is often taken for granted, but by recalling the underlying concepts, new applications involving relative GNSS positioning can be envisioned and/or developed. Similarly, differencing measurements over time can provide useful information about the motion of the user, which can be important for many applications.
(For Mark Petovello’s complete answer to this question, including formulas and tables, please download the full article using the pdf link above.)
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